research article periodic solutions for nonlinear integro

Research ArticlePeriodic Solutions for Nonlinear Integro-Differential Systemswith Piecewise Constant Argument

Kuo-Shou Chiu

Departamento de Matematica Facultad de Ciencias Basicas Universidad Metropolitana de Ciencias de la Educacion Santiago Chile

Correspondence should be addressed to Kuo-Shou Chiu kschiuumcecl

Received 31 August 2013 Accepted 10 October 2013 Published 12 January 2014

Academic Editors R Adams A Ibeas and M Inc

Copyright copy 2014 Kuo-Shou Chiu This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

We investigate the existence of the periodic solutions of a nonlinear integro-differential system with piecewise alternately advancedand retarded argument of generalized type in short DEPCAG that is the argument is a general step function We consider thecritical case when associated linear homogeneous system admits nontrivial periodic solutions Criteria of existence of periodicsolutions of such equations are obtained In the process we use Greenrsquos function for periodic solutions and convert the givenDEPCAG into an equivalent integral equation Then we construct appropriate mappings and employ Krasnoselskiirsquos fixed pointtheorem to show the existence of a periodic solution of this type of nonlinear differential equations We also use the contractionmapping principle to show the existence of a unique periodic solution Appropriate examples are given to show the feasibility ofour results

1 Introduction

Among the functional differential equationsMyshkis [1] pro-posed to study differential equations with piecewise constantarguments DEPCAThe theory of scalar DEPCA of the type

119889119909 (119905)

119889119905= 119891 (119905 119909 (119905) 119909 (120574 (119905)))

120574 (119905) = [119905] or 120574 (119905) = 2 [119905 + 1

2]

(1)

where [sdot] signifies the greatest integer function was initiatedin [2ndash4] inWiener [5] the first book inDEPCA andhas beendeveloped by many authors [6ndash21] Applications of DEPCAare discussed in [5 22ndash25] They are hybrid equations theycombine the properties of both continuous systems anddiscrete equations Over the years great attention has beenpaid to the study of the existence of periodic solutions ofseveral different types of differential equations For specificreferences see [5ndash7 13 17 18 23 24 26ndash34]

Let Z N R and C be the set of all integers natural realand complex numbers respectively Denote by | sdot | a norm inR119899 119899 isin N Fix two real sequences 119905

119894 120574119894 119894 isin Z such that

119905119894lt 119905119894+1

and 119905119894le 120574119894le 119905119894+1

for all 119894 isin Z and 119905119894rarr plusmninfin as

119894 rarr plusmninfinLet 120574 R rarr R be a step function given by 120574(119905) = 120574

119894for

119905 isin 119868119894= [119905119894 119905119894+1) and consider theDEPCA (1) with this general

120574 In this case we speak of DEPCA of general type in shortDEPCAG Indeed 120574(119905) = [119905] corresponds to 120574

119894= 119905119894= 119894 isin Z

and 120574(119905) = 2[(119905 + 1)2] corresponds to 119905119894= 2119894 minus 1 120574

119894= 2119894

119894 isin Z The particular case of DEPCAG when 120574119894= 119905119894 119894 isin

Z an only delayed situation is considered by first time inAkhmet [8] The other extreme case is the only advancedsituation 120574

119894= 119905119894+1

Any other situation means an alternatelyadvanced anddelayed situationwith 119868+

119894= [119905119894 120574119894] the advanced

intervals and 119868minus119894= [120574119894 119905119894+1) the delayed intervals In [15 16]

Pinto has cleared the importance of the advanced and delayedintervalsThis decompositionwill be present in all our resultsSee [12 13 15 16 23 24 35] The integration or solutionof a DEPCA as proposed by its founders [2ndash4 6] is basedon the reduction of DEPCA to discrete equations To studynonlinear DEPCAG we will use the approach proposed byAkhmet in [9] based on the construction of an equivalentintegral equation but we also remark the clear influence ofthe discrete part and the corresponding difference equationswill be fundamental

Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 514854 14 pageshttpdxdoiorg1011552014514854

2 The Scientific World Journal

In 2008 Akhmet et al [10] obtained some sufficientconditions for the existence and uniqueness of periodicsolutions for the following system

1199091015840

(119905) = 119860 (119905) 119909 (119905) + ℎ (119905) + 120583119892 (119905 119909 (119905) 119909 (120574 (119905)) 120583) (2)

where119860 R rarr R119899times119899 ℎ R rarr R and 119892 RtimesR119899timesR119899times119868 rarr

R119899 are continuous functions 120574(119905) = 119905119894if 119905119894le 119905 lt 119905

119894+1 and 120583 is

a small parameter belonging to an interval 119868 sub R with 0 isin 119868Recently Chiu and Pinto [23] using Poincare operator a

new Gronwall type lemma and fixed point theory obtainedsome sufficient conditions for the existence and uniquenessof periodic (or harmonic) and subharmonic solutions ofquasilinear differential equation with a general piecewiseconstant argument of the form

1199101015840

(119905) = 119860 (119905) 119910 (119905) + 119891 (119905 119910 (119905) 119910 (120574 (119905))) (3)

where 119905 isin R 119910 isin C119901 119860(119905) is a 119901 times 119901 matrix for 119901 isin N119891(119905 119909 119910) is a 119901 dimensional vector and 119891 is continuous inthe first argument and 120574(119905) = 120574

119894 if 119905119894le 119905 lt 119905

119894+1 119894 isin Z

In this paper comparing the three DEPCAG inequalities ofGronwall type and remarked new Gronwall lemma not onlyrequests a weaker condition than the other Gronwall lemmasbut also has a better estimate

It is well-known that there are many subjects in physicsand technology using mathematical methods that depend onthe linear and nonlinear integro-differential equations andit became clear that the existence of the periodic solutionsand its algorithm structure from more important problemsin the present time Where many of studies and researches[36ndash40] dedicates for treatment the autonomous and non-autonomous periodic systems and specially with the integralequations and differential equations and the linear andnonlinear differential and which is dealing in general shapewith the problems about periodic solutions theory and themodern methods in its quality treatment for the periodicdifferential equations

Samoilenko and Ronto [41] assume the numerical-analytic method to study the periodic solutions for ordinarydifferential equations and its algorithm structure and thismethod includes uniform sequences of periodic functionsand the results of that study are using the periodic solutionsonwide range in the difference of new processes industry andtechnology For example Samoilenko and Ronto [41] inves-tigated the existence and approximation of periodic solutionfor nonlinear system of integro-differential equations whichhas the form

1199091015840

(119905) = 119891(119905 119909 (119905) int

119905+119879

119905

119892 (119904 119909 (119904)) 119889119904) (4)

where 119909 isin 119863 sub R119899 119863 is a closed and bounded domainThe vectors functions 119891(119905 119909 119910) and 119892(119905 119909) are continuousfunctions in 119905 119909 119910 and periodic in 119905 of period 119879

Butris [42] investigated the periodic solution of nonlinearsystem of integro-differential equations depending on the

gamma distribution by using the numerical analyticmethodwhich has the form

1199091015840

(119905) = 119891(119905 120573 (119905 120572) 119909 (119905) int

119905+119879

119905

119892 (119904 120573 (119904 120572) 119909 (119904)) 119889119904)

119905 isin R

(5)

where 119909 isin 119863 sub R119899 119863 is a closed and bounded domain Thevector functions119891(119905 120573(119905 120572) 119909) and119892(119905 120573(119905 120572) 119909) are definedon the domain (119905 120573(119905 120572) 119909) isin R times [0 119879] times 119863 times 119863

1

In the current paper we study the existence of periodicsolutions of a nonlinear integro-differential system withpiecewise alternately advanced and retarded argument

1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574 (119905))) 119905 isin R

(6)

where 119860 R rarr R119899times119899 119891 R times R119899 times R119899 rarr R119899 and119892 R times R119899 times R119899 rarr R119899 are continuous in their respectivearguments In the analysis we use the idea of Greenrsquos functionfor periodic solutions and convert the nonlinear integro-differential systems with DEPCAG (6) into an equivalentintegral equationThen we employ Krasnoselskiirsquos fixed pointtheorem and show the existence of a periodic solution of thenonlinear integro-differential systems with DEPCAG (6) inTheorem 12We also obtain the existence of a unique periodicsolution in Theorem 14 employing the contraction mappingprinciple as the basicmathematical tool Furthermore appro-priate examples are given to show the feasibility of our results

In our paperwe assume that the solutions of the nonlinearintegro-differential systems with DEPCAG (6) are continu-ous functions But the deviating argument 120574(119905) is discontin-uous Thus in general the right-hand side of the DEPCAGsystem (6) has discontinuities at moments 119905

119894isin R 119894 isin Z As a

result we consider the solutions of theDEPCAGas functionswhich are continuous and continuously differentiable withinintervals [119905

119894 119905119894+1) 119894 isin Z In other words by a solution 119911(119905)

of the DEPCAG system (6) we mean a continuous functionon R such that the derivative 1199111015840(119905) exists at each point 119905 isinR with the possible exception of the points 119905

119894isin R 119894 isin

Z where a one-sided derivative exists and the nonlinearintegro-differential systemswithDEPCAG (6) are satisfied by119911(119905) on each interval (119905

119894 119905119894+1) 119894 isin Z as well

The rest of the paper is organized as follows In Section 2some definitions and preliminary results are introduced Weshow double 120596-periodicity of Greenrsquos function Section 3 isdevoted to establishing some criteria for the existence anduniqueness of periodic solutions of the DEPCAG system (6)Greenrsquos function and Banach Schauder and Krasnoselskiirsquosfixed point theorems below are fundamental to obtain themain results Furthermore appropriate examples are pro-vided in Section 4 to show the feasibility of our results

The Scientific World Journal 3

2 Greenrsquos Function and Periodicity

In this section we state and define Greenrsquos function for peri-odic solutions of the nonlinear integro-differential systemwith piecewise alternately advanced and retarded argument(6)

Let 119868 be the 119899 times 119899 identity matrix Denote byΦ(119905 119904) Φ(119904 119904) = 119868 119905 119904 isin R the fundamental matrixof solutions of the homogeneous system (7)

For every 119905 isin R let 119894 = 119894(119905) isin Z be the unique integersuch that 119905 isin 119868

119894= [119905119894 119905119894+1)

From now on the following assumption will be needed

(119873120596) The homogenous equation

1199101015840

(119905) = 119860 (119905) 119910 (119905) (7)

does not admit any nontrivial 120596-periodic solution

Remark 1 For 120591 isin R the condition (119873120596) equivalent to the

matrix (119868 minus Φ(120591 + 120596 120591)) is nonsingular

Now we solve the DEPCAG system (6) on 119868119894(120591)

=

[119905119894(120591) 119905119894(120591)+1

)

1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574119894(120591))) 119905 isin [119905

119894(120591) 119905119894(120591)+1

)

(8)

which has the solution given by

119911 (119905) = Φ (119905 120591) 119911 (120591)

+ int

119905

120591

Φ (119905 119904) [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

(9)

For 119905 rarr 119905119894(120591)+1

in (9) we have

119911 (119905119894(120591)+1

) = Φ (119905119894(120591)+1

120591) 119911 (120591)

+ int

119905119894(120591)+1

120591

Φ(119905119894(120591)+1

119904)

times [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+ int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

(10)

and in general by induction for any 119894(119905) ge 119894(120591)

119911 (119905) = Φ (119905 120591) 119911 (120591)

+ int

119905119894(120591)+1

120591

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

+

119894(119905)minus1

sum

119895=119894(120591)+1

int

119905119895+1

119905119895

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119905119895+1

119904

119862 (119904 119906 119911 (120574119895)) 119889119906

+

119894(119904+120596)minus1

sum

119896=119895+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119895)) ] 119889119904

+ int

119905

119905119894(119905)

Φ (119905 119904) [119891(119904 119911 (119904) int

119905119894(119905)+1

119904

119862 (119904 119906 119911 (120574119894(119905))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(119905)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(119905))) ] 119889119904

(11)


On the other hand one can easily see that

int

119905

120591

119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904

= int

119905119894(120591)+1

120591

119892 (119904 119911 (119904) 119911 (120574119894(120591))) 119889119904

+

119894(119905)minus1

sum

119895=119894(120591)+1

int

119905119895+1

119905119895

119892 (119904 119911 (119904) 119911 (120574119895)) 119889119904

+ int

119905

119905119894(119905)

