• Home

  • Documents

  • Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary...
9
Hindawi Publishing Corporation Discrete Dynamics in Nature and Society Volume 2013, Article ID 679290, 8 pages http://dx.doi.org/10.1155/2013/679290 Research Article Solvability for Discrete Fractional Boundary Value Problems with a -Laplacian Operator Weidong Lv School of Mathematics and Statistics, Longdong University, Qingyang, Gansu 745000, China Correspondence should be addressed to Weidong Lv; [email protected] Received 5 July 2013; Accepted 3 September 2013 Academic Editor: Jehad Alzabut Copyright © 2013 Weidong Lv. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. is paper is concerned with the solvability for a discrete fractional -Laplacian boundary value problem. Some existence and uniqueness results are obtained by means of the Banach contraction mapping principle. Additionally, two representative examples are presented to illustrate the effectiveness of the main results. 1. Introduction For any number ∈ R and each interval I of R, we denote N = {, + 1, + 2, . . .} and I N = I ∩ N throughout this paper. It is also worth noting that, in what follows, we appeal to the convention that the empty sum is taken to be 0. In this paper, we will consider the existence and unique- ness of solutions for the following discrete fractional bound- ary value problem involving a -Laplacian operator Δ [ (Δ )] () = ( + − 1, ( + − 1)) , ∈ [0, ] N 0 , ( − 2) = 1 ( + + 1) , Δ ( − 2) = Δ ( − 1) = 2 Δ ( + ) , (1) where 1 < ≤ 2, ∈ N 1 , 1 ̸ =1, 2 ̸ =1, Δ is the forward difference operator with stepsize 1, Δ denotes the discrete Caputo fractional difference of order , : [ − 1, + − 1] N −1 × R → R is a continuous function, and is the -Laplacian operator; that is, () = || −2 , >1. Obviously, is invertible and its inverse operator is , where >1 is a constant such that 1/ + 1/ = 1. e theory of fractional differential equations has become a new important branch of mathematics (see, e.g., [1–8]). At the same time, boundary value problems for fractional differential equations have received considerable attention [9–18]. It is well known that discrete analogues of differential equations can be very useful in applications [19, 20], espe- cially for using computer to simulate the behavior of solutions for certain dynamic equations. Compared to continuous case, significantly less is known about the discrete fractional calculus. However, within the recent years, a lot of papers have appeared on discrete fractional calculus and discrete frac- tional boundary value problems; see [21–37]. For example, in [25], Atıcı and Eloe explored a discrete fractional con- jugate boundary value problem with the Riemann-Liouville fractional difference. To the best of our knowledge, this is pioneering work on discussing boundary value problems in discrete fractional calculus. Aſter that, Goodrich stud- ied discrete fractional boundary value problems involving the Riemann-Liouville fractional difference intensively and obtained a series of excellent results; see [26–31]. In [33, 34], Bastos et al. considered the discrete fractional calculus of variations and established several necessary optimality con- ditions for fractional difference variational problems. Abdel- jawad introduced the conception of Caputo fractional differ- ence and developed some useful properties of it in [35]. Fer- reira in [37] initially investigated the existence and unique- ness of solutions for some discrete fractional boundary value problems of order less than one by the Banach fixed point theorem. Very recently, some authors have focused their attention on the existence of solutions for fractional boundary value problems with the -Laplacian operator in continuous case [38–44]. However, as far as we know, few papers can be found

Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

  • Upload
    others

  • View
    6

  • Download
    0

Embed Size (px)

Citation preview

Page 1: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

Hindawi Publishing CorporationDiscrete Dynamics in Nature and SocietyVolume 2013 Article ID 679290 8 pageshttpdxdoiorg1011552013679290

Research ArticleSolvability for Discrete Fractional Boundary ValueProblems with a 119901-Laplacian Operator

Weidong Lv

School of Mathematics and Statistics Longdong University Qingyang Gansu 745000 China

Correspondence should be addressed to Weidong Lv lvweidong2004163com

Received 5 July 2013 Accepted 3 September 2013

Academic Editor Jehad Alzabut

Copyright copy 2013 Weidong Lv This is an open access article distributed under the Creative Commons Attribution License whichpermits unrestricted use distribution and reproduction in any medium provided the original work is properly cited

This paper is concerned with the solvability for a discrete fractional 119901-Laplacian boundary value problem Some existence anduniqueness results are obtained by means of the Banach contraction mapping principle Additionally two representative examplesare presented to illustrate the effectiveness of the main results

1 Introduction

For any number 119886 isin R and each interval I of R we denoteN119886= 119886 119886 + 1 119886 + 2 and IN

119886

= I cap N119886throughout this

paper It is also worth noting that in what follows we appealto the convention that the empty sum is taken to be 0

In this paper we will consider the existence and unique-ness of solutions for the following discrete fractional bound-ary value problem involving a 119901-Laplacian operator

Δ [120601119901(Δ120572

119862119906)] (119905)=119891 (119905 + 120572 minus 1 119906 (119905 + 120572 minus 1)) 119905 isin [0 119887]N

0

119906 (120572 minus 2) = 1205731119906 (120572 + 119887 + 1)

Δ119906 (120572 minus 2) = Δ119906 (120572 minus 1) = 1205732Δ119906 (120572 + 119887)

(1)

where 1 lt 120572 le 2 119887 isin N1 1205731

= 1 1205732

= 1 Δ is theforward difference operator with stepsize 1 Δ120572

119862denotes

the discrete Caputo fractional difference of order 120572 119891

[120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR rarr R is a continuous function and120601119901is the119901-Laplacian operator that is 120601

119901(119906) = |119906|

119901minus2

119906119901 gt 1Obviously120601

119901is invertible and its inverse operator is120601

119902 where

119902 gt 1 is a constant such that 1119901 + 1119902 = 1The theory of fractional differential equations has become

a new important branch of mathematics (see eg [1ndash8])At the same time boundary value problems for fractionaldifferential equations have received considerable attention[9ndash18] It is well known that discrete analogues of differential

equations can be very useful in applications [19 20] espe-cially for using computer to simulate the behavior of solutionsfor certain dynamic equations Compared to continuous casesignificantly less is known about the discrete fractionalcalculusHowever within the recent years a lot of papers haveappeared on discrete fractional calculus and discrete frac-tional boundary value problems see [21ndash37] For examplein [25] Atıcı and Eloe explored a discrete fractional con-jugate boundary value problem with the Riemann-Liouvillefractional difference To the best of our knowledge this ispioneering work on discussing boundary value problemsin discrete fractional calculus After that Goodrich stud-ied discrete fractional boundary value problems involvingthe Riemann-Liouville fractional difference intensively andobtained a series of excellent results see [26ndash31] In [33 34]Bastos et al considered the discrete fractional calculus ofvariations and established several necessary optimality con-ditions for fractional difference variational problems Abdel-jawad introduced the conception of Caputo fractional differ-ence and developed some useful properties of it in [35] Fer-reira in [37] initially investigated the existence and unique-ness of solutions for some discrete fractional boundary valueproblems of order less than one by the Banach fixed pointtheorem

Very recently some authors have focused their attentionon the existence of solutions for fractional boundary valueproblems with the 119901-Laplacian operator in continuous case[38ndash44] However as far as we know few papers can be found

2 Discrete Dynamics in Nature and Society

in the literature for the discrete fractional boundary valueproblems with the 119901-Laplacian operator [45]

Inspired by the aforementioned results wewill investigatethe discrete fractional 119901-Laplacian boundary value problem(1) and establish some sufficient conditions for the existenceand uniqueness of solutions to it by using the Banachcontraction mapping principle

The remainder of this paper is organized as followsSection 2 preliminarily provides some necessary basic knowl-edge for the theory of discrete fractional calculus InSection 3 the existence and uniqueness results for the solu-tion to problem (1) will be established with the help ofthe contraction mapping principle Finally in Section 4 twoconcrete examples are provided to illustrate the possibleapplications of the established analytical results

2 Preliminaries

For the convenience of the reader we begin by presentinghere some necessary basic definitions and lemmas on discretefractional calculus theory

Definition 1 (see [21]) For any 119905 and ] the falling factorialfunction is defined as

119905]=

Γ (119905 + 1)

Γ (119905 + 1 minus ])(2)

provided that the right-hand side is well defined We appealto the convention that if 119905 + 1 minus ] is a pole of the Gammafunction and 119905 + 1 is not a pole then 119905

]= 0

Definition 2 (see [46]) The ]th fractional sum of a function119891 N119886rarr R for ] gt 0 is defined by

Δminus]119891 (119905) =

1

Γ (])

119905minus]

sum119904=119886

(119905 minus 119904 minus 1)]minus1

119891 (119904) for 119905 isin N119886+] (3)

Definition 3 (see [35]) The ]th Caputo fractional differenceof a function 119891 N

119886rarr R for ] gt 0 ] notin N is defined by

Δ]119862119891 (119905) = Δ

minus(119899minus])Δ119899

119891 (119905)

=1

Γ (119899 minus ])

119905minus119899+]

sum119904=119886

(119905 minus 119904 minus 1)119899minus]minus1

Δ119899

119891 (119904)

for 119905 isin N119886+119899minus]

