research article existence and multiple positive solutions for...

Research ArticleExistence and Multiple Positive Solutions forBoundary Value Problem of Fractional Differential Equationwith 119901-Laplacian Operator

Min Jiang and Shouming Zhong

School of Mathematics Science University of Electronic Science and Technology of China Chengdu 611731 China

Correspondence should be addressed to Min Jiang jiangmin10701126com

Received 28 February 2014 Accepted 26 April 2014 Published 15 May 2014

Academic Editor Bashir Ahmad

Copyright copy 2014 M Jiang and S Zhong This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

This paper investigates the existence multiplicity nonexistence and uniqueness of positive solutions to a kind of two-pointboundary value problem for nonlinear fractional differential equations with 119901-Laplacian operator By using fixed point techniquescombining with partially ordered structure of Banach space we establish some criteria for existence and uniqueness of positivesolution of fractional differential equations with 119901-Laplacian operator in terms of different value of parameter In particular thedependence of positive solution on the parameter was obtained Finally several illustrative examples are given to support theobtained new results The study of illustrative examples shows that the obtained results are applicable

1 Introduction

In this paper we consider boundary value problem for afractional differential equation with 119901-Laplacian operator

(120601119901 (119863120572

0+119909 (119905)))

1015840

+ 120582120596 (119905) 119891 (119905 119909 (119905)) = 0 0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) = 120578int

1

0

119892 (119904) 119909 (119904) 119889119904

(1)

where 2 lt 120572 le 3 0 lt 120578 lt 1 is a constant and 120582 gt 0 is aparameter 119901 gt 1 120601119901(119904) is the p-Laplacian operator that is120601119901(119904) = |119904|

119901minus2119904 120601minus1

119901(119904) = 120601119902(119904) 1119901 + 1119902 = 1 119863 denotes

the Caputo fractional derivativeFractional differential equations have been of great inter-

est recently It is caused both by the intensive developmentof the theory of fractional calculus itself and by the appli-cations of such constructions in various fields of sciencesand engineering such as physics chemistry aerodynamicselectrodynamics of complex medium electrical circuits andbiology (see [1ndash7] and their references) Differential equa-tions with p-Laplacian arise naturally in non-Newtonian

mechanics nonlinear elasticity glaciology population biol-ogy combustion theory and nonlinear flow laws Since thep-Laplacian operator and fractional calculus arise from somany applied fields the fractional p-Laplacian differentialequations are worth studying Recently there have appeareda very large number of papers which are devoted to theexistence of solutions of boundary value problems and initialvalue problems for the fractional differential equations (see[8ndash12]) and the existence of solutions of boundary valueproblems for the fractional p-Laplacian differential equationshas just begun in recent years (see [13ndash22]) On the otherhand there are few papers that consider the eigenvalueintervals of fractional boundary value problems (see [23 24])In [23] the author discussed the following system

119863120572

0+119909 (119905) + 120582ℎ (119905) 119891 (119909 (119905)) = 0 0 lt 119905 lt 1

2 lt 120572 le 3

119909 (0) = 1199091015840(0) = 119909

10158401015840(0) = 119909

10158401015840(1) = 0

(2)

By the use of appropriate conditions with respect tolim119905rarr0(119891(119905)119905) = 1198910 and lim119905rarrinfin(119891(119905)119905) = 119891infin the authorproved that the above problem has at least one or two positive

Hindawi Publishing CorporationAbstract and Applied AnalysisVolume 2014 Article ID 512426 18 pageshttpdxdoiorg1011552014512426

2 Abstract and Applied Analysis

solutions for some 120582 where 1198910 = 0 119897infin 119891infin = 0 119897infin 0 lt

119897 lt infin But almost all the results which the author obtaineddepend on both 1198910 and 119891infin the case depends on one of 1198910and 119891infin the author only discussed 1198910 = 0infin and 119891infin = 0infinthe case 1198910 = 119897 or 119891infin = 119897 has not been discussed On theother hand there exist several results on the existence of onesolution to fractional p-Laplacian boundary value problems(BVPs) there are to the best of our knowledge relatively fewresults on the nonexistence and the uniqueness of positivesolutions to fractional p-Laplacian differential equation withparameter

Motivated by the above questions in this paper we willestablish several sufficient conditions for the existence ofpositive solutions of (1) by using fixed point theorem andfixed point index theory

The work is organized in the following fashion In Sec-tion 2 we provide some necessary background In particularwe will introduce some lemmas and definitions associatedwith fixed point index theory The main results will be statedand proved in Section 3 Two examples are given in Section 4

2 Preliminaries

In this section we introduce definitions and preliminary factswhich are used throughout this paper

Definition 1 (see [1]) The Riemann-Liouville fractional inte-gral of order 120572 gt 0 of a function 119909 (0 +infin) rarr R is givenby

119868120572

0+119909 (119905) =

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119909 (119904) 119889119904 (3)

provided that the right side is point-wise defined on (0 +infin)

and Γ is the Gamma function

Definition 2 (see [1]) The Caputo fractional derivative oforder 120572 gt 0 of a continuous function 119909 (0 +infin) rarr R isgiven by

119863120572

0+119909 (119905) =

1

Γ (119899 minus 120572)int

119905

0

(119905 minus 119904)119899minus120572minus1

119909(119899)

(119904) 119889119904 (4)

where 119899 = [120572] + 1 provided that the right side is point-wisedefined on (0 +infin)

Lemma 3 (see [1]) Let 120572 gt 0 and assume that 119909 isin 119862[0 1]and then the fractional differential equation 119863

120572

0119909(119905) = 0 has

unique solutions

119906 (119905) = 1198880 + 1198881119905 + 11988821199052+ sdot sdot sdot + 119888119899minus1119905

119899minus1

119888119894 isin R 119894 = 0 1 2 119899 minus 1 119899 minus 1 lt 120572 le 119899

(5)

Lemma 4 (see [1]) Let 120572 gt 0 Then

119868120572

0+119863120572

0+119909 (119905)

= 119909 (119905) + 1198880 + 1198881119905 + 11988821199052+ sdot sdot sdot + 119888119899minus1119905

119899minus1

(6)

where 119888119894 isin R 119894 = 0 1 2 119899 minus 1 119899 minus 1 lt 120572 le 119899

Lemma 5 (see [25]) Let 119875 be a cone in a real Banach space119864 and letΩ be a bounded open set of 119864 Assume that operator119860 119875 cap Ω rarr 119875 is completely continuous If there exists a1199090 gt 0 such that

119909 minus 119860119909 = 1199051199090 forall119909 isin 119875 cap 120597Ω 119905 ge 0 (7)

then 119894(119860 119875 cap Ω 119875) = 0

Lemma 6 (see [26]) Let 119864 be a Banach space and 119875 sube 119864 acone and Ω1 and Ω2 are open set with 0 isin Ω1 Ω1 sube Ω2 andlet 119879 119875 cap (Ω2 Ω1) rarr 119875 be completely continuous operatorsuch that either

(i) 119879119906 le 119906 119906 isin 119875cap120597Ω1 and 119879119906 ge 119906 119906 isin 119875cap120597Ω2

or

(ii) 119879119906 ge 119906 119906 isin 119875cap120597Ω1 and 119879119906 le 119906 119906 isin 119875cap120597Ω2

holds Then 119879 has a fixed point in 119875 cap (Ω2 Ω1)

3 Main Results

In this section we present some new results on the existencemultiplicity nonexistence and the uniqueness of positivesolution of problem (1) and dependence of the positivesolution 119909120582(119905) on the parameter 120582

Let 119864 = 119862[0 1] then 119864 is a real Banach space with thenorm sdot defined by 119909 = max0le119905le1|119909(119905)|

Lemma 7 Let 119910(119905) isin 119862(0 1) cap 1198711(0 1) and 119910(119905) ge 0 then the

solution of the problem

(120601119901 (119863120572

0+119909 (119905)))

1015840

+ 120582119910 (119905) = 0 0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) = 120578int

1

0

119892 (119904) 119909 (119904) 119889119904

(8)

is given by

119909 (119905) =1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

(9)

Abstract and Applied Analysis 3

119909 (119905) ge 0 min0le119905le1

119909 (119905) ge 120590 119909 (10)

where 2 lt 120572 le 3 0 lt 120578 lt 1 1 minus 120578 int1

0119892(119904)119889119904 gt 0 and 120590 =

120578 int1

0(1 minus 119904)119892(119904)119889119904(1 minus 120578 int

1

0119904119892(119904)119889119904)

Proof It is easy to see by integration of (8) that

120601119901 (119863120572

0+119909 (119905)) minus 120601119901 (119863

120572

0+119909 (0)) = minus120582int

119905

0

119910 (119904) 119889119904 (11)

By the boundary condition 1199091015840(0) = 119909

10158401015840(0) = 0 we can easily

get that1198631205720+

119909(0) = 0 and we obtain that

120601119901 (119863120572

0+119909 (119905)) = minus120582int

119905

0

119910 (119904) 119889119904 (12)

that is

119863120572

0+119909 (119905) = minus120601119902 (120582int

119905

0

119910 (119904) 119889119904) (13)

By Lemmas 3 and 4 we get that

119909 (119905) = minus119868120572120601119902 (120582int

119905

0

119910 (119904) 119889119904) + 1198880 + 1198881119905 + 11988821199052 (14)

Using the boundary condition 1199091015840(0) = 119909

10158401015840(0) = 0 119909(1) =

120578 int1

0119892(119904)119909(119904)119889119904 we get 1198881 = 1198882 = 0 and

1198880 =1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minus120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119905) (int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904) 119889119905

(15)

Thus

119909 (119905) = minus1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minus120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119905) (int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904) 119889119905

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

(16)

Direct differentiation of (9) implies

1199091015840(119905)

= minus1

Γ (120572 minus 1)int

119905

0

(119905 minus 119904)120572minus2

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904 le 0

(17)

By differentiation of (17) we get

11990910158401015840(119905)

= minus1


119905

0

(119905 minus 119904)120572minus3

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904 le 0

(18)