119892 (119904 119911 (119904) 119911 (120574119894(119905))) 119889119904

(12)Then any solution of the DEPCAG system (6) with the initialcondition 119911(120591) = 120585 can be written as

119911 (119905) = Φ (119905 120591) 120585

+ int

119905

120591

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904 120591 isin R

(13)Amongst these solutions that one will be 120596-periodic forwhich 119911(120591) = 120585 = 119911(120591 + 120596) by the condition (119873

120596) and using

(13) we get

120585 = (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(14)

A substitution of (14) into (13) yields

119911 (119905) = Φ (119905 120591) (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

119905

120591

Φ (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(15)

It is easy to check the following identity using properties ofthe functionΦ(119905)

Φminus1

(120591) (119868 minus Φ (120591 + 120596 120591))minus1

Φ (120591 + 120596)

= (Φminus1

(120591 + 120596)Φ(120591) minus 119868)minus1

(16)

It follows that

119911 (119905) = int

119905

120591

Φ (119905) (119868 + (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

)Φ(119904)minus1

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

120591+120596

119905

Φ (119905) (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

Φminus1

(119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(17)

In such a case the DEPCAG system (6) has 120596-periodicsolution 119911(119905) given by the integral equation (17) Beforestudying the existence of solutions of integral equation (17)in the next section firstly we define Greenrsquos function forperiodic solutions of the DEPCAG system (6)

Definition 2 Suppose that the condition (119873120596) holds For each

119905 119904 isin [120591 120591 + 120596] Greenrsquos function for (6) is given by

119866 (119905 119904) = Φ (119905) (119868 + 119863)Φ

minus1

(119904) 120591 le 119904 le 119905 le 120591 + 120596

Φ (119905)119863Φminus1

(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)

where Φ(119905) is a fundamental solution of (7) and

119863 = ((Φminus1

(120591)Φ (120591 + 120596))minus1

minus 119868)minus1

(19)

We note that the condition (119873120596) implies the existence of

the matrix119863To prove double 120596-periodicity of Greenrsquos function119866(119905 119904)

we first give the following lemma

Lemma 3 Suppose that the condition (119873120596) holds Let the

matrix 119863 be defined by (19) and then one has the followingidentities

(119868 + 119863) = (Φminus1

(120591)Φ (120591 + 120596))minus1

119863

= (119868 minus Φminus1

(120591)Φ (120591 + 120596))minus1

Φ (119905)119863Φminus1

(119905 + 120596) minus Φ (119905)119863Φminus1

(119905) = 119868

(119868 + 119863)Φminus1

(120591 minus 120596)Φ (120591) = 119863

(20)


For Lemma 3we can prove an important property double120596-periodicity of Greenrsquos function 119866(119905 119904) to study after theexistence of periodic solutions

Lemma 4 Suppose that the condition (119873120596) holds Then

Greenrsquos function119866(119905 119904) is double120596-periodic that is119866(119905+120596 119904+120596) = 119866(119905 119904)

3 Existence of Periodic Solutions

In this section we prove the main theorems of this paperso we recall the nonlinear integro-differential systems withDEPCAG (6)

1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574 (119905))) 119905 isin R

(21)

For this a natural Banach space is

P120596= 120601 R 997888rarr R

119899

| 120601 is 120596-periodic

continuous function(22)

with the supremum norm

119911 = sup119905isinR

|119911 (119905)| = sup119905isin[120591120591+120596]

|119911 (119905)| (23)

Consider 119891 RtimesR119899 timesR119899 rarr R119899 and 119892 RtimesR119899 timesR119899 rarr

R119899 are continuous functions Moreover we will refer to thefollowing specific conditions

Continuous Condition(119862) Let 119877 gt 0 119905 119904 isin R and 119910

1 1199102isin R119899 |119910

119894| le 119877 119894 = 1 2

For any 120598 gt 0 there exist 120575 gt 0 and 120582 R2 rarr [0infin)

a function such that |1199101minus 1199102| le 120575 implies

1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120598120582 (119905 119904) 119905 119904 isin R (24)

where sup119905isinR int

119905+120596

119905

120582(119905 119904)119889119904 =

Lipschitz Conditions(119871119862) There exists a continuous function 120582 R2 rarr [0infin)

such that 119905 119904 isin R and for any 1199101 1199102isin R119899 we have

1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 (25)

Moreover sup119905isinR int119905+120596

119905

120582(119905 119904)119889119904 = and

sup119905isinR int119905+120596

119905

|119862(119905 119904 0)|119889119904 = 0

(119871119891) For 119905 isin R and119909

1119910111990921199102isin R119899 there exist functions

1199011 1199012 R rarr [0infin) such that

1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 1199092 1199102)1003816100381610038161003816

le 1199011(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1199012 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(26)

Moreover 120572 = max119905isinR

|119891(119905 0 0)|

int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904 le 119871

1

1205781= sdot max119905isin[120591120591+120596]

10038161003816100381610038161199012 (119905)1003816100381610038161003816

120596 isin R+ 120591 isin R

(27)

(119871119892) For 119905 isin R and119909



1003816100381610038161003816119892 (119905 1199091 1199101) minus 119892 (119905 1199092 1199102)1003816100381610038161003816

le 1205921(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1205922 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(28)


|119892(119905 0 0)| and

int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904 le 119871

2 120596 isin R

+ 120591 isin R (29)

Invariance Conditions

(119872119862) For every 119877 gt 0 119905 119904 isin R |119910| le 119877 there exist 120582 ℎ

R2 rarr [0infin) functions and positive constants ℎfor which

1003816100381610038161003816119862 (119905 119904 119910)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101003816100381610038161003816 + ℎ (119905 119904) (30)

where sup119905isinR int119905+120596

119905

120582(119905 119904)119889119904 = and sup119905isinR int119905+120596

119905

ℎ(119905 119904)119889119904 = ℎ

(119872119891) For every 119877 gt 0 119905 isin R |119909| |119910| le 119877 there existfunctions 119898

1 1198982

R rarr [0infin) and positiveconstants 120588

1 1198621 1205782for which

1003816100381610038161003816119891 (119905 119909 119910)1003816100381610038161003816 le 1198981 (119905) |119909| + 1198982 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205881 (31)

where int120591+120596

120591

[1198981(119904) + 119898

2(119904)]119889119904 le 119862

1and 120578

2= ℎ sdot

max119905isin[120591120591+120596]

|1198982(119905)| 120596 isin R

+ 120591 isin R


1 1205812 R rarr [0infin) and positive constants

1205882 1198622for which

1003816100381610038161003816119892 (119905 119909 119910)1003816100381610038161003816 le 1205811 (119905) |119909| + 1205812 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205882 (32)

where int120591+120596120591

[1205811(119904) + 120581

2(119904)]119889119904 le 119862

2 120596 isin R

+ 120591 isin R

Periodic Conditions

(119875) There exists 120596 gt 0 such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909

2 1199102) are periodic

functions in 119905 with a period 120596 for all 119905 ge 120591(2) 119862(119905 + 120596 119904 + 120596 119909

1) = 119862(119905 119904 119909

1) for all 119905 ge 120591


(3) there exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596 119901) condition that

is

119905119894+119901

= 119905119894+ 120596 120574

119894+119901= 120574119894+ 120596 for 119894 isin Z (33)

Remark 5 Note that (120596 119901) condition is a discrete relationwhich moves the interval 119868

119894into 119868

119894+119901 Then we have the

following consequences

(i) For any 120591 isin R the interval [120591 120591 + 120596] can bedecomposed as follows

[120591 119905119894(120591)+1

] cup

119894(120591)+119901minus1

⋃

119895=119894(120591)+1

119868119895cup [119905119894(120591)+119901

120591 + 120596] (34)

(ii) For 119905 isin [119905119894 119905119894+1) we have

(a) 119905 + 120596 isin [119905119894+119901 119905119894+119901+1

) (b) 120574 (119905) + 120596 isin [119905119894+119901 119905119894+119901+1

)

(35)

Then

120574 (119905 + 120596) = 120574119894(119905+120596)

= 120574119894(119905)+119901

= 120574119894(119905)

+ 120596 = 120574 (119905) + 120596 (36)

Using Definition 2 Remark 5 and double 120596-periodicityof Greenrsquos function similar formula is given by (17) So wehave obtained the following result

Proposition 6 Suppose that the conditions (119873120596) and (P) hold

Let (120591 119911(120591)) isin R times R119899 Then 119911(119905) = 119911(119905 120591 119911(120591)) is 120596-periodicsolution on R of the DEPCAG system (6) if and only if 119911(119905) is120596-periodic solution of the integral equation

119911 (119905) = int

120591+120596

120591

119866 (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(37)

where Greenrsquos function 119866(119905 119904) is defined by (18)

Consider the operatorI P120596rarr P120596by

(I119911) (119905) = int120591+120596

120591

119866 (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(38)

It is easy to see that the DEPCAG system (6) has 120596-periodic solution if and only if the operator I has one fixedpoint in P

120596

To prove some existence criteria for 120596-periodic solutionsof the DEPCAG system (6) we use the Banach Schauder andKrasnoselskiirsquos fixed point theorems

Next we state first Krasnoselskiirsquos fixed point theoremwhich enables us to prove the existence of a periodic solutionFor the proof of Krasnoselskiirsquos fixed point theorem we referthe reader to [43]

Theorem A (Krasnoselskiirsquos fixed point theorem) Let 119878 bea closed convex nonempty subset of a Banach space (119864 sdot )Suppose thatA andBmap 119878 into 119864 such that

(i) 119909 119910 isin 119878 impliesA119909 +B119910 isin 119878

(ii) A is a contraction mapping

(iii) B is completely continuous

Then there exists 119911 isin 119878 with 119911 = A119911 +B119911

Remark 7 Krasnoselskiirsquos theorem may be combined withBanach and Schauderrsquos fixed point theorems In a certainsense we can interpret this as follows if a compact operatorhas the fixed point property under a small perturbationthen this property can be inherited The theorem is usefulin establishing the existence results for perturbed operatorequations It also has awide range of applications to nonlinearintegral equations of mixed type for proving the existence ofsolutions Thus the existence of fixed points for the sum oftwo operators has attracted tremendous interest and theirapplications are frequent in nonlinear analysis See [32 3343ndash46]

We note that to apply Krasnoselskiirsquos fixed point theoremwe need to construct two mappings one is contraction andthe other is compact Therefore we express (38) as

(I119911) (119905) = (A119911) (119905) + (B119911) (119905) (39)

whereA B P120596rarr P120596are given by

(A119911) (119905)

= int

120591+120596

120591

119866 (119905 119904) 119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)119889119904

(40)

(B119911) (119905) = int

120591+120596

120591

119866 (119905 119904) 119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904 (41)

To simplify notations we introduce the following constantand sets

119888119866= max119905119904isin[120591120591+120596]

|119866 (119905 119904)| S = 119911 isin P120596 119911 le 119877

CR = 119911 isin R119899

119911 le 119877

(42)

Lemma8 If (119873120596) (119875) (119871

119891) and (119871

119862) holdA is given by (40)

with 1198881198661198711lt 1 and thenA is a contraction mapping

Proof Let A be defined by (40) First we want to show that(A120593)(119905 + 120596) = (A120593)(119905)


Let 120593 isin P120596 Then using (40) the periodicity of Greenrsquos

function and (120596 119901) condition we arrive at

(A120593) (119905 + 120596)

= int

120591+2120596

120591+120596

119866 (119905 + 120596 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+2120596

119904+120596

119862 (119904 + 120596 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906 + 120596))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906) + 120596)) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= (A120593) (119905)

(43)

Secondly we show thatA is a contractionmapping Let120593 120577 isinP120596 then we have

1003817100381710038171003817A120593 minusA1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816A120593 (119905) minusA120577 (119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904) int

119904+120596

119904

120582 (119904 119906)

times1003816100381610038161003816120593 (120574 (119906)) minus 120577 (120574 (119906))

1003816100381610038161003816 119889119906] 119889119904

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904) int

119904+120596

119904

120582 (119904 119906) 119889119906] 119889119904)

times1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 1198881198661198711

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

(44)

