Hindawi Publishing CorporationJournal of Function Spaces and ApplicationsVolume 2013 Article ID 793263 5 pageshttpdxdoiorg1011552013793263

Research ArticleGeneralized Quasilinearization for the System of FractionalDifferential Equations

Peiguang Wang1 and Ying Hou2

1 College of Electronic and Information Engineering Hebei University Baoding 071002 China2 College of Mathematics and Computer Science Hebei University Baoding 071002 China

Correspondence should be addressed to Peiguang Wang pgwangmailhbueducn

Received 9 December 2012 Accepted 8 February 2013

Academic Editor Yongsheng S Han

Copyright copy 2013 P Wang and Y HouThis 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

This paper considers the initial value problems of the system of fractional differential equations and constructs two monotonesequences of upper and lower solutions By using quasilinearization technique monotone sequences of approximate solutions thatconverge quadratically to a solution are obtained

1 Introduction

In this paper we consider the system of Caputo fractionaldifferential equations

119888

119863119902

119909 = 119891 (119905 119909) 119909 (1199050) = 1199090 (1)

where 0 lt 119902 lt 1 119891 isin 119862[Ω 119877119899] Ω = [(119905 119909) 120572(119905) le 119909 le 120573(119905)120572 120573 isin 119862

119902

[119869 119877119899

] 119905 isin 119869] and 119869 = [1199050 119879]

The theories and properties of fractional differentialequations have received attention from some researchersbecause many mathematical modeling appeared in the fieldsof physics chemistry engineering and biological sciencesand so on For examples and details we can refer to themonographs of Miller and Ross [1] Podlubny [2] Kilbas etal [3] and West et al [4] and the papers of Debnath [5]Rossikhin and Shitikova [6] and Ferreira et al [7] There aremany results on the basic theory of initial value and boundaryvalue problems for fractional differential equations whichcan be found in [8ndash10]Meanwhile there are some qualitativeand numerical solutions for various fractional equations withdelay and impulsive effects For details see some recentpapers [11ndash18] and the references therein

It is well known that the monotone iterative technique isan ingenious method providing a constructive approach tofind solutions for the nonlinear problem via linear iteratesLakshmikantham and Vatsala [19] and McRae [20] inves-tigated the existence of minimal and maximal solutions of

fractional differential equations by establishing a comparisonresult and using the monotone method respectively Ben-chohra andHamani [21] used amonotone iterative techniquein the presence of lower and upper solutions to discussthe existence of solutions to impulsive fractional differentialinclusions

Quasilinearization [22] provides an elegant and easierapproach to obtain a sequence of approximate solutions withquadratic convergence and the method has been extendedto fractional differential equations in [23 24] However tothe best of our knowledge there are few results for thesystem of fractional differential equations especially resultson the convergence of the system In the present paperwe will discuss the approximate solutions of the system offractional differential equations through the application ofquasilinearization The significance of this work lies in thefact that the system of fractional differential equations canalso obtain a monotone sequence of approximate solutionsconverging uniformly to the solution of the problem andpossessing quadratic convergence

The nonhomogeneous linear system of Caputo fractionaldifferential equations is given by

119888

119863119902

119909 = 119861119909 + 119892 (119905 119910) 119909 (1199050) = 1199090 (2)

where 119861 is an nth order matrix over complex field and 119892 is ann-dimensional locally integrable column vector function on119869

2 Journal of Function Spaces and Applications

Using the method of successive approximations we getthe solution of (2) as

119909 (119905) = 1199090119864119902(119861(119905 minus 119905

0)119902

) + (119905 minus 1199050)119902minus1

119864119902119902

times (119861(119905 minus 1199050)119902

)119896

119892 (119905 119910) 119905 isin 119869

(3)

where

119864119902(119861) =

infin

sum

119896=0

119861119896

Γ (119902119896 + 1) 119864

119902119902(119861) =

infin

sum

119896=0

119861119896

Γ (119902119896 + 119902)(4)

areMittag-Leffler functions of one parameter and twoparam-eters respectively

2 Preliminaries

Now we present the following definition and lemma whichhelp to prove our main result

Definition 1 Let 120572 120573 isin 119862119902[119869 119877119899] be lower and uppersolutions of (1) if they satisfy the inequalities

119888

119863119902

120572 le 119891 (119905 120572) 1205720le 1199090

119888

119863119902

120573 ge 119891 (119905 120573) 1205730ge 1199090

(5)

respectively for 119905 isin 119869

Lemma 2 Suppose that 120572 120573 isin 119862119902[119869 119877119899] are lower and uppersolutions of (1) and

(H1) 119891 isin 119862[Ω 119877119899] are quasimonotone nondecreasing in 119909for each 119905 isin 119869 and for each 119894 isin 119868 = 1 2 119899

119891119894(119905 119909) minus 119891

119894(119905 119910) le 119871

119899

sum

119895=1

(119909119895minus 119910119895) (6)

where 119909 ge 119910 119871 ge 0 is a constantThen 120572(119905

0) le 120573(119905

0) implies that 120572(119905) le 120573(119905)

Proof Firstly suppose that119888

119863119902

120573 gt 119891 (119905 120573) 119888

119863119902

120572 le 119891 (119905 120572) 119905 isin 119869 (7)

and 120572(1199050) lt 120573(119905

0) We will prove that 120572(119905) lt 120573(119905) 119905 isin 119869

Suppose the conclusion is not true then the set

Z =119899

⋃

119894=1

[119905 isin 119869 120573119894(119905) ge 120572

119894(119905)] (8)

is nonemptyLet 1199051= inf Z Certainly 119905

1gt 0 Since the set Z is closed

1199051isin Z and consequently there exists a 119895 such that 120573

119895(1199051) =

120572119895(1199051) Moreover 120573

119894(1199051) ge 120572119894(1199051) for 119894 = 119895 and

120572119895(119905) lt 120573

119895(119905) 119905 isin [0 119905

1) (9)

Hence it easily follows that119888

119863119902

120572119895(1199051) ge119888

119863119902

120573119895(1199051) (10)

This together with the quasimonotonicity of 119891 yields

119891119895(1199051 120572 (1199051)) ge119888

119863119902

120572119895(1199051) ge119888

119863119902

119909120573119895(1199051)

gt 119891119895(1199051 120573119895(1199051)) ge 119891

119895(1199051 120572 (1199051))

(11)

which leads to a contradictionIn order to prove the case of nonstrict inequalities

consider the functions

119894(119905) = 120573

119894(119905) + 120576119890

(119899+1)119871 119894119905 (12)

where 120576 gt 0 is sufficiently small constant Then using (6) wehave119888

119863119902

119894(119905) =

119888

119863119902

120573119894(119905) + (119899 + 1) 120576119871

119894119890(119899+1)119871 119894119905

ge 119891119894(119905 120573 (119905)) + (119899 + 1) 120576119871

119894119890(119899+1)119871 119894119905

ge 119891119894(119905 (119905)) minus 119899120576119871

119894119890(119899+1)119871 119894119905 + (119899 + 1) 120576119871

119894119890(119899+1)119871 119894119905

gt 119891119894(119905 (119905))

(13)

Also 119894(1199050) gt 120572

119894(1199050) Now using the result corresponding to

strict inequalities we get

120572119894(119905) lt

119894(119905) = 120573

119894(119905) + 120576119890

(119899+1)119871 119894119905 119905 isin 119869 (14)

Letting 120576 rarr 0 we obtain the required result and the proof iscomplete

Corollary 3 The function 119891(119905 120572) = 120590(119905)120572 where 120590(119905) le 119871 isadmissible in Lemma 2 to yield 120572(119905) le 0 119905 isin 119869

3 Main Result

Theorem 4 Assume that 1205720 1205730isin 119862119902

[119869 119877119899

] are lower andupper solutions of (1) such that 120572

0(119905) le 120573

0(119905) 119905 isin 119869 and

(H1) 119891 120601 isin 119862[Ω 119877119899] are quasimonotone nondecreasing in119909 for 119905 isin 119869 119891

119909 120601119909 119891119909119909 120601119909119909

exist and are continuous onΩ satisfying 119891

119909119909+ 120601119909119909ge 0 120601

119909119909ge 0

(H2) 119886119894119895(119905 1205720 1205730) ge 0 119894 = 119895 where 119860(119905 120572

0 1205730) =

[119886119894119895(119905 1205720 1205730)] is an 119899 times 119899matrix given by

119860 (119905 1205720 1205730) equiv 119891119909(119905 1205720) + 120601119909(119905 1205720) minus 120601119909(119905 1205730)

= 119865119909(119905 1205720) minus 120601119909(119905 1205730)

(15)

where 119865119909= 119891119909+ 120601119909

Then there existmonotone sequences 120572119899(119905) 120573

119899(119905)which

converge uniformly to the solution of (1) and the convergenceis quadratic

Proof (H1) in Theorem 4 implies for any 119909 ge 119910 119909 119910 isin Ω

119891 (119905 119909) ge 119891 (119905 119910) + [119891119909(119905 119910) + 120601