Thus the solution of problem (8) is nonincreasing andconcave on [0 1] and

119909 = 119909 (0) min0le119905le1

119909 (119905) = 119909 (1) (19)

On the other hand as we assume that 119910(119905) ge 0 we see that

119909 (1) =120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

119904

0

[(1 minus 120591)120572minus1

minus (119904 minus 120591)120572minus1

]


times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

+int

1

119904

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

ge 0

(20)

Therefore 119909(119905) ge 0 and is concave for 119905 isin [0 1] So for every119905 isin [0 1]

119909 (119905) ge 119905119909 (1) + (1 minus 119905) 119909 (0) (21)

Therefore

120578int

1

0

119909 (119905) 119892 (119905) 119889119905 ge 120578119909 (1) int

1

0

119905119892 (119905) 119889119905

+ 120578119909 (0) int

1

0

(1 minus 119905) 119892 (119905) 119889119905

(22)

Since 119909(1) = 120578 int1

0119909(119905)119892(119905)119889119905 1 minus 120578 int

1

0119892(119905)119889119905 gt 0 we have

119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) (23)

that is

min0le119905le1

119909 (119905) = 119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) = 120590 119909 (24)

The Lemma is proved

We construct a cone 119875 in 119864 by

119875 = 119909 isin 119864 119909 ge 0 min0le119905le1

119909 (119905) ge 120590 119909 (25)

where 120590 is defined in Lemma 7 It is easy to see 119875 is a closedconvex cone of 119864

Define 119879 119875 rarr 119864 by

(119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

(26)

It is clear that the fixed points of the operator 120601119902(120582)119879 arethe solutions of the boundary value problems (1)

We make the following hypotheses

(H1) 119891 [0 1] times [0infin) rarr [0infin) is continuous(H2) 120596 (0 1) rarr [0infin) is continuous and not identical

zero on any closed subinterval of (0 1) with 0 lt

int1

0120596(119905)119889119905 lt +infin

(H3) 1 minus 120578 int1

0119892(119904)119889119904 gt 0

(H4) 1198910

120601isin [0infin] where 119891

0

120601= lim119909rarr0(119891(119905 119909)120601119901(119909))

uniformly for 119905 isin [0 1](H5) 119891

infin


infin

120601= lim119909rarrinfin(119891(119905 119909)120601119901(119909))

uniformly for 119905 isin [0 1](H6) 119891(119905 119909) gt 0 for any 119905 isin [0 1] and 119909 gt 0

Lemma 8 Assume that (H1)ndash(H3) hold Then 119879 119875 rarr 119864 iscompletely continuous

Proof First we show that 119879 is continuous It is easy to see119879(119875) sub 119875 For 119909119899(119905) isin 119864 and 119909119899(119905) rarr 119909(119905) as 119899 rarr infin bythe continuous of 119891(119905 119909(119905)) we get 119891(119905 119909119899(119905)) rarr 119891(119905 119909(119905))as 119899 rarr infin This implies that

120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

997888rarr 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

(27)

So we have

sup119905isin[01]

10038161003816100381610038161003816100381610038161003816120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

minus 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

10038161003816100381610038161003816100381610038161003816997888rarr 0 119899 997888rarr infin

(28)

Denote Π = |(119879119909119899)(119905) minus (119879119909)(119905)| then

Π le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904


+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)minus120601119902

times (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583)

minus 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583) minus 120601119902

times (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le sup119904isin[01]

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816

times2

Γ (120572 + 1)(1 +

120578 int1

0119892 (119904) 119889119904

1 minus 120578 int1

0119892 (119904) 119889119904

) 997888rarr 0

119899 997888rarr infin

(29)

Hence 119879 is continuousSecond we show that 119879 is compactFor 119909(119905) isin 119861119903 = 119909 isin 119875 119909 le 119903 by condition (H1)

119891(119905 119909(119905)) lt infin Denote119872 = max0le119905le1119909le119903119891(119905 119909(119905))Then

|119879119909 (119905)|

le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)119889120591

10038161003816100381610038161003816100381610038161003816

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

119889120591 + int

119904

0

(119904 minus 120591)120572minus1

119889120591)119889119904

+1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

(1 minus 119904)120572minus1

119889119904 + int

119905

0

(119905 minus 119904)120572minus1

119889119904

le

2120601119902 (119872int1

0120596 (120591) 119889120591)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (120591) 119889120591)

lt infin

(30)

that is 119879 maps bounded sets into bounded sets in 119875For 119909(119905) isin 119861119903 0 lt 1199051 lt 1199052 lt 1

1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816int

1199052

0

(1199052 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


minus int

1199051

0

(1199051 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times

100381610038161003816100381610038161003816100381610038161003816

int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

+int

1199052

1199051

(1199052 minus 119904)120572minus1

119889119904

100381610038161003816100381610038161003816100381610038161003816

(31)

By mean value theorem we obtain that

int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

le int

1199051

0

(120572 minus 1) (1 minus 119904)120572minus2

(1199052 minus 1199051) 119889119904

le int

1

0


(1199052 minus 1199051) 119889119904

= 1199052 minus 1199051

(32)

Thus1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)

1003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591) [(1199052 minus 1199051) +1

120572(1199052 minus 1199051)

120572]

(33)

This shows that |119879119909(1199052) minus 119879119909(1199051)| rarr 0 as 1199052 minus 1199051 rarr 0so 119879119909 119909 isin 119861119903 is equicontinuous Therefore the operator119879 119875 rarr 119875 is completely continuous by the Arzela-Ascolitheorem

Now for convenience we introduce the following nota-tions Let

int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 = 1205741

120601119902 (int

1

0

120596 (120591) 119889120591) = 1205742

119898 (119903) = minmin119905

119891 (119905 119909)

120601119901 (119903) 119909 isin [120590119903 119903]

(34)

Theorem 9 Assume that (H1)ndash(H5) hold

(i) If 0 lt 1198910

120601lt +infin then there exists 1205850 gt 0 such that for

every 0 lt 119903 lt 1205850 problem (1) has a positive solution119909119903(119905) satisfying 119909119903 = 119903 associated with

120582 = 120582119903 isin [1205820 1205820] (35)

where 1205820 and 1205820 are two positive finite numbers

(ii) If 0 lt 119891infin

120601lt +infin then there exists 120585lowast

0gt 0 such that for

every 119877 gt 120585lowast

0 problem (1) has a positive solution 119909119877(119905)

satisfying 119909119877 = 119877 for any

120582 = 120582119877 isin [120582lowast

0 120582lowast

0] (36)

where 120582lowast

0and 120582

lowast

0are two positive finite numbers

(iii) If 1198910120601

= +infin then there exists 1205851 gt 0 such that for any0 lt 119903lowast

lt 1205851 problem (1) has a positive solution 119909119903lowast(119905)

satisfying 119909119903lowast = 119903lowast for any

120582 = 120582119903lowast isin (0 120582lowast] (37)

where 120582lowast is a positive finite number

(iv) If 119891infin

120601= +infin then there exists 1205851 gt 0 such that

for every 119877lowast gt 1205851 problem (1) has a positive solution119909119877lowast

(119905) satisfying 119909119877lowast

= 119877lowast for any

120582 = 120582119877lowast

isin (0 120582lowast] (38)

where 120582lowast is a positive finite number(v) If there exists 120585 gt 0 such that 119898(119903) gt 120585 then problem

(1) has positive solution 119909119903(119905) satisfying 119909119903 = 119903 forany

120582 = 120582119903 isin (0 ] (39)

where is a positive finite number

Proof (i) It follows from 0 lt 1198910

120601lt +infin that there exists

1198971 1198972 120588 gt 0 such that

1198971120601119901 (119909) le 119891 (119905 119909) le 1198972120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 120588)

(40)

Let 1205850 = 120588120590 we show that 1205850 is required When 119909 isin 119875 cap

120597Ω119903 we have

119909 (119905) ge 120590 119909 = 120590119903 (41)

we need 119909(119905) lt 120588 this implies that 120590119903 lt 120588 as 119903 lt 1205850 so120590119903 lt 1205901205850 = 120588

Let 1205820 = (11198971)(Γ(120572)1205901205741)119901minus1 Then we may assume that

119909 minus 120601119902 (1205820) 119879119909 = 0 (forall119909 isin 119875 cap 120597Ω119903) (42)

if not then there exists119909119903 isin 119875cap120597Ω119903 such that120601119902(1205820)119879119909119903 = 119909119903and then (35) already holds for 120582119903 = 1205820

Define 120595(119905) equiv 1 forall119905 isin [0 1] then 120595(119905) isin 119875 and 120595 equiv 1We now show that

119909 minus 120601119902 (1205820) 119879119909 = 120581120595 (forall119909 isin 119875 cap 120597Ω119903 120581 ge 0) (43)

In fact if there exists 1199091 isin 119875 cap 120597Ω119903 1205811 ge 0 such that 1199091 minus120601119902(1205820)1198791199091 = 1205811120595 then (42) implies that 1205811 gt 0 On the otherhand 1199091 = 120601119902(1205820)1198791199091 + 1205811120595 ge 1205811120595 we may choose 120581

lowast=

max120581 | 1199091 ge 120581120595 then 1205811 le 120581lowastlt +infin 1199091 ge 120581

lowast120595 Therefore

120581lowast= 120581lowast 1003817100381710038171003817120595

1003817100381710038171003817 le10038171003817100381710038171199091

1003817100381710038171003817 = 119903 (44)


Consequently for any 119905 isin [0 1] (26) and (44) imply that

1199091 (119905)

= 120601119902 (1205820)1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+120578120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+ 1205811120595

ge120578120590120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (120581lowast120595 (120591)) 119889120591) 119889119904

+ 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

0

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 + 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

01205741

Γ (120572)+ 1205811120595

= 120581lowast+ 120581120595

(45)

which implies that 1199091(119905) ge (120581lowast+ 120581)120595(119905) 119905 isin [0 1] which is a

contradiction to the definition of 120581lowast Thus (42) holds and byLemma 5 the fixed point index