HenceA defines a contraction mapping

Similarly B is given by (41) which may be also acontraction operator

Lemma 9 If (119873120596) (119875) and (119871

119892) holdB is given by (41)with

1198881198661198712lt 1 and thenB is a contraction mapping

Lemma 10 If (119873120596) holdsB is defined by (41) and thenB is

completely continuous that isB is continuous and the imageofB is contained in a compact set

Proof Step 1 First we prove that B P120596

rarr P120596is

continuousAs the operator A a change of variable in (41) we have

(B120593)(119905 + 120596) = (B120593)(119905) Now we want to show B iscontinuous

The function 119892(119905 119909 119910) is uniformly continuous on [120591 120591 +120596] times CR times CR and by the periodicity in 119905 the function119892(119905 119909 119910) is uniformly continuous on R times CR times CR Thusfor any 1205981015840 = (120598119888

119866120596) gt 0 there exists 120575 = 120575(120598) gt 0 such

that 1199111 1199112isin S 119911

1minus 1199112 le 120575 implies |119892(119905 119911

1(119905) 1199111(120574(119905))) minus

119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598

1015840 for 119905 isin [120591 120591 + 120596] Then B1199111minus

B1199112 le 120598 In fact by the continuity of 119892

1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840

for 119905 isin [120591 120591 + 120596](45)


and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)

and the continuity ofB is proved

Step 2 We show that the image of B is contained in acompact set

Let 119909 119910 isin CR and 119904 isin [120591 120591 + 120596] for the continuity of thefunction 119892(119904 119909 119910) there exists119872 gt 0 such that |119892(119904 119909 119910)| le119872 Let 120593

119899isin S where 119899 is a positive integer then we have

1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)

Moreover a direct calculation (B120593119899(119905))1015840 shows that

(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904

minus Φ (119905)119863Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)

As 119860(119905) is bounded on [120591 120591 + 120596] and B120593119899(119905)

119892(119905 120593119899(119905) 120593119899(120574(119905)) are bounded on [120591 120591 + 120596] times S times S

Thus the above expression yields (B120593119899)1015840

le 119871 for some

positive constant 119871 Hence the sequence (B120593119899) is uniformly

bounded and equicontinuous Ascoli-Arzelarsquos theoremimplies that a subsequence (B120593

119899119896) of (B120593

119899) converges

uniformly to a continuous 120596-periodic function Thus B iscontinuous andB(S) is a compact set

In a similar way forA we obtain the following

Lemma 11 If (119873120596) and (119862) hold A is defined by (40) and

thenA is completely continuous

Theorem 12 Suppose the hypotheses (119873120596) (119875) (119871

119891) (119871119862)

(119872119892) hold Let119877 be a positive constant satisfying the inequality

119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)

Then the DEPCAG system (6) has at least one 120596-periodicsolution in S

Proof By Lemma 8 the mapping A is a contraction and itis clear that A P

120596rarr P

120596 Also from Lemma 10 B is

completely continuousNext we prove that if 120593 120577 isin S with 120593 le 119877 and 120577 le 119877

then A120601 +B120577 le 119877Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then

1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)

We now see that all the conditions of Krasnoselskiirsquostheorem are satisfied Thus there exists a fixed point 119911 in S

such that 119911 = A119911 +B119911 By Proposition 6 this fixed point isa solution of the DEPCAG system (6) Hence the DEPCAGsystem (6) has 120596-periodic solution

By the symmetry of the conditions we will obtain asTheorem 12

Theorem 13 Suppose the hypothesis (119873120596) (119875) (119871

119892) (119872119891)

(119872119862) (119862) hold Let 119877 be a positive constant satisfying the

inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)


By Lemma 9 the mapping B is a contraction and it isclear that B P

120596rarr P

120596 Also from Lemma 11 A is


then A120601 +B120577 le 119877

Let 120593 120577 isin S with 120593 le 119877 and 120577 le 119877 Then

1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)

Applying Banachrsquos fixed point theorem we have thefollowing



119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)

then the DEPCAG system (6) has a unique120596-periodic solution

Proof Let the mappingI be given by (38) For 120593 120577 isin P120596 in

view of (38) we have1003817100381710038171003817I120593 minusI120577

1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)

This completes the proof by invoking the contractionmapping principle

As a direct consequence of the method Schauderrsquos theo-rem implies the following


119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)


Remark 16 Considering the DEPCAG system (6)

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)

Krasnoselskii (see [47]) proved that if 119860 is a stable constantmatrix without piecewise alternately advanced and retardedargument and lim

|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then

the system (56) has at least one periodic solution In our caseapplyingTheorem 12 this result is also valid for the DEPCAGsystem (56) requiring the hypothesis (119873

120596) (1198753) (119872119892) 119860(119905)

and 119892(119905 119909 119910) are periodic functions in 119905 with a period 120596 forall 119905 ge 120591 and |119892(119905 119909 119910)| le 119888

1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2

and 1205882constants for |119909| + |119910| le 119877 119877 gt 0

Remark 17 Considering the nonlinear system of differentialequations with a general piecewise alternately advanced andretarded argument

1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)

In this particular case applyingTheorem 13 this result is alsovalid for the DEPCAG system (57) requiring the hypothesis(119873120596) (1198751) (1198753) (119871119892) (119872119891) and 119888

119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1

Remark 18 Suppose that (1198751) is satisfied by 120596 = 120596

1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a

rational number for all 119894 119895 = 1 2 3 then (1198751) (1198752) and

(1198753) are simultaneously satisfied by 120596 = lcm120596

1 1205962 1205963

where lcm1205961 1205962 1205963 denotes the least common multiple

between 1205961 1205962and 120596

3 In the general case it is possible

that there exist five possible periods 1205961for 119860 120596

2for 119891 120596

3

for 119892 1205964for 119862 and the sequences 119905

119894119894isinZ 120574119894119894isinZ satisfy the

(1205965 119901) condition If 120596

119894120596119895is a rational number for all 119894 119895 =

1 2 3 4 5 so in this situation our results insure the existenceof 120596-periodic solution with 120596 = lcm120596

1 1205962 1205963 1205964 1205965

Therefore the above results insure the existence of 120596-periodicsolutions of the DEPCAG system (6) These solutions arecalled subharmonic solutions See Corollaries 19ndash22

To determine criteria for the existence and uniqueness ofsubharmonic solutions of theDEPCAG system (6) fromnowon we make the following assumption

(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895

which is a rational number for all 119894 119895 = 1 2 3 4 5

such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909


functions in 119905 with a period 1205961 1205962 and 120596

3

respectively for all 119905 ge 120591(2) 119862(119905 + 120596

4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591

(3) There exists 119901 isin Z+ for which the sequences119905119894119894isinZ 120574119894119894isinZ satisfy the (120596

5 119901) condition

As immediate corollaries of Theorems 12ndash15 and Remark 18the following results are true


Corollary 19 Suppose the hypotheses (119873120596) (119875120596) (119871119891) (119871119862)

(119872119892) and (49) hold Then the DEPCAG system (6) has at least

one subharmonic solution in S


(119872119862) (119862) and (51) hold Then the DEPCAG system (6) has at

least one subharmonic solution in S


(119871119892) and (53) holdThen theDEPCAG system (6) has a unique

subharmonic solution

Corollary 22 Suppose the hypotheses (119873120596) (119875120596) (119872

119891)

(119872119862) (119872119892) (119862) and (55) hold Then the DEPCAG system (6)

has at least one subharmonic solution in S

4 Applications and Illustrative Examples

Wewill introduce appropriate examples in this sectionTheseexamples will show the feasibility of our theory

Mathematical modelling of real-life problems usuallyresults in functional equations like ordinary or partial differ-ential equations integral and integro-differential equationsand stochastic equations Many mathematical formulationsof physical phenomena contain integro-differential equa-tions these equations arise inmany fields like fluid dynamicsbiologicalmodels and chemical kinetics So we first considernonlinear integro-differential equations with a general piece-wise constant argument mentioned in the introduction andobtain some new sufficient conditions for the existence of theperiodic solutions of these systems

Example 1 Let 120582 R2 rarr [0infin) and ℎ R2 rarr [0infin) betwo functions satisfying

sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)

Consider the following nonlinear integro-differential equa-tions with piecewise alternately advanced and retarded argu-ment of generalized type

1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)

where the sequences 119905119894119894isinZ and 120574

119894119894isinZ satisfy the (2120587 119901) con-

dition 120582 and ℎ are double 2120587-periodic continuous functionsand 119886 is a 2120587-periodic continuous function satisfying (119873

120596)

The conditions of Theorem 15 are fulfilled Indeed

(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|

(ii) 119892(119905 119909 119910) = (sin 119905)1199094 + (1 + cos2119905)1199109 satisfies (119872119892)

|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588

2for every 119909 119910 such

that |119909| |119910| le 119877 uniformly in 119905 isin R where 41205871198881(119877) le

1198622

(iii) 119862(119905 119904 119910) = ln[1 + |1199103|120582(119905 119904) + ℎ(119905 119904)] satisfies (119872119862)

|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621

(iv) 119892 R times R119899 times R119899 rarr R119899 is continuous and 119862(119905 119904 119910)satisfies (119862)

Indeed for any 120576 gt 0 there exists 120575 gt 0 such that |1199101minus1199102| le 120575

implies

1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 1205761198882 (119877) 120582 (119905 119904) for 119905 119904 isin R

(60)

Furthermore there exists 119877 such that

exp (2119886lowast120587)exp (2119886

lowast120587) minus 1

[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)

where 119886lowast = sup119905isinR |119886(119905)| and 119886

lowast= inf119905isinR|119886(119905)|

Then by Theorem 15 the DEPCAG system (59) has atleast one 2120587-periodic solution

Example 2 Thus many examples can be constructed whereour results can be applied

Let Λ R rarr R119899times119899 and 120583 R rarr R119899 be two functionssatisfying

sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)

Now consider the integro-differential system with piecewisealternately advanced and retarded argument

1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)


119894119894isinZ satisfy the (120596 119901)

condition 119860 119861 Λ 120581 and 120583 are 120596-periodic continuousfunctions and

(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)

(ii) 119861 is 120596-periodic matrix function |119861(119905)| le 119887 119892 R119899 times

R119899 rarr R119899 is a continuous function and |119892(119909 119910)| le1198881(119877)|119909| + 119888

2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2

(iii) |120581(119909) minus 120581(119910)| le 1198883(119877)|119909 minus 119910| where 119888

3(119877)Λ120596 le 119871

1

Thehypotheses ofTheorem 12 are fulfilledThen if there exists119877 such that

119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)

Theorem 12 implies that there exists at least a 120596-periodicsolution of the DEPCAG system (63)


Note that similar results can be obtained under (119871119892)

and (119872119862) On the other hand the periodic situation of the

DEPCAGsystem (56) and (57) can be treated in the samewayLet us consider another example for second-order differ-

ential equations with a general piecewise constant argumentIn this case we can show the existence and uniquenessof periodic solutions of the following nonlinear DEPCAGsystem

Example 3 Consider the following nonlinear DEPCAG sys-tem

11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1

119894 isin Z and thesequences 119905

119894119894isinZ and 120574

119894119894isinZ satisfy the (2120587120596 119901) condition

We write the DEPCAG system (65) in the system form

1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)

It is easy to see that the linear homogenous system1199111015840

(119905) = 119860119911(119905) does not admit any nontrivial 120596-periodicsolution that is the condition (119873

120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R

+satisfies the condition

2119888119866119877 (1205811+ 1205812119877) lt 1 (68)

Then for 120593 120595 isin S we have1003817100381710038171003817119892 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119892 (sdot 120595 (sdot) 120595 (120574 (sdot)))

1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)

In a similar way for 119891 we have1003817100381710038171003817119891 (sdot 120593 (sdot) 120593 (120574 (sdot))) minus 119891 (sdot 120595 (sdot) 120595 (120574 (sdot)))

1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)

ByTheorem 14 the DEPCAG system (65) has a unique 2120587120596-periodic solution in S

Conflict of Interests

The author declares that there is no conflict of interests re-garding the publication of this paper

Acknowledgment

This research was in part supported by FIBE 01-12 DIUMCE

References

[1] ADMyshkis ldquoOn certain problems in the theory of differentialequations with deviating argumentsrdquo Uspekhi Matematich-eskikh Nauk vol 32 pp 173ndash202 1977