119909(119905 119910)]

times (119909 minus 119910) minus [120601 (119905 119909) minus 120601 (119905 119910)]

(16)

Journal of Function Spaces and Applications 3

And for any

1205720(119905) le 119910 le 119909 le 120573

0(119905) (17)

we have

119886119894119895(119905 119910 119909) ge 0 119894 = 119895 (18)

since 119886119894119895(119905 119910 119909) ge 119886

119894119895(119905 1205720 1205730) by assumption (H

1)

It is also clear that for 1205720(119905) le 119910 le 119909 le 120573

0(119905)

119891119894(119905 1199091 119909

119899) minus 119891119894(119905 1199101 119910

119899)

le 119871119894

119899

sum

119894=1

(119909119894minus 119910119894)

(19)

Let 1205721 1205731be the solutions of IVPs

119888

119863119902

1205721= 119891 (119905 120572

0) + [119860 (119905 120572

0 1205730)]

times (1205721minus 1205720) 120572

1(0) = 119909

0

119888

119863119902

1205731= 119891 (119905 120573

0) + [119860 (119905 120572

0 1205730)]

times (1205731minus 1205730) 120573

1(0) = 119909

0

(20)

where 1205720(0) le 119909

0le 1205730(0) We will prove that 120572

0le 1205721 119905 isin 119869

To do this let 119901 = 1205720minus 1205721 so that 119901(0) le 0 Then using (20)

we obtain119888

119863119902

119901 =119888

119863119902

1205720minus119888

119863119902

1205721

le 119891 (119905 1205720) minus [119891 (119905 120572

0) + 119860 (119905 120572

0 1205730) (1205721minus 1205720)]

= 119860 (119905 1205720 1205730) 119901

(21)

Since 119860(119905 1205720 1205730) is quasimonotone nondecreasing by

assumption (H1) to (H

2) and it follows from Corollary 3 that

119901(119905) le 0 119905 isin 119869 proving that 1205720le 1205721 119905 isin 119869

Now we let 119901 = 1205721minus 1205730and note that 119901(0) le 0 Also

119888

119863119902

119901 =119888

119863119902

1205721minus119888

119863119902

1205730

le 119891 (119905 1205720) + 119860 (119905 120572

0 1205730) (1205721minus 1205720) minus 119891 (119905 120573

0)

(22)

since 1205730ge 1205720 using (16) we get

119891 (119905 1205730) ge 119891 (119905 120572

0) + 119865119909(119905 1205720) (1205730minus 1205720)

minus [120601 (119905 1205730) minus 120601 (119905 120572

0)]

(23)

In view of (H1) we have

120601 (119905 1205730) minus 120601 (119905 120572

0) le 120601119909(119905 1205730) (1205730minus 1205720) (24)

which yields

119891 (119905 1205730) ge 119891 (119905 120572

0) + [119865

119909(119905 1205720) minus120601119909(119905 1205730)] (1205730minus 1205720)

(25)

Hence we obtain119888

119863119902

119901 le 119860 (119905 1205720 1205730) 119901 (26)

this implies that 1205721(119905) le 120573

0(119905) 119905 isin 119869 using Corollary 3 As a

result we have

1205720(119905) le 120572

1(119905) le 120573

0(119905) 119905 isin 119869 (27)

In a similar way we can prove that

1205720(119905) le 120573

1(119905) le 120573

0(119905) 119905 isin 119869 (28)

To show 1205721(119905) le 120573

1(119905) we use (16) (19) and (H

1)

119888

119863119902

1205721= 119891 (119905 120572

0) + [119865

119909(119905 1205720) minus 120601119909(119905 1205730)] (1205721minus 1205720)

le 119891 (119905 1205721) + 120601 (119905 120572

1) minus 120601 (119905 120572

0)

minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721) + [int

1

0

120601119909(119905 1199041205721+ (1 minus 119904) 120572

0) 119889119904]

times (1205721minus 1205720) minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721) + 120601119909(119905 1205721) (1205721minus 1205720)

minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721)

(29)

Using a similar argument it is easy to show that119888

119863119902

1205731ge 119891 (119905 120573

1) (30)

And therefore by Lemma 2 and (19) it follows that

1205721(119905) le 120573

1(119905) 119905 isin 119869 (31)

by (19) we have 119891 is Lipschitzian in 119909 onΩ This proves that

1205720(119905) le 120572

1(119905) le 120573

1(119905) le 120573

0(119905) 119905 isin 119869 (32)

Now assume that for some 119896 gt 1

120572119896minus1(119905) le 120572

119896(119905) le 120573

119896(119905) le 120573

119896minus1(119905) 119905 isin 119869 (33)

We now aim to show that

120572119896(119905) le 120572

119896+1(119905) le 120573

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (34)

where 120572119896+1(119905) and 120573

119896+1(119905) are the solutions of linear IVPs

119888

119863119902

120572119896+1= 119891 (119905 120572

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120572119896+1minus 120572119896) 120572

119896+1(0) = 119909

0

119888

119863119902

120573119896+1= 119891 (119905 120573

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120573119896+1minus 120573119896) 120573

119896+1(0) = 119909

0

(35)

Now set 119901 = 120572119896minus 120572119896+1

so that119888

119863119902

119901 =119888

119863119902

120572119896minus119888

119863119902

120572119896+1

le 119891 (119905 120572119896) minus [119891 (119905 120572

119896) + (119865

119909(119905 120572119896) minus 120601119909(119905 120573119896))

times (120572119896+1minus 120572119896)]

= 119865119909(119905 120572119896) minus 120601119909(119905 120573119896) 119901

= 119860 (119905 120572119896 120573119896) 119901

(36)

4 Journal of Function Spaces and Applications

and 119901(0) = 0 It follows from Corollary 3 and using (16) that

120572119896(119905) le 120572

119896+1(119905) 119905 isin 119869 (37)

On the other hand letting 119901 = 120572119896+1minus 120573119896yields

119888

119863119902

119901 =119888

119863119902

120572119896+1minus119888

119863119902

120573119896

le 119891 (119905 120572119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120572119896+1minus 120572119896) minus 119891 (119905 120573

119896)

(38)

Since 120573119896ge 120572119896 (16) and (H

1) give as before

119891 (119905 120573119896) ge 119891 (119905 120572

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)] (120573119896minus 120572119896)

(39)

which shows that119888

119863119902

119901 le [119865119909(119905 120572119896) minus 120601119909(119905 120573119896)] 119901 = 119860 (119905 120572

119896 120573119896) 119901 (40)

This proves that 119901(119905) le 0 using Corollary 3 and (16) since119901(0) = 0 Hence we get

120572119896(119905) le 120572

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (41)

Similarly we can prove that

120572119896(119905) le 120573

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (42)

Also by (16) (35) and the fact that 120572119896+1ge 120572119896 we obtain

119888

119863119902

120572119896+1le 119891 (119905 120572

119896+1) + 120601 (119905 120572

119896+1) minus 120601 (119905 120572

119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + [int

1

0

120601119909(119905 119904120572119896+1+ (1 minus 119904) 120572

119896) 119889119904]

times (120572119896+1minus 120572119896) minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + 120601119909(119905 120572119896+1) (120572119896+1minus 120572119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1)

(43)

Using a similar argument we have as before 119888119863119902120573119896+1ge

119891(119905 120573119896+1) and hence Lemma 2 shows that

120572119896+1(119905) le 120573

119896+1(119905) 119905 isin 119869 (44)

By (19) we have that 119891 is Lipschitzian in 119909 on Ω andis quasimonotone nondecreasing in 119909 This proves (34)Therefore by induction we have for all 119899

1205720(119905) le 120572

1(119905) le sdot sdot sdot le 120572

119899(119905) le 120573

119899(119905)

le sdot sdot sdot 1205731(119905) le 120573

0(119905) 119905 isin 119869

(45)

Employing the standard procedure it is now easy to provethat the sequences 120572

119899(119905) and 120573

119899(119905) converge uniformly and

monotonically to the unique solution of (1) on 119869

We will now show that the convergence of 120572119899(119905) and

120573119899(119905) to 119909(119905) is quadratic First set

119901119899+1= 119909 minus 120572

119899+1 119903119899+1= 120573119899+1minus 119909 119905 isin 119869 (46)

so that 119901119899+1(0) = 119903

119899+1(0) = 0 Then using integral mean value

theorem and the fact that 119909 ge 120572119899+1

and (H1) we get

119888

119863119902

119901119899+1=119888

119863119902

119909 minus119888

119863119902

120572119899+1

= 119891 (119905 119909) minus [119891 (119905 120572119899) + (119865

119909(119905 120572119899) minus 120601119909(119905 120573119899))

times (120572119899+1minus 120572119899)]