119894 (120601119902 (1205820) 119879 119875 cap 120597Ω119903 119875) = 0 (46)

On the other hand by the fact that the fixed point index ofconstants operator is 1 so

119894 (120599 119875 cap 120597Ω119903 119875) = 1 (47)

where 120599 is the zero operator It follows therefore from (46)and (47) and the homotopy invariance property that thereexist 119909119903 isin 119875 cap 120597Ω119903 and 0 lt ]0 lt 1 such that ]0120601119902(1205820)119879119909119903 =119909119903 but 119909119903 = 120601119902(120582119903)119879119909119903 which implies that ]0120601119902(1205820)119879119909119903 =

120601119902(120582119903)119879119909119903 it follows that we get ]0120601119902(1205820) = 120601119902(120582119903)) that is

]0120582119902minus1

0= 120582119902minus1

119903 (48)

and then

120582119903 = ]1(119902minus1)0

1205820 = ]119901minus10

1205820 lt 1205820 (49)

From the proof above for any 119903 lt 1205850 there exists a positivesolution 119909119903 isin 119875 cap 120597Ω119903 associated with 120582 = 120582119903 gt 0


Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)

with 119909119903 = 119903Next we show that 120582119903 ge 1205820 In fact

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)

it follows that we get

120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)

In conclusion 120582119903 isin [1205820 1205820](ii) It follows from 0 lt 119891

infin

120601lt infin that there exist

119897lowast

1 119897lowast

2 120588lowastgt 0 such that

119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 120588lowast)

(55)

Let 120585lowast0= 120588lowast120590 the following proof are similar to (i)

(iii) It follows from 1198910

120601= +infin there exists 119897lowast gt 0 120588

lowastgt 0

such that

1198910

120601ge 119897lowast120601119901 (119909) (forall119905 isin [0 1] 0 le 119909 le 120588

lowast) (56)

Let 1205851 = 120588lowast120590 we show that 1205851 is required When 119909 isin 119875 cap

120597Ω119903lowast we have 119909(119905) ge 120590119909 = 120590119903lowast since 119909 le 120588

lowast then we need120590119903lowastlt 120588lowast as 119903lowast lt 1205851 so 120590119903

lowastlt 120588lowast holds

Let 120582lowast = (1119897lowast)(Γ(120572)1205901205741)

1(119902minus1) we proceed in the sameway as in the proof of (i) replacing (42) we may assume that119909 minus 120601119902(120582

lowast)119879119909 = 0 (forall119909 isin 119875 cap 120597Ω119903lowast) and replacing (43) we

can prove 119909 minus 120601119902(120582lowast)119879119909 = 120581120595 (forall119909 isin 119875 cap 120597Ω119903lowast 120581 ge 0) It

follows from Lemma 5 that 119894(120601119902(120582lowast)119879 119875 cap Ω119903lowast 119875) = 0 Note

that 119894(120579 119875 cap Ω119903lowast 119875) = 1 we can easily show that there exists119909119903lowast isin 119875cap120597Ω119903lowast and 0 lt ]119903lowast lt 1 such that ]119903lowast120601119902(120582

lowast)119879119909119903lowast = 119909119903lowast

Hence (37) holds for 120582119903lowast = 120582lowast]1minus119901119903lowast

lt 120582lowast

The proof ofTheorem 9(iv) follows by themethod similarto Theorem 9(iii) we omit it here

(v) It follows that 119898(119903) gt 120585 for any 119909 isin 119875 cap 120597Ω119903 we have120590119903 le 119909 le 119903 and 119891(119905 119909)120601119901(119909) ge 119891(119905 119909)120601119901(119903) ge 119898(119903) ge 120585then

119891 (119905 119909) ge 120585120601119901 (119909) forall119909 isin [120590119903 119903] 119905 isin [0 1] (57)

Let = (1120585)(Γ(120572)1205901205741)119901minus1 The following proof is similar to

that of (iv) This finished the proof of (v)

Remark 10 InTheorem 9 all the criteria obtained depend onone of 1198910

120601and 119891

infin

120601

Let Ψ(119909) = 120601119902(120582)119879119909 the following theorems give out themultiply nonexistence and the dependence of parameter


(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim

119909rarrinfin119891(119905 119909) = 119891infin isin

(0 +infin] uniformly for 119905 isin [0 1] then there exists120582lowast

gt 0 such that the problem (1) has two solutions forany 120582 gt 120582

lowast

(ii) If 1198910120601

= 0 and 119891infin

120601= 0 then there exists 120582 gt 0 such

that for any 120582 lt 120582 problem (1) has no solution

Proof (i) It follows from 119891infin isin (0 +infin] that for any 120577 gt 0there exists 120588 gt 0 such that

119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that

119891 (119905 119909) le 1205761120601119901 (119909) (forall119905 isin [0 1] 0 le 119909 le 1205881) (59)

where 1205761 satisfied 120601119902(1205821205761)1205742Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904) le 1

For forall119909 isin 119875 cap 120597Ω1205881

we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied

Ψ119909 le 119909 forall119909 isin 119875 cap 120597Ω1205881

(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)

119901minus1 thenforall119909 isin 119875 cap 120597Ω120588

3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies

Ψ119909 gt 119909 forall119909 isin 119875 cap 120597Ω1205883

(63)

Next for 119891infin

120601= 0 there exist 1205762 gt 0 120588

2gt 1205883gt 0 such

that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1

Similar to the above proof we get


(65)

Applying Lemma 7 to (61) (65) and (63) yields that Ψ

has two fixed points 1199091 1199092 such that 1199091 isin 119875cap (Ω1205883

Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883

)(ii) It follows from 119891

0

120601= 0 and 119891

infin

120601= 0 that there exists

1205761 1205762 1205881 1205882 gt 0 such that

119891 (119905 119909) le 1205761120601119901 (119909) (forall119905 isin [0 1] 0 le 119909 le 1205881)

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)

then for 119909 isin [1205881 1205882] by the continuous of 119891(119905 119909(119905)) there

exists 119909 isin [1205881 1205882] such that

max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then

119891 (119905 119909) le Υ120601119901 (119909) (forall0 le 119905 le 1 119909 ge 0) (68)

Assuming 119909(119905) is a positive solution of problem (1) we willshow that this leads to a contradiction for 120582 lt 120582 where

120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1 It follows from(26) that

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)

This implies that 1 le 120601119902(120582Υ)1205742Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1

0119892(119904)119889119904) = 1 which is a contradiction This finishes the

proof of (ii)

Remark 12 In (i) the condition lim119909rarrinfin

119891(119905 119909) = 119891infin isin

(0 +infin] uniformly for 119905 isin [0 1] can be replaced withcondition (H6) The same methods can be used to prove theresults

Remark 13 From the above proof the condition 119891infin isin

(0 +infin] is important to keep the problem (1) having at leasttwo positive solutions if this condition is not satisfied thenthere exists 120582 gt 0 small enough such that the problem (1) hasno positive solutions for all 120582 lt 120582

Theorem 14 Assume that (H1)ndash(H6) hold(i) If 1198910

120601= +infin and 119891

infin

120601= +infin then there exists 120582lowast0 gt 0

such that problem (1) has at least two solutions for 120582 isin

(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin

120601= +infin then there exists 120582 gt 0

such that problem (1) has no solution for all 120582 gt 120582

Proof (i) It follows from 1198910

120601= +infin that there exists 1199031 gt 0

such that

119891 (119905 119909) ge 119871lowast1120601119901 (119909) (forall119905 isin [0 1] 0 le 119909 le 1199031) (70)

where 119871lowast1 satisfies 1205902120601119902(120582119871lowast1)1205741Γ(120572) ge 1

Then for 119909 isin 119875cap120597Ω1199031

we have 120590119909 le 119909 le 1199031 for 119905 isin [0 1]

and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that

Ψ119909 ge 119909 forall119909 isin 119875 cap 120597Ω1199031

(72)

Next from 119891infin

120601= +infin there exists 1199032 gt 0 such that

119891 (119905 119909) ge 119871lowast2120601119901 (119909) forall119909 ge 1199032 (73)

where 119871lowast2 satisfies

1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)

Let 1199032 gt max1199031 1199032120590 then for 119909 isin 119875 cap 120597Ω1199032

we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)

Similar to the above proof we can get


(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1Thenfor 119909 isin 119875 cap 120597Ω119903

3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that

Ψ119909 lt 119909 forall119909 isin 119875 cap 120597Ω1199033

(79)

Applying Lemma 7 to (72) (76) and (79) yields that Ψ hastwo fixed points 1199091 1199092 such that 1199091 isin 119875 cap (Ω119903

3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033

)(ii) By the proof of (i) and the continuity of 119891(119905 119909) there

exists 119909 isin [1199031 1199032] such that min119909isin[11990311199032](119891(119905 119909)120601119901(119909)) =

119891(119905 119909)120601119901(119909)Let Υ = min119871lowast1 119871lowast2 119891(119905 119909)120601119901(119909) since (H6) holds

then Υ gt 0 and

119891 (119905 119909) ge Υ120601119901 (119909) forall119909 ge 0 119905 isin [0 1] (80)

Assuming 119909(119905) is a positive solution of problem (1) we willshow that this leads to contradiction for all 120582 gt 120582 where 120582 =

(1Υ)(Γ(120572)1205741120590)119901minus1 It follows form (26) that

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)

This implies that 1 ge 120601119902(120582Υ)1205741120590Γ(120572) = (120582Υ)119902minus1

1205741120590Γ(120572) gt

(120582Υ)119902minus1

1205741120590Γ(120572) = 1 which is a contradiction This com-pletes the proof of (ii)

Corollary 15 Assume that (H1)ndash(H5) holds

(i) If lim119909rarrinfin

119891(119905 119909) = 119891infin isin (0 +infin] uniformly for 119905 isin

[0 1] and one of 1198910120601

= 0 and 119891infin

120601= 0 is satisfied then

there exists 120582lowast gt 0 such that the problem (1) has at leastone positive solution for any 120582 gt 120582

lowast(ii) If one of 1198910

120601= +infin and 119891

infin

120601= +infin holds then there

exists 120582lowast0 gt 0 such that problem (1) has at least onepositive solutions for any 120582 isin (0 120582lowast0)