[2] K L Cooke and JWiener ldquoRetarded differential equations withpiecewise constant delaysrdquo Journal of Mathematical Analysisand Applications vol 99 no 1 pp 265ndash297 1984

[3] K L Cooke and J Wiener ldquoAn equation alternately of retardedand advanced typerdquo Proceedings of the American MathematicalSociety vol 99 pp 726ndash732 1987

[4] S M Shah and J Wiener ldquoAdvanced differential equations withpiecewise constant argument deviationsrdquo International JournalofMathematics andMathematical Sciences vol 6 no 4 pp 671ndash703 1983

[5] J WienerGeneralized Solutions of Functional Differential Equa-tions World Scientific Singapore 1993

[6] A R Aftabizadeh and J Wiener ldquoOscillatory and periodicsolutions of an equation alternately of retarded and advancedtypesrdquo Applicable Analysis vol 23 no 3 pp 219ndash231 1986

[7] A R Aftabizadeh J Wiener and J M Xu ldquoOscillatory andperiodic solutions of delay differential equations with piecewiseconstant argumentrdquo Proceedings of the American MathematicalSociety vol 99 pp 673ndash679 1987

[8] M U Akhmet ldquoIntegral manifolds of differential equationswith piecewise constant argument of generalized typerdquo Nonlin-ear Analysis Theory Methods and Applications vol 66 no 2pp 367ndash383 2007

[9] M U Akhmet Nonlinear Hybrid ContinuousDiscrete-TimeModels Atlantis Press Amsterdam The Netherlands 2011

[10] M U Akhmet C Buyukadali and T Ergenc ldquoPeriodic solu-tions of the hybrid system with small parameterrdquo NonlinearAnalysis Hybrid Systems vol 2 no 2 pp 532ndash543 2008

[11] A Cabada J B Ferreiro and J J Nieto ldquoGreenrsquos functionand comparison principles for first order periodic differentialequations with piecewise constant argumentsrdquo Journal of Math-ematical Analysis and Applications vol 291 no 2 pp 690ndash6972004

[12] K S Chiu and M Pinto ldquoVariation of parameters formula andGronwall inequality for differential equations with a generalpiecewise constant argumentrdquo Acta Mathematicae ApplicataeSinica vol 27 no 4 pp 561ndash568 2011


[13] K S Chiu and M Pinto ldquoOscillatory and periodic solutionsin alternately advanced and delayed differential equationsrdquoCarpathian Journal of Mathematics vol 29 no 2 pp 149ndash1582013

[14] J J Nieto and R Rodrıguez-Lopez ldquoGreenrsquos function forsecond-order periodic boundary value problemswith piecewiseconstant argumentsrdquo Journal of Mathematical Analysis andApplications vol 304 no 1 pp 33ndash57 2005

[15] M Pinto ldquoAsymptotic equivalence of nonlinear and quasi lin-ear differential equations with piecewise constant argumentsrdquoMathematical and Computer Modelling vol 49 no 9-10 pp1750ndash1758 2009

[16] M Pinto ldquoCauchy and Green matrices type and stability inalternately advanced and delayed differential systemsrdquo Journalof Difference Equations and Applications vol 17 no 2 pp 235ndash254 2011

[17] G Q Wang ldquoExistence theorem of periodic solutions for adelay nonlinear differential equation with piecewise constantargumentsrdquo Journal of Mathematical Analysis and Applicationsvol 298 no 1 pp 298ndash307 2004

[18] G Q Wang and S S Cheng ldquoExistence and uniqueness ofperiodic solutions for a second-order nonlinear differentialequationwith piecewise constant argumentrdquo International Jour-nal ofMathematics andMathematical Sciences vol 2009 ArticleID 950797 14 pages 2009

[19] Y H Xia Z Huang and M Han ldquoExistence of almost periodicsolutions for forced perturbed systems with piecewise constantargumentrdquo Journal of Mathematical Analysis and Applicationsvol 333 no 2 pp 798ndash816 2007

[20] P Yang Y Liu and W Ge ldquoGreenrsquos function for second orderdifferential equations with piecewise constant argumentsrdquoNon-linear Analysis Theory Methods and Applications vol 64 no 8pp 1812ndash1830 2006

[21] R Yuan ldquoThe existence of almost periodic solutions of retardeddifferential equations with piecewise constant argumentrdquo Non-linear Analysis Theory Methods and Applications vol 48 no 7pp 1013ndash1032 2002

[22] A I Alonso J Hong and R Obaya ldquoAlmost periodic typesolutions of differential equations with piecewise constant argu-ment via almost periodic type sequencesrdquo Applied MathematicsLetters vol 13 no 2 pp 131ndash137 2000

[23] K S Chiu and M Pinto ldquoPeriodic solutions of differentialequations with a general piecewise constant argument andapplicationsrdquo Electronic Journal of Qualitative Theory of Differ-ential Equations vol 46 pp 1ndash19 2010

[24] K S Chiu M Pinto and J C Jeng ldquoExistence and globalconvergence of periodic solutions in recurrent neural networkmodels with a general piecewise alternately advanced andretarded argumentrdquo Acta Applicandae Mathematicae 2013

[25] L Dai Nonlinear Dynamics of Piecewise of Constant Systemsand Implememtation of Piecewise Constants Arguments WorldScientific Singapore 2008

[26] T A Burton Stability and Periodic Solutions of Ordinary andFunctional Differential Equations Academic Press New YorkNY USA 1985

[27] T A Burton and T Furumochi ldquoPeriodic and asymptoticallyperiodic solutions of neutral integral equationsrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 10pp 1ndash19 2000

[28] T A Burton and T Furumochi ldquoExistence theorems andperiodic solutions of neutral integral equationsrdquo NonlinearAnalysis Theory Methods and Applications vol 43 no 4 pp527ndash546 2001

[29] F Karakoc H Bereketoglu and G Seyhan ldquoOscillatory andperiodic solutions of impulsive differential equations withpiecewise constant argumentrdquoActa ApplicandaeMathematicaevol 110 no 1 pp 499ndash510 2010

[30] Y Liu P Yang and W Ge ldquoPeriodic solutions of higher-order delay differential equationsrdquo Nonlinear Analysis TheoryMethods and Applications vol 63 no 1 pp 136ndash152 2005

[31] X L Liu and W T Li ldquoPeriodic solutions for dynamic equa-tions on time scalesrdquo Nonlinear Analysis Theory Methods andApplications vol 67 no 5 pp 1457ndash1463 2007

[32] M Pinto ldquoDichotomy and existence of periodic solutions ofquasilinear functional differential equationsrdquo Nonlinear Analy-sis Theory Methods and Applications vol 72 no 3-4 pp 1227ndash1234 2010

[33] Y N Raffoul ldquoPeriodic solutions for neutral nonlinear differ-ential equations with functional delayrdquo Electronic Journal ofDifferential Equations vol 2003 no 102 pp 1ndash7 2003

[34] G Q Wang and S S Cheng ldquoPeriodic solutions of discreteRayleigh equations with deviating argumentsrdquo Taiwanese Jour-nal of Mathematics vol 13 no 6 pp 2051ndash2067 2009

[35] K S Chiu ldquoStability of oscillatory solutions of differential equa-tions with a general piecewise constant argumentrdquo ElectronicJournal of Qualitative Theory of Differential Equations vol 88pp 1ndash15 2011

[36] R N Butris and M A Aziz ldquoSome theorems in the existenceand uniqueness for system of nonlinear integro-differentialequationsrdquo Journal Education and Science vol 18 pp 76ndash892006

[37] A YMitropolsky andD IMortynyukPeriodic Solutions for theOscillations Systems with Retarded Argument General SchoolKiev Ukraine 1979

[38] L L Tai ldquoNumerical-analytic method for the investigation ofautonomous systems of differential equationsrdquo Ukrainskii Ma-tematicheskii Zhurnal vol 30 no 3 pp 309ndash317 1978

[39] Yu D Shlapak ldquoPeriodic solutions of ordinary differentialequations of first order that are not solved with respect to thederivativerdquo Ukrainian Mathematical Journal vol 32 no 5 pp420ndash425 1980

[40] A M Samoilenko ldquoNumerical-analytic method for the investi-gation of periodic systems of ordinary differential equationsrdquoUkrainskii Matematicheskii Zhurnal vol 17 no 4 pp 16ndash231965

[41] AM Samoilenko andN I RontoNumerical-Analytic Methodsfor Investigations of Periodic Solutions Mir Publishers MoscowRussia 1979

[42] R N Butris ldquoPeriodic solution of nonlinear system of integro-differential equations depending on the gamma distributionrdquoGeneral Mathematics Notes vol 15 no 1 pp 56ndash71 2013

[43] D R Smart Fixed Points Theorems Cambridge UniversityPress Cambridge UK 1980

[44] T A Burton ldquoKrasnoselskiirsquos inversion principle and fixedpointsrdquo Nonlinear Analysis Theory Methods and Applicationsvol 30 no 7 pp 3975ndash3986 1997

[45] T A Burton ldquoA fixed-point theorem of Krasnoselskiirdquo AppliedMathematics Letters vol 11 no 1 pp 85ndash88 1998


[46] M Pinto and D Sepulveda ldquoh-asymptotic stability by fixedpoint in neutral nonlinear differential equations with delayrdquoNonlinear Analysis Theory Methods and Applications vol 74no 12 pp 3926ndash3933 2011

[47] M A Krasnoselskii Translation Along Trajectories of Differen-tial Equations Nauka Moscow Russia 1966 (Russian) Englishtranslation American Mathematical Society Providence RIUSA 1968

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of


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Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

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Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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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


In 2008 Akhmet et al [10] obtained some sufficientconditions for the existence and uniqueness of periodicsolutions for the following system

1199091015840

(119905) = 119860 (119905) 119909 (119905) + ℎ (119905) + 120583119892 (119905 119909 (119905) 119909 (120574 (119905)) 120583) (2)

where119860 R rarr R119899times119899 ℎ R rarr R and 119892 RtimesR119899timesR119899times119868 rarr

R119899 are continuous functions 120574(119905) = 119905119894if 119905119894le 119905 lt 119905

119894+1 and 120583 is

a small parameter belonging to an interval 119868 sub R with 0 isin 119868Recently Chiu and Pinto [23] using Poincare operator a

new Gronwall type lemma and fixed point theory obtainedsome sufficient conditions for the existence and uniquenessof periodic (or harmonic) and subharmonic solutions ofquasilinear differential equation with a general piecewiseconstant argument of the form

1199101015840

(119905) = 119860 (119905) 119910 (119905) + 119891 (119905 119910 (119905) 119910 (120574 (119905))) (3)

where 119905 isin R 119910 isin C119901 119860(119905) is a 119901 times 119901 matrix for 119901 isin N119891(119905 119909 119910) is a 119901 dimensional vector and 119891 is continuous inthe first argument and 120574(119905) = 120574

119894 if 119905119894le 119905 lt 119905

119894+1 119894 isin Z

In this paper comparing the three DEPCAG inequalities ofGronwall type and remarked new Gronwall lemma not onlyrequests a weaker condition than the other Gronwall lemmasbut also has a better estimate

It is well-known that there are many subjects in physicsand technology using mathematical methods that depend onthe linear and nonlinear integro-differential equations andit became clear that the existence of the periodic solutionsand its algorithm structure from more important problemsin the present time Where many of studies and researches[36ndash40] dedicates for treatment the autonomous and non-autonomous periodic systems and specially with the integralequations and differential equations and the linear andnonlinear differential and which is dealing in general shapewith the problems about periodic solutions theory and themodern methods in its quality treatment for the periodicdifferential equations

Samoilenko and Ronto [41] assume the numerical-analytic method to study the periodic solutions for ordinarydifferential equations and its algorithm structure and thismethod includes uniform sequences of periodic functionsand the results of that study are using the periodic solutionsonwide range in the difference of new processes industry andtechnology For example Samoilenko and Ronto [41] inves-tigated the existence and approximation of periodic solutionfor nonlinear system of integro-differential equations whichhas the form

1199091015840

(119905) = 119891(119905 119909 (119905) int

119905+119879

119905

119892 (119904 119909 (119904)) 119889119904) (4)

where 119909 isin 119863 sub R119899 119863 is a closed and bounded domainThe vectors functions 119891(119905 119909 119910) and 119892(119905 119909) are continuousfunctions in 119905 119909 119910 and periodic in 119905 of period 119879