= 119865 (119905 119909) minus 119865 (119905 120572119899) minus [119865

119909(119905 120572119899) minus 120601119909(119905 120573119899)]

times (120572119899+1minus 120572119899) minus [120601 (119905 119909) minus 120601 (119905 120572

119899)]

= (int

1

0

119865119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904) 119901

119899

minus [119865119909(119905 120572119899) minus 120601119909(119905 120573119899)] (minus119901

119899+1+ 119901119899)

minus [int

1

0

120601119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] (119909 minus 120572

119899)

le (119865119909(119905 119909) minus 119865

119909(119905 120572119899)) 119901119899+ (120601119909(119905 120573119899)

minus120601119909(119905 120572119899)) 119901119899+ 119891119909(119905 120572119899) 119901119899+1

(47)

So we get

119888

119863119902

119901119899+1le 119901119879

119899[int

1

0

119865119909119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] 119901

119899

+ (119903119899+ 119901119899)119879

[int

1

0

120601119909119909(119905 119904120573119899+ (1 minus 119904) 120572

119899) 119889119904]

+ 119891119909(119905 120572119899) 119901119899+1

le 119872119901119899+1+119882(119901

119899sdot 119901119899) + 119873 (119903

119899+ 119901119899) 119901119899

le 119872119901119899+1+ (119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

(48)

where |119901119899|2

= (|1199011198991|2

|1199011198992|2

|119901119899119899|2

) |119903119899| = (|119903

1198991|2

|1199031198992|2

|119903119899119899|2

) 119891119909le 119872 sum119899

119894=1119891119894

119909119909le 119882 and sum119899

119894=1120601119894

119909119909le 119873 in Ω

and119872119882 and119873 are 119899times119899 positive matrices Using (2) to (4)we can get

119901119899+1(119905) le [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

times (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896

le 119881[(119882 +3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

(49)

where 119881 = (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896 Thus we have

1003816100381610038161003816119901119899+11003816100381610038161003816 le 119881 [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

] (50)

Journal of Function Spaces and Applications 5

Similarly we have

1003816100381610038161003816119903119899+11003816100381610038161003816 le 119881 [

119882

2

10038161003816100381610038161199011198991003816100381610038161003816

2

+ (3119882

2+5119873

2)10038161003816100381610038161199031198991003816100381610038161003816

2

] (51)

The proof is complete

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This work is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations John Wiley amp Sons NewYork NY USA 1993

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

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

[4] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[5] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[6] Y Rossikhin andMV Shitikova ldquoApplications of fractional cal-culus to dynamic problems of linear and nonlinear hereditarymechanics of solidsrdquo Applied Mechanics Reviews vol 50 no 1pp 15ndash67 1997

[7] N M Fonseca Ferreira F B Duarte M F M Lima M GMarcos and J A Tenreiro Machado ldquoApplication of fractionalcalculus in the dynamical analysis and control of mechanicalmanipulatorsrdquo Fractional Calculus amp Applied Analysis vol 11no 1 pp 91ndash113 2008

[8] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis vol 69 no 8pp 2677ndash2682 2008

[9] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[10] C B Zhai W P Yan and C Yang ldquoA sum operator method forthe existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value prob-lemsrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 4 pp 858ndash866 2013

[11] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquoNonlinear Analysis vol 69 no 10 pp 3337ndash33432008

[12] T Jankowski ldquoFractional differential equations with deviatingargumentsrdquo Dynamic Systems and Applications vol 17 no 3-4pp 677ndash684 2008

[13] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[14] B Ahmad and S Sivasundaram ldquoExistence of solutions forimpulsive integral boundary value problems of fractionalorderrdquo Nonlinear Analysis vol 4 no 1 pp 134ndash141 2010

[15] W Deng ldquoNumerical algorithm for the time fractional Fokker-Planck equationrdquo Journal of Computational Physics vol 227 no2 pp 1510ndash1522 2007

[16] T Jankowski ldquoFractional equations of Volterra type involvinga Riemann-Liouville derivativerdquo Applied Mathematics Lettersvol 26 no 3 pp 344ndash350 2013

[17] M M El-Borai K E-S El-Nadi and H A Fouad ldquoOn somefractional stochastic delay differential equationsrdquo Computers ampMathematics withApplications vol 59 no 3 pp 1165ndash1170 2010

[18] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[19] V Lakshmikantham and A S Vatsala ldquoGeneral uniquenessand monotone iterative technique for fractional differentialequationsrdquo Applied Mathematics Letters vol 21 no 8 pp 828ndash834 2008

[20] F A McRae ldquoMonotone iterative technique and existenceresults for fractional differential equationsrdquo Nonlinear Analysisvol 71 no 12 pp 6093ndash6096 2009

[21] M Benchohra and S Hamani ldquoThe method of upper andlower solutions and impulsive fractional differential inclusionsrdquoNonlinear Analysis vol 3 no 4 pp 433ndash440 2009

[22] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems Kluwer Academic Publish-ers Dordrecht The Netherlands 1998

[23] J Vasundhara Devi and Ch Suseela ldquoQuasilinearization forfractional differential equationsrdquo Communications in AppliedAnalysis vol 12 no 4 pp 407ndash418 2008

[24] J Vasundhara Devi F A McRae and Z Drici ldquoGeneralizedquasilinearization for fractional differential equationsrdquo Com-puters ampMathematics with Applications vol 59 no 3 pp 1057ndash1062 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

2 Journal of Function Spaces and Applications

Using the method of successive approximations we getthe solution of (2) as

119909 (119905) = 1199090119864119902(119861(119905 minus 119905

0)119902

) + (119905 minus 1199050)119902minus1

119864119902119902

times (119861(119905 minus 1199050)119902

)119896

119892 (119905 119910) 119905 isin 119869

(3)

where

119864119902(119861) =

infin

sum

119896=0

119861119896

Γ (119902119896 + 1) 119864

119902119902(119861) =

infin

sum

119896=0

119861119896

Γ (119902119896 + 119902)(4)

areMittag-Leffler functions of one parameter and twoparam-eters respectively

2 Preliminaries

Now we present the following definition and lemma whichhelp to prove our main result

Definition 1 Let 120572 120573 isin 119862119902[119869 119877119899] be lower and uppersolutions of (1) if they satisfy the inequalities

119888

119863119902

120572 le 119891 (119905 120572) 1205720le 1199090

119888

119863119902

120573 ge 119891 (119905 120573) 1205730ge 1199090

(5)

respectively for 119905 isin 119869

Lemma 2 Suppose that 120572 120573 isin 119862119902[119869 119877119899] are lower and uppersolutions of (1) and

(H1) 119891 isin 119862[Ω 119877119899] are quasimonotone nondecreasing in 119909for each 119905 isin 119869 and for each 119894 isin 119868 = 1 2 119899

119891119894(119905 119909) minus 119891

119894(119905 119910) le 119871

119899

sum

119895=1

(119909119895minus 119910119895) (6)

where 119909 ge 119910 119871 ge 0 is a constantThen 120572(119905

0) le 120573(119905

0) implies that 120572(119905) le 120573(119905)

Proof Firstly suppose that119888

119863119902

120573 gt 119891 (119905 120573) 119888

119863119902

120572 le 119891 (119905 120572) 119905 isin 119869 (7)

and 120572(1199050) lt 120573(119905

0) We will prove that 120572(119905) lt 120573(119905) 119905 isin 119869

Suppose the conclusion is not true then the set

Z =119899

⋃

119894=1

[119905 isin 119869 120573119894(119905) ge 120572

119894(119905)] (8)

is nonemptyLet 1199051= inf Z Certainly 119905

1gt 0 Since the set Z is closed

1199051isin Z and consequently there exists a 119895 such that 120573

119895(1199051) =

120572119895(1199051) Moreover 120573

119894(1199051) ge 120572119894(1199051) for 119894 = 119895 and

120572119895(119905) lt 120573

119895(119905) 119905 isin [0 119905

1) (9)

Hence it easily follows that119888

119863119902

120572119895(1199051) ge119888

119863119902

120573119895(1199051) (10)

This together with the quasimonotonicity of 119891 yields

119891119895(1199051 120572 (1199051)) ge119888

119863119902

120572119895(1199051) ge119888

119863119902

119909120573119895(1199051)

gt 119891119895(1199051 120573119895(1199051)) ge 119891

119895(1199051 120572 (1199051))

(11)

which leads to a contradictionIn order to prove the case of nonstrict inequalities

consider the functions

119894(119905) = 120573

119894(119905) + 120576119890

(119899+1)119871 119894119905 (12)

where 120576 gt 0 is sufficiently small constant Then using (6) wehave119888

119863119902

119894(119905) =

119888

119863119902

120573119894(119905) + (119899 + 1) 120576119871

119894119890(119899+1)119871 119894119905

ge 119891119894(119905 120573 (119905)) + (119899 + 1) 120576119871

119894119890(119899+1)119871 119894119905

ge 119891119894(119905 (119905)) minus 119899120576119871

119894119890(119899+1)119871 119894119905 + (119899 + 1) 120576119871

119894119890(119899+1)119871 119894119905

gt 119891119894(119905 (119905))