(4)

where 119899 is the smallest integer greater than or equal to ] andΔ119899 is the 119899th forward difference operator If ] = 119899 isin N then

Δ]119862119891(119905) = Δ

119899

119891(119905)

Lemma 4 (see [45]) Assume that ] gt 0 and 119891 is defined onN119886 Then

Δminus]Δ]119862119891 (119905) = 119891 (119905) + 119888

0+ 1198881119905 + sdot sdot sdot + 119888

119899minus1119905119899minus1

(5)

where 119888119894isin R 119894 = 1 2 119899 minus 1 and 119899 is the smallest integer

greater than or equal to ]

Now we state and prove the following lemma whichprovides a representation for the solution to (1) if the solutionexists

Lemma5 Let ℎ [120572minus1 120572+119887minus1]N120572minus1

rarr R and let1205731 1205732

= 1Then the following problem

Δ [120601119901(Δ120572

119862119906)] (119905) = ℎ (119905 + 120572 minus 1) 119905 isin [0 119887]N

0

(6)

119906 (120572 minus 2) = 1205731119906 (120572 + 119887 + 1)

Δ119906 (120572 minus 2) = Δ119906 (120572 minus 1) = 1205732Δ119906 (120572 + 119887)

(7)

has a unique solution

119906 (119905) =119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+1205731

(1 minus 1205731) Γ (120572)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

(8)

where 119886(119905) = (1205732[1205731(120572 + 119887 + 1) + 2 minus 120572] + (1 minus 120573

1)119905)((1 minus

1205731)(1 minus 120573

2))

Proof The definition of the discrete Caputo fractional differ-ence together with condition Δ119906(120572 minus 2) = Δ119906(120572 minus 1) impliesthat Δ120572

119862119906(0) = 0 So from (6) we have

120601119901(Δ120572

119862119906 (119905)) = 120601

119901(Δ120572

119862119906 (0)) +

119905minus1

sum

119904=0

ℎ (119904 + 120572 minus 1)

=

119905minus1

sum

119904=0

ℎ (119904 + 120572 minus 1)

(9)

and then

Δ120572

119862119906 (119905) = 120601

119902(

119905minus1

sum

119904=0

ℎ (119904 + 120572 minus 1)) 119905 isin [0 119887 + 1]N0

(10)

Hence in view of Lemma 4 we can get

119906 (119905) =1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+ 1198880+ 1198881119905

(11)

where 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

1198880 1198881isin R

Discrete Dynamics in Nature and Society 3

Furthermore we have

Δ119906 (119905) =1

Γ (120572 minus 1)

119905minus(120572minus1)

sum

119904=0

(119905 minus 119904 minus 1)120572minus2

120601119902

times (

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1)) + 1198881 119905 isin [120572 minus 2 120572 + 119887]N

120572minus2

(12)

Then by conditions 119906(120572 minus 2) = 1205731119906(120572 + 119887 + 1) Δ119906(120572 minus 2) =

1205732Δ119906(120572 + 119887) we can get

1198880=

1205731

(1 minus 1205731) Γ (120572)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+1205732[1205731(120572 + 119887 + 1) + 2 minus 120572]

(1 minus 1205731) (1 minus 120573

2) Γ (120572 minus 1)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

1198881=

1205732

(1 minus 1205732) Γ (120572 minus 1)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

(13)

Substituting the values of 1198880and 1198881in (11) we get (8) This

completes the proof

Finally we list below the following basic properties ofthe 119901-Laplacian operator which will be used in the sequel

(1) If 1 lt 119901 lt 2 119906V gt 0 and |119906| |V| ge 119898 gt 0 then

10038161003816100381610038161003816120601119901(V) minus 120601

119901(119906)

10038161003816100381610038161003816le (119901 minus 1)119898

119901minus2

|V minus 119906| (14)

(2) If 119901 gt 2 |119906| |V| le 119872 then

10038161003816100381610038161003816120601119901(V) minus 120601

119901(119906)

10038161003816100381610038161003816le (119901 minus 1)119872

119901minus2

|V minus 119906| (15)

3 Main Results

In this section we will use the Banach contraction mappingprinciple to prove the existence and uniqueness for the sol-ution to problem (1)

Let E denote the Banach space of all functions from [120572 minus

2 120572 + 119887 + 1]N120572minus2

into R endowed with the norm defined by119906 = max|119906(119905)| 119905 isin [120572 minus 2 120572 + 119887 + 1]N

120572minus2

For the sake of convenience to the following discussion

we set

119886 = max |119886 (119905)| 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

(16)

where 119886(119905) is as given in Lemma 5 Also for any 119906 V isin E wedenote

119860 (V 119906) (119905) = 120601119902(

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1)))

minus 120601119902(

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1)))

(17)

for 119905 isin [0 119887 + 1]N0

Obviously 119860(V 119906)(0) = 0In view of Lemma 5 we transform problem (1) as

119906 = F119906 (18)

whereF E rarr E is defined by

(F119906) (119905) =119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

(19)

for 119905 isin [120572minus2 120572+119887+1]N120572minus2

It is clear to see that 119906 is a solutionof the problem (1) if and only if 119906 is a fixed point ofF

Now we state the main results as follows

Theorem 6 Suppose p gt 2 1205731

= 1 1205732

= 1 and the followingcondition holds

(H1) there exist positive numbers 120582 and 119896 with

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887) times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(20)

4 Discrete Dynamics in Nature and Society

such that

120582120572119905120572minus1

le 119891 (119905 119906) 119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

(21)

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(22)

Then the problem (1) has a unique solution

Proof For any 119906 isin E by (21) we can get that

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

ge 120582

119905minus1

sum

119904=0

120572(119904 + 120572 minus 1)120572minus1

= 120582(119905 + 120572 minus 1)120572

ge 120582Γ (120572 + 1) 119905 isin [1 119887 + 1]N0

(23)

Due to 119901 gt 2 and 1119901 + 1119902 = 1 we know that 1 lt 119902 lt 2 By(14) and (22) for any 119906 V isin E 119905 isin [1 119887 + 1]N

1

we have

|119860 (V 119906) (119905)| le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus 119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

(24)

Next for any 119906 V isin E and for each 119905 isin [120572minus2 120572+119887+1]N120572minus2

together with the fact that 119860(V 119906)(0) = 0 we obtain

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

le 119886 (120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

=10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 [119886120572 + 120572 + 119887] +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887) Γ (120572 + 119887)10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 Γ (120572 + 1) Γ (119887 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

= ((119896 (119902 minus 1) (119887 + 1) [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886 + 120572 + 119887)

+10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)] [120582Γ (120572 + 1)]119902minus2

times

119887minus1

prod

119894=1

(120572 + 119894)) times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906

= 119871 V minus 119906

(25)

where 119871 = (119896(119902 minus 1)(119887 + 1)[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 +

119887)][120582Γ(120572 + 1)]119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) From (20) we

get that 0 lt 119871 lt 1 which implies that F is a contractionmapping By means of the Banach contraction mappingprinciple we get that F has a unique fixed point in E thatis the problem (1) has a unique solution This completes theproof

With a similar proof to that of Theorem 6 we can get thefollowing theorem

Theorem 7 Suppose 119901 gt 2 1205731

= 1 1205732

= 1 and the followingcondition holds

Discrete Dynamics in Nature and Society 5

(H2) there exist constants 120582 gt 0 and 119896 with

0 lt 119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(26)

such that119891 (119905 119906)

le minus120582120572119905120572minus1

119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 119887 + 120572 minus 1]N120572minus1

119906 V isin R

(27)

Then the problem (1) has a unique solution

Theorem 8 Suppose 1 lt 119901 lt 2 1205731

= 1 1205732

= 1 and thefollowing condition holds

(H3) there exists a nonnegative function 119892 [120572 minus 1 120572 + 119887 minus

1]N120572minus1

rarr R and sum119887

119904=0119892(119904 + 120572 minus 1) = 119872 gt 0 such that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N

120572minus1

timesR (28)

and there exists a positive constant 119896 such that1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)

1003816100381610038161003816

le 119896 |V minus 119906| for 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(29)

Then the problem (1) has a unique solution provided that

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(30)

Proof By (28) we can get that for 119905 isin [1 119887 + 1]N1

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le

119887

sum

119904=0

119892 (119904 + 120572 minus 1) = 119872

(31)

In view of 1 lt 119901 lt 2 and 1119901 + 1119902 = 1 we can get 119902 gt 2From (15) and (29) for any V 119906 isin E we have

|119860 (V 119906) (119905)| le (119902 minus 1)119872119902minus2

times

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le (119902 minus 1)119872119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1)119872119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1)119872119902minus2

119905 V minus 119906 119905 isin [1 119887 + 1]N1

(32)

Hence for any 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

by 119860(119906 V)(0) = 0 wehave

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

le 119886(120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

6 Discrete Dynamics in Nature and Society

= ((119896 [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906 = 119871 V minus 119906

(33)

where 119871 = (119896[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 + 119887)](119902 minus

1)(119887 + 1)119872119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) In view of (30) F

is a contraction Thus the conclusion of the theorem followsby the contraction mapping principle This completes theproof

4 Examples

In this section we will illustrate the possible application ofthe above established analytical results with the following twoconcrete examples