Butris [42] investigated the periodic solution of nonlinearsystem of integro-differential equations depending on the

gamma distribution by using the numerical analyticmethodwhich has the form

1199091015840

(119905) = 119891(119905 120573 (119905 120572) 119909 (119905) int

119905+119879

119905

119892 (119904 120573 (119904 120572) 119909 (119904)) 119889119904)

119905 isin R

(5)

where 119909 isin 119863 sub R119899 119863 is a closed and bounded domain Thevector functions119891(119905 120573(119905 120572) 119909) and119892(119905 120573(119905 120572) 119909) are definedon the domain (119905 120573(119905 120572) 119909) isin R times [0 119879] times 119863 times 119863

1

In the current paper we study the existence of periodicsolutions of a nonlinear integro-differential system withpiecewise alternately advanced and retarded argument

1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574 (119905))) 119905 isin R

(6)

where 119860 R rarr R119899times119899 119891 R times R119899 times R119899 rarr R119899 and119892 R times R119899 times R119899 rarr R119899 are continuous in their respectivearguments In the analysis we use the idea of Greenrsquos functionfor periodic solutions and convert the nonlinear integro-differential systems with DEPCAG (6) into an equivalentintegral equationThen we employ Krasnoselskiirsquos fixed pointtheorem and show the existence of a periodic solution of thenonlinear integro-differential systems with DEPCAG (6) inTheorem 12We also obtain the existence of a unique periodicsolution in Theorem 14 employing the contraction mappingprinciple as the basicmathematical tool Furthermore appro-priate examples are given to show the feasibility of our results

In our paperwe assume that the solutions of the nonlinearintegro-differential systems with DEPCAG (6) are continu-ous functions But the deviating argument 120574(119905) is discontin-uous Thus in general the right-hand side of the DEPCAGsystem (6) has discontinuities at moments 119905

119894isin R 119894 isin Z As a

result we consider the solutions of theDEPCAGas functionswhich are continuous and continuously differentiable withinintervals [119905

119894 119905119894+1) 119894 isin Z In other words by a solution 119911(119905)

of the DEPCAG system (6) we mean a continuous functionon R such that the derivative 1199111015840(119905) exists at each point 119905 isinR with the possible exception of the points 119905

119894isin R 119894 isin

Z where a one-sided derivative exists and the nonlinearintegro-differential systemswithDEPCAG (6) are satisfied by119911(119905) on each interval (119905

119894 119905119894+1) 119894 isin Z as well

The rest of the paper is organized as follows In Section 2some definitions and preliminary results are introduced Weshow double 120596-periodicity of Greenrsquos function Section 3 isdevoted to establishing some criteria for the existence anduniqueness of periodic solutions of the DEPCAG system (6)Greenrsquos function and Banach Schauder and Krasnoselskiirsquosfixed point theorems below are fundamental to obtain themain results Furthermore appropriate examples are pro-vided in Section 4 to show the feasibility of our results






119894= [119905119894 119905119894+1)



1199101015840

(119905) = 119860 (119905) 119910 (119905) (7)





=

[119905119894(120591) 119905119894(120591)+1

)

1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574119894(120591))) 119905 isin [119905

119894(120591) 119905119894(120591)+1

)

(8)


119911 (119905) = Φ (119905 120591) 119911 (120591)

+ int

119905

120591

Φ (119905 119904) [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

(9)

For 119905 rarr 119905119894(120591)+1

in (9) we have

119911 (119905119894(120591)+1

) = Φ (119905119894(120591)+1

120591) 119911 (120591)

+ int

119905119894(120591)+1

120591

Φ(119905119894(120591)+1

119904)

times [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+ int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

(10)


119911 (119905) = Φ (119905 120591) 119911 (120591)

+ int

119905119894(120591)+1

120591

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

+

119894(119905)minus1

sum

119895=119894(120591)+1

int

119905119895+1

119905119895

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119905119895+1

119904

119862 (119904 119906 119911 (120574119895)) 119889119906

+

119894(119904+120596)minus1

sum

119896=119895+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119895)) ] 119889119904

+ int

119905

119905119894(119905)

Φ (119905 119904) [119891(119904 119911 (119904) int

119905119894(119905)+1

119904

119862 (119904 119906 119911 (120574119894(119905))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(119905)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(119905))) ] 119889119904

(11)



int

119905

120591

119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904

= int

119905119894(120591)+1

120591

119892 (119904 119911 (119904) 119911 (120574119894(120591))) 119889119904

+

119894(119905)minus1

sum

119895=119894(120591)+1

int

119905119895+1

119905119895

119892 (119904 119911 (119904) 119911 (120574119895)) 119889119904

+ int

119905

119905119894(119905)

119892 (119904 119911 (119904) 119911 (120574119894(119905))) 119889119904


119911 (119905) = Φ (119905 120591) 120585

+ int

119905

120591

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904 120591 isin R


120596) and using

(13) we get

120585 = (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(14)


119911 (119905) = Φ (119905 120591) (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

119905

120591

Φ (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(15)


Φminus1

(120591) (119868 minus Φ (120591 + 120596 120591))minus1

Φ (120591 + 120596)

= (Φminus1

(120591 + 120596)Φ(120591) minus 119868)minus1

(16)

It follows that

119911 (119905) = int

119905

120591

Φ (119905) (119868 + (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

)Φ(119904)minus1

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

120591+120596

119905

Φ (119905) (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

Φminus1

(119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(17)




119866 (119905 119904) = Φ (119905) (119868 + 119863)Φ

minus1

(119904) 120591 le 119904 le 119905 le 120591 + 120596

Φ (119905)119863Φminus1

(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)


119863 = ((Φminus1

(120591)Φ (120591 + 120596))minus1

minus 119868)minus1

(19)






(119868 + 119863) = (Φminus1

(120591)Φ (120591 + 120596))minus1

119863


(120591)Φ (120591 + 120596))minus1

Φ (119905)119863Φminus1

(119905 + 120596) minus Φ (119905)119863Φminus1

(119905) = 119868

(119868 + 119863)Φminus1

(120591 minus 120596)Φ (120591) = 119863

(20)







1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574 (119905))) 119905 isin R

(21)


P120596= 120601 R 997888rarr R

119899

| 120601 is 120596-periodic



119911 = sup119905isinR

|119911 (119905)| = sup119905isin[120591120591+120596]

|119911 (119905)| (23)




1 1199102isin R119899 |119910

119894| le 119877 119894 = 1 2



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120598120582 (119905 119904) 119905 119904 isin R (24)


119905+120596

119905

120582(119905 119904)119889119904 =



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 (25)


119905

120582(119905 119904)119889119904 = and

sup119905isinR int119905+120596

119905

|119862(119905 119904 0)|119889119904 = 0

(119871119891) For 119905 isin R and119909



1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 1199092 1199102)1003816100381610038161003816

le 1199011(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1199012 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(26)


|119891(119905 0 0)|

int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904 le 119871

1

1205781= sdot max119905isin[120591120591+120596]

10038161003816100381610038161199012 (119905)1003816100381610038161003816

120596 isin R+ 120591 isin R

(27)

(119871119892) For 119905 isin R and119909



1003816100381610038161003816119892 (119905 1199091 1199101) minus 119892 (119905 1199092 1199102)1003816100381610038161003816

le 1205921(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1205922 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(28)


|119892(119905 0 0)| and

int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904 le 119871

2 120596 isin R

+ 120591 isin R (29)




1003816100381610038161003816119862 (119905 119904 119910)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101003816100381610038161003816 + ℎ (119905 119904) (30)


119905

120582(119905 119904)119889119904 = and sup119905isinR int119905+120596

119905

ℎ(119905 119904)119889119904 = ℎ


1 1198982


1 1198621 1205782for which

1003816100381610038161003816119891 (119905 119909 119910)1003816100381610038161003816 le 1198981 (119905) |119909| + 1198982 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205881 (31)

where int120591+120596

120591

[1198981(119904) + 119898

2(119904)]119889119904 le 119862

1and 120578

2= ℎ sdot

max119905isin[120591120591+120596]

|1198982(119905)| 120596 isin R

+ 120591 isin R



1205882 1198622for which

1003816100381610038161003816119892 (119905 119909 119910)1003816100381610038161003816 le 1205811 (119905) |119909| + 1205812 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205882 (32)

where int120591+120596120591

[1205811(119904) + 120581

2(119904)]119889119904 le 119862

2 120596 isin R

+ 120591 isin R

Periodic Conditions


(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



1) = 119862(119905 119904 119909

1) for all 119905 ge 120591



is

119905119894+119901

= 119905119894+ 120596 120574

119894+119901= 120574119894+ 120596 for 119894 isin Z (33)


119894into 119868




[120591 119905119894(120591)+1

] cup

119894(120591)+119901minus1

⋃

119895=119894(120591)+1

119868119895cup [119905119894(120591)+119901

120591 + 120596] (34)


(a) 119905 + 120596 isin [119905119894+119901 119905119894+119901+1

) (b) 120574 (119905) + 120596 isin [119905119894+119901 119905119894+119901+1

)

(35)

Then

120574 (119905 + 120596) = 120574119894(119905+120596)

= 120574119894(119905)+119901

= 120574119894(119905)

+ 120596 = 120574 (119905) + 120596 (36)




119911 (119905) = int

120591+120596

120591

119866 (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(37)



(I119911) (119905) = int120591+120596

120591

119866 (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(38)


120596










(I119911) (119905) = (A119911) (119905) + (B119911) (119905) (39)


(A119911) (119905)

= int

120591+120596

120591

119866 (119905 119904) 119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)119889119904

(40)

(B119911) (119905) = int

120591+120596

120591

119866 (119905 119904) 119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904 (41)


119888119866= max119905119904isin[120591120591+120596]

|119866 (119905 119904)| S = 119911 isin P120596 119911 le 119877

CR = 119911 isin R119899

119911 le 119877

(42)

Lemma8 If (119873120596) (119875) (119871

119891) and (119871







(A120593) (119905 + 120596)

= int

120591+2120596

120591+120596

119866 (119905 + 120596 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+2120596

119904+120596

119862 (119904 + 120596 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906 + 120596))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906) + 120596)) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= (A120593) (119905)

(43)


1003817100381710038171003817A120593 minusA1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816A120593 (119905) minusA120577 (119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904) int

119904+120596

119904

120582 (119904 119906)

times1003816100381610038161003816120593 (120574 (119906)) minus 120577 (120574 (119906))

1003816100381610038161003816 119889119906] 119889119904

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904) int

119904+120596

119904

120582 (119904 119906) 119889119906] 119889119904)

times1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 1198881198661198711

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

(44)



Lemma 9 If (119873120596) (119875) and (119871






rarr P120596is





that 1199111 1199112isin S 119911

1minus 1199112 le 120575 implies |119892(119905 119911

1(119905) 1199111(120574(119905))) minus

119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598



1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840

for 119905 isin [120591 120591 + 120596](45)


and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)





1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)


(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904


(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)




le 119871 for some



119899119896) of (B120593

119899) converges






119891) (119871119862)


119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)



120596rarr P




1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra














119894= [119905119894 119905119894+1)



1199101015840

(119905) = 119860 (119905) 119910 (119905) (7)





=

[119905119894(120591) 119905119894(120591)+1

)

1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574119894(120591))) 119905 isin [119905

119894(120591) 119905119894(120591)+1

)

(8)


119911 (119905) = Φ (119905 120591) 119911 (120591)

+ int

119905

120591

Φ (119905 119904) [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

(9)

For 119905 rarr 119905119894(120591)+1

in (9) we have

119911 (119905119894(120591)+1

) = Φ (119905119894(120591)+1

120591) 119911 (120591)

+ int

119905119894(120591)+1

120591

Φ(119905119894(120591)+1

119904)

times [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+ int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

(10)


119911 (119905) = Φ (119905 120591) 119911 (120591)

+ int

119905119894(120591)+1

120591

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119905119894(120591)+1

119904

119862 (119904 119906 119911 (120574119894(120591))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(120591)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(120591))) ] 119889119904

+

119894(119905)minus1

sum

119895=119894(120591)+1

int

119905119895+1

119905119895

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119905119895+1

119904

119862 (119904 119906 119911 (120574119895)) 119889119906

+

119894(119904+120596)minus1

sum

119896=119895+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119895)) ] 119889119904