(13)

Also 119894(1199050) gt 120572

119894(1199050) Now using the result corresponding to

strict inequalities we get

120572119894(119905) lt

119894(119905) = 120573

119894(119905) + 120576119890

(119899+1)119871 119894119905 119905 isin 119869 (14)

Letting 120576 rarr 0 we obtain the required result and the proof iscomplete

Corollary 3 The function 119891(119905 120572) = 120590(119905)120572 where 120590(119905) le 119871 isadmissible in Lemma 2 to yield 120572(119905) le 0 119905 isin 119869

3 Main Result

Theorem 4 Assume that 1205720 1205730isin 119862119902

[119869 119877119899

] are lower andupper solutions of (1) such that 120572

0(119905) le 120573

0(119905) 119905 isin 119869 and

(H1) 119891 120601 isin 119862[Ω 119877119899] are quasimonotone nondecreasing in119909 for 119905 isin 119869 119891

119909 120601119909 119891119909119909 120601119909119909

exist and are continuous onΩ satisfying 119891

119909119909+ 120601119909119909ge 0 120601

119909119909ge 0

(H2) 119886119894119895(119905 1205720 1205730) ge 0 119894 = 119895 where 119860(119905 120572

0 1205730) =

[119886119894119895(119905 1205720 1205730)] is an 119899 times 119899matrix given by

119860 (119905 1205720 1205730) equiv 119891119909(119905 1205720) + 120601119909(119905 1205720) minus 120601119909(119905 1205730)

= 119865119909(119905 1205720) minus 120601119909(119905 1205730)

(15)

where 119865119909= 119891119909+ 120601119909

Then there existmonotone sequences 120572119899(119905) 120573

119899(119905)which

converge uniformly to the solution of (1) and the convergenceis quadratic

Proof (H1) in Theorem 4 implies for any 119909 ge 119910 119909 119910 isin Ω

119891 (119905 119909) ge 119891 (119905 119910) + [119891119909(119905 119910) + 120601

119909(119905 119910)]

times (119909 minus 119910) minus [120601 (119905 119909) minus 120601 (119905 119910)]

(16)

Journal of Function Spaces and Applications 3

And for any

1205720(119905) le 119910 le 119909 le 120573

0(119905) (17)

we have

119886119894119895(119905 119910 119909) ge 0 119894 = 119895 (18)

since 119886119894119895(119905 119910 119909) ge 119886

119894119895(119905 1205720 1205730) by assumption (H

1)

It is also clear that for 1205720(119905) le 119910 le 119909 le 120573

0(119905)

119891119894(119905 1199091 119909

119899) minus 119891119894(119905 1199101 119910

119899)

le 119871119894

119899

sum

119894=1

(119909119894minus 119910119894)

(19)

Let 1205721 1205731be the solutions of IVPs

119888

119863119902

1205721= 119891 (119905 120572

0) + [119860 (119905 120572

0 1205730)]

times (1205721minus 1205720) 120572

1(0) = 119909

0

119888

119863119902

1205731= 119891 (119905 120573

0) + [119860 (119905 120572

0 1205730)]

times (1205731minus 1205730) 120573

1(0) = 119909

0

(20)

where 1205720(0) le 119909

0le 1205730(0) We will prove that 120572

0le 1205721 119905 isin 119869

To do this let 119901 = 1205720minus 1205721 so that 119901(0) le 0 Then using (20)

we obtain119888

119863119902

119901 =119888

119863119902

1205720minus119888

119863119902

1205721

le 119891 (119905 1205720) minus [119891 (119905 120572

0) + 119860 (119905 120572

0 1205730) (1205721minus 1205720)]

= 119860 (119905 1205720 1205730) 119901

(21)

Since 119860(119905 1205720 1205730) is quasimonotone nondecreasing by

assumption (H1) to (H

2) and it follows from Corollary 3 that

119901(119905) le 0 119905 isin 119869 proving that 1205720le 1205721 119905 isin 119869

Now we let 119901 = 1205721minus 1205730and note that 119901(0) le 0 Also

119888

119863119902

119901 =119888

119863119902

1205721minus119888

119863119902

1205730

le 119891 (119905 1205720) + 119860 (119905 120572

0 1205730) (1205721minus 1205720) minus 119891 (119905 120573

0)

(22)

since 1205730ge 1205720 using (16) we get

119891 (119905 1205730) ge 119891 (119905 120572

0) + 119865119909(119905 1205720) (1205730minus 1205720)

minus [120601 (119905 1205730) minus 120601 (119905 120572

0)]

(23)

In view of (H1) we have

120601 (119905 1205730) minus 120601 (119905 120572

0) le 120601119909(119905 1205730) (1205730minus 1205720) (24)

which yields

119891 (119905 1205730) ge 119891 (119905 120572

0) + [119865

119909(119905 1205720) minus120601119909(119905 1205730)] (1205730minus 1205720)

(25)

Hence we obtain119888

119863119902

119901 le 119860 (119905 1205720 1205730) 119901 (26)

this implies that 1205721(119905) le 120573

0(119905) 119905 isin 119869 using Corollary 3 As a

result we have

1205720(119905) le 120572

1(119905) le 120573

0(119905) 119905 isin 119869 (27)

In a similar way we can prove that

1205720(119905) le 120573

1(119905) le 120573

0(119905) 119905 isin 119869 (28)

To show 1205721(119905) le 120573

1(119905) we use (16) (19) and (H

1)

119888

119863119902

1205721= 119891 (119905 120572

0) + [119865

119909(119905 1205720) minus 120601119909(119905 1205730)] (1205721minus 1205720)

le 119891 (119905 1205721) + 120601 (119905 120572

1) minus 120601 (119905 120572

0)

minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721) + [int

1

0

120601119909(119905 1199041205721+ (1 minus 119904) 120572

0) 119889119904]

times (1205721minus 1205720) minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721) + 120601119909(119905 1205721) (1205721minus 1205720)

minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721)

(29)

Using a similar argument it is easy to show that119888

119863119902

1205731ge 119891 (119905 120573

1) (30)

And therefore by Lemma 2 and (19) it follows that

1205721(119905) le 120573

1(119905) 119905 isin 119869 (31)

by (19) we have 119891 is Lipschitzian in 119909 onΩ This proves that

1205720(119905) le 120572

1(119905) le 120573

1(119905) le 120573

0(119905) 119905 isin 119869 (32)

Now assume that for some 119896 gt 1

120572119896minus1(119905) le 120572

119896(119905) le 120573

119896(119905) le 120573

119896minus1(119905) 119905 isin 119869 (33)

We now aim to show that

120572119896(119905) le 120572

119896+1(119905) le 120573

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (34)

where 120572119896+1(119905) and 120573

119896+1(119905) are the solutions of linear IVPs

119888

119863119902

120572119896+1= 119891 (119905 120572

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120572119896+1minus 120572119896) 120572

119896+1(0) = 119909

0

119888

119863119902

120573119896+1= 119891 (119905 120573

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120573119896+1minus 120573119896) 120573

119896+1(0) = 119909

0

(35)

Now set 119901 = 120572119896minus 120572119896+1

so that119888

119863119902

119901 =119888

119863119902

120572119896minus119888

119863119902

120572119896+1

le 119891 (119905 120572119896) minus [119891 (119905 120572

119896) + (119865

119909(119905 120572119896) minus 120601119909(119905 120573119896))

times (120572119896+1minus 120572119896)]

= 119865119909(119905 120572119896) minus 120601119909(119905 120573119896) 119901

= 119860 (119905 120572119896 120573119896) 119901

(36)

4 Journal of Function Spaces and Applications

and 119901(0) = 0 It follows from Corollary 3 and using (16) that

120572119896(119905) le 120572

119896+1(119905) 119905 isin 119869 (37)

On the other hand letting 119901 = 120572119896+1minus 120573119896yields

119888

119863119902

119901 =119888

119863119902

120572119896+1minus119888

119863119902

120573119896

le 119891 (119905 120572119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120572119896+1minus 120572119896) minus 119891 (119905 120573

119896)

(38)

Since 120573119896ge 120572119896 (16) and (H

1) give as before

119891 (119905 120573119896) ge 119891 (119905 120572

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)] (120573119896minus 120572119896)

(39)

which shows that119888

119863119902

119901 le [119865119909(119905 120572119896) minus 120601119909(119905 120573119896)] 119901 = 119860 (119905 120572

119896 120573119896) 119901 (40)

This proves that 119901(119905) le 0 using Corollary 3 and (16) since119901(0) = 0 Hence we get

120572119896(119905) le 120572

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (41)

Similarly we can prove that

120572119896(119905) le 120573

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (42)