Proof (i) The conclusion is a direct consequence of Theo-rem 11(i)

(ii) The conclusion is a direct consequence of Theo-rem 14(i)

Theorem 16 Assume that (H1)ndash(H5) holdThen the followingconclusions hold

(i) If 1198910120601

= 0 and 119891infin

120601= infin then the problem (1) has a

positive solution 119909120582(119905) for all 120582 gt 0(ii) If 1198910

120601= infin and 119891

infin

120601= 0 then the problem (1) has a

positive solution 119909120582(119905) for all 120582 gt 0

Proof Weonly prove (i) the proof of (ii) is similar sowe omitit here

Let 120582 gt 0 since 1198910

120601= 0 there exists 119903 gt 0 119897 gt 0 such that

119891 (119905 119909) le 119897120601119901 (119909) (forall119905 isin [0 1] 119909 isin [0 119903]) (82)

where 119897 satisfied

1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)

Similar to the proof of Theorem 14 we obtain that

Ψ119909 le 119909 forall119909 isin 119875 cap 120597Ω119903 (84)


From 119891infin

120601= infin there exists 119877 gt 119903 gt 0 119871 gt 0 such that

119891 (119905 119909) ge 119871120601119901 (119909) (forall119905 isin [0 1] 119909 ge 119877) (85)where 119871 satisfied

1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)

We get thatΨ119909 ge 119909 forall119909 isin 119875 cap 120597Ω119877 (87)

Applying Lemma 7 to (84) (87) yields thatΨ has a fixed point119909 such that 119909 isin 119875 cap (Ω119877 Ω119903)


If 0 lt 1198910

120601lt infin and 0 lt 119891

infin

120601lt infin then there exists 1 2 gt

0 such that problem (1) has no positive solution for any 120582 gt 1

and 0 lt 120582 lt 2

Proof It follows from 0 lt 1198910


infin

120601lt infin that

there exist 1 gt 0 2 gt 0 1 gt 0 2 gt 0 and 1199032 gt 1199031 gt 0

such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =

min119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 By the condition(H6) gt 0 Then forall119909 ge 0 we have

120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)

where = max1 2 gt 0 and = min1198971 2 gt 0Let 1 = (1)(Γ(120572)1205741120590)

119901minus1 and 2 =

(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1

If problem (1)has positive solution 119909(119905) in 119875 then

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)

which is a contradiction This finishes the proof of Theo-rem 17

4 Examples

In this section we give out same examples to illustrate ourmain results can be used in practice

Example 1 Consider the following fractional differentialequation

(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12

By simple computation 119897lowast1= 1205872 119897

lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1asymp 14222 Then it is easy to see

all the conditions of Theorem 9(ii) are satisfied thus forall120582 isin

[14222 25149]] the problem (91) has at least one positivesolution

Example 2 Consider the boundary value problem

(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0

forall119905 isin [0 1]

1198910

120601= lim119909rarrinfin

119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904

forall119905 isin [0 1] 119909 isin [4infin)

(94)

so 1205761 = 1radic2119890 1205762 = 4radic21198904 and forall119909 isin

[12 4] max12le119909le4(119891(119905 119909)120601119901(119909)) = radic2119890minus1 We

have Υ = max1radic2119890 4radic21198904 radic2119890minus1

= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904

By Theorem 11(ii) for any 120582 lt 120582 = 3radic512058714

1198904 the problemhas no positive solution

5 Conclusions

Recently differential equations with p-Laplacian operatorwere widely discussed by several authors In this paperby using fixed point techniques combining with partiallyordered structure of Banach space we obtained the existencemultiply and the dependence of parameter These newresults we presented can be used in numerical computationand analyze mathematical models of physical phenomenamechanics nonlinear dynamics and many other relatedfields

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

The authors are very grateful to anonymous referees for theirvaluable and detailed suggestions and comments to improvethe original paper This work is supported by the Programfor New Century Excellent Talents in University (NCET-10-0097)

References

[1] I Podlubny Fractional Differential Equations vol 198 Aca-demic Press San Diego Calif USA 1999

[2] A A Kilbas H M Srivastava and J J Trujillo Theoryand Applications of Fractional Differential Equations vol 204of North-Holland Mathematics Studies Elsevier Science BVAmsterdam The Netherlands 2006

[3] Y A Rossikhin and M V Shitikova ldquoApplication of fractionalderivatives to the analysis of damped vibrations of viscoelasticsingle mass systemsrdquoActa Mechanica vol 120 no 1ndash4 pp 109ndash125 1997

[4] J Sabatier O Agrawal and J T Machado Advances in Frac-tional Calculus Theoretical developments and applications inphysics and engineering Springer Berlin Germany 2007

[5] N Engheta ldquoOn fractional calculus and fractional multipolesin electromagnetismrdquo IEEE Transactions on Antennas andPropagation vol 44 no 4 pp 554ndash566 1996

[6] R Hilfer Applications of Fractional Calculus in Physics WorldScientific Singapore 2000

[7] R L Bagley and R A Calico ldquoFractional order state equationsfor the control of viscoelastically damped structuresrdquo Journalof Guidance Control and Dynamics vol 14 no 2 pp 304ndash3111991

[8] W-X Zhou andY-D Chu ldquoExistence of solutions for fractionaldifferential equations with multi-point boundary conditionsrdquoCommunications in Nonlinear Science and Numerical Simula-tion vol 17 no 3 pp 1142ndash1148 2012

[9] B Ahmad and G Wang ldquoA study of an impulsive four-pointnonlocal boundary value problemof nonlinear fractional differ-ential equationsrdquo Computers amp Mathematics with Applicationsvol 62 no 3 pp 1341ndash1349 2011

[10] B Ahmad and S Sivasundaram ldquoOn four-point nonlocalboundary value problems of nonlinear integro-differentialequations of fractional orderrdquo Applied Mathematics and Com-putation vol 217 no 2 pp 480ndash487 2010

[11] N Nyamoradi ldquoPositive solutions for multi-point boundaryvalue problems for nonlinear fractional differential equationsrdquoJournal of Contemporary Mathematical Analysis vol 48 no 4pp 145ndash157 2013

[12] B Ahmad and J J Nieto ldquoExistence of solutions for nonlocalboundary value problems of higher-order nonlinear fractionaldifferential equationsrdquo Abstract and Applied Analysis vol 2009Article ID 494720 9 pages 2009

[13] X Liu M Jia and X Xiang ldquoOn the solvability of a fractionaldifferential equationmodel involving the119901-Laplacian operatorrdquoComputers ampMathematics with Applications vol 64 no 10 pp3267ndash3275 2012

[14] J Wang H Xiang and Z Liu ldquoExistence of concave positivesolutions for boundary value problem of nonlinear fractionaldifferential equation with 119901-Laplacian operatorrdquo InternationalJournal of Mathematics and Mathematical Sciences vol 2010Article ID 495138 17 pages 2010

[15] J Wang and H Xiang ldquoUpper and lower solutions method fora class of singular fractional boundary value problems with 119901-Laplacian operatorrdquo Abstract and Applied Analysis vol 2010Article ID 971824 12 pages 2010

[16] H Lu andZHan ldquoExistence of positive solutions for boundary-value problem of fractional differential equation with 119901-laplacian operatorrdquo The American Jouranl of Engineering andTechnology Research vol 11 pp 3757ndash3764 2011

[17] G Chai ldquoPositive solutions for boundary value problem offractional differential equation with 119901-Laplacian operatorrdquoBoundary Value Problems vol 2012 article 18 2012

[18] T Chen W Liu and Z Hu ldquoA boundary value problem forfractional differential equation with 119901-Laplacian operator atresonancerdquoNonlinear AnalysisTheoryMethodsampApplicationsvol 75 no 6 pp 3210ndash3217 2012

[19] T Chen andW Liu ldquoAn anti-periodic boundary value problemfor the fractional differential equation with a 119901-LaplacianoperatorrdquoAppliedMathematics Letters of Rapid Publication vol25 no 11 pp 1671ndash1675 2012

[20] S J Li X G Zhang Y H Wu and L Caccetta ldquoExtremalsolutions for 119901-Laplacian differential systems via iterative com-putationrdquo Applied Mathematics and Computation vol 26 pp1151ndash1158 2013

[21] Z Liu and L Lu ldquoA class of BVPs for nonlinear fractionaldifferential equations with 119901-Laplacian operatorrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 70pp 1ndash16 2012

[22] Z Liu L Lu and I Szanto ldquoExistence of solutions for fractionalimpulsive differential equations with 119901-Laplacian operatorrdquoActa Mathematica Hungarica vol 141 no 3 pp 203ndash219 2013

[23] Z Bai ldquoEigenvalue intervals for a class of fractional boundaryvalue problemrdquo Computers amp Mathematics with Applicationsvol 64 no 10 pp 3253ndash3257 2012

[24] X Zhang L Wang and Q Sun ldquoExistence of positive solutionsfor a class of nonlinear fractional differential equations withintegral boundary conditions and a parameterrdquo Applied Mathe-matics and Computation vol 226 pp 708ndash718 2014


[25] D J Guo and V Lakshmikantham Nonlinear Problems inAbstract Cones vol 5 Academic Press New York NY USA1988

[26] M A Krasnoselrsquoskii Topological Methods in the Theory ofNonlinear Integral Equations Translated by A H ArmstrongElmsford Pergamon Turkey 1964

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of


Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of


Mathematical PhysicsAdvances in

Complex AnalysisJournal of


OptimizationJournal of


CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of


Operations ResearchAdvances in

Journal of


Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences


The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014


Algebra

Discrete Dynamics in Nature and Society



Decision SciencesAdvances in

Discrete MathematicsJournal of


Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of


solutions for some 120582 where 1198910 = 0 119897infin 119891infin = 0 119897infin 0 lt