+ int

119905

119905119894(119905)

Φ (119905 119904) [119891(119904 119911 (119904) int

119905119894(119905)+1

119904

119862 (119904 119906 119911 (120574119894(119905))) 119889119906

+

119894(119904+120596)minus1

sum

119896=119894(119905)+1

int

119905119896+1

119905119896

119862 (119904 119906 119911 (120574119896)) 119889119906

+int

119904+120596

119905119894(119904+120596)

119862 (119904 119906 119911 (120574119894(119904+120596)

)) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574119894(119905))) ] 119889119904

(11)



int

119905

120591

119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904

= int

119905119894(120591)+1

120591

119892 (119904 119911 (119904) 119911 (120574119894(120591))) 119889119904

+

119894(119905)minus1

sum

119895=119894(120591)+1

int

119905119895+1

119905119895

119892 (119904 119911 (119904) 119911 (120574119895)) 119889119904

+ int

119905

119905119894(119905)

119892 (119904 119911 (119904) 119911 (120574119894(119905))) 119889119904


119911 (119905) = Φ (119905 120591) 120585

+ int

119905

120591

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904 120591 isin R


120596) and using

(13) we get

120585 = (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(14)


119911 (119905) = Φ (119905 120591) (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

119905

120591

Φ (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(15)


Φminus1

(120591) (119868 minus Φ (120591 + 120596 120591))minus1

Φ (120591 + 120596)

= (Φminus1

(120591 + 120596)Φ(120591) minus 119868)minus1

(16)

It follows that

119911 (119905) = int

119905

120591

Φ (119905) (119868 + (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

)Φ(119904)minus1

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

120591+120596

119905

Φ (119905) (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

Φminus1

(119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(17)




119866 (119905 119904) = Φ (119905) (119868 + 119863)Φ

minus1

(119904) 120591 le 119904 le 119905 le 120591 + 120596

Φ (119905)119863Φminus1

(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)


119863 = ((Φminus1

(120591)Φ (120591 + 120596))minus1

minus 119868)minus1

(19)






(119868 + 119863) = (Φminus1

(120591)Φ (120591 + 120596))minus1

119863


(120591)Φ (120591 + 120596))minus1

Φ (119905)119863Φminus1

(119905 + 120596) minus Φ (119905)119863Φminus1

(119905) = 119868

(119868 + 119863)Φminus1

(120591 minus 120596)Φ (120591) = 119863

(20)







1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574 (119905))) 119905 isin R

(21)


P120596= 120601 R 997888rarr R

119899

| 120601 is 120596-periodic



119911 = sup119905isinR

|119911 (119905)| = sup119905isin[120591120591+120596]

|119911 (119905)| (23)




1 1199102isin R119899 |119910

119894| le 119877 119894 = 1 2



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120598120582 (119905 119904) 119905 119904 isin R (24)


119905+120596

119905

120582(119905 119904)119889119904 =



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 (25)


119905

120582(119905 119904)119889119904 = and

sup119905isinR int119905+120596

119905

|119862(119905 119904 0)|119889119904 = 0

(119871119891) For 119905 isin R and119909



1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 1199092 1199102)1003816100381610038161003816

le 1199011(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1199012 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(26)


|119891(119905 0 0)|

int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904 le 119871

1

1205781= sdot max119905isin[120591120591+120596]

10038161003816100381610038161199012 (119905)1003816100381610038161003816

120596 isin R+ 120591 isin R

(27)

(119871119892) For 119905 isin R and119909



1003816100381610038161003816119892 (119905 1199091 1199101) minus 119892 (119905 1199092 1199102)1003816100381610038161003816

le 1205921(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1205922 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(28)


|119892(119905 0 0)| and

int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904 le 119871

2 120596 isin R

+ 120591 isin R (29)




1003816100381610038161003816119862 (119905 119904 119910)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101003816100381610038161003816 + ℎ (119905 119904) (30)


119905

120582(119905 119904)119889119904 = and sup119905isinR int119905+120596

119905

ℎ(119905 119904)119889119904 = ℎ


1 1198982


1 1198621 1205782for which

1003816100381610038161003816119891 (119905 119909 119910)1003816100381610038161003816 le 1198981 (119905) |119909| + 1198982 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205881 (31)

where int120591+120596

120591

[1198981(119904) + 119898

2(119904)]119889119904 le 119862

1and 120578

2= ℎ sdot

max119905isin[120591120591+120596]

|1198982(119905)| 120596 isin R

+ 120591 isin R



1205882 1198622for which

1003816100381610038161003816119892 (119905 119909 119910)1003816100381610038161003816 le 1205811 (119905) |119909| + 1205812 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205882 (32)

where int120591+120596120591

[1205811(119904) + 120581

2(119904)]119889119904 le 119862

2 120596 isin R

+ 120591 isin R

Periodic Conditions


(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



1) = 119862(119905 119904 119909

1) for all 119905 ge 120591



is

119905119894+119901

= 119905119894+ 120596 120574

119894+119901= 120574119894+ 120596 for 119894 isin Z (33)


119894into 119868




[120591 119905119894(120591)+1

] cup

119894(120591)+119901minus1

⋃

119895=119894(120591)+1

119868119895cup [119905119894(120591)+119901

120591 + 120596] (34)


(a) 119905 + 120596 isin [119905119894+119901 119905119894+119901+1

) (b) 120574 (119905) + 120596 isin [119905119894+119901 119905119894+119901+1

)

(35)

Then

120574 (119905 + 120596) = 120574119894(119905+120596)

= 120574119894(119905)+119901

= 120574119894(119905)

+ 120596 = 120574 (119905) + 120596 (36)




119911 (119905) = int

120591+120596

120591

119866 (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(37)



(I119911) (119905) = int120591+120596

120591

119866 (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(38)


120596










(I119911) (119905) = (A119911) (119905) + (B119911) (119905) (39)


(A119911) (119905)

= int

120591+120596

120591

119866 (119905 119904) 119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)119889119904

(40)

(B119911) (119905) = int

120591+120596

120591

119866 (119905 119904) 119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904 (41)


119888119866= max119905119904isin[120591120591+120596]

|119866 (119905 119904)| S = 119911 isin P120596 119911 le 119877

CR = 119911 isin R119899

119911 le 119877

(42)

Lemma8 If (119873120596) (119875) (119871

119891) and (119871







(A120593) (119905 + 120596)

= int

120591+2120596

120591+120596

119866 (119905 + 120596 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+2120596

119904+120596

119862 (119904 + 120596 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906 + 120596))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906) + 120596)) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= (A120593) (119905)

(43)


1003817100381710038171003817A120593 minusA1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816A120593 (119905) minusA120577 (119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904) int

119904+120596

119904

120582 (119904 119906)

times1003816100381610038161003816120593 (120574 (119906)) minus 120577 (120574 (119906))

1003816100381610038161003816 119889119906] 119889119904

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904) int

119904+120596

119904

120582 (119904 119906) 119889119906] 119889119904)

times1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 1198881198661198711

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

(44)



Lemma 9 If (119873120596) (119875) and (119871






rarr P120596is





that 1199111 1199112isin S 119911

1minus 1199112 le 120575 implies |119892(119905 119911

1(119905) 1199111(120574(119905))) minus

119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598



1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840

for 119905 isin [120591 120591 + 120596](45)


and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)





1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)


(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904


(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)




le 119871 for some



119899119896) of (B120593

119899) converges






119891) (119871119862)


119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)



120596rarr P




1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











int

119905

120591

119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904

= int

119905119894(120591)+1

120591

119892 (119904 119911 (119904) 119911 (120574119894(120591))) 119889119904

+

119894(119905)minus1

sum

119895=119894(120591)+1

int

119905119895+1

119905119895

119892 (119904 119911 (119904) 119911 (120574119895)) 119889119904

+ int

119905

119905119894(119905)

119892 (119904 119911 (119904) 119911 (120574119894(119905))) 119889119904


119911 (119905) = Φ (119905 120591) 120585

+ int

119905

120591

Φ (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904 120591 isin R


120596) and using

(13) we get

120585 = (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(14)


119911 (119905) = Φ (119905 120591) (119868 minus Φ (120591 + 120596 120591))minus1

times int

120591+120596

120591

Φ (120591 + 120596 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

119905

120591

Φ (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(15)


Φminus1

(120591) (119868 minus Φ (120591 + 120596 120591))minus1

Φ (120591 + 120596)

= (Φminus1

(120591 + 120596)Φ(120591) minus 119868)minus1

(16)

It follows that

119911 (119905) = int

119905

120591

Φ (119905) (119868 + (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

)Φ(119904)minus1

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

+ int

120591+120596

119905

Φ (119905) (Φminus1

(120591 + 120596)Φ (120591) minus 119868)minus1

Φminus1

(119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(17)




119866 (119905 119904) = Φ (119905) (119868 + 119863)Φ

minus1

(119904) 120591 le 119904 le 119905 le 120591 + 120596

Φ (119905)119863Φminus1

(119904) 120591 le 119905 lt 119904 le 120591 + 120596(18)


119863 = ((Φminus1

(120591)Φ (120591 + 120596))minus1

minus 119868)minus1

(19)






(119868 + 119863) = (Φminus1

(120591)Φ (120591 + 120596))minus1

119863


(120591)Φ (120591 + 120596))minus1

Φ (119905)119863Φminus1

(119905 + 120596) minus Φ (119905)119863Φminus1

(119905) = 119868

(119868 + 119863)Φminus1

(120591 minus 120596)Φ (120591) = 119863

(20)







1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574 (119905))) 119905 isin R

(21)


P120596= 120601 R 997888rarr R

119899

| 120601 is 120596-periodic



119911 = sup119905isinR

|119911 (119905)| = sup119905isin[120591120591+120596]

|119911 (119905)| (23)




1 1199102isin R119899 |119910

119894| le 119877 119894 = 1 2



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120598120582 (119905 119904) 119905 119904 isin R (24)


119905+120596

119905

120582(119905 119904)119889119904 =



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 (25)


119905

120582(119905 119904)119889119904 = and

sup119905isinR int119905+120596

119905

|119862(119905 119904 0)|119889119904 = 0

(119871119891) For 119905 isin R and119909



1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 1199092 1199102)1003816100381610038161003816

le 1199011(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1199012 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(26)


|119891(119905 0 0)|

int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904 le 119871

1

1205781= sdot max119905isin[120591120591+120596]

10038161003816100381610038161199012 (119905)1003816100381610038161003816

120596 isin R+ 120591 isin R

(27)

(119871119892) For 119905 isin R and119909



1003816100381610038161003816119892 (119905 1199091 1199101) minus 119892 (119905 1199092 1199102)1003816100381610038161003816

le 1205921(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1205922 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(28)


|119892(119905 0 0)| and

int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904 le 119871

2 120596 isin R

+ 120591 isin R (29)




1003816100381610038161003816119862 (119905 119904 119910)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101003816100381610038161003816 + ℎ (119905 119904) (30)


119905

120582(119905 119904)119889119904 = and sup119905isinR int119905+120596

119905

ℎ(119905 119904)119889119904 = ℎ


1 1198982


1 1198621 1205782for which

1003816100381610038161003816119891 (119905 119909 119910)1003816100381610038161003816 le 1198981 (119905) |119909| + 1198982 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205881 (31)

where int120591+120596

120591

[1198981(119904) + 119898

2(119904)]119889119904 le 119862

1and 120578

2= ℎ sdot

max119905isin[120591120591+120596]

|1198982(119905)| 120596 isin R

+ 120591 isin R



1205882 1198622for which

1003816100381610038161003816119892 (119905 119909 119910)1003816100381610038161003816 le 1205811 (119905) |119909| + 1205812 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205882 (32)

where int120591+120596120591

[1205811(119904) + 120581

2(119904)]119889119904 le 119862

2 120596 isin R

+ 120591 isin R

Periodic Conditions


(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



1) = 119862(119905 119904 119909

1) for all 119905 ge 120591



is

119905119894+119901

= 119905119894+ 120596 120574

119894+119901= 120574119894+ 120596 for 119894 isin Z (33)


119894into 119868




[120591 119905119894(120591)+1

] cup

119894(120591)+119901minus1

⋃

119895=119894(120591)+1

119868119895cup [119905119894(120591)+119901

120591 + 120596] (34)