Also by (16) (35) and the fact that 120572119896+1ge 120572119896 we obtain

119888

119863119902

120572119896+1le 119891 (119905 120572

119896+1) + 120601 (119905 120572

119896+1) minus 120601 (119905 120572

119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + [int

1

0

120601119909(119905 119904120572119896+1+ (1 minus 119904) 120572

119896) 119889119904]

times (120572119896+1minus 120572119896) minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + 120601119909(119905 120572119896+1) (120572119896+1minus 120572119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1)

(43)

Using a similar argument we have as before 119888119863119902120573119896+1ge

119891(119905 120573119896+1) and hence Lemma 2 shows that

120572119896+1(119905) le 120573

119896+1(119905) 119905 isin 119869 (44)

By (19) we have that 119891 is Lipschitzian in 119909 on Ω andis quasimonotone nondecreasing in 119909 This proves (34)Therefore by induction we have for all 119899

1205720(119905) le 120572

1(119905) le sdot sdot sdot le 120572

119899(119905) le 120573

119899(119905)

le sdot sdot sdot 1205731(119905) le 120573

0(119905) 119905 isin 119869

(45)

Employing the standard procedure it is now easy to provethat the sequences 120572

119899(119905) and 120573

119899(119905) converge uniformly and

monotonically to the unique solution of (1) on 119869

We will now show that the convergence of 120572119899(119905) and

120573119899(119905) to 119909(119905) is quadratic First set

119901119899+1= 119909 minus 120572

119899+1 119903119899+1= 120573119899+1minus 119909 119905 isin 119869 (46)

so that 119901119899+1(0) = 119903

119899+1(0) = 0 Then using integral mean value

theorem and the fact that 119909 ge 120572119899+1

and (H1) we get

119888

119863119902

119901119899+1=119888

119863119902

119909 minus119888

119863119902

120572119899+1

= 119891 (119905 119909) minus [119891 (119905 120572119899) + (119865

119909(119905 120572119899) minus 120601119909(119905 120573119899))

times (120572119899+1minus 120572119899)]

= 119865 (119905 119909) minus 119865 (119905 120572119899) minus [119865

119909(119905 120572119899) minus 120601119909(119905 120573119899)]

times (120572119899+1minus 120572119899) minus [120601 (119905 119909) minus 120601 (119905 120572

119899)]

= (int

1

0

119865119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904) 119901

119899

minus [119865119909(119905 120572119899) minus 120601119909(119905 120573119899)] (minus119901

119899+1+ 119901119899)

minus [int

1

0

120601119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] (119909 minus 120572

119899)

le (119865119909(119905 119909) minus 119865

119909(119905 120572119899)) 119901119899+ (120601119909(119905 120573119899)

minus120601119909(119905 120572119899)) 119901119899+ 119891119909(119905 120572119899) 119901119899+1

(47)

So we get

119888

119863119902

119901119899+1le 119901119879

119899[int

1

0

119865119909119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] 119901

119899

+ (119903119899+ 119901119899)119879

[int

1

0

120601119909119909(119905 119904120573119899+ (1 minus 119904) 120572

119899) 119889119904]

+ 119891119909(119905 120572119899) 119901119899+1

le 119872119901119899+1+119882(119901

119899sdot 119901119899) + 119873 (119903

119899+ 119901119899) 119901119899

le 119872119901119899+1+ (119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

(48)

where |119901119899|2

= (|1199011198991|2

|1199011198992|2

|119901119899119899|2

) |119903119899| = (|119903

1198991|2

|1199031198992|2

|119903119899119899|2

) 119891119909le 119872 sum119899

119894=1119891119894

119909119909le 119882 and sum119899

119894=1120601119894

119909119909le 119873 in Ω

and119872119882 and119873 are 119899times119899 positive matrices Using (2) to (4)we can get

119901119899+1(119905) le [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

times (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896

le 119881[(119882 +3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

(49)

where 119881 = (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896 Thus we have

1003816100381610038161003816119901119899+11003816100381610038161003816 le 119881 [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

] (50)

Journal of Function Spaces and Applications 5

Similarly we have

1003816100381610038161003816119903119899+11003816100381610038161003816 le 119881 [

119882

2

10038161003816100381610038161199011198991003816100381610038161003816

2

+ (3119882

2+5119873

2)10038161003816100381610038161199031198991003816100381610038161003816

2

] (51)

The proof is complete

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This work is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations John Wiley amp Sons NewYork NY USA 1993

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

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

[4] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[5] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[6] Y Rossikhin andMV Shitikova ldquoApplications of fractional cal-culus to dynamic problems of linear and nonlinear hereditarymechanics of solidsrdquo Applied Mechanics Reviews vol 50 no 1pp 15ndash67 1997

[7] N M Fonseca Ferreira F B Duarte M F M Lima M GMarcos and J A Tenreiro Machado ldquoApplication of fractionalcalculus in the dynamical analysis and control of mechanicalmanipulatorsrdquo Fractional Calculus amp Applied Analysis vol 11no 1 pp 91ndash113 2008

[8] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis vol 69 no 8pp 2677ndash2682 2008

[9] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[10] C B Zhai W P Yan and C Yang ldquoA sum operator method forthe existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value prob-lemsrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 4 pp 858ndash866 2013

[11] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquoNonlinear Analysis vol 69 no 10 pp 3337ndash33432008

[12] T Jankowski ldquoFractional differential equations with deviatingargumentsrdquo Dynamic Systems and Applications vol 17 no 3-4pp 677ndash684 2008

[13] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[14] B Ahmad and S Sivasundaram ldquoExistence of solutions forimpulsive integral boundary value problems of fractionalorderrdquo Nonlinear Analysis vol 4 no 1 pp 134ndash141 2010

[15] W Deng ldquoNumerical algorithm for the time fractional Fokker-Planck equationrdquo Journal of Computational Physics vol 227 no2 pp 1510ndash1522 2007

[16] T Jankowski ldquoFractional equations of Volterra type involvinga Riemann-Liouville derivativerdquo Applied Mathematics Lettersvol 26 no 3 pp 344ndash350 2013

[17] M M El-Borai K E-S El-Nadi and H A Fouad ldquoOn somefractional stochastic delay differential equationsrdquo Computers ampMathematics withApplications vol 59 no 3 pp 1165ndash1170 2010

[18] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[19] V Lakshmikantham and A S Vatsala ldquoGeneral uniquenessand monotone iterative technique for fractional differentialequationsrdquo Applied Mathematics Letters vol 21 no 8 pp 828ndash834 2008

[20] F A McRae ldquoMonotone iterative technique and existenceresults for fractional differential equationsrdquo Nonlinear Analysisvol 71 no 12 pp 6093ndash6096 2009

[21] M Benchohra and S Hamani ldquoThe method of upper andlower solutions and impulsive fractional differential inclusionsrdquoNonlinear Analysis vol 3 no 4 pp 433ndash440 2009

[22] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems Kluwer Academic Publish-ers Dordrecht The Netherlands 1998

[23] J Vasundhara Devi and Ch Suseela ldquoQuasilinearization forfractional differential equationsrdquo Communications in AppliedAnalysis vol 12 no 4 pp 407ndash418 2008

[24] J Vasundhara Devi F A McRae and Z Drici ldquoGeneralizedquasilinearization for fractional differential equationsrdquo Com-puters ampMathematics with Applications vol 59 no 3 pp 1057ndash1062 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 3

And for any

1205720(119905) le 119910 le 119909 le 120573

0(119905) (17)

we have

119886119894119895(119905 119910 119909) ge 0 119894 = 119895 (18)

since 119886119894119895(119905 119910 119909) ge 119886

119894119895(119905 1205720 1205730) by assumption (H

1)

It is also clear that for 1205720(119905) le 119910 le 119909 le 120573

0(119905)

119891119894(119905 1199091 119909

119899) minus 119891119894(119905 1199101 119910

119899)

le 119871119894

119899

sum

119894=1

(119909119894minus 119910119894)

(19)

Let 1205721 1205731be the solutions of IVPs

119888

119863119902

1205721= 119891 (119905 120572

0) + [119860 (119905 120572

0 1205730)]

times (1205721minus 1205720) 120572

1(0) = 119909

0

119888

119863119902

1205731= 119891 (119905 120573

0) + [119860 (119905 120572

0 1205730)]

times (1205731minus 1205730) 120573

1(0) = 119909

0

(20)

where 1205720(0) le 119909

0le 1205730(0) We will prove that 120572

0le 1205721 119905 isin 119869

To do this let 119901 = 1205720minus 1205721 so that 119901(0) le 0 Then using (20)

we obtain119888

119863119902

119901 =119888

119863119902

1205720minus119888

119863119902

1205721

le 119891 (119905 1205720) minus [119891 (119905 120572

0) + 119860 (119905 120572

0 1205730) (1205721minus 1205720)]

= 119860 (119905 1205720 1205730) 119901

(21)