119897 lt infin But almost all the results which the author obtaineddepend on both 1198910 and 119891infin the case depends on one of 1198910and 119891infin the author only discussed 1198910 = 0infin and 119891infin = 0infinthe case 1198910 = 119897 or 119891infin = 119897 has not been discussed On theother hand there exist several results on the existence of onesolution to fractional p-Laplacian boundary value problems(BVPs) there are to the best of our knowledge relatively fewresults on the nonexistence and the uniqueness of positivesolutions to fractional p-Laplacian differential equation withparameter

Motivated by the above questions in this paper we willestablish several sufficient conditions for the existence ofpositive solutions of (1) by using fixed point theorem andfixed point index theory

The work is organized in the following fashion In Sec-tion 2 we provide some necessary background In particularwe will introduce some lemmas and definitions associatedwith fixed point index theory The main results will be statedand proved in Section 3 Two examples are given in Section 4

2 Preliminaries

In this section we introduce definitions and preliminary factswhich are used throughout this paper

Definition 1 (see [1]) The Riemann-Liouville fractional inte-gral of order 120572 gt 0 of a function 119909 (0 +infin) rarr R is givenby

119868120572

0+119909 (119905) =

1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

119909 (119904) 119889119904 (3)

provided that the right side is point-wise defined on (0 +infin)

and Γ is the Gamma function

Definition 2 (see [1]) The Caputo fractional derivative oforder 120572 gt 0 of a continuous function 119909 (0 +infin) rarr R isgiven by

119863120572

0+119909 (119905) =

1

Γ (119899 minus 120572)int

119905

0

(119905 minus 119904)119899minus120572minus1

119909(119899)

(119904) 119889119904 (4)

where 119899 = [120572] + 1 provided that the right side is point-wisedefined on (0 +infin)

Lemma 3 (see [1]) Let 120572 gt 0 and assume that 119909 isin 119862[0 1]and then the fractional differential equation 119863

120572

0119909(119905) = 0 has

unique solutions

119906 (119905) = 1198880 + 1198881119905 + 11988821199052+ sdot sdot sdot + 119888119899minus1119905

119899minus1

119888119894 isin R 119894 = 0 1 2 119899 minus 1 119899 minus 1 lt 120572 le 119899

(5)

Lemma 4 (see [1]) Let 120572 gt 0 Then

119868120572

0+119863120572

0+119909 (119905)

= 119909 (119905) + 1198880 + 1198881119905 + 11988821199052+ sdot sdot sdot + 119888119899minus1119905

119899minus1

(6)

where 119888119894 isin R 119894 = 0 1 2 119899 minus 1 119899 minus 1 lt 120572 le 119899

Lemma 5 (see [25]) Let 119875 be a cone in a real Banach space119864 and letΩ be a bounded open set of 119864 Assume that operator119860 119875 cap Ω rarr 119875 is completely continuous If there exists a1199090 gt 0 such that

119909 minus 119860119909 = 1199051199090 forall119909 isin 119875 cap 120597Ω 119905 ge 0 (7)

then 119894(119860 119875 cap Ω 119875) = 0

Lemma 6 (see [26]) Let 119864 be a Banach space and 119875 sube 119864 acone and Ω1 and Ω2 are open set with 0 isin Ω1 Ω1 sube Ω2 andlet 119879 119875 cap (Ω2 Ω1) rarr 119875 be completely continuous operatorsuch that either

(i) 119879119906 le 119906 119906 isin 119875cap120597Ω1 and 119879119906 ge 119906 119906 isin 119875cap120597Ω2

or

(ii) 119879119906 ge 119906 119906 isin 119875cap120597Ω1 and 119879119906 le 119906 119906 isin 119875cap120597Ω2

holds Then 119879 has a fixed point in 119875 cap (Ω2 Ω1)

3 Main Results

In this section we present some new results on the existencemultiplicity nonexistence and the uniqueness of positivesolution of problem (1) and dependence of the positivesolution 119909120582(119905) on the parameter 120582

Let 119864 = 119862[0 1] then 119864 is a real Banach space with thenorm sdot defined by 119909 = max0le119905le1|119909(119905)|

Lemma 7 Let 119910(119905) isin 119862(0 1) cap 1198711(0 1) and 119910(119905) ge 0 then the

solution of the problem

(120601119901 (119863120572

0+119909 (119905)))

1015840

+ 120582119910 (119905) = 0 0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) = 120578int

1

0

119892 (119904) 119909 (119904) 119889119904

(8)

is given by

119909 (119905) =1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

(9)


119909 (119905) ge 0 min0le119905le1

119909 (119905) ge 120590 119909 (10)


0119892(119904)119889119904 gt 0 and 120590 =

120578 int1

0(1 minus 119904)119892(119904)119889119904(1 minus 120578 int

1

0119904119892(119904)119889119904)


120601119901 (119863120572

0+119909 (119905)) minus 120601119901 (119863

120572

0+119909 (0)) = minus120582int

119905

0

119910 (119904) 119889119904 (11)


10158401015840(0) = 0 we can easily

get that1198631205720+


120601119901 (119863120572

0+119909 (119905)) = minus120582int

119905

0

119910 (119904) 119889119904 (12)

that is

119863120572

0+119909 (119905) = minus120601119902 (120582int

119905

0

119910 (119904) 119889119904) (13)


119909 (119905) = minus119868120572120601119902 (120582int

119905

0

119910 (119904) 119889119904) + 1198880 + 1198881119905 + 11988821199052 (14)


10158401015840(0) = 0 119909(1) =

120578 int1

0119892(119904)119909(119904)119889119904 we get 1198881 = 1198882 = 0 and

1198880 =1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minus120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119905) (int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904) 119889119905

(15)

Thus

119909 (119905) = minus1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minus120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119905) (int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904) 119889119905

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

(16)


1199091015840(119905)

= minus1


119905

0

(119905 minus 119904)120572minus2

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904 le 0

(17)


11990910158401015840(119905)

= minus1


119905

0

(119905 minus 119904)120572minus3

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904 le 0

(18)


119909 = 119909 (0) min0le119905le1

119909 (119905) = 119909 (1) (19)


119909 (1) =120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

119904

0

[(1 minus 120591)120572minus1


]


times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

+int

1

119904

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

ge 0

(20)


119909 (119905) ge 119905119909 (1) + (1 minus 119905) 119909 (0) (21)

Therefore

120578int

1

0

119909 (119905) 119892 (119905) 119889119905 ge 120578119909 (1) int

1

0

119905119892 (119905) 119889119905

+ 120578119909 (0) int

1

0

(1 minus 119905) 119892 (119905) 119889119905

(22)

Since 119909(1) = 120578 int1

0119909(119905)119892(119905)119889119905 1 minus 120578 int

1

0119892(119905)119889119905 gt 0 we have

119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) (23)

that is

min0le119905le1

119909 (119905) = 119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) = 120590 119909 (24)

The Lemma is proved


119875 = 119909 isin 119864 119909 ge 0 min0le119905le1

119909 (119905) ge 120590 119909 (25)


Define 119879 119875 rarr 119864 by

(119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

(26)





int1

0120596(119905)119889119905 lt +infin

(H3) 1 minus 120578 int1

0119892(119904)119889119904 gt 0

(H4) 1198910


0

120601= lim119909rarr0(119891(119905 119909)120601119901(119909))


infin


infin

120601= lim119909rarrinfin(119891(119905 119909)120601119901(119909))




120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

997888rarr 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

(27)

So we have

sup119905isin[01]

10038161003816100381610038161003816100381610038161003816120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

minus 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

10038161003816100381610038161003816100381610038161003816997888rarr 0 119899 997888rarr infin

(28)


Π le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904


+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)minus120601119902

times (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583)

minus 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583) minus 120601119902

times (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904


10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816

times2

Γ (120572 + 1)(1 +

120578 int1

0119892 (119904) 119889119904

1 minus 120578 int1

0119892 (119904) 119889119904

) 997888rarr 0

119899 997888rarr infin

(29)



|119879119909 (119905)|

le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)119889120591

10038161003816100381610038161003816100381610038161003816

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

119889120591 + int

119904

0

(119904 minus 120591)120572minus1

119889120591)119889119904

+1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

(1 minus 119904)120572minus1

119889119904 + int

119905

0

(119905 minus 119904)120572minus1

119889119904

le

2120601119902 (119872int1

0120596 (120591) 119889120591)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (120591) 119889120591)

lt infin

(30)


1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816int

1199052

0

(1199052 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


minus int

1199051

0

(1199051 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times

100381610038161003816100381610038161003816100381610038161003816

int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

+int

1199052

1199051

(1199052 minus 119904)120572minus1

119889119904

100381610038161003816100381610038161003816100381610038161003816

(31)


int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

le int

1199051

0


(1199052 minus 1199051) 119889119904

le int

1

0


(1199052 minus 1199051) 119889119904

= 1199052 minus 1199051

(32)

Thus1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)

1003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591) [(1199052 minus 1199051) +1

120572(1199052 minus 1199051)

120572]

(33)



int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 = 1205741

120601119902 (int

1

0

120596 (120591) 119889120591) = 1205742

119898 (119903) = minmin119905

119891 (119905 119909)

120601119901 (119903) 119909 isin [120590119903 119903]

(34)


(i) If 0 lt 1198910



120582 = 120582119903 isin [1205820 1205820] (35)




0gt 0 such that for




120582 = 120582119877 isin [120582lowast

0 120582lowast

0] (36)

where 120582lowast

0and 120582

lowast


(iii) If 1198910120601






(iv) If 119891infin





120582 = 120582119877lowast

isin (0 120582lowast] (38)



120582 = 120582119903 isin (0 ] (39)




1198971 1198972 120588 gt 0 such that

1198971120601119901 (119909) le 119891 (119905 119909) le 1198972120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 120588)

(40)


120597Ω119903 we have

119909 (119905) ge 120590 119909 = 120590119903 (41)








lowast=



120581lowast= 120581lowast 1003817100381710038171003817120595

1003817100381710038171003817 le10038171003817100381710038171199091

1003817100381710038171003817 = 119903 (44)



1199091 (119905)