Example 1 Consider the discrete fractional boundary valueproblem

Δ [1206013(Δ32

119862119906)] (119905)

= 3(119905 +1

2)

12

times [1

2+ sin2 ( 119906 (119905 + 12)

10radic3+ 120579)

+1

390

1003816100381610038161003816100381610038161003816119906 (119905 +

1

2)1003816100381610038161003816100381610038161003816] 119905 isin [0 2]N

0

10119906 (minus1

2) = 119906 (

9

2)

10Δ119906 (minus1

2) = 10Δ119906 (

1

2) = Δ119906(

7

2)

(34)

here 120579 is a real number

Conclusion Problem (34) has a unique nonnegative solution

Proof Corresponding to problem (1) 119901 = 3 gt 2 119902 = 32 120572 =

32 1205731= 110 120573

2= 110 119887 = 2 and 119891(119905 119906) = 3119905

12

[12 +

sin2((11990610radic3) + 120579) + (1390)|119906|] (119905 119906) isin [12 52]N12

timesRChoosing 120582 = 1 and 119896 = 3100 by direct calculation we

can verify that

119896 =3

100lt (

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=18radic3120587

14

325asymp 01277

(35)

It is easy to verify that

120582120572119905120572minus1

=3

211990512

le 311990512

[1

2+ sin2 ( 119906

10radic3+ 120579) +

1

390|119906|]

= 119891 (119905 119906) (119905 119906) isin [1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

= 311990512

10038161003816100381610038161003816100381610038161003816sin2 ( V

10radic3+ 120579)

minus sin2 ( 119906

10radic3+ 120579)

+1

390|V| minus

1

390|119906|

10038161003816100381610038161003816100381610038161003816

le 3(5

2)

12

(1

300+

1

390) |V minus 119906|

=69

4160radic120587 |V minus 119906|

asymp 00294 |V minus 119906| lt 119896 |V minus 119906|

(36)

for 119905 isin [12 52]N12

119906 V isin R Therefore by Theorem 6the boundary value problem (34) has a unique solutionFurthermore from the nonnegativeness of 119891 and the expres-sion of F we also get that the unique solution of (34) isnonnegative

Example 2 Consider the nonlinear discrete fractionalboundary value problem

Δ [12060132

(Δ32

119862119906)] (119905)

=3

2(119905 +

1

2)

12

times sin2 (119906 (119905 + 12)

40+ 120596) 119905 isin [0 2]N

0

2119906 (minus1

2) = minus119906(

9

2)

2Δ119906 (minus1

2) = 2Δ119906(

1

2) = Δ119906(

7

2)

(37)

where 120596 is a real number

Conclusion Problem (37) has a unique solution

Proof The problem (37) can be regarded as problem (1)where 119901 = 32 lt 2 119902 = 3 gt 2 120572 = 32 120573

1= minus12

1205732= 12 119887 = 2 and119891(119905 119906) = (32)119905

12sin2((11990640)+120596) (119905 119906)

Discrete Dynamics in Nature and Society 7

isin [12 52]N12

timesR Taking 119892(119905) = (32)11990512 119905 isin [12 52]N

12

then119872 = (105Γ(12))32 Let 119896 = 1600 asymp 00017 we have

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=64

15225radic120587asymp 00024

(38)

Moreover we can verify that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [

1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816 le

3

211990512

1

402|V minus 119906|

le3

2(5

2)

12 1

402|V minus 119906|

=9radic120587

10240|V minus 119906|

asymp 00016 |V minus 119906| lt 119896 |V minus 119906|

(39)

for 119905 isin [12 52]N12

V 119906 isin RTherefore byTheorem 8 problem (37) has a unique solu-

tion

Acknowledgment

This work was supported by the Longdong University GrantXYZK-1010 and XYZK-1007

References

[1] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[2] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[3] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

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

[6] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[7] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[8] J Sabatier O P Agrawal and J A T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer Heidelberg Germany2007

[9] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[10] Z Bai ldquoOn positive solutions of a nonlocal fractional boundaryvalue problemrdquoNonlinear AnalysisTheory Methods amp Applica-tions A vol 72 no 2 pp 916ndash924 2010

[11] Z Bai and Y Zhang ldquoSolvability of fractional three-pointboundary value problems with nonlinear growthrdquo AppliedMathematics and Computation vol 218 no 5 pp 1719ndash17252011

[12] C Bai ldquoTriple positive solutions for a boundary value problemof nonlinear fractional differential equationrdquo Electronic Journalof Qualitative Theory of Differential Equations vol 2008 article24 2008

[13] X Xu D Jiang and C Yuan ldquoMultiple positive solutions for theboundary value problem of a nonlinear fractional differentialequationrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 71 no 10 pp 4676ndash4688 2009

[14] B Ahmad ldquoExistence of solutions for irregular boundary valueproblems of nonlinear fractional differential equationsrdquoAppliedMathematics Letters of Rapid Publication vol 23 no 4 pp 390ndash394 2010

[15] S Zhang ldquoPositive solutions to singular boundary value prob-lem for nonlinear fractional differential equationrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1300ndash13092010

[16] W Jiang ldquoThe existence of solutions to boundary value prob-lems of fractional differential equations at resonancerdquoNonlinearAnalysis Theory Methods amp Applications A vol 74 no 5 pp1987ndash1994 2011

[17] Y Chen and X Tang ldquoSolvability of sequential fractional ordermulti-point boundary value problems at resonancerdquo AppliedMathematics and Computation vol 218 no 14 pp 7638ndash76482012

[18] A Guezane-Lakoud and R Khaldi ldquoSolvability of a fractionalboundary value problem with fractional integral conditionrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 4 pp 2692ndash2700 2012

[19] R Hilscher and V Zeidan ldquoNonnegativity and positivity ofquadratic functionals in discrete calculus of variations surveyrdquoJournal of Difference Equations and Applications vol 11 no 9pp 857ndash875 2005

[20] W G Kelley and A C Peterson Difference Equations AnIntroduction with Applications Academic Press New York NYUSA 1991

[21] F M Atici and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[22] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[23] F M Atıcı and P W Eloe ldquoDiscrete fractional calculus withthe nabla operatorrdquo Electronic Journal of Qualitative Theory ofDifferential Equations vol 2009 article 3 12 pages 2009

[24] F M Atıcı and S Sengul ldquoModeling with fractional differenceequationsrdquo Journal of Mathematical Analysis and Applicationsvol 369 no 1 pp 1ndash9 2010

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 2: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

2 Discrete Dynamics in Nature and Society

in the literature for the discrete fractional boundary valueproblems with the 119901-Laplacian operator [45]

Inspired by the aforementioned results wewill investigatethe discrete fractional 119901-Laplacian boundary value problem(1) and establish some sufficient conditions for the existenceand uniqueness of solutions to it by using the Banachcontraction mapping principle

The remainder of this paper is organized as followsSection 2 preliminarily provides some necessary basic knowl-edge for the theory of discrete fractional calculus InSection 3 the existence and uniqueness results for the solu-tion to problem (1) will be established with the help ofthe contraction mapping principle Finally in Section 4 twoconcrete examples are provided to illustrate the possibleapplications of the established analytical results

2 Preliminaries

For the convenience of the reader we begin by presentinghere some necessary basic definitions and lemmas on discretefractional calculus theory

Definition 1 (see [21]) For any 119905 and ] the falling factorialfunction is defined as

119905]=

Γ (119905 + 1)

Γ (119905 + 1 minus ])(2)

provided that the right-hand side is well defined We appealto the convention that if 119905 + 1 minus ] is a pole of the Gammafunction and 119905 + 1 is not a pole then 119905

]= 0

Definition 2 (see [46]) The ]th fractional sum of a function119891 N119886rarr R for ] gt 0 is defined by

Δminus]119891 (119905) =

1

Γ (])

119905minus]

sum119904=119886

(119905 minus 119904 minus 1)]minus1

119891 (119904) for 119905 isin N119886+] (3)

Definition 3 (see [35]) The ]th Caputo fractional differenceof a function 119891 N

119886rarr R for ] gt 0 ] notin N is defined by

Δ]119862119891 (119905) = Δ

minus(119899minus])Δ119899

119891 (119905)

=1

Γ (119899 minus ])

119905minus119899+]

sum119904=119886

(119905 minus 119904 minus 1)119899minus]minus1

Δ119899

119891 (119904)

for 119905 isin N119886+119899minus]

(4)

where 119899 is the smallest integer greater than or equal to ] andΔ119899 is the 119899th forward difference operator If ] = 119899 isin N then

Δ]119862119891(119905) = Δ

119899

119891(119905)

Lemma 4 (see [45]) Assume that ] gt 0 and 119891 is defined onN119886 Then

Δminus]Δ]119862119891 (119905) = 119891 (119905) + 119888

0+ 1198881119905 + sdot sdot sdot + 119888

119899minus1119905119899minus1

(5)

where 119888119894isin R 119894 = 1 2 119899 minus 1 and 119899 is the smallest integer

greater than or equal to ]

Now we state and prove the following lemma whichprovides a representation for the solution to (1) if the solutionexists

Lemma5 Let ℎ [120572minus1 120572+119887minus1]N120572minus1

rarr R and let1205731 1205732

= 1Then the following problem

Δ [120601119901(Δ120572

119862119906)] (119905) = ℎ (119905 + 120572 minus 1) 119905 isin [0 119887]N

0

(6)