(a) 119905 + 120596 isin [119905119894+119901 119905119894+119901+1

) (b) 120574 (119905) + 120596 isin [119905119894+119901 119905119894+119901+1

)

(35)

Then

120574 (119905 + 120596) = 120574119894(119905+120596)

= 120574119894(119905)+119901

= 120574119894(119905)

+ 120596 = 120574 (119905) + 120596 (36)




119911 (119905) = int

120591+120596

120591

119866 (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(37)



(I119911) (119905) = int120591+120596

120591

119866 (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(38)


120596










(I119911) (119905) = (A119911) (119905) + (B119911) (119905) (39)


(A119911) (119905)

= int

120591+120596

120591

119866 (119905 119904) 119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)119889119904

(40)

(B119911) (119905) = int

120591+120596

120591

119866 (119905 119904) 119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904 (41)


119888119866= max119905119904isin[120591120591+120596]

|119866 (119905 119904)| S = 119911 isin P120596 119911 le 119877

CR = 119911 isin R119899

119911 le 119877

(42)

Lemma8 If (119873120596) (119875) (119871

119891) and (119871







(A120593) (119905 + 120596)

= int

120591+2120596

120591+120596

119866 (119905 + 120596 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+2120596

119904+120596

119862 (119904 + 120596 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906 + 120596))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906) + 120596)) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= (A120593) (119905)

(43)


1003817100381710038171003817A120593 minusA1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816A120593 (119905) minusA120577 (119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904) int

119904+120596

119904

120582 (119904 119906)

times1003816100381610038161003816120593 (120574 (119906)) minus 120577 (120574 (119906))

1003816100381610038161003816 119889119906] 119889119904

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904) int

119904+120596

119904

120582 (119904 119906) 119889119906] 119889119904)

times1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 1198881198661198711

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

(44)



Lemma 9 If (119873120596) (119875) and (119871






rarr P120596is





that 1199111 1199112isin S 119911

1minus 1199112 le 120575 implies |119892(119905 119911

1(119905) 1199111(120574(119905))) minus

119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598



1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840

for 119905 isin [120591 120591 + 120596](45)


and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)





1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)


(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904


(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)




le 119871 for some



119899119896) of (B120593

119899) converges






119891) (119871119862)


119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)



120596rarr P




1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119891(119905 119911 (119905) int

119905+120596

119905

119862 (119905 119904 119911 (120574 (119904))) 119889119904)

+ 119892 (119905 119911 (119905) 119911 (120574 (119905))) 119905 isin R

(21)


P120596= 120601 R 997888rarr R

119899

| 120601 is 120596-periodic



119911 = sup119905isinR

|119911 (119905)| = sup119905isin[120591120591+120596]

|119911 (119905)| (23)




1 1199102isin R119899 |119910

119894| le 119877 119894 = 1 2



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120598120582 (119905 119904) 119905 119904 isin R (24)


119905+120596

119905

120582(119905 119904)119889119904 =



1003816100381610038161003816119862 (119905 119904 1199101) minus 119862 (119905 119904 1199102)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816 (25)


119905

120582(119905 119904)119889119904 = and

sup119905isinR int119905+120596

119905

|119862(119905 119904 0)|119889119904 = 0

(119871119891) For 119905 isin R and119909



1003816100381610038161003816119891 (119905 1199091 1199101) minus 119891 (119905 1199092 1199102)1003816100381610038161003816

le 1199011(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1199012 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(26)


|119891(119905 0 0)|

int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904 le 119871

1

1205781= sdot max119905isin[120591120591+120596]

10038161003816100381610038161199012 (119905)1003816100381610038161003816

120596 isin R+ 120591 isin R

(27)

(119871119892) For 119905 isin R and119909



1003816100381610038161003816119892 (119905 1199091 1199101) minus 119892 (119905 1199092 1199102)1003816100381610038161003816

le 1205921(119905)

10038161003816100381610038161199091 minus 11990921003816100381610038161003816 + 1205922 (119905)

10038161003816100381610038161199101 minus 11991021003816100381610038161003816

(28)


|119892(119905 0 0)| and

int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904 le 119871

2 120596 isin R

+ 120591 isin R (29)




1003816100381610038161003816119862 (119905 119904 119910)1003816100381610038161003816 le 120582 (119905 119904)

10038161003816100381610038161199101003816100381610038161003816 + ℎ (119905 119904) (30)


119905

120582(119905 119904)119889119904 = and sup119905isinR int119905+120596

119905

ℎ(119905 119904)119889119904 = ℎ


1 1198982


1 1198621 1205782for which

1003816100381610038161003816119891 (119905 119909 119910)1003816100381610038161003816 le 1198981 (119905) |119909| + 1198982 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205881 (31)

where int120591+120596

120591

[1198981(119904) + 119898

2(119904)]119889119904 le 119862

1and 120578

2= ℎ sdot

max119905isin[120591120591+120596]

|1198982(119905)| 120596 isin R

+ 120591 isin R



1205882 1198622for which

1003816100381610038161003816119892 (119905 119909 119910)1003816100381610038161003816 le 1205811 (119905) |119909| + 1205812 (119905)

10038161003816100381610038161199101003816100381610038161003816 + 1205882 (32)

where int120591+120596120591

[1205811(119904) + 120581

2(119904)]119889119904 le 119862

2 120596 isin R

+ 120591 isin R

Periodic Conditions


(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



1) = 119862(119905 119904 119909

1) for all 119905 ge 120591



is

119905119894+119901

= 119905119894+ 120596 120574

119894+119901= 120574119894+ 120596 for 119894 isin Z (33)


119894into 119868




[120591 119905119894(120591)+1

] cup

119894(120591)+119901minus1

⋃

119895=119894(120591)+1

119868119895cup [119905119894(120591)+119901

120591 + 120596] (34)


(a) 119905 + 120596 isin [119905119894+119901 119905119894+119901+1

) (b) 120574 (119905) + 120596 isin [119905119894+119901 119905119894+119901+1

)

(35)

Then

120574 (119905 + 120596) = 120574119894(119905+120596)

= 120574119894(119905)+119901

= 120574119894(119905)

+ 120596 = 120574 (119905) + 120596 (36)




119911 (119905) = int

120591+120596

120591

119866 (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(37)



(I119911) (119905) = int120591+120596

120591

119866 (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(38)


120596










(I119911) (119905) = (A119911) (119905) + (B119911) (119905) (39)


(A119911) (119905)

= int

120591+120596

120591

119866 (119905 119904) 119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)119889119904

(40)

(B119911) (119905) = int

120591+120596

120591

119866 (119905 119904) 119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904 (41)


119888119866= max119905119904isin[120591120591+120596]

|119866 (119905 119904)| S = 119911 isin P120596 119911 le 119877

CR = 119911 isin R119899

119911 le 119877

(42)

Lemma8 If (119873120596) (119875) (119871

119891) and (119871







(A120593) (119905 + 120596)

= int

120591+2120596

120591+120596

119866 (119905 + 120596 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+2120596

119904+120596

119862 (119904 + 120596 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906 + 120596))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906) + 120596)) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= (A120593) (119905)

(43)


1003817100381710038171003817A120593 minusA1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816A120593 (119905) minusA120577 (119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904) int

119904+120596

119904

120582 (119904 119906)

times1003816100381610038161003816120593 (120574 (119906)) minus 120577 (120574 (119906))

1003816100381610038161003816 119889119906] 119889119904

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904) int

119904+120596

119904

120582 (119904 119906) 119889119906] 119889119904)

times1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 1198881198661198711

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

(44)



Lemma 9 If (119873120596) (119875) and (119871






rarr P120596is





that 1199111 1199112isin S 119911

1minus 1199112 le 120575 implies |119892(119905 119911

1(119905) 1199111(120574(119905))) minus

119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598



1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840

for 119905 isin [120591 120591 + 120596](45)


and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)





1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)


(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904


(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)




le 119871 for some



119899119896) of (B120593

119899) converges






119891) (119871119862)


119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)



120596rarr P




1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











is

119905119894+119901

= 119905119894+ 120596 120574

119894+119901= 120574119894+ 120596 for 119894 isin Z (33)


119894into 119868




[120591 119905119894(120591)+1

] cup

119894(120591)+119901minus1

⋃

119895=119894(120591)+1

119868119895cup [119905119894(120591)+119901

120591 + 120596] (34)


(a) 119905 + 120596 isin [119905119894+119901 119905119894+119901+1

) (b) 120574 (119905) + 120596 isin [119905119894+119901 119905119894+119901+1

)

(35)

Then

120574 (119905 + 120596) = 120574119894(119905+120596)

= 120574119894(119905)+119901

= 120574119894(119905)

+ 120596 = 120574 (119905) + 120596 (36)




119911 (119905) = int

120591+120596

120591

119866 (119905 119904) [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(37)



(I119911) (119905) = int120591+120596

120591

119866 (119905 119904)

times [119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)

+119892 (119904 119911 (119904) 119911 (120574 (119904))) ] 119889119904

(38)


120596










(I119911) (119905) = (A119911) (119905) + (B119911) (119905) (39)


(A119911) (119905)

= int

120591+120596

120591

119866 (119905 119904) 119891(119904 119911 (119904) int

119904+120596

119904

119862 (119904 119906 119911 (120574 (119906))) 119889119906)119889119904

(40)

(B119911) (119905) = int

120591+120596

120591

119866 (119905 119904) 119892 (119904 119911 (119904) 119911 (120574 (119904))) 119889119904 (41)


119888119866= max119905119904isin[120591120591+120596]

|119866 (119905 119904)| S = 119911 isin P120596 119911 le 119877

CR = 119911 isin R119899

119911 le 119877

(42)

Lemma8 If (119873120596) (119875) (119871

119891) and (119871







(A120593) (119905 + 120596)

= int

120591+2120596

120591+120596

119866 (119905 + 120596 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+2120596

119904+120596

119862 (119904 + 120596 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906 + 120596))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906) + 120596)) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= (A120593) (119905)

(43)


1003817100381710038171003817A120593 minusA1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816A120593 (119905) minusA120577 (119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904) int

119904+120596

119904

120582 (119904 119906)

times1003816100381610038161003816120593 (120574 (119906)) minus 120577 (120574 (119906))

1003816100381610038161003816 119889119906] 119889119904

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904) int

119904+120596

119904

120582 (119904 119906) 119889119906] 119889119904)

times1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 1198881198661198711

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

(44)



Lemma 9 If (119873120596) (119875) and (119871






rarr P120596is





that 1199111 1199112isin S 119911

1minus 1199112 le 120575 implies |119892(119905 119911

1(119905) 1199111(120574(119905))) minus

119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598



1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840

for 119905 isin [120591 120591 + 120596](45)


and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)





1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)


(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904


(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)




le 119871 for some



119899119896) of (B120593

119899) converges






119891) (119871119862)


119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)



120596rarr P




1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra












(A120593) (119905 + 120596)

= int

120591+2120596

120591+120596

119866 (119905 + 120596 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+2120596

119904+120596

119862 (119904 + 120596 119906 120593 (120574 (119906))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906 + 120596))) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 + 120596 119904 + 120596)

times 119891(119904 + 120596 120593 (119904 + 120596)

int

119904+120596

119904

119862 (119904 + 120596 119906 + 120596 120593 (120574 (119906) + 120596)) 119889119906)119889119904

= int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

= (A120593) (119905)

(43)


1003817100381710038171003817A120593 minusA1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816A120593 (119905) minusA120577 (119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904) int

119904+120596

119904

120582 (119904 119906)

times1003816100381610038161003816120593 (120574 (119906)) minus 120577 (120574 (119906))

1003816100381610038161003816 119889119906] 119889119904

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904) int

119904+120596

119904

120582 (119904 119906) 119889119906] 119889119904)

times1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

le (119888119866int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 1198881198661198711

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

(44)



Lemma 9 If (119873120596) (119875) and (119871






rarr P120596is





that 1199111 1199112isin S 119911

1minus 1199112 le 120575 implies |119892(119905 119911

1(119905) 1199111(120574(119905))) minus

119892(119905 1199112(119905) 1199112(120574(119905)))| le 120598



1003816100381610038161003816119892 (119905 1199111 (119905) 1199111 (120574 (119905))) minus 119892 (119905 1199112 (119905) 1199112 (120574 (119905)))1003816100381610038161003816 le 1205981015840

for 119905 isin [120591 120591 + 120596](45)


and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)