= 120601119902 (1205820)1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+120578120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+ 1205811120595

ge120578120590120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (120581lowast120595 (120591)) 119889120591) 119889119904

+ 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

0

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 + 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

01205741

Γ (120572)+ 1205811120595

= 120581lowast+ 120581120595

(45)



119894 (120601119902 (1205820) 119879 119875 cap 120597Ω119903 119875) = 0 (46)


119894 (120599 119875 cap 120597Ω119903 119875) = 1 (47)



]0120582119902minus1

0= 120582119902minus1

119903 (48)

and then

120582119903 = ]1(119902minus1)0

1205820 = ]119901minus10

1205820 lt 1205820 (49)



Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)


119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










119909 (119905) ge 0 min0le119905le1

119909 (119905) ge 120590 119909 (10)


0119892(119904)119889119904 gt 0 and 120590 =

120578 int1

0(1 minus 119904)119892(119904)119889119904(1 minus 120578 int

1

0119904119892(119904)119889119904)


120601119901 (119863120572

0+119909 (119905)) minus 120601119901 (119863

120572

0+119909 (0)) = minus120582int

119905

0

119910 (119904) 119889119904 (11)


10158401015840(0) = 0 we can easily

get that1198631205720+


120601119901 (119863120572

0+119909 (119905)) = minus120582int

119905

0

119910 (119904) 119889119904 (12)

that is

119863120572

0+119909 (119905) = minus120601119902 (120582int

119905

0

119910 (119904) 119889119904) (13)


119909 (119905) = minus119868120572120601119902 (120582int

119905

0

119910 (119904) 119889119904) + 1198880 + 1198881119905 + 11988821199052 (14)


10158401015840(0) = 0 119909(1) =

120578 int1

0119892(119904)119909(119904)119889119904 we get 1198881 = 1198882 = 0 and

1198880 =1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minus120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119905) (int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904) 119889119905

(15)

Thus

119909 (119905) = minus1

Γ (120572)int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minus120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119905) (int

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904) 119889119905

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

(16)


1199091015840(119905)

= minus1


119905

0

(119905 minus 119904)120572minus2

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904 le 0

(17)


11990910158401015840(119905)

= minus1


119905

0

(119905 minus 119904)120572minus3

120601119902 (120582int

119904

0

119910 (120591) 119889120591) 119889119904 le 0

(18)


119909 = 119909 (0) min0le119905le1

119909 (119905) = 119909 (1) (19)


119909 (1) =120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

119904

0

[(1 minus 120591)120572minus1


]


times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

+int

1

119904

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

ge 0

(20)


119909 (119905) ge 119905119909 (1) + (1 minus 119905) 119909 (0) (21)

Therefore

120578int

1

0

119909 (119905) 119892 (119905) 119889119905 ge 120578119909 (1) int

1

0

119905119892 (119905) 119889119905

+ 120578119909 (0) int

1

0

(1 minus 119905) 119892 (119905) 119889119905

(22)

Since 119909(1) = 120578 int1

0119909(119905)119892(119905)119889119905 1 minus 120578 int

1

0119892(119905)119889119905 gt 0 we have

119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) (23)

that is

min0le119905le1

119909 (119905) = 119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) = 120590 119909 (24)

The Lemma is proved


119875 = 119909 isin 119864 119909 ge 0 min0le119905le1

119909 (119905) ge 120590 119909 (25)


Define 119879 119875 rarr 119864 by

(119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

(26)





int1

0120596(119905)119889119905 lt +infin

(H3) 1 minus 120578 int1

0119892(119904)119889119904 gt 0

(H4) 1198910


0

120601= lim119909rarr0(119891(119905 119909)120601119901(119909))


infin


infin

120601= lim119909rarrinfin(119891(119905 119909)120601119901(119909))




120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

997888rarr 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

(27)

So we have

sup119905isin[01]

10038161003816100381610038161003816100381610038161003816120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

minus 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

10038161003816100381610038161003816100381610038161003816997888rarr 0 119899 997888rarr infin

(28)


Π le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904


+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)minus120601119902

times (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583)

minus 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583) minus 120601119902

times (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904


10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816

times2

Γ (120572 + 1)(1 +

120578 int1

0119892 (119904) 119889119904

1 minus 120578 int1

0119892 (119904) 119889119904

) 997888rarr 0

119899 997888rarr infin

(29)



|119879119909 (119905)|

le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)119889120591

10038161003816100381610038161003816100381610038161003816

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

119889120591 + int

119904

0

(119904 minus 120591)120572minus1

119889120591)119889119904

+1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

(1 minus 119904)120572minus1

119889119904 + int

119905

0

(119905 minus 119904)120572minus1

119889119904

le

2120601119902 (119872int1

0120596 (120591) 119889120591)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (120591) 119889120591)

lt infin

(30)


1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816int

1199052

0

(1199052 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


minus int

1199051

0

(1199051 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times

100381610038161003816100381610038161003816100381610038161003816

int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

+int

1199052

1199051

(1199052 minus 119904)120572minus1

119889119904

100381610038161003816100381610038161003816100381610038161003816

(31)


int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

le int

1199051

0


(1199052 minus 1199051) 119889119904

le int

1

0


(1199052 minus 1199051) 119889119904

= 1199052 minus 1199051

(32)

Thus1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)

1003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591) [(1199052 minus 1199051) +1

120572(1199052 minus 1199051)

120572]

(33)



int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 = 1205741

120601119902 (int

1

0

120596 (120591) 119889120591) = 1205742

119898 (119903) = minmin119905

119891 (119905 119909)

120601119901 (119903) 119909 isin [120590119903 119903]

(34)


(i) If 0 lt 1198910



120582 = 120582119903 isin [1205820 1205820] (35)




0gt 0 such that for




120582 = 120582119877 isin [120582lowast

0 120582lowast

0] (36)

where 120582lowast

0and 120582

lowast


(iii) If 1198910120601






(iv) If 119891infin





120582 = 120582119877lowast

isin (0 120582lowast] (38)



120582 = 120582119903 isin (0 ] (39)




1198971 1198972 120588 gt 0 such that

1198971120601119901 (119909) le 119891 (119905 119909) le 1198972120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 120588)

(40)


120597Ω119903 we have

119909 (119905) ge 120590 119909 = 120590119903 (41)








lowast=



120581lowast= 120581lowast 1003817100381710038171003817120595

1003817100381710038171003817 le10038171003817100381710038171199091

1003817100381710038171003817 = 119903 (44)



1199091 (119905)

= 120601119902 (1205820)1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+120578120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+ 1205811120595

ge120578120590120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (120581lowast120595 (120591)) 119889120591) 119889119904

+ 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

0

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 + 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

01205741

Γ (120572)+ 1205811120595

= 120581lowast+ 120581120595

(45)



119894 (120601119902 (1205820) 119879 119875 cap 120597Ω119903 119875) = 0 (46)


119894 (120599 119875 cap 120597Ω119903 119875) = 1 (47)



]0120582119902minus1

0= 120582119902minus1

119903 (48)

and then

120582119903 = ]1(119902minus1)0

1205820 = ]119901minus10

1205820 lt 1205820 (49)



Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)


119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times 120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591

+int

1

119904

(1 minus 120591)120572minus1

120601119902 (120582int

120591

0

119910 (120583) 119889120583)119889120591)119889119904

ge 0

(20)


119909 (119905) ge 119905119909 (1) + (1 minus 119905) 119909 (0) (21)

Therefore

120578int

1

0

119909 (119905) 119892 (119905) 119889119905 ge 120578119909 (1) int

1

0

119905119892 (119905) 119889119905

+ 120578119909 (0) int

1

0

(1 minus 119905) 119892 (119905) 119889119905

(22)

Since 119909(1) = 120578 int1

0119909(119905)119892(119905)119889119905 1 minus 120578 int

1

0119892(119905)119889119905 gt 0 we have

119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) (23)

that is

min0le119905le1

119909 (119905) = 119909 (1) ge120578 int1

0(1 minus 119905) 119892 (119905) 119889119905

1 minus 120578 int1

0119905119892 (119905) 119889119905

119909 (0) = 120590 119909 (24)

The Lemma is proved


119875 = 119909 isin 119864 119909 ge 0 min0le119905le1

119909 (119905) ge 120590 119909 (25)


Define 119879 119875 rarr 119864 by

(119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minusint

119905

0

(119905 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

(26)





int1

0120596(119905)119889119905 lt +infin

(H3) 1 minus 120578 int1

0119892(119904)119889119904 gt 0

(H4) 1198910


0

120601= lim119909rarr0(119891(119905 119909)120601119901(119909))


infin


infin

120601= lim119909rarrinfin(119891(119905 119909)120601119901(119909))




120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

997888rarr 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

(27)

So we have

sup119905isin[01]

10038161003816100381610038161003816100381610038161003816120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909119899 (119904)) 119889119904)

minus 120601119902 (int

119905

0

119908 (119904) 119891 (119904 119909 (119904)) 119889119904)

10038161003816100381610038161003816100381610038161003816997888rarr 0 119899 997888rarr infin

(28)


Π le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904


+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)minus120601119902

times (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583)

minus 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583) minus 120601119902

times (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904


10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816

times2

Γ (120572 + 1)(1 +

120578 int1

0119892 (119904) 119889119904

1 minus 120578 int1

0119892 (119904) 119889119904

) 997888rarr 0

119899 997888rarr infin

(29)



|119879119909 (119905)|

le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)119889120591

10038161003816100381610038161003816100381610038161003816

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

119889120591 + int

119904

0

(119904 minus 120591)120572minus1

119889120591)119889119904

+1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

(1 minus 119904)120572minus1

119889119904 + int

119905

0

(119905 minus 119904)120572minus1

119889119904

le

2120601119902 (119872int1

0120596 (120591) 119889120591)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (120591) 119889120591)

lt infin

(30)


1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816int

1199052

0

(1199052 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


minus int

1199051

0

(1199051 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times

100381610038161003816100381610038161003816100381610038161003816

int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

+int

1199052

1199051

(1199052 minus 119904)120572minus1

119889119904

100381610038161003816100381610038161003816100381610038161003816

(31)


int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

le int

1199051

0


(1199052 minus 1199051) 119889119904

le int

1

0


(1199052 minus 1199051) 119889119904

= 1199052 minus 1199051

(32)