Since 119860(119905 1205720 1205730) is quasimonotone nondecreasing by

assumption (H1) to (H

2) and it follows from Corollary 3 that

119901(119905) le 0 119905 isin 119869 proving that 1205720le 1205721 119905 isin 119869

Now we let 119901 = 1205721minus 1205730and note that 119901(0) le 0 Also

119888

119863119902

119901 =119888

119863119902

1205721minus119888

119863119902

1205730

le 119891 (119905 1205720) + 119860 (119905 120572

0 1205730) (1205721minus 1205720) minus 119891 (119905 120573

0)

(22)

since 1205730ge 1205720 using (16) we get

119891 (119905 1205730) ge 119891 (119905 120572

0) + 119865119909(119905 1205720) (1205730minus 1205720)

minus [120601 (119905 1205730) minus 120601 (119905 120572

0)]

(23)

In view of (H1) we have

120601 (119905 1205730) minus 120601 (119905 120572

0) le 120601119909(119905 1205730) (1205730minus 1205720) (24)

which yields

119891 (119905 1205730) ge 119891 (119905 120572

0) + [119865

119909(119905 1205720) minus120601119909(119905 1205730)] (1205730minus 1205720)

(25)

Hence we obtain119888

119863119902

119901 le 119860 (119905 1205720 1205730) 119901 (26)

this implies that 1205721(119905) le 120573

0(119905) 119905 isin 119869 using Corollary 3 As a

result we have

1205720(119905) le 120572

1(119905) le 120573

0(119905) 119905 isin 119869 (27)

In a similar way we can prove that

1205720(119905) le 120573

1(119905) le 120573

0(119905) 119905 isin 119869 (28)

To show 1205721(119905) le 120573

1(119905) we use (16) (19) and (H

1)

119888

119863119902

1205721= 119891 (119905 120572

0) + [119865

119909(119905 1205720) minus 120601119909(119905 1205730)] (1205721minus 1205720)

le 119891 (119905 1205721) + 120601 (119905 120572

1) minus 120601 (119905 120572

0)

minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721) + [int

1

0

120601119909(119905 1199041205721+ (1 minus 119904) 120572

0) 119889119904]

times (1205721minus 1205720) minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721) + 120601119909(119905 1205721) (1205721minus 1205720)

minus 120601119909(119905 1205730) (1205721minus 1205720)

le 119891 (119905 1205721)

(29)

Using a similar argument it is easy to show that119888

119863119902

1205731ge 119891 (119905 120573

1) (30)

And therefore by Lemma 2 and (19) it follows that

1205721(119905) le 120573

1(119905) 119905 isin 119869 (31)

by (19) we have 119891 is Lipschitzian in 119909 onΩ This proves that

1205720(119905) le 120572

1(119905) le 120573

1(119905) le 120573

0(119905) 119905 isin 119869 (32)

Now assume that for some 119896 gt 1

120572119896minus1(119905) le 120572

119896(119905) le 120573

119896(119905) le 120573

119896minus1(119905) 119905 isin 119869 (33)

We now aim to show that

120572119896(119905) le 120572

119896+1(119905) le 120573

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (34)

where 120572119896+1(119905) and 120573

119896+1(119905) are the solutions of linear IVPs

119888

119863119902

120572119896+1= 119891 (119905 120572

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120572119896+1minus 120572119896) 120572

119896+1(0) = 119909

0

119888

119863119902

120573119896+1= 119891 (119905 120573

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120573119896+1minus 120573119896) 120573

119896+1(0) = 119909

0

(35)

Now set 119901 = 120572119896minus 120572119896+1

so that119888

119863119902

119901 =119888

119863119902

120572119896minus119888

119863119902

120572119896+1

le 119891 (119905 120572119896) minus [119891 (119905 120572

119896) + (119865

119909(119905 120572119896) minus 120601119909(119905 120573119896))

times (120572119896+1minus 120572119896)]

= 119865119909(119905 120572119896) minus 120601119909(119905 120573119896) 119901

= 119860 (119905 120572119896 120573119896) 119901

(36)

4 Journal of Function Spaces and Applications

and 119901(0) = 0 It follows from Corollary 3 and using (16) that

120572119896(119905) le 120572

119896+1(119905) 119905 isin 119869 (37)

On the other hand letting 119901 = 120572119896+1minus 120573119896yields

119888

119863119902

119901 =119888

119863119902

120572119896+1minus119888

119863119902

120573119896

le 119891 (119905 120572119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120572119896+1minus 120572119896) minus 119891 (119905 120573

119896)

(38)

Since 120573119896ge 120572119896 (16) and (H

1) give as before

119891 (119905 120573119896) ge 119891 (119905 120572

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)] (120573119896minus 120572119896)

(39)

which shows that119888

119863119902

119901 le [119865119909(119905 120572119896) minus 120601119909(119905 120573119896)] 119901 = 119860 (119905 120572

119896 120573119896) 119901 (40)

This proves that 119901(119905) le 0 using Corollary 3 and (16) since119901(0) = 0 Hence we get

120572119896(119905) le 120572

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (41)

Similarly we can prove that

120572119896(119905) le 120573

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (42)

Also by (16) (35) and the fact that 120572119896+1ge 120572119896 we obtain

119888

119863119902

120572119896+1le 119891 (119905 120572

119896+1) + 120601 (119905 120572

119896+1) minus 120601 (119905 120572

119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + [int

1

0

120601119909(119905 119904120572119896+1+ (1 minus 119904) 120572

119896) 119889119904]

times (120572119896+1minus 120572119896) minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + 120601119909(119905 120572119896+1) (120572119896+1minus 120572119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1)

(43)

Using a similar argument we have as before 119888119863119902120573119896+1ge

119891(119905 120573119896+1) and hence Lemma 2 shows that

120572119896+1(119905) le 120573

119896+1(119905) 119905 isin 119869 (44)

By (19) we have that 119891 is Lipschitzian in 119909 on Ω andis quasimonotone nondecreasing in 119909 This proves (34)Therefore by induction we have for all 119899

1205720(119905) le 120572

1(119905) le sdot sdot sdot le 120572

119899(119905) le 120573

119899(119905)

le sdot sdot sdot 1205731(119905) le 120573

0(119905) 119905 isin 119869

(45)

Employing the standard procedure it is now easy to provethat the sequences 120572

119899(119905) and 120573

119899(119905) converge uniformly and

monotonically to the unique solution of (1) on 119869

We will now show that the convergence of 120572119899(119905) and

120573119899(119905) to 119909(119905) is quadratic First set

119901119899+1= 119909 minus 120572

119899+1 119903119899+1= 120573119899+1minus 119909 119905 isin 119869 (46)

so that 119901119899+1(0) = 119903

119899+1(0) = 0 Then using integral mean value

theorem and the fact that 119909 ge 120572119899+1

and (H1) we get

119888

119863119902

119901119899+1=119888

119863119902

119909 minus119888

119863119902

120572119899+1

= 119891 (119905 119909) minus [119891 (119905 120572119899) + (119865

119909(119905 120572119899) minus 120601119909(119905 120573119899))

times (120572119899+1minus 120572119899)]

= 119865 (119905 119909) minus 119865 (119905 120572119899) minus [119865

119909(119905 120572119899) minus 120601119909(119905 120573119899)]

times (120572119899+1minus 120572119899) minus [120601 (119905 119909) minus 120601 (119905 120572

119899)]

= (int

1

0

119865119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904) 119901

119899

minus [119865119909(119905 120572119899) minus 120601119909(119905 120573119899)] (minus119901

119899+1+ 119901119899)

minus [int

1

0

120601119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] (119909 minus 120572

119899)

le (119865119909(119905 119909) minus 119865

119909(119905 120572119899)) 119901119899+ (120601119909(119905 120573119899)

minus120601119909(119905 120572119899)) 119901119899+ 119891119909(119905 120572119899) 119901119899+1

(47)

So we get

119888

119863119902

119901119899+1le 119901119879

119899[int

1

0

119865119909119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] 119901

119899

+ (119903119899+ 119901119899)119879

[int

1

0

120601119909119909(119905 119904120573119899+ (1 minus 119904) 120572

119899) 119889119904]

+ 119891119909(119905 120572119899) 119901119899+1

le 119872119901119899+1+119882(119901

119899sdot 119901119899) + 119873 (119903

119899+ 119901119899) 119901119899

le 119872119901119899+1+ (119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

(48)

where |119901119899|2

= (|1199011198991|2

|1199011198992|2

|119901119899119899|2

) |119903119899| = (|119903

1198991|2

|1199031198992|2

|119903119899119899|2

) 119891119909le 119872 sum119899

119894=1119891119894

119909119909le 119882 and sum119899

119894=1120601119894

119909119909le 119873 in Ω

and119872119882 and119873 are 119899times119899 positive matrices Using (2) to (4)we can get