1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)


(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904


(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)




le 119871 for some



119899119896) of (B120593

119899) converges






119891) (119871119862)


119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)



120596rarr P




1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










and then1003817100381710038171003817B1199111minusB1199112

1003817100381710038171003817

= sup119905isin[120591120591+120596]

1003816100381610038161003816B1199111(119905) minusB119911

2(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 1199111 (119904) 1199111 (120574 (119904)))

minus119892 (119904 1199112(119904) 1199112(120574 (119904)))

1003816100381610038161003816 119889119904

le int

120591+120596

120591

1198881198661205981015840

119889119904 le 1198881198661205981015840

120596 = 120598

(46)





1003817100381710038171003817B120593119899

1003817100381710038171003817 = sup119905isin[120591120591+120596]

1003816100381610038161003816B120593119899(119905)1003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593119899 (119904) 120593119899 (120574 (119904)))1003816100381610038161003816 119889119904 le 119888119866119872120596

(47)


(B120593119899(119905))1015840

= (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

1015840

= Φ1015840

(119905) int

119905

120591

(119868 + 119863)Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904

+ Φ (119905) (119868 + 119863)Φminus1

(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

+ Φ1015840

(119905) int

120591+120596

119905

[119863Φminus1

(119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904)))] 119889119904


(119905) 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905) (int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120593119899(119904) 120593119899(120574 (119904))) 119889119904)

+ 119892 (119905 120593119899(119905) 120593119899(120574 (119905)))

= 119860 (119905)B120593119899(119905) + 119892 (119905 120593

119899(119905) 120593119899(120574 (119905)))

(48)




le 119871 for some



119899119896) of (B120593

119899) converges






119891) (119871119862)


119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596 le 119877 (49)



120596rarr P




1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times [119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0) + 119891 (119904 0 0) ] 119889119904

+int

120591+120596

120591

119866 (119905 119904) 119892 (119904 120577 (119904) 120577 (120574 (119904))) 119889119904

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus119891 (119904 0 0)

1003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119891 (119904 0 0)

1003816100381610038161003816 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120577 (119904) 120577 (120574 (119904)))

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816

+ 1199012(119904)

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596


le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra










le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[119862 (119904 119906 120593 (120574 (119906)))

minus119862 (119904 119906 0) + 119862 (119904 119906 0) ] 119889119906

1003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866int

120591+120596

120591

1199011(119904)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |119862 (119904 119906 0)|] 119889119906

10038161003816100381610038161003816100381610038161003816119889119904

+ 120572119888119866120596 + 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 1205882119888119866120596

le 119888119866(int

120591+120596

120591

1199011(119904) + 119901

2(119904) 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866(int

120591+120596

120591

[1205811(119904) + 120581

1(119904)] 119889119904)

10038171003817100381710038171205771003817100381710038171003817

+ 1198881198660int

120591+120596

120591

1199012(119904) 119889119904 + 120572119888

119866120596 + 1205882119888119866120596

le 119888119866(1198711+ 1198622) 119877 + 119888

119866(120572 + 120578

1+ 1205882) 120596

(50)





119892) (119872119891)


inequality

119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596 le 119877 (51)



120596rarr P



then A120601 +B120577 le 119877


1003817100381710038171003817A120593 +B1205771003817100381710038171003817

= sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816int

120591+120596

120591

119866 (119905 119904)

times 119891(119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)119889119904

+ int

120591+120596

120591

119866 (119905 119904) [(119892 (119904 120577 (119904) 120577 (120574 (119904)))

minus119892 (119904 0 0) + 119892 (119904 0 0) )] 119889119904

1003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816+ 1205881)119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times (1205921(119904) 120577 (119904) + 120592

2(119904) 120577 (120574 (119904))) 119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 0 0)

1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

(1198981(119905)

1003816100381610038161003816120593 (119904)1003816100381610038161003816 + 1198982 (119905)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

[120582 (119904 119906)1003816100381610038161003816120593 (120574 (119906))

1003816100381610038161003816 + |ℎ (119904 119906)|] 119889119906

10038161003816100381610038161003816100381610038161003816

+1205881)119889119904

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(int

120591+120596

120591

[1198981(119904) + 119898

2(119904)] 119889119904)

10038171003817100381710038171205931003817100381710038171003817

+ 119888119866ℎ int

120591+120596

120591

1198982(119904) 119889119904 + 120588

1119888119866120596

+ 119888119866(int

120591+120596

120591

(1205921(119904) + 120592

2(119904)) 119889119904)

10038171003817100381710038171205771003817100381710038171003817 + 120573119888119866120596

le 119888119866(1198621+ 1198712) 119877 + 119888

119866(120573 + 120578

2+ 1205881) 120596

(52)




119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra











119891) (119871119862)

(119871119892) hold If

119888119866(1198711+ 1198712) lt 1 (53)




1003817100381710038171003817

le sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|

times

10038161003816100381610038161003816100381610038161003816119891 (119904 120593 (119904) int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906)

minus 119891(119904 120577 (119904) int

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906)

10038161003816100381610038161003816100381610038161003816119889119904

+ sup119905isin[120591120591+120596]

int

120591+120596

120591

|119866 (119905 119904)|1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866int

120591+120596

120591

[1199011(119904)

1003816100381610038161003816120593 (119904) minus 120577 (119904)1003816100381610038161003816 + 1199012 (119904)

times

10038161003816100381610038161003816100381610038161003816int

119904+120596

119904

119862 (119904 119906 120593 (120574 (119906))) 119889119906

minusint

119904+120596

119904

119862 (119904 119906 120577 (120574 (119906))) 119889119906

10038161003816100381610038161003816100381610038161003816] 119889119904

+ 119888119866int

120591+120596

120591

1003816100381610038161003816119892 (119904 120593 (119904) 120593 (120574 (119904)))

minus119892 (119904 120577 (119904) 120577 (120574 (119904)))1003816100381610038161003816 119889119904

le 119888119866(int

120591+120596

120591

[1199011(119904) + 119901

2(119904)] 119889119904

+int

120591+120596

120591

[1205921(119904) + 120592

2(119904)] 119889119904)

1003817100381710038171003817120593 minus 1205771003817100381710038171003817

le 119888119866(1198711+ 1198712)1003817100381710038171003817120593 minus 120577

1003817100381710038171003817

(54)




119891) (119872119862)


inequality

119888119866(1198621+ 1198622) 119877 + 119888

119866(1205881+ 1205882+ 1205782) 120596 le 119877 (55)



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119892 (119905 119909 (119905) 119909 (120574 (119905))) (56)


|119909|+|119910|rarr+infin(|119892(119905 119909 119910)|(|119909|+|119910|)) = 0 then


120596) (1198753) (119872119892) 119860(119905)


1(|119909| + |119910|) + 120588

2 with 2119888

1120596 lt 119862

2



1199091015840

(119905) = 119860 (119905) 119909 (119905) + 119891 (119905 119909 (119905) 119909 (120574 (119905)))

+ 119892 (119905 119909 (119905) 119909 (120574 (119905)))

(57)


119866(1198621+1198712)119877+119888119866(120573+1205881)120596 le 119877

with = 1


1

(1198752) by 120596 = 120596

2 and (119875

3) by 120596 = 120596

3 if 120596

119894120596119895is a



1 1205962 1205963


between 1205961 1205962and 120596



2for 119891 120596

3



(1205965 119901) condition If 120596



1 1205962 1205963 1205964 1205965



(119875120596) There exists 120596 = lcm120596

1 1205962 1205963 1205964 1205965 gt 0 120596

119894120596119895


such that

(1) 119860(119905) 119891(119905 1199091 1199101) and 119892(119905 119909



3


4 119904 + 120596

4 1199093) = 119862(119905 119904 119909

3) for all 119905 ge 120591


5 119901) condition













119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra




















119891)







sup119905isinR

int

119905+2120587

119905

120582 (119905 119904) 119889119904 le sup119905isinR

int

119905+2120587

119905

ℎ (119905 119904) 119889119904 le ℎ (58)


1199111015840

(119905) = 119886 (119905) 119911 (119905)

+ int

119905+2120587

119905

ln [1 + 1003816100381610038161003816119911 (120574 (119904))10038161003816100381610038163

120582 (119905 119904) + ℎ (119905 119904)] 119889119904

+ (sin 119905) 1199114 (119905) + (1 + cos2119905) 1199119 (120574 (119905)) 119905 isin R

(59)




120596)


(i) 119891(119905 119909 119910) = 119910 satisfies (119872119891) |119891(119905 119909 119910)| le |119910|


|119892(119905 119909 119910)| le 1198881(119877)(|119909| + |119910|) + 120588



1198622


|119862(119905 119904 119910)| le 1198882(119877)120582(119905 119904)|119910| +ℎ(119905 119904) for |119910| le 119877 where

2120587 le 1198621



implies


(60)


exp (2119886lowast120587)exp (2119886


[(1198621+ 1198622) 119877 + 2120587 (120588

2+ ℎ)] le 119877 (61)


lowast= inf119905isinR|119886(119905)|




sup119905isinR

int

119905+120596

119905

|Λ (119904)| 119889119904 = Λ lt infin

sup119905isinR

int

119905+120596

119905

1003816100381610038161003816120583 (119905 minus 119904)1003816100381610038161003816 119889119904 = 120583 lt infin

(62)


1199111015840

(119905) = 119860 (119905) 119911 (119905) + 119861 (119905) 119892 (119911 (119905) 119911 (120574 (119905)))

+int

119905+120596

119905

[Λ (119904) 120581 (119911 (120574 (119904))) + 120583 (119905 minus 119904)] 119889119904

(63)




(i) 1199111015840(119905) = 119860(119905)119911(119905) satisfies (119873120596)



2(119877)|119910| + 120599

1for |119909| |119910| le 119877 where (119888

1(119877) +

1198881(119877))120596 le 119862

2


3(119877)Λ120596 le 119871

1


119862119866(1198711+ 1198871198622) 119877 + 119862

119866(1198871205991+ 120581 (0) Λ + 120583)120596 lt 119877 (64)








11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra















11991010158401015840

(119905) + (12058121199102

(119905) minus 2) 1199101015840

(119905) minus 8119910 (119905)

minus 1205811sin (120596119905) 1199102 (120574 (119905)) minus 120581

2cos (120596119905) = 0

(65)

where 1205811 1205812isin R 120574(119905) = 120574

119894if 119905119894le 120574119894lt 119905119894+1


119894119894isinZ and 120574



1199111015840

(119905) = (0 1

8 2) 119911 (119905) + (

0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

+ (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(66)

where

119860 = (0 1

8 2)

119891 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205811sin (120596119905) 1199112

1(120574 (119905))

)

119892 (119905 119911 (119905) 119911 (120574 (119905))) = (0

1205812cos (120596119905) minus 120581

21199112

1(119905) 1199112(119905))

(67)



120596) is satisfied Let 120593(119905) =

(1205931(119905) 1205932(119905)) 120595(119905) = (120595

1(119905) 1205952(119905)) and define S = 119911 isin

P120596 119911 le 119877 where 119877 isin R


2119888119866119877 (1205811+ 1205812119877) lt 1 (68)


1003817100381710038171003817

le sup119905isin[120591120591+120596]

10038161003816100381610038161003816100381610038161003816(

0

1205812cos (120596119905) minus 1205812120593

2

1(119905) 1205932 (119905)

)

minus (0

1205812cos (120596119905) minus 1205812120595

2

1(119905) 1205952 (119905)

)

10038161003816100381610038161003816100381610038161003816

le sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205812(1205951 (119905) + 1205931 (119905)) 1205932 (119905) 120581

21205952

1(119905)) (

1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816

le 212058121198772 sup119905isin[120591120591+120596]

1003816100381610038161003816100381610038161003816(1205951 (119905) minus 1205931 (119905)

1205952 (119905) minus 1205932 (119905)

)

1003816100381610038161003816100381610038161003816= 212058121198772 1003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(69)


1003817100381710038171003817

le 212058111198771003817100381710038171003817120593 minus 120595

1003817100381710038171003817

(70)




Acknowledgment


References

























































Volume 2014




Journal of











Journal of


Function Spaces






Algebra





















































Volume 2014




Journal of











Journal of


Function Spaces






Algebra









research article periodic solutions for nonlinear integro

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