Thus1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)

1003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591) [(1199052 minus 1199051) +1

120572(1199052 minus 1199051)

120572]

(33)



int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 = 1205741

120601119902 (int

1

0

120596 (120591) 119889120591) = 1205742

119898 (119903) = minmin119905

119891 (119905 119909)

120601119901 (119903) 119909 isin [120590119903 119903]

(34)


(i) If 0 lt 1198910



120582 = 120582119903 isin [1205820 1205820] (35)




0gt 0 such that for




120582 = 120582119877 isin [120582lowast

0 120582lowast

0] (36)

where 120582lowast

0and 120582

lowast


(iii) If 1198910120601






(iv) If 119891infin





120582 = 120582119877lowast

isin (0 120582lowast] (38)



120582 = 120582119903 isin (0 ] (39)




1198971 1198972 120588 gt 0 such that

1198971120601119901 (119909) le 119891 (119905 119909) le 1198972120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 120588)

(40)


120597Ω119903 we have

119909 (119905) ge 120590 119909 = 120590119903 (41)








lowast=



120581lowast= 120581lowast 1003817100381710038171003817120595

1003817100381710038171003817 le10038171003817100381710038171199091

1003817100381710038171003817 = 119903 (44)



1199091 (119905)

= 120601119902 (1205820)1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+120578120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+ 1205811120595

ge120578120590120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (120581lowast120595 (120591)) 119889120591) 119889119904

+ 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

0

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 + 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

01205741

Γ (120572)+ 1205811120595

= 120581lowast+ 120581120595

(45)



119894 (120601119902 (1205820) 119879 119875 cap 120597Ω119903 119875) = 0 (46)


119894 (120599 119875 cap 120597Ω119903 119875) = 1 (47)



]0120582119902minus1

0= 120582119902minus1

119903 (48)

and then

120582119903 = ]1(119902minus1)0

1205820 = ]119901minus10

1205820 lt 1205820 (49)



Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)


119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119899 (120591)) 119889120591)minus120601119902

times (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583)

minus 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119899 (120583)) 119889120583) minus 120601119902

times (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904


10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909119899 (120591)) 119889120591)

minus 120601119902 (int

119904

0

119908 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816

times2

Γ (120572 + 1)(1 +

120578 int1

0119892 (119904) 119889119904

1 minus 120578 int1

0119892 (119904) 119889119904

) 997888rarr 0

119899 997888rarr infin

(29)



|119879119909 (119905)|

le1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+int

119905

0

(119905 minus 119904)120572minus1

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

119904

0

120596 (120591) 119889120591)

10038161003816100381610038161003816100381610038161003816119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)119889120591

10038161003816100381610038161003816100381610038161003816

+ int

119904

0

(119904 minus 120591)120572minus1

times

10038161003816100381610038161003816100381610038161003816120601119902 (119872int

120591

0

120596 (120583) 119889120583)

10038161003816100381610038161003816100381610038161003816119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

119892 (119904) (int

1

0

(1 minus 120591)120572minus1

119889120591 + int

119904

0

(119904 minus 120591)120572minus1

119889120591)119889119904

+1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times int

1

0

(1 minus 119904)120572minus1

119889119904 + int

119905

0

(119905 minus 119904)120572minus1

119889119904

le

2120601119902 (119872int1

0120596 (120591) 119889120591)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (120591) 119889120591)

lt infin

(30)


1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)1003816100381610038161003816

=1

Γ (120572)

10038161003816100381610038161003816100381610038161003816int

1199052

0

(1199052 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


minus int

1199051

0

(1199051 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times

100381610038161003816100381610038161003816100381610038161003816

int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

+int

1199052

1199051

(1199052 minus 119904)120572minus1

119889119904

100381610038161003816100381610038161003816100381610038161003816

(31)


int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

le int

1199051

0


(1199052 minus 1199051) 119889119904

le int

1

0


(1199052 minus 1199051) 119889119904

= 1199052 minus 1199051

(32)

Thus1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)

1003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591) [(1199052 minus 1199051) +1

120572(1199052 minus 1199051)

120572]

(33)



int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 = 1205741

120601119902 (int

1

0

120596 (120591) 119889120591) = 1205742

119898 (119903) = minmin119905

119891 (119905 119909)

120601119901 (119903) 119909 isin [120590119903 119903]

(34)


(i) If 0 lt 1198910



120582 = 120582119903 isin [1205820 1205820] (35)




0gt 0 such that for




120582 = 120582119877 isin [120582lowast

0 120582lowast

0] (36)

where 120582lowast

0and 120582

lowast


(iii) If 1198910120601






(iv) If 119891infin





120582 = 120582119877lowast

isin (0 120582lowast] (38)



120582 = 120582119903 isin (0 ] (39)




1198971 1198972 120588 gt 0 such that

1198971120601119901 (119909) le 119891 (119905 119909) le 1198972120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 120588)

(40)


120597Ω119903 we have

119909 (119905) ge 120590 119909 = 120590119903 (41)








lowast=



120581lowast= 120581lowast 1003817100381710038171003817120595

1003817100381710038171003817 le10038171003817100381710038171199091

1003817100381710038171003817 = 119903 (44)



1199091 (119905)

= 120601119902 (1205820)1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+120578120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+ 1205811120595

ge120578120590120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (120581lowast120595 (120591)) 119889120591) 119889119904

+ 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

0

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 + 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

01205741

Γ (120572)+ 1205811120595

= 120581lowast+ 120581120595

(45)



119894 (120601119902 (1205820) 119879 119875 cap 120597Ω119903 119875) = 0 (46)


119894 (120599 119875 cap 120597Ω119903 119875) = 1 (47)



]0120582119902minus1

0= 120582119902minus1

119903 (48)

and then

120582119903 = ]1(119902minus1)0

1205820 = ]119901minus10

1205820 lt 1205820 (49)



Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)


119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










minus int

1199051

0

(1199051 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

10038161003816100381610038161003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591)

times

100381610038161003816100381610038161003816100381610038161003816

int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

+int

1199052

1199051

(1199052 minus 119904)120572minus1

119889119904

100381610038161003816100381610038161003816100381610038161003816

(31)


int

1199051

0

((1199052 minus 119904)120572minus1


) 119889119904

le int

1199051

0


(1199052 minus 1199051) 119889119904

le int

1

0


(1199052 minus 1199051) 119889119904

= 1199052 minus 1199051

(32)

Thus1003816100381610038161003816119879119909 (1199052) minus 119879119909 (1199051)

1003816100381610038161003816

le1

Γ (120572)120601119902 (119872int

1

0

120596 (120591) 119889120591) [(1199052 minus 1199051) +1

120572(1199052 minus 1199051)

120572]

(33)



int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 = 1205741

120601119902 (int

1

0

120596 (120591) 119889120591) = 1205742

119898 (119903) = minmin119905

119891 (119905 119909)

120601119901 (119903) 119909 isin [120590119903 119903]

(34)


(i) If 0 lt 1198910



120582 = 120582119903 isin [1205820 1205820] (35)




0gt 0 such that for




120582 = 120582119877 isin [120582lowast

0 120582lowast

0] (36)

where 120582lowast

0and 120582

lowast


(iii) If 1198910120601






(iv) If 119891infin





120582 = 120582119877lowast

isin (0 120582lowast] (38)



120582 = 120582119903 isin (0 ] (39)




1198971 1198972 120588 gt 0 such that

1198971120601119901 (119909) le 119891 (119905 119909) le 1198972120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 120588)

(40)


120597Ω119903 we have

119909 (119905) ge 120590 119909 = 120590119903 (41)








lowast=



120581lowast= 120581lowast 1003817100381710038171003817120595

1003817100381710038171003817 le10038171003817100381710038171199091

1003817100381710038171003817 = 119903 (44)



1199091 (119905)

= 120601119902 (1205820)1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+120578120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+ 1205811120595

ge120578120590120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (120581lowast120595 (120591)) 119889120591) 119889119904

+ 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

0

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 + 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

01205741

Γ (120572)+ 1205811120595

= 120581lowast+ 120581120595

(45)



119894 (120601119902 (1205820) 119879 119875 cap 120597Ω119903 119875) = 0 (46)


119894 (120599 119875 cap 120597Ω119903 119875) = 1 (47)



]0120582119902minus1

0= 120582119902minus1

119903 (48)

and then

120582119903 = ]1(119902minus1)0

1205820 = ]119901minus10

1205820 lt 1205820 (49)



Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)


119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











1199091 (119905)

= 120601119902 (1205820)1

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+120578120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+ 1205811120595

ge120578120590120601119902 (1205820)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times (int

120591

0

120596 (120583) 119891 (120583 1199091 (120583)) 119889120583)119889120591)119889119904

+120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119891 (120591 1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (1199091 (120591)) 119889120591) 119889119904

+ 1205811120595

ge120590120601119902 (1205820)

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (1198971 int

119904

0

120596 (120591) 120601119901 (120581lowast120595 (120591)) 119889120591) 119889119904

+ 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

0

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904 + 1205811120595

=120590120581lowast119897119902minus1

1120582119902minus1

01205741

Γ (120572)+ 1205811120595

= 120581lowast+ 120581120595

(45)



119894 (120601119902 (1205820) 119879 119875 cap 120597Ω119903 119875) = 0 (46)


119894 (120599 119875 cap 120597Ω119903 119875) = 1 (47)



]0120582119902minus1

0= 120582119902minus1

119903 (48)

and then

120582119903 = ]1(119902minus1)0

1205820 = ]119901minus10

1205820 lt 1205820 (49)



Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)


119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










Thus

119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

(50)


119909119903 (119905)

=120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591)119889119904

le120578120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904) int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

120591

0

120596 (120583) 119891 (120583 119909119903 (120583)) 119889120583)119889120591 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