119901119899+1(119905) le [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

times (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896

le 119881[(119882 +3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

(49)

where 119881 = (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896 Thus we have

1003816100381610038161003816119901119899+11003816100381610038161003816 le 119881 [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

] (50)

Journal of Function Spaces and Applications 5

Similarly we have

1003816100381610038161003816119903119899+11003816100381610038161003816 le 119881 [

119882

2

10038161003816100381610038161199011198991003816100381610038161003816

2

+ (3119882

2+5119873

2)10038161003816100381610038161199031198991003816100381610038161003816

2

] (51)

The proof is complete

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This work is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations John Wiley amp Sons NewYork NY USA 1993

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

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

[4] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[5] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[6] Y Rossikhin andMV Shitikova ldquoApplications of fractional cal-culus to dynamic problems of linear and nonlinear hereditarymechanics of solidsrdquo Applied Mechanics Reviews vol 50 no 1pp 15ndash67 1997

[7] N M Fonseca Ferreira F B Duarte M F M Lima M GMarcos and J A Tenreiro Machado ldquoApplication of fractionalcalculus in the dynamical analysis and control of mechanicalmanipulatorsrdquo Fractional Calculus amp Applied Analysis vol 11no 1 pp 91ndash113 2008

[8] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis vol 69 no 8pp 2677ndash2682 2008

[9] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[10] C B Zhai W P Yan and C Yang ldquoA sum operator method forthe existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value prob-lemsrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 4 pp 858ndash866 2013

[11] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquoNonlinear Analysis vol 69 no 10 pp 3337ndash33432008

[12] T Jankowski ldquoFractional differential equations with deviatingargumentsrdquo Dynamic Systems and Applications vol 17 no 3-4pp 677ndash684 2008

[13] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[14] B Ahmad and S Sivasundaram ldquoExistence of solutions forimpulsive integral boundary value problems of fractionalorderrdquo Nonlinear Analysis vol 4 no 1 pp 134ndash141 2010

[15] W Deng ldquoNumerical algorithm for the time fractional Fokker-Planck equationrdquo Journal of Computational Physics vol 227 no2 pp 1510ndash1522 2007

[16] T Jankowski ldquoFractional equations of Volterra type involvinga Riemann-Liouville derivativerdquo Applied Mathematics Lettersvol 26 no 3 pp 344ndash350 2013

[17] M M El-Borai K E-S El-Nadi and H A Fouad ldquoOn somefractional stochastic delay differential equationsrdquo Computers ampMathematics withApplications vol 59 no 3 pp 1165ndash1170 2010

[18] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[19] V Lakshmikantham and A S Vatsala ldquoGeneral uniquenessand monotone iterative technique for fractional differentialequationsrdquo Applied Mathematics Letters vol 21 no 8 pp 828ndash834 2008

[20] F A McRae ldquoMonotone iterative technique and existenceresults for fractional differential equationsrdquo Nonlinear Analysisvol 71 no 12 pp 6093ndash6096 2009

[21] M Benchohra and S Hamani ldquoThe method of upper andlower solutions and impulsive fractional differential inclusionsrdquoNonlinear Analysis vol 3 no 4 pp 433ndash440 2009

[22] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems Kluwer Academic Publish-ers Dordrecht The Netherlands 1998

[23] J Vasundhara Devi and Ch Suseela ldquoQuasilinearization forfractional differential equationsrdquo Communications in AppliedAnalysis vol 12 no 4 pp 407ndash418 2008

[24] J Vasundhara Devi F A McRae and Z Drici ldquoGeneralizedquasilinearization for fractional differential equationsrdquo Com-puters ampMathematics with Applications vol 59 no 3 pp 1057ndash1062 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Journal of Function Spaces and Applications

and 119901(0) = 0 It follows from Corollary 3 and using (16) that

120572119896(119905) le 120572

119896+1(119905) 119905 isin 119869 (37)

On the other hand letting 119901 = 120572119896+1minus 120573119896yields

119888

119863119902

119901 =119888

119863119902

120572119896+1minus119888

119863119902

120573119896

le 119891 (119905 120572119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)]

times (120572119896+1minus 120572119896) minus 119891 (119905 120573

119896)

(38)

Since 120573119896ge 120572119896 (16) and (H

1) give as before

119891 (119905 120573119896) ge 119891 (119905 120572

119896) + [119865

119909(119905 120572119896) minus 120601119909(119905 120573119896)] (120573119896minus 120572119896)

(39)

which shows that119888

119863119902

119901 le [119865119909(119905 120572119896) minus 120601119909(119905 120573119896)] 119901 = 119860 (119905 120572

119896 120573119896) 119901 (40)

This proves that 119901(119905) le 0 using Corollary 3 and (16) since119901(0) = 0 Hence we get

120572119896(119905) le 120572

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (41)

Similarly we can prove that

120572119896(119905) le 120573

119896+1(119905) le 120573

119896(119905) 119905 isin 119869 (42)

Also by (16) (35) and the fact that 120572119896+1ge 120572119896 we obtain

119888

119863119902

120572119896+1le 119891 (119905 120572

119896+1) + 120601 (119905 120572

119896+1) minus 120601 (119905 120572

119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + [int

1

0

120601119909(119905 119904120572119896+1+ (1 minus 119904) 120572

119896) 119889119904]

times (120572119896+1minus 120572119896) minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1) + 120601119909(119905 120572119896+1) (120572119896+1minus 120572119896)

minus 120601119909(119905 120573119896) (120572119896+1minus 120572119896)

le 119891 (119905 120572119896+1)

(43)

Using a similar argument we have as before 119888119863119902120573119896+1ge

119891(119905 120573119896+1) and hence Lemma 2 shows that

120572119896+1(119905) le 120573

119896+1(119905) 119905 isin 119869 (44)

By (19) we have that 119891 is Lipschitzian in 119909 on Ω andis quasimonotone nondecreasing in 119909 This proves (34)Therefore by induction we have for all 119899

1205720(119905) le 120572

1(119905) le sdot sdot sdot le 120572

119899(119905) le 120573

119899(119905)

le sdot sdot sdot 1205731(119905) le 120573

0(119905) 119905 isin 119869

(45)

Employing the standard procedure it is now easy to provethat the sequences 120572

119899(119905) and 120573

119899(119905) converge uniformly and

monotonically to the unique solution of (1) on 119869

We will now show that the convergence of 120572119899(119905) and

120573119899(119905) to 119909(119905) is quadratic First set

119901119899+1= 119909 minus 120572

119899+1 119903119899+1= 120573119899+1minus 119909 119905 isin 119869 (46)

so that 119901119899+1(0) = 119903

119899+1(0) = 0 Then using integral mean value

theorem and the fact that 119909 ge 120572119899+1

and (H1) we get

119888

119863119902

119901119899+1=119888

119863119902

119909 minus119888

119863119902

120572119899+1

= 119891 (119905 119909) minus [119891 (119905 120572119899) + (119865

119909(119905 120572119899) minus 120601119909(119905 120573119899))

times (120572119899+1minus 120572119899)]

= 119865 (119905 119909) minus 119865 (119905 120572119899) minus [119865

119909(119905 120572119899) minus 120601119909(119905 120573119899)]

times (120572119899+1minus 120572119899) minus [120601 (119905 119909) minus 120601 (119905 120572

119899)]

= (int

1

0

119865119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904) 119901

119899

minus [119865119909(119905 120572119899) minus 120601119909(119905 120573119899)] (minus119901

119899+1+ 119901119899)

minus [int

1

0

120601119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] (119909 minus 120572

119899)

le (119865119909(119905 119909) minus 119865

119909(119905 120572119899)) 119901119899+ (120601119909(119905 120573119899)

minus120601119909(119905 120572119899)) 119901119899+ 119891119909(119905 120572119899) 119901119899+1

(47)

So we get

119888

119863119902

119901119899+1le 119901119879

119899[int

1

0

119865119909119909(119905 119904119909 + (1 minus 119904) 120572

119899) 119889119904] 119901

119899

+ (119903119899+ 119901119899)119879

[int

1

0

120601119909119909(119905 119904120573119899+ (1 minus 119904) 120572

119899) 119889119904]

+ 119891119909(119905 120572119899) 119901119899+1

le 119872119901119899+1+119882(119901

119899sdot 119901119899) + 119873 (119903

119899+ 119901119899) 119901119899

le 119872119901119899+1+ (119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

(48)

where |119901119899|2

= (|1199011198991|2

|1199011198992|2

|119901119899119899|2

) |119903119899| = (|119903

1198991|2

|1199031198992|2

|119903119899119899|2

) 119891119909le 119872 sum119899

119894=1119891119894

119909119909le 119882 and sum119899

119894=1120601119894

119909119909le 119873 in Ω

and119872119882 and119873 are 119899times119899 positive matrices Using (2) to (4)we can get

119901119899+1(119905) le [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

times (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896

le 119881[(119882 +3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

]