119906 (120572 minus 2) = 1205731119906 (120572 + 119887 + 1)

Δ119906 (120572 minus 2) = Δ119906 (120572 minus 1) = 1205732Δ119906 (120572 + 119887)

(7)

has a unique solution

119906 (119905) =119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+1205731

(1 minus 1205731) Γ (120572)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

(8)

where 119886(119905) = (1205732[1205731(120572 + 119887 + 1) + 2 minus 120572] + (1 minus 120573

1)119905)((1 minus

1205731)(1 minus 120573

2))

Proof The definition of the discrete Caputo fractional differ-ence together with condition Δ119906(120572 minus 2) = Δ119906(120572 minus 1) impliesthat Δ120572

119862119906(0) = 0 So from (6) we have

120601119901(Δ120572

119862119906 (119905)) = 120601

119901(Δ120572

119862119906 (0)) +

119905minus1

sum

119904=0

ℎ (119904 + 120572 minus 1)

=

119905minus1

sum

119904=0

ℎ (119904 + 120572 minus 1)

(9)

and then

Δ120572

119862119906 (119905) = 120601

119902(

119905minus1

sum

119904=0

ℎ (119904 + 120572 minus 1)) 119905 isin [0 119887 + 1]N0

(10)

Hence in view of Lemma 4 we can get

119906 (119905) =1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+ 1198880+ 1198881119905

(11)

where 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

1198880 1198881isin R

Discrete Dynamics in Nature and Society 3

Furthermore we have

Δ119906 (119905) =1

Γ (120572 minus 1)

119905minus(120572minus1)

sum

119904=0

(119905 minus 119904 minus 1)120572minus2

120601119902

times (

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1)) + 1198881 119905 isin [120572 minus 2 120572 + 119887]N

120572minus2

(12)

Then by conditions 119906(120572 minus 2) = 1205731119906(120572 + 119887 + 1) Δ119906(120572 minus 2) =

1205732Δ119906(120572 + 119887) we can get

1198880=

1205731

(1 minus 1205731) Γ (120572)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+1205732[1205731(120572 + 119887 + 1) + 2 minus 120572]

(1 minus 1205731) (1 minus 120573

2) Γ (120572 minus 1)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

1198881=

1205732

(1 minus 1205732) Γ (120572 minus 1)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

(13)

Substituting the values of 1198880and 1198881in (11) we get (8) This

completes the proof

Finally we list below the following basic properties ofthe 119901-Laplacian operator which will be used in the sequel

(1) If 1 lt 119901 lt 2 119906V gt 0 and |119906| |V| ge 119898 gt 0 then

10038161003816100381610038161003816120601119901(V) minus 120601

119901(119906)

10038161003816100381610038161003816le (119901 minus 1)119898

119901minus2

|V minus 119906| (14)

(2) If 119901 gt 2 |119906| |V| le 119872 then

10038161003816100381610038161003816120601119901(V) minus 120601

119901(119906)

10038161003816100381610038161003816le (119901 minus 1)119872

119901minus2

|V minus 119906| (15)

3 Main Results

In this section we will use the Banach contraction mappingprinciple to prove the existence and uniqueness for the sol-ution to problem (1)

Let E denote the Banach space of all functions from [120572 minus

2 120572 + 119887 + 1]N120572minus2

into R endowed with the norm defined by119906 = max|119906(119905)| 119905 isin [120572 minus 2 120572 + 119887 + 1]N

120572minus2

For the sake of convenience to the following discussion

we set

119886 = max |119886 (119905)| 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

(16)

where 119886(119905) is as given in Lemma 5 Also for any 119906 V isin E wedenote

119860 (V 119906) (119905) = 120601119902(

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1)))

minus 120601119902(

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1)))

(17)

for 119905 isin [0 119887 + 1]N0

Obviously 119860(V 119906)(0) = 0In view of Lemma 5 we transform problem (1) as

119906 = F119906 (18)

whereF E rarr E is defined by

(F119906) (119905) =119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

(19)

for 119905 isin [120572minus2 120572+119887+1]N120572minus2

It is clear to see that 119906 is a solutionof the problem (1) if and only if 119906 is a fixed point ofF

Now we state the main results as follows

Theorem 6 Suppose p gt 2 1205731

= 1 1205732

= 1 and the followingcondition holds

(H1) there exist positive numbers 120582 and 119896 with

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887) times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(20)

4 Discrete Dynamics in Nature and Society

such that

120582120572119905120572minus1

le 119891 (119905 119906) 119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

(21)

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(22)

Then the problem (1) has a unique solution

Proof For any 119906 isin E by (21) we can get that

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

ge 120582

119905minus1

sum

119904=0

120572(119904 + 120572 minus 1)120572minus1

= 120582(119905 + 120572 minus 1)120572

ge 120582Γ (120572 + 1) 119905 isin [1 119887 + 1]N0

(23)

Due to 119901 gt 2 and 1119901 + 1119902 = 1 we know that 1 lt 119902 lt 2 By(14) and (22) for any 119906 V isin E 119905 isin [1 119887 + 1]N

1

we have

|119860 (V 119906) (119905)| le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus 119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

(24)

Next for any 119906 V isin E and for each 119905 isin [120572minus2 120572+119887+1]N120572minus2

together with the fact that 119860(V 119906)(0) = 0 we obtain

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

le 119886 (120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

=10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 [119886120572 + 120572 + 119887] +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887) Γ (120572 + 119887)10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 Γ (120572 + 1) Γ (119887 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

= ((119896 (119902 minus 1) (119887 + 1) [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886 + 120572 + 119887)

+10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)] [120582Γ (120572 + 1)]119902minus2

times

119887minus1

prod

119894=1

(120572 + 119894)) times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906

= 119871 V minus 119906

(25)

where 119871 = (119896(119902 minus 1)(119887 + 1)[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 +

119887)][120582Γ(120572 + 1)]119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) From (20) we

get that 0 lt 119871 lt 1 which implies that F is a contractionmapping By means of the Banach contraction mappingprinciple we get that F has a unique fixed point in E thatis the problem (1) has a unique solution This completes theproof

With a similar proof to that of Theorem 6 we can get thefollowing theorem

Theorem 7 Suppose 119901 gt 2 1205731

= 1 1205732

= 1 and the followingcondition holds

Discrete Dynamics in Nature and Society 5

(H2) there exist constants 120582 gt 0 and 119896 with

0 lt 119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(26)

such that119891 (119905 119906)

le minus120582120572119905120572minus1

119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 119887 + 120572 minus 1]N120572minus1

119906 V isin R

(27)

Then the problem (1) has a unique solution

Theorem 8 Suppose 1 lt 119901 lt 2 1205731

= 1 1205732

= 1 and thefollowing condition holds

(H3) there exists a nonnegative function 119892 [120572 minus 1 120572 + 119887 minus

1]N120572minus1

rarr R and sum119887

119904=0119892(119904 + 120572 minus 1) = 119872 gt 0 such that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N

120572minus1

timesR (28)

and there exists a positive constant 119896 such that1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)

1003816100381610038161003816

le 119896 |V minus 119906| for 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(29)

Then the problem (1) has a unique solution provided that

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(30)

Proof By (28) we can get that for 119905 isin [1 119887 + 1]N1

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le

119887

sum

119904=0

119892 (119904 + 120572 minus 1) = 119872

(31)

In view of 1 lt 119901 lt 2 and 1119901 + 1119902 = 1 we can get 119902 gt 2From (15) and (29) for any V 119906 isin E we have

|119860 (V 119906) (119905)| le (119902 minus 1)119872119902minus2

times

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le (119902 minus 1)119872119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1)119872119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1)119872119902minus2

119905 V minus 119906 119905 isin [1 119887 + 1]N1

(32)

Hence for any 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

by 119860(119906 V)(0) = 0 wehave

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

le 119886(120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

6 Discrete Dynamics in Nature and Society

= ((119896 [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906 = 119871 V minus 119906

(33)

where 119871 = (119896[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 + 119887)](119902 minus

1)(119887 + 1)119872119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) In view of (30) F

is a contraction Thus the conclusion of the theorem followsby the contraction mapping principle This completes theproof

4 Examples

In this section we will illustrate the possible application ofthe above established analytical results with the following twoconcrete examples

Example 1 Consider the discrete fractional boundary valueproblem

Δ [1206013(Δ32

119862119906)] (119905)

= 3(119905 +1

2)

12

times [1

2+ sin2 ( 119906 (119905 + 12)

10radic3+ 120579)

+1

390

1003816100381610038161003816100381610038161003816119906 (119905 +

1

2)1003816100381610038161003816100381610038161003816] 119905 isin [0 2]N

0

10119906 (minus1

2) = 119906 (

9

2)

10Δ119906 (minus1

2) = 10Δ119906 (

1

2) = Δ119906(

7

2)

(34)

here 120579 is a real number

Conclusion Problem (34) has a unique nonnegative solution

Proof Corresponding to problem (1) 119901 = 3 gt 2 119902 = 32 120572 =

32 1205731= 110 120573

2= 110 119887 = 2 and 119891(119905 119906) = 3119905

12

[12 +

sin2((11990610radic3) + 120579) + (1390)|119906|] (119905 119906) isin [12 52]N12

timesRChoosing 120582 = 1 and 119896 = 3100 by direct calculation we

can verify that

119896 =3

100lt (

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=18radic3120587

14

325asymp 01277

(35)

It is easy to verify that

120582120572119905120572minus1

=3

211990512

le 311990512

[1

2+ sin2 ( 119906

10radic3+ 120579) +

1

390|119906|]

= 119891 (119905 119906) (119905 119906) isin [1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

= 311990512

10038161003816100381610038161003816100381610038161003816sin2 ( V

10radic3+ 120579)

minus sin2 ( 119906

10radic3+ 120579)

+1

390|V| minus

1

390|119906|

10038161003816100381610038161003816100381610038161003816

le 3(5

2)

12

(1

300+

1

390) |V minus 119906|

=69

4160radic120587 |V minus 119906|

asymp 00294 |V minus 119906| lt 119896 |V minus 119906|

(36)

for 119905 isin [12 52]N12

119906 V isin R Therefore by Theorem 6the boundary value problem (34) has a unique solutionFurthermore from the nonnegativeness of 119891 and the expres-sion of F we also get that the unique solution of (34) isnonnegative

Example 2 Consider the nonlinear discrete fractionalboundary value problem

Δ [12060132

(Δ32

119862119906)] (119905)

=3

2(119905 +

1

2)

12

times sin2 (119906 (119905 + 12)

40+ 120596) 119905 isin [0 2]N

0

2119906 (minus1

2) = minus119906(

9

2)

2Δ119906 (minus1

2) = 2Δ119906(

1

2) = Δ119906(

7

2)

(37)

where 120596 is a real number

Conclusion Problem (37) has a unique solution

Proof The problem (37) can be regarded as problem (1)where 119901 = 32 lt 2 119902 = 3 gt 2 120572 = 32 120573

1= minus12

1205732= 12 119887 = 2 and119891(119905 119906) = (32)119905

12sin2((11990640)+120596) (119905 119906)

Discrete Dynamics in Nature and Society 7

isin [12 52]N12

timesR Taking 119892(119905) = (32)11990512 119905 isin [12 52]N

12

then119872 = (105Γ(12))32 Let 119896 = 1600 asymp 00017 we have

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=64

15225radic120587asymp 00024

(38)

Moreover we can verify that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [

1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816 le

3

211990512

1

402|V minus 119906|

le3

2(5

2)

12 1

402|V minus 119906|

=9radic120587

10240|V minus 119906|

asymp 00016 |V minus 119906| lt 119896 |V minus 119906|

(39)

for 119905 isin [12 52]N12

V 119906 isin RTherefore byTheorem 8 problem (37) has a unique solu-

tion

Acknowledgment

This work was supported by the Longdong University GrantXYZK-1010 and XYZK-1007

References

[1] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[2] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[3] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

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

[6] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[7] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[8] J Sabatier O P Agrawal and J A T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer Heidelberg Germany2007

[9] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[10] Z Bai ldquoOn positive solutions of a nonlocal fractional boundaryvalue problemrdquoNonlinear AnalysisTheory Methods amp Applica-tions A vol 72 no 2 pp 916ndash924 2010

[11] Z Bai and Y Zhang ldquoSolvability of fractional three-pointboundary value problems with nonlinear growthrdquo AppliedMathematics and Computation vol 218 no 5 pp 1719ndash17252011

[12] C Bai ldquoTriple positive solutions for a boundary value problemof nonlinear fractional differential equationrdquo Electronic Journalof Qualitative Theory of Differential Equations vol 2008 article24 2008

[13] X Xu D Jiang and C Yuan ldquoMultiple positive solutions for theboundary value problem of a nonlinear fractional differentialequationrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 71 no 10 pp 4676ndash4688 2009

[14] B Ahmad ldquoExistence of solutions for irregular boundary valueproblems of nonlinear fractional differential equationsrdquoAppliedMathematics Letters of Rapid Publication vol 23 no 4 pp 390ndash394 2010

[15] S Zhang ldquoPositive solutions to singular boundary value prob-lem for nonlinear fractional differential equationrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1300ndash13092010

[16] W Jiang ldquoThe existence of solutions to boundary value prob-lems of fractional differential equations at resonancerdquoNonlinearAnalysis Theory Methods amp Applications A vol 74 no 5 pp1987ndash1994 2011

[17] Y Chen and X Tang ldquoSolvability of sequential fractional ordermulti-point boundary value problems at resonancerdquo AppliedMathematics and Computation vol 218 no 14 pp 7638ndash76482012

[18] A Guezane-Lakoud and R Khaldi ldquoSolvability of a fractionalboundary value problem with fractional integral conditionrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 4 pp 2692ndash2700 2012

[19] R Hilscher and V Zeidan ldquoNonnegativity and positivity ofquadratic functionals in discrete calculus of variations surveyrdquoJournal of Difference Equations and Applications vol 11 no 9pp 857ndash875 2005

[20] W G Kelley and A C Peterson Difference Equations AnIntroduction with Applications Academic Press New York NYUSA 1991

[21] F M Atici and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[22] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[23] F M Atıcı and P W Eloe ldquoDiscrete fractional calculus withthe nabla operatorrdquo Electronic Journal of Qualitative Theory ofDifferential Equations vol 2009 article 3 12 pages 2009

[24] F M Atıcı and S Sengul ldquoModeling with fractional differenceequationsrdquo Journal of Mathematical Analysis and Applicationsvol 369 no 1 pp 1ndash9 2010

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 3: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

Discrete Dynamics in Nature and Society 3

Furthermore we have

Δ119906 (119905) =1

Γ (120572 minus 1)

119905minus(120572minus1)

sum

119904=0

(119905 minus 119904 minus 1)120572minus2

120601119902

times (

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1)) + 1198881 119905 isin [120572 minus 2 120572 + 119887]N

120572minus2

(12)

Then by conditions 119906(120572 minus 2) = 1205731119906(120572 + 119887 + 1) Δ119906(120572 minus 2) =

1205732Δ119906(120572 + 119887) we can get

1198880=

1205731

(1 minus 1205731) Γ (120572)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

+1205732[1205731(120572 + 119887 + 1) + 2 minus 120572]

(1 minus 1205731) (1 minus 120573

2) Γ (120572 minus 1)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

1198881=

1205732

(1 minus 1205732) Γ (120572 minus 1)

times

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

120601119902(

119904minus1

sum

120591=0

ℎ (120591 + 120572 minus 1))

(13)

Substituting the values of 1198880and 1198881in (11) we get (8) This

completes the proof

Finally we list below the following basic properties ofthe 119901-Laplacian operator which will be used in the sequel

(1) If 1 lt 119901 lt 2 119906V gt 0 and |119906| |V| ge 119898 gt 0 then

10038161003816100381610038161003816120601119901(V) minus 120601

119901(119906)

10038161003816100381610038161003816le (119901 minus 1)119898

119901minus2

|V minus 119906| (14)

(2) If 119901 gt 2 |119906| |V| le 119872 then

10038161003816100381610038161003816120601119901(V) minus 120601

119901(119906)

10038161003816100381610038161003816le (119901 minus 1)119872

119901minus2

|V minus 119906| (15)

3 Main Results

In this section we will use the Banach contraction mappingprinciple to prove the existence and uniqueness for the sol-ution to problem (1)

Let E denote the Banach space of all functions from [120572 minus

2 120572 + 119887 + 1]N120572minus2

into R endowed with the norm defined by119906 = max|119906(119905)| 119905 isin [120572 minus 2 120572 + 119887 + 1]N

120572minus2

For the sake of convenience to the following discussion

we set

119886 = max |119886 (119905)| 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

(16)

where 119886(119905) is as given in Lemma 5 Also for any 119906 V isin E wedenote

119860 (V 119906) (119905) = 120601119902(

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1)))

minus 120601119902(

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1)))

(17)

for 119905 isin [0 119887 + 1]N0

Obviously 119860(V 119906)(0) = 0In view of Lemma 5 we transform problem (1) as

119906 = F119906 (18)

whereF E rarr E is defined by

(F119906) (119905) =119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

times 120601119902(

119904minus1

sum

120591=0

119891 (120591 + 120572 minus 1 119906 (120591 + 120572 minus 1)))

(19)

for 119905 isin [120572minus2 120572+119887+1]N120572minus2

It is clear to see that 119906 is a solutionof the problem (1) if and only if 119906 is a fixed point ofF

Now we state the main results as follows

Theorem 6 Suppose p gt 2 1205731

= 1 1205732

= 1 and the followingcondition holds

(H1) there exist positive numbers 120582 and 119896 with

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887) times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(20)

4 Discrete Dynamics in Nature and Society

such that

120582120572119905120572minus1

le 119891 (119905 119906) 119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

(21)

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(22)

Then the problem (1) has a unique solution

Proof For any 119906 isin E by (21) we can get that

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

ge 120582

119905minus1

sum

119904=0

120572(119904 + 120572 minus 1)120572minus1

= 120582(119905 + 120572 minus 1)120572

ge 120582Γ (120572 + 1) 119905 isin [1 119887 + 1]N0

(23)

Due to 119901 gt 2 and 1119901 + 1119902 = 1 we know that 1 lt 119902 lt 2 By(14) and (22) for any 119906 V isin E 119905 isin [1 119887 + 1]N

1

we have

|119860 (V 119906) (119905)| le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus 119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

(24)

Next for any 119906 V isin E and for each 119905 isin [120572minus2 120572+119887+1]N120572minus2

together with the fact that 119860(V 119906)(0) = 0 we obtain

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

le 119886 (120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

=10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 [119886120572 + 120572 + 119887] +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887) Γ (120572 + 119887)10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 Γ (120572 + 1) Γ (119887 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

= ((119896 (119902 minus 1) (119887 + 1) [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886 + 120572 + 119887)

+10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)] [120582Γ (120572 + 1)]119902minus2

times

119887minus1

prod

119894=1

(120572 + 119894)) times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906

= 119871 V minus 119906

(25)

where 119871 = (119896(119902 minus 1)(119887 + 1)[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 +

119887)][120582Γ(120572 + 1)]119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) From (20) we

get that 0 lt 119871 lt 1 which implies that F is a contractionmapping By means of the Banach contraction mappingprinciple we get that F has a unique fixed point in E thatis the problem (1) has a unique solution This completes theproof

With a similar proof to that of Theorem 6 we can get thefollowing theorem

Theorem 7 Suppose 119901 gt 2 1205731

= 1 1205732

= 1 and the followingcondition holds

Discrete Dynamics in Nature and Society 5

(H2) there exist constants 120582 gt 0 and 119896 with

0 lt 119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(26)

such that119891 (119905 119906)

le minus120582120572119905120572minus1

119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 119887 + 120572 minus 1]N120572minus1

119906 V isin R

(27)

Then the problem (1) has a unique solution

Theorem 8 Suppose 1 lt 119901 lt 2 1205731

= 1 1205732

= 1 and thefollowing condition holds

(H3) there exists a nonnegative function 119892 [120572 minus 1 120572 + 119887 minus

1]N120572minus1

rarr R and sum119887

119904=0119892(119904 + 120572 minus 1) = 119872 gt 0 such that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N

120572minus1

timesR (28)

and there exists a positive constant 119896 such that1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)

1003816100381610038161003816

le 119896 |V minus 119906| for 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(29)

Then the problem (1) has a unique solution provided that

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(30)

Proof By (28) we can get that for 119905 isin [1 119887 + 1]N1

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le

119887

sum

119904=0

119892 (119904 + 120572 minus 1) = 119872

(31)

In view of 1 lt 119901 lt 2 and 1119901 + 1119902 = 1 we can get 119902 gt 2From (15) and (29) for any V 119906 isin E we have

|119860 (V 119906) (119905)| le (119902 minus 1)119872119902minus2

times

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le (119902 minus 1)119872119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1)119872119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1)119872119902minus2

119905 V minus 119906 119905 isin [1 119887 + 1]N1

(32)

Hence for any 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

by 119860(119906 V)(0) = 0 wehave

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

le 119886(120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

6 Discrete Dynamics in Nature and Society

= ((119896 [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906 = 119871 V minus 119906

(33)

where 119871 = (119896[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 + 119887)](119902 minus

1)(119887 + 1)119872119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) In view of (30) F

is a contraction Thus the conclusion of the theorem followsby the contraction mapping principle This completes theproof

4 Examples

In this section we will illustrate the possible application ofthe above established analytical results with the following twoconcrete examples

Example 1 Consider the discrete fractional boundary valueproblem

Δ [1206013(Δ32

119862119906)] (119905)

= 3(119905 +1

2)

12

times [1

2+ sin2 ( 119906 (119905 + 12)

10radic3+ 120579)

+1

390

1003816100381610038161003816100381610038161003816119906 (119905 +

1

2)1003816100381610038161003816100381610038161003816] 119905 isin [0 2]N

0

10119906 (minus1

2) = 119906 (

9

2)

10Δ119906 (minus1

2) = 10Δ119906 (

1

2) = Δ119906(

7

2)

(34)

here 120579 is a real number

Conclusion Problem (34) has a unique nonnegative solution

Proof Corresponding to problem (1) 119901 = 3 gt 2 119902 = 32 120572 =

32 1205731= 110 120573

2= 110 119887 = 2 and 119891(119905 119906) = 3119905

12

[12 +

sin2((11990610radic3) + 120579) + (1390)|119906|] (119905 119906) isin [12 52]N12

timesRChoosing 120582 = 1 and 119896 = 3100 by direct calculation we

can verify that

119896 =3

100lt (

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=18radic3120587

14

325asymp 01277

(35)

It is easy to verify that

120582120572119905120572minus1

=3

211990512

le 311990512

[1

2+ sin2 ( 119906

10radic3+ 120579) +

1

390|119906|]

= 119891 (119905 119906) (119905 119906) isin [1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

= 311990512

10038161003816100381610038161003816100381610038161003816sin2 ( V

10radic3+ 120579)

minus sin2 ( 119906

10radic3+ 120579)

+1

390|V| minus

1

390|119906|

10038161003816100381610038161003816100381610038161003816

le 3(5

2)

12

(1

300+

1

390) |V minus 119906|

=69

4160radic120587 |V minus 119906|

asymp 00294 |V minus 119906| lt 119896 |V minus 119906|

(36)

for 119905 isin [12 52]N12

119906 V isin R Therefore by Theorem 6the boundary value problem (34) has a unique solutionFurthermore from the nonnegativeness of 119891 and the expres-sion of F we also get that the unique solution of (34) isnonnegative

Example 2 Consider the nonlinear discrete fractionalboundary value problem

Δ [12060132

(Δ32

119862119906)] (119905)

=3

2(119905 +

1

2)

12

times sin2 (119906 (119905 + 12)

40+ 120596) 119905 isin [0 2]N

0

2119906 (minus1

2) = minus119906(

9

2)

2Δ119906 (minus1

2) = 2Δ119906(

1

2) = Δ119906(

7

2)

(37)

where 120596 is a real number

Conclusion Problem (37) has a unique solution

Proof The problem (37) can be regarded as problem (1)where 119901 = 32 lt 2 119902 = 3 gt 2 120572 = 32 120573

1= minus12

1205732= 12 119887 = 2 and119891(119905 119906) = (32)119905

12sin2((11990640)+120596) (119905 119906)

Discrete Dynamics in Nature and Society 7

isin [12 52]N12

timesR Taking 119892(119905) = (32)11990512 119905 isin [12 52]N

12

then119872 = (105Γ(12))32 Let 119896 = 1600 asymp 00017 we have

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=64

15225radic120587asymp 00024

(38)

Moreover we can verify that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [

1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816 le

3

211990512

1

402|V minus 119906|

le3

2(5

2)

12 1

402|V minus 119906|

=9radic120587

10240|V minus 119906|

asymp 00016 |V minus 119906| lt 119896 |V minus 119906|

(39)

for 119905 isin [12 52]N12

V 119906 isin RTherefore byTheorem 8 problem (37) has a unique solu-

tion

Acknowledgment

This work was supported by the Longdong University GrantXYZK-1010 and XYZK-1007

References

[1] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[2] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[3] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

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

[6] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[7] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[8] J Sabatier O P Agrawal and J A T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer Heidelberg Germany2007

[9] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[10] Z Bai ldquoOn positive solutions of a nonlocal fractional boundaryvalue problemrdquoNonlinear AnalysisTheory Methods amp Applica-tions A vol 72 no 2 pp 916ndash924 2010

[11] Z Bai and Y Zhang ldquoSolvability of fractional three-pointboundary value problems with nonlinear growthrdquo AppliedMathematics and Computation vol 218 no 5 pp 1719ndash17252011

[12] C Bai ldquoTriple positive solutions for a boundary value problemof nonlinear fractional differential equationrdquo Electronic Journalof Qualitative Theory of Differential Equations vol 2008 article24 2008

[13] X Xu D Jiang and C Yuan ldquoMultiple positive solutions for theboundary value problem of a nonlinear fractional differentialequationrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 71 no 10 pp 4676ndash4688 2009

[14] B Ahmad ldquoExistence of solutions for irregular boundary valueproblems of nonlinear fractional differential equationsrdquoAppliedMathematics Letters of Rapid Publication vol 23 no 4 pp 390ndash394 2010

[15] S Zhang ldquoPositive solutions to singular boundary value prob-lem for nonlinear fractional differential equationrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1300ndash13092010

[16] W Jiang ldquoThe existence of solutions to boundary value prob-lems of fractional differential equations at resonancerdquoNonlinearAnalysis Theory Methods amp Applications A vol 74 no 5 pp1987ndash1994 2011

[17] Y Chen and X Tang ldquoSolvability of sequential fractional ordermulti-point boundary value problems at resonancerdquo AppliedMathematics and Computation vol 218 no 14 pp 7638ndash76482012

[18] A Guezane-Lakoud and R Khaldi ldquoSolvability of a fractionalboundary value problem with fractional integral conditionrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 4 pp 2692ndash2700 2012

[19] R Hilscher and V Zeidan ldquoNonnegativity and positivity ofquadratic functionals in discrete calculus of variations surveyrdquoJournal of Difference Equations and Applications vol 11 no 9pp 857ndash875 2005

[20] W G Kelley and A C Peterson Difference Equations AnIntroduction with Applications Academic Press New York NYUSA 1991

[21] F M Atici and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[22] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[23] F M Atıcı and P W Eloe ldquoDiscrete fractional calculus withthe nabla operatorrdquo Electronic Journal of Qualitative Theory ofDifferential Equations vol 2009 article 3 12 pages 2009

[24] F M Atıcı and S Sengul ldquoModeling with fractional differenceequationsrdquo Journal of Mathematical Analysis and Applicationsvol 369 no 1 pp 1ndash9 2010

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 4: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

4 Discrete Dynamics in Nature and Society

such that

120582120572119905120572minus1

le 119891 (119905 119906) 119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

(21)

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(22)

Then the problem (1) has a unique solution

Proof For any 119906 isin E by (21) we can get that

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

ge 120582

119905minus1

sum

119904=0

120572(119904 + 120572 minus 1)120572minus1

= 120582(119905 + 120572 minus 1)120572

ge 120582Γ (120572 + 1) 119905 isin [1 119887 + 1]N0

(23)

Due to 119901 gt 2 and 1119901 + 1119902 = 1 we know that 1 lt 119902 lt 2 By(14) and (22) for any 119906 V isin E 119905 isin [1 119887 + 1]N

1

we have

|119860 (V 119906) (119905)| le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus 119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1) [120582Γ (120572 + 1)]119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

(24)

Next for any 119906 V isin E and for each 119905 isin [120572minus2 120572+119887+1]N120572minus2

together with the fact that 119860(V 119906)(0) = 0 we obtain

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=0

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=0

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=0

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

le 119886 (120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

=10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 [119886120572 + 120572 + 119887] +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887) Γ (120572 + 119887)10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 Γ (120572 + 1) Γ (119887 + 1)

times 119896 (119902 minus 1) (119887 + 1) [120582Γ (120572 + 1)]119902minus2

V minus 119906

= ((119896 (119902 minus 1) (119887 + 1) [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886 + 120572 + 119887)

+10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)] [120582Γ (120572 + 1)]119902minus2

times

119887minus1

prod

119894=1

(120572 + 119894)) times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906

= 119871 V minus 119906

(25)

where 119871 = (119896(119902 minus 1)(119887 + 1)[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 +

119887)][120582Γ(120572 + 1)]119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) From (20) we

get that 0 lt 119871 lt 1 which implies that F is a contractionmapping By means of the Banach contraction mappingprinciple we get that F has a unique fixed point in E thatis the problem (1) has a unique solution This completes theproof

With a similar proof to that of Theorem 6 we can get thefollowing theorem

Theorem 7 Suppose 119901 gt 2 1205731

= 1 1205732

= 1 and the followingcondition holds

Discrete Dynamics in Nature and Society 5

(H2) there exist constants 120582 gt 0 and 119896 with

0 lt 119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(26)

such that119891 (119905 119906)

le minus120582120572119905120572minus1

119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 119887 + 120572 minus 1]N120572minus1

119906 V isin R

(27)

Then the problem (1) has a unique solution

Theorem 8 Suppose 1 lt 119901 lt 2 1205731

= 1 1205732

= 1 and thefollowing condition holds

(H3) there exists a nonnegative function 119892 [120572 minus 1 120572 + 119887 minus

1]N120572minus1

rarr R and sum119887

119904=0119892(119904 + 120572 minus 1) = 119872 gt 0 such that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N

120572minus1

timesR (28)

and there exists a positive constant 119896 such that1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)

1003816100381610038161003816

le 119896 |V minus 119906| for 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(29)

Then the problem (1) has a unique solution provided that

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(30)

Proof By (28) we can get that for 119905 isin [1 119887 + 1]N1

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le

119887

sum

119904=0

119892 (119904 + 120572 minus 1) = 119872

(31)

In view of 1 lt 119901 lt 2 and 1119901 + 1119902 = 1 we can get 119902 gt 2From (15) and (29) for any V 119906 isin E we have

|119860 (V 119906) (119905)| le (119902 minus 1)119872119902minus2

times

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le (119902 minus 1)119872119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1)119872119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1)119872119902minus2

119905 V minus 119906 119905 isin [1 119887 + 1]N1

(32)

Hence for any 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

by 119860(119906 V)(0) = 0 wehave

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

le 119886(120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

6 Discrete Dynamics in Nature and Society

= ((119896 [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906 = 119871 V minus 119906

(33)

where 119871 = (119896[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 + 119887)](119902 minus

1)(119887 + 1)119872119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) In view of (30) F

is a contraction Thus the conclusion of the theorem followsby the contraction mapping principle This completes theproof

4 Examples

In this section we will illustrate the possible application ofthe above established analytical results with the following twoconcrete examples

Example 1 Consider the discrete fractional boundary valueproblem

Δ [1206013(Δ32

119862119906)] (119905)

= 3(119905 +1

2)

12

times [1

2+ sin2 ( 119906 (119905 + 12)

10radic3+ 120579)

+1

390

1003816100381610038161003816100381610038161003816119906 (119905 +

1

2)1003816100381610038161003816100381610038161003816] 119905 isin [0 2]N

0

10119906 (minus1

2) = 119906 (

9

2)

10Δ119906 (minus1

2) = 10Δ119906 (

1

2) = Δ119906(

7

2)

(34)

here 120579 is a real number

Conclusion Problem (34) has a unique nonnegative solution

Proof Corresponding to problem (1) 119901 = 3 gt 2 119902 = 32 120572 =

32 1205731= 110 120573

2= 110 119887 = 2 and 119891(119905 119906) = 3119905

12

[12 +

sin2((11990610radic3) + 120579) + (1390)|119906|] (119905 119906) isin [12 52]N12

timesRChoosing 120582 = 1 and 119896 = 3100 by direct calculation we

can verify that

119896 =3

100lt (

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=18radic3120587

14

325asymp 01277

(35)

It is easy to verify that

120582120572119905120572minus1

=3

211990512

le 311990512

[1

2+ sin2 ( 119906

10radic3+ 120579) +

1

390|119906|]

= 119891 (119905 119906) (119905 119906) isin [1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

= 311990512

10038161003816100381610038161003816100381610038161003816sin2 ( V

10radic3+ 120579)

minus sin2 ( 119906

10radic3+ 120579)

+1

390|V| minus

1

390|119906|

10038161003816100381610038161003816100381610038161003816

le 3(5

2)

12

(1

300+

1

390) |V minus 119906|

=69

4160radic120587 |V minus 119906|

asymp 00294 |V minus 119906| lt 119896 |V minus 119906|

(36)

for 119905 isin [12 52]N12

119906 V isin R Therefore by Theorem 6the boundary value problem (34) has a unique solutionFurthermore from the nonnegativeness of 119891 and the expres-sion of F we also get that the unique solution of (34) isnonnegative

Example 2 Consider the nonlinear discrete fractionalboundary value problem

Δ [12060132

(Δ32

119862119906)] (119905)

=3

2(119905 +

1

2)

12

times sin2 (119906 (119905 + 12)

40+ 120596) 119905 isin [0 2]N

0

2119906 (minus1

2) = minus119906(

9

2)

2Δ119906 (minus1

2) = 2Δ119906(

1

2) = Δ119906(

7

2)

(37)

where 120596 is a real number

Conclusion Problem (37) has a unique solution

Proof The problem (37) can be regarded as problem (1)where 119901 = 32 lt 2 119902 = 3 gt 2 120572 = 32 120573

1= minus12

1205732= 12 119887 = 2 and119891(119905 119906) = (32)119905

12sin2((11990640)+120596) (119905 119906)

Discrete Dynamics in Nature and Society 7

isin [12 52]N12

timesR Taking 119892(119905) = (32)11990512 119905 isin [12 52]N

12

then119872 = (105Γ(12))32 Let 119896 = 1600 asymp 00017 we have

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=64

15225radic120587asymp 00024

(38)

Moreover we can verify that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [

1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816 le

3

211990512

1

402|V minus 119906|

le3

2(5

2)

12 1

402|V minus 119906|

=9radic120587

10240|V minus 119906|

asymp 00016 |V minus 119906| lt 119896 |V minus 119906|

(39)

for 119905 isin [12 52]N12

V 119906 isin RTherefore byTheorem 8 problem (37) has a unique solu-

tion

Acknowledgment

This work was supported by the Longdong University GrantXYZK-1010 and XYZK-1007

References

[1] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[2] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[3] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

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

[6] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[7] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[8] J Sabatier O P Agrawal and J A T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer Heidelberg Germany2007

[9] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[10] Z Bai ldquoOn positive solutions of a nonlocal fractional boundaryvalue problemrdquoNonlinear AnalysisTheory Methods amp Applica-tions A vol 72 no 2 pp 916ndash924 2010

[11] Z Bai and Y Zhang ldquoSolvability of fractional three-pointboundary value problems with nonlinear growthrdquo AppliedMathematics and Computation vol 218 no 5 pp 1719ndash17252011

[12] C Bai ldquoTriple positive solutions for a boundary value problemof nonlinear fractional differential equationrdquo Electronic Journalof Qualitative Theory of Differential Equations vol 2008 article24 2008

[13] X Xu D Jiang and C Yuan ldquoMultiple positive solutions for theboundary value problem of a nonlinear fractional differentialequationrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 71 no 10 pp 4676ndash4688 2009

[14] B Ahmad ldquoExistence of solutions for irregular boundary valueproblems of nonlinear fractional differential equationsrdquoAppliedMathematics Letters of Rapid Publication vol 23 no 4 pp 390ndash394 2010

[15] S Zhang ldquoPositive solutions to singular boundary value prob-lem for nonlinear fractional differential equationrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1300ndash13092010

[16] W Jiang ldquoThe existence of solutions to boundary value prob-lems of fractional differential equations at resonancerdquoNonlinearAnalysis Theory Methods amp Applications A vol 74 no 5 pp1987ndash1994 2011

[17] Y Chen and X Tang ldquoSolvability of sequential fractional ordermulti-point boundary value problems at resonancerdquo AppliedMathematics and Computation vol 218 no 14 pp 7638ndash76482012

[18] A Guezane-Lakoud and R Khaldi ldquoSolvability of a fractionalboundary value problem with fractional integral conditionrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 4 pp 2692ndash2700 2012

[19] R Hilscher and V Zeidan ldquoNonnegativity and positivity ofquadratic functionals in discrete calculus of variations surveyrdquoJournal of Difference Equations and Applications vol 11 no 9pp 857ndash875 2005

[20] W G Kelley and A C Peterson Difference Equations AnIntroduction with Applications Academic Press New York NYUSA 1991

[21] F M Atici and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[22] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[23] F M Atıcı and P W Eloe ldquoDiscrete fractional calculus withthe nabla operatorrdquo Electronic Journal of Qualitative Theory ofDifferential Equations vol 2009 article 3 12 pages 2009

[24] F M Atıcı and S Sengul ldquoModeling with fractional differenceequationsrdquo Journal of Mathematical Analysis and Applicationsvol 369 no 1 pp 1ndash9 2010

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 5: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

Discrete Dynamics in Nature and Society 5

(H2) there exist constants 120582 gt 0 and 119896 with

0 lt 119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(26)

such that119891 (119905 119906)

le minus120582120572119905120572minus1

119891119900119903 (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

le 119896 |V minus 119906| 119891119900119903 119905 isin [120572 minus 1 119887 + 120572 minus 1]N120572minus1

119906 V isin R

(27)

Then the problem (1) has a unique solution

Theorem 8 Suppose 1 lt 119901 lt 2 1205731

= 1 1205732

= 1 and thefollowing condition holds

(H3) there exists a nonnegative function 119892 [120572 minus 1 120572 + 119887 minus

1]N120572minus1

rarr R and sum119887

119904=0119892(119904 + 120572 minus 1) = 119872 gt 0 such that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [120572 minus 1 120572 + 119887 minus 1]N

120572minus1

timesR (28)

and there exists a positive constant 119896 such that1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)

1003816100381610038161003816

le 119896 |V minus 119906| for 119905 isin [120572 minus 1 120572 + 119887 minus 1]N120572minus1

119906 V isin R

(29)

Then the problem (1) has a unique solution provided that

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

(30)

Proof By (28) we can get that for 119905 isin [1 119887 + 1]N1

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le

119887

sum

119904=0

119892 (119904 + 120572 minus 1) = 119872

(31)

In view of 1 lt 119901 lt 2 and 1119901 + 1119902 = 1 we can get 119902 gt 2From (15) and (29) for any V 119906 isin E we have

|119860 (V 119906) (119905)| le (119902 minus 1)119872119902minus2

times

1003816100381610038161003816100381610038161003816100381610038161003816

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus

119905minus1

sum

119904=0

119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))

1003816100381610038161003816100381610038161003816100381610038161003816

le (119902 minus 1)119872119902minus2

times

119905minus1

sum

119904=0

1003816100381610038161003816119891 (119904 + 120572 minus 1 V (119904 + 120572 minus 1))

minus119891 (119904 + 120572 minus 1 119906 (119904 + 120572 minus 1))1003816100381610038161003816

le (119902 minus 1)119872119902minus2

119905minus1

sum

119904=0

119896 V minus 119906

le 119896 (119902 minus 1)119872119902minus2

119905 V minus 119906 119905 isin [1 119887 + 1]N1

(32)

Hence for any 119905 isin [120572 minus 2 120572 + 119887 + 1]N120572minus2

by 119860(119906 V)(0) = 0 wehave

|(FV) (119905) minus (F119906) (119905)|

=

1003816100381610038161003816100381610038161003816100381610038161003816

119886 (119905)

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

119860 (V 119906) (119904)

+1205731

(1 minus 1205731) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

119860 (V 119906) (119904)

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

119860 (V 119906) (119904)1003816100381610038161003816100381610038161003816100381610038161003816

le |119886 (119905)|

Γ (120572 minus 1)

119887+1

sum

119904=1

(120572 + 119887 minus 119904 minus 1)120572minus2

+

100381610038161003816100381612057311003816100381610038161003816

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572)

119887+1

sum

119904=1

(120572 + 119887 minus 119904)120572minus1

+1

Γ (120572)

119905minus120572

sum

119904=1

(119905 minus 119904 minus 1)120572minus1

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

le 119886(120572 + 119887 minus 1)

120572minus1

Γ (120572)

+

100381610038161003816100381612057311003816100381610038161003816 (120572 + 119887)

120572

(10038161003816100381610038161 minus 120573

1

1003816100381610038161003816) Γ (120572 + 1)+

(120572 + 119887)120572

Γ (120572 + 1)

times 119896 (119902 minus 1) (119887 + 1)119872119902minus2

V minus 119906

6 Discrete Dynamics in Nature and Society

= ((119896 [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906 = 119871 V minus 119906

(33)

where 119871 = (119896[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 + 119887)](119902 minus

1)(119887 + 1)119872119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) In view of (30) F

is a contraction Thus the conclusion of the theorem followsby the contraction mapping principle This completes theproof

4 Examples

In this section we will illustrate the possible application ofthe above established analytical results with the following twoconcrete examples

Example 1 Consider the discrete fractional boundary valueproblem

Δ [1206013(Δ32

119862119906)] (119905)

= 3(119905 +1

2)

12

times [1

2+ sin2 ( 119906 (119905 + 12)

10radic3+ 120579)

+1

390

1003816100381610038161003816100381610038161003816119906 (119905 +

1

2)1003816100381610038161003816100381610038161003816] 119905 isin [0 2]N

0

10119906 (minus1

2) = 119906 (

9

2)

10Δ119906 (minus1

2) = 10Δ119906 (

1

2) = Δ119906(

7

2)

(34)

here 120579 is a real number

Conclusion Problem (34) has a unique nonnegative solution

Proof Corresponding to problem (1) 119901 = 3 gt 2 119902 = 32 120572 =

32 1205731= 110 120573

2= 110 119887 = 2 and 119891(119905 119906) = 3119905

12

[12 +

sin2((11990610radic3) + 120579) + (1390)|119906|] (119905 119906) isin [12 52]N12

timesRChoosing 120582 = 1 and 119896 = 3100 by direct calculation we

can verify that

119896 =3

100lt (

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=18radic3120587

14

325asymp 01277

(35)

It is easy to verify that

120582120572119905120572minus1

=3

211990512

le 311990512

[1

2+ sin2 ( 119906

10radic3+ 120579) +

1

390|119906|]

= 119891 (119905 119906) (119905 119906) isin [1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

= 311990512

10038161003816100381610038161003816100381610038161003816sin2 ( V

10radic3+ 120579)

minus sin2 ( 119906

10radic3+ 120579)

+1

390|V| minus

1

390|119906|

10038161003816100381610038161003816100381610038161003816

le 3(5

2)

12

(1

300+

1

390) |V minus 119906|

=69

4160radic120587 |V minus 119906|

asymp 00294 |V minus 119906| lt 119896 |V minus 119906|

(36)

for 119905 isin [12 52]N12

119906 V isin R Therefore by Theorem 6the boundary value problem (34) has a unique solutionFurthermore from the nonnegativeness of 119891 and the expres-sion of F we also get that the unique solution of (34) isnonnegative

Example 2 Consider the nonlinear discrete fractionalboundary value problem

Δ [12060132

(Δ32

119862119906)] (119905)

=3

2(119905 +

1

2)

12

times sin2 (119906 (119905 + 12)

40+ 120596) 119905 isin [0 2]N

0

2119906 (minus1

2) = minus119906(

9

2)

2Δ119906 (minus1

2) = 2Δ119906(

1

2) = Δ119906(

7

2)

(37)

where 120596 is a real number

Conclusion Problem (37) has a unique solution

Proof The problem (37) can be regarded as problem (1)where 119901 = 32 lt 2 119902 = 3 gt 2 120572 = 32 120573

1= minus12

1205732= 12 119887 = 2 and119891(119905 119906) = (32)119905

12sin2((11990640)+120596) (119905 119906)

Discrete Dynamics in Nature and Society 7

isin [12 52]N12

timesR Taking 119892(119905) = (32)11990512 119905 isin [12 52]N

12

then119872 = (105Γ(12))32 Let 119896 = 1600 asymp 00017 we have

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=64

15225radic120587asymp 00