=120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119891 (120591 119909119903 (120591)) 119889120591) 119889119904

le120601119902 (120582119903) 120578 int

1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

+120601119902 (120582119903)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972120582119903 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904

=120601119902 (120582119903)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (1198972 int

119904

0

120596 (120591) 119909119903 (120591) 119889120591) 119889119904


le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972)

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(51)

which implies that

10038171003817100381710038171199091199031003817100381710038171003817 le

10038171003817100381710038171199091199031003817100381710038171003817 120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

(52)

that is

120601119902 (1205821199031198972) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

ge 1 (53)


120582119903 ge1

1198972(

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

1205742)

119901minus1

= 1205820(54)


infin


119897lowast

1 119897lowast


119897lowast

1120601119901 (119909) le 119891 (119905 119909) le 119897

lowast

2120601119901 (119909)


(55)




lowastgt 0

such that

1198910


lowast) (56)













lt 120582lowast







120601and 119891

infin

120601



(i) If 1198910

120601= 0 and 119891

infin

120601= 0 lim




lowast

(ii) If 1198910120601

= 0 and 119891infin




119891 (119905 119909) gt 120577 (forall119905 isin [0 1] 119909 ge 120588) (58)

Since 1198910

120601= 0 there exists 1205761 gt 0 120588

1gt 0 such that



0119892(119904)119889119904) le 1


we have

Ψ (119909) = (120601119902 (120582) 119879119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)


times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 1205761119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578 int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

1

0

120596 (120583) 1205761120601119901 (119909 (120583)) 119889120583)119889120591

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902(120582int

1

0

120596 (120591) 1205761120601119901 (119909 (120591)) 119889120591) 119889119904

le1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

times 120601119902 (120582int

1

0

120596 (120591) 1205761120601119901 (119909) 119889120591)

=120601119902 (1205821205761) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 le 119909

(60)

This implied


(61)

Let 1205883gt max120588120590 120588

1 and 120582

lowast= (1120577)(120588

3Γ(120572)1205901205741)


3

we have 120590119909 le 119909 le 1205883and

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

gt120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120577 119889120591) 119889119904

gt120590120601119902 (120582120577)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120590120601119902 (120582120577) 1205741

Γ (120572)= 1205883

(62)

which implies


(63)


120601= 0 there exist 1205762 gt 0 120588


that

119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882) (64)


0119892(119904)119889119904) le 1



(65)



Ω1205881

) and1199092 isin 119875 cap (Ω120588

2

Ω1205883


0

120601= 0 and 119891

infin


1205761 1205762 1205881 1205882 gt 0 such that


119891 (119905 119909) le 1205762120601119901 (119909) (forall119905 isin [0 1] 119909 ge 1205882)

(66)



max1205881le119909le120588

2

119891 (119905 119909)

120601119901 (119909)=

119891 (119905 119909)

120601119901 (119909) (67)

Let Υ = max1205761 1205762 119891(119905 119909)120601119901(119909) gt 0 then



120582 = (1Υ)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1


times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










times 120601119902 (120582int

1

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582Υ) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

(69)


0119892(119904)119889119904) =

(120582Υ)119902minus1

1205742Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904) lt (120582 Υ)

119902minus11205742Γ(120572+1)(1minus

120578 int1


proof of (ii)


119891(119905 119909) = 119891infin isin





120601= +infin and 119891

infin



(0 120582lowast0)(ii) If 1198910

120601= +infin and 119891

infin





such that





and

(Ψ119909) (119905)

=1

Γ (120572)

times int

1

0

(1 minus 119904 )120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

ge120590120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge120590

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119871lowast1120601119901 (119909 (120591)) 119889120591) 119889119904

ge1205902120601119902 (120582119871lowast1) 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=1205902120601119902 (120582119871lowast1) 1205742

Γ (120572)119909

(71)


which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










which implies that


(72)





1205902120601119902 (120582119871lowast2) 1205742

Γ (120572)ge 1 (74)


we have

119909 (119905) ge 120590 119909 = 1205901199032 gt 1199032 (75)



(76)

Let 1199031 lt 1199033 lt 11990321198721199033

= max119891(119905 119909) | 119905 isin [0 1] 119909 le 1199033+1 gt

0 and 120582lowast0 = (11198721199033

)(Γ(120572+1)1199033(1minus120578 int1

0119892(119904)119889119904)1205742)


3

we have

119891 (119905 119909) le 1198721199033

forall119905 isin [0 1] (77)

Hence

(Ψ119909) (119905)

=1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

minus int

119905

0

(119905 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

+120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583)1198721199033

119889120583)119889120591119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591)1198721199033

119889120591)119889119904

le120578120601119902 (120582119872119903

3

) int1

0119892 (119904) 119889119904

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (int

1

0

120596 (120583) 119889120583)119889120591

+120601119902 (120582119872119903

3

)

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

1

0

120596 (120591) 119889120591) 119889119904

=1205742120601119902 (120582119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

lt1205742120601119902 (120582lowast0119872119903

3

)

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

= 1199033

(78)


This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










This implies that


(79)


3

Ω1199031

) and1199092 isin 119875 cap (Ω119903

2

Ω1199033




then Υ gt 0 and




119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) Υ120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582Υ) 120590 119909

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582Υ) 1205741120590

Γ (120572)119909

(81)


1205741120590Γ(120572) gt

(120582Υ)119902minus1





[0 1] and one of 1198910120601

= 0 and 119891infin




120601= +infin and 119891

infin






(i) If 1198910120601

= 0 and 119891infin



120601= infin and 119891

infin




Let 120582 gt 0 since 1198910




1205742120582119902minus1

119897119902minus1

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

le 1 (83)




From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










From 119891infin



1205902120582119902minus1

119871119902minus1

1205741

Γ (120572)ge 1 (86)




If 0 lt 1198910


infin



and 0 lt 120582 lt 2



infin

120601lt infin that


such that1120601119901 (119909) le 119891 (119905 119909) le 1120601119901 (119909)

(forall119905 isin [0 1] 0 le 119909 le 1199031)

2120601119901 (119909) le 119891 (119905 119909) le 2120601119901 (119909)

(forall119905 isin [0 1] 119909 ge 1199032)

(88)

Let = max119891(119905 119909)120601119901(119909) 1199031 le 119909 le 1199032 0 le 119905 le 1 =


120601119901 (119909) le 119891 (119905 119909) le 120601119901 (119909) (89)



(1)(Γ(120572 + 1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)

119901minus1


119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

times120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

ge1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

times 120601119902 (120582int

119904

0

120596 (120591) 120601119901 (120590 119909 119889120591)) 119889119904

ge120601119902 (120582) 120590 119909

Γ (120572)

times int

1

0

(1 minus 119904)120572minus1

120601119902 (int

119904

0

120596 (120591) 119889120591) 119889119904

=120601119902 (120582) 1205741120590

Γ (120572)119909

gt120601119902 (1 ) 1205741120590

Γ (120572)119909 = 119909

119909 = 119909 (0)

=120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times (int

1

0

(1 minus 120591)120572minus1

120601119902

times (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591

minus int

119904

0

(119904 minus 120591)120572minus1

120601119902

times(120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591)119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904


le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le120578

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

119892 (119904)

times int

1

0

(1 minus 120591)120572minus1

times 120601119902 (120582int

120591

0

120596 (120583) 119891 (120583 119909 (120583)) 119889120583)119889120591 119889119904

+1

Γ (120572)int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

=1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

119904

0

120596 (120591) 119891 (120591 119909 (120591)) 119889120591) 119889119904

le1

Γ (120572) (1 minus 120578 int1

0119892 (119904) 119889119904)

times int

1

0

(1 minus 119904)120572minus1

120601119902

times (120582int

1

0

120596 (120591) 120601119901 (119909 (120591)) 119889120591) 119889119904

=120601119902 (120582) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909

lt120601119902 (2) 1205742

Γ (120572 + 1) (1 minus 120578 int1

0119892 (119904) 119889119904)

119909 = 119909

(90)


4 Examples



(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901minus1 arctan119909 = 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(91)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901minus1 arctan119909 120578 = 12 and 119892(119905) = 12


lowast

2= radic21205872 1205741 = int

1

0(1minus

119904)32

1206013(int119904

0120591 119889120591)119889119904 asymp 00043 1205742 = 1206013(int

1

0120591 119889120591) = 025 120590 asymp

01429 120582lowast

0= (1119897

lowast

1)(Γ(120572)1205901205741)

119901minus1asymp 25149 120582

lowast

0= (1119897

lowast

2)(Γ(120572+

1)(1 minus 120578 int1

0119892(119904)119889119904)1205742)





(120601119901 (11986352

0+119909 (119905)))

1015840

+ 120582119905(1 + 1199052)12

119909119901119890minus119909

= 0

0 lt 119905 lt 1

1199091015840(0) = 119909

10158401015840(0) = 0

119909 (1) =1

2int

1

0

1

2119909 (119904) 119889119904

(92)

where 120572 = 52 119901 = 32 119902 = 3 120596(119905) = 119905 119891(119905 119909) = (1 +

1199052)12

119909119901119890minus119909

120578 = 12 and 119892(119905) = 12

1198910

120601= lim119909rarr0

119891 (119905 119909)

120601119901 (119909)= lim119909rarr0

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


1198910


119891 (119905 119909)

120601119901 (119909)= lim119909rarrinfin

(1 + 1199052)12

119909119901119890minus119909

119909119901minus1= 0


(93)

Let 1205881= 12 120588

2= 4 then

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 1

2radic119890le

1

radic2119890

forall119905 isin [0 1] 119909 isin [01

2]

119891 (119905 119909)

120601119901 (119909)le (1 + 119905

2)12 4

1198904le

4radic2

1198904


(94)




= radic2119890minus1 1205742 = 025


and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and 120582 = (1Υ)(Γ(120572+1)(1minus120578 int1

0119892(119904)119889119904)1205742)

119901minus1= 3radic5120587

141198904



5 Conclusions




Acknowledgments


References



































Volume 2014




Journal of











Journal of


Function Spaces






Algebra



















Volume 2014




Journal of











Journal of


Function Spaces






Algebra









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