(49)

where 119881 = (119879 minus 1199050)119902minus1

119864119902119902(119872(119879 minus 119905

0)119902

)119896 Thus we have

1003816100381610038161003816119901119899+11003816100381610038161003816 le 119881 [(119882 +

3119873

2)10038161003816100381610038161199011198991003816100381610038161003816

2

+119873

2

10038161003816100381610038161199031198991003816100381610038161003816

2

] (50)

Journal of Function Spaces and Applications 5

Similarly we have

1003816100381610038161003816119903119899+11003816100381610038161003816 le 119881 [

119882

2

10038161003816100381610038161199011198991003816100381610038161003816

2

+ (3119882

2+5119873

2)10038161003816100381610038161199031198991003816100381610038161003816

2

] (51)

The proof is complete

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This work is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations John Wiley amp Sons NewYork NY USA 1993

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

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

[4] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[5] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[6] Y Rossikhin andMV Shitikova ldquoApplications of fractional cal-culus to dynamic problems of linear and nonlinear hereditarymechanics of solidsrdquo Applied Mechanics Reviews vol 50 no 1pp 15ndash67 1997

[7] N M Fonseca Ferreira F B Duarte M F M Lima M GMarcos and J A Tenreiro Machado ldquoApplication of fractionalcalculus in the dynamical analysis and control of mechanicalmanipulatorsrdquo Fractional Calculus amp Applied Analysis vol 11no 1 pp 91ndash113 2008

[8] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis vol 69 no 8pp 2677ndash2682 2008

[9] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[10] C B Zhai W P Yan and C Yang ldquoA sum operator method forthe existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value prob-lemsrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 4 pp 858ndash866 2013

[11] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquoNonlinear Analysis vol 69 no 10 pp 3337ndash33432008

[12] T Jankowski ldquoFractional differential equations with deviatingargumentsrdquo Dynamic Systems and Applications vol 17 no 3-4pp 677ndash684 2008

[13] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[14] B Ahmad and S Sivasundaram ldquoExistence of solutions forimpulsive integral boundary value problems of fractionalorderrdquo Nonlinear Analysis vol 4 no 1 pp 134ndash141 2010

[15] W Deng ldquoNumerical algorithm for the time fractional Fokker-Planck equationrdquo Journal of Computational Physics vol 227 no2 pp 1510ndash1522 2007

[16] T Jankowski ldquoFractional equations of Volterra type involvinga Riemann-Liouville derivativerdquo Applied Mathematics Lettersvol 26 no 3 pp 344ndash350 2013

[17] M M El-Borai K E-S El-Nadi and H A Fouad ldquoOn somefractional stochastic delay differential equationsrdquo Computers ampMathematics withApplications vol 59 no 3 pp 1165ndash1170 2010

[18] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[19] V Lakshmikantham and A S Vatsala ldquoGeneral uniquenessand monotone iterative technique for fractional differentialequationsrdquo Applied Mathematics Letters vol 21 no 8 pp 828ndash834 2008

[20] F A McRae ldquoMonotone iterative technique and existenceresults for fractional differential equationsrdquo Nonlinear Analysisvol 71 no 12 pp 6093ndash6096 2009

[21] M Benchohra and S Hamani ldquoThe method of upper andlower solutions and impulsive fractional differential inclusionsrdquoNonlinear Analysis vol 3 no 4 pp 433ndash440 2009

[22] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems Kluwer Academic Publish-ers Dordrecht The Netherlands 1998

[23] J Vasundhara Devi and Ch Suseela ldquoQuasilinearization forfractional differential equationsrdquo Communications in AppliedAnalysis vol 12 no 4 pp 407ndash418 2008

[24] J Vasundhara Devi F A McRae and Z Drici ldquoGeneralizedquasilinearization for fractional differential equationsrdquo Com-puters ampMathematics with Applications vol 59 no 3 pp 1057ndash1062 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Journal of Function Spaces and Applications 5

Similarly we have

1003816100381610038161003816119903119899+11003816100381610038161003816 le 119881 [

119882

2

10038161003816100381610038161199011198991003816100381610038161003816

2

+ (3119882

2+5119873

2)10038161003816100381610038161199031198991003816100381610038161003816

2

] (51)

The proof is complete

Acknowledgments

The authors would like to thank the reviewers for their valu-able suggestions and comments This work is supported bythe National Natural Science Foundation of China (11271106)and the Natural Science Foundation of Hebei Province ofChina (A2013201232)

References

[1] K S Miller and B Ross An Introduction to the FractionalCalculus and Differential Equations John Wiley amp Sons NewYork NY USA 1993

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

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

[4] B J West M Bologna and P Grigolini Physics of FractalOperators Springer New York NY USA 2003

[5] L Debnath ldquoRecent applications of fractional calculus toscience and engineeringrdquo International Journal of Mathematicsand Mathematical Sciences vol 2003 no 54 pp 3413ndash34422003

[6] Y Rossikhin andMV Shitikova ldquoApplications of fractional cal-culus to dynamic problems of linear and nonlinear hereditarymechanics of solidsrdquo Applied Mechanics Reviews vol 50 no 1pp 15ndash67 1997

[7] N M Fonseca Ferreira F B Duarte M F M Lima M GMarcos and J A Tenreiro Machado ldquoApplication of fractionalcalculus in the dynamical analysis and control of mechanicalmanipulatorsrdquo Fractional Calculus amp Applied Analysis vol 11no 1 pp 91ndash113 2008

[8] V Lakshmikantham and A S Vatsala ldquoBasic theory of frac-tional differential equationsrdquo Nonlinear Analysis vol 69 no 8pp 2677ndash2682 2008

[9] Z Denton and A S Vatsala ldquoFractional integral inequalitiesand applicationsrdquo Computers amp Mathematics with Applicationsvol 59 no 3 pp 1087ndash1094 2010

[10] C B Zhai W P Yan and C Yang ldquoA sum operator method forthe existence and uniqueness of positive solutions to Riemann-Liouville fractional differential equation boundary value prob-lemsrdquo Communications in Nonlinear Science and NumericalSimulation vol 18 no 4 pp 858ndash866 2013

[11] V Lakshmikantham ldquoTheory of fractional functional differen-tial equationsrdquoNonlinear Analysis vol 69 no 10 pp 3337ndash33432008

[12] T Jankowski ldquoFractional differential equations with deviatingargumentsrdquo Dynamic Systems and Applications vol 17 no 3-4pp 677ndash684 2008

[13] M Benchohra J Henderson S K Ntouyas and A OuahabldquoExistence results for fractional order functional differentialequations with infinite delayrdquo Journal of Mathematical Analysisand Applications vol 338 no 2 pp 1340ndash1350 2008

[14] B Ahmad and S Sivasundaram ldquoExistence of solutions forimpulsive integral boundary value problems of fractionalorderrdquo Nonlinear Analysis vol 4 no 1 pp 134ndash141 2010

[15] W Deng ldquoNumerical algorithm for the time fractional Fokker-Planck equationrdquo Journal of Computational Physics vol 227 no2 pp 1510ndash1522 2007

[16] T Jankowski ldquoFractional equations of Volterra type involvinga Riemann-Liouville derivativerdquo Applied Mathematics Lettersvol 26 no 3 pp 344ndash350 2013

[17] M M El-Borai K E-S El-Nadi and H A Fouad ldquoOn somefractional stochastic delay differential equationsrdquo Computers ampMathematics withApplications vol 59 no 3 pp 1165ndash1170 2010

[18] J Henderson and A Ouahab ldquoImpulsive differential inclusionswith fractional orderrdquo Computers amp Mathematics with Applica-tions vol 59 no 3 pp 1191ndash1226 2010

[19] V Lakshmikantham and A S Vatsala ldquoGeneral uniquenessand monotone iterative technique for fractional differentialequationsrdquo Applied Mathematics Letters vol 21 no 8 pp 828ndash834 2008

[20] F A McRae ldquoMonotone iterative technique and existenceresults for fractional differential equationsrdquo Nonlinear Analysisvol 71 no 12 pp 6093ndash6096 2009

[21] M Benchohra and S Hamani ldquoThe method of upper andlower solutions and impulsive fractional differential inclusionsrdquoNonlinear Analysis vol 3 no 4 pp 433ndash440 2009

[22] V Lakshmikantham and A S Vatsala Generalized Quasilin-earization for Nonlinear Problems Kluwer Academic Publish-ers Dordrecht The Netherlands 1998

[23] J Vasundhara Devi and Ch Suseela ldquoQuasilinearization forfractional differential equationsrdquo Communications in AppliedAnalysis vol 12 no 4 pp 407ndash418 2008

[24] J Vasundhara Devi F A McRae and Z Drici ldquoGeneralizedquasilinearization for fractional differential equationsrdquo Com-puters ampMathematics with Applications vol 59 no 3 pp 1057ndash1062 2010

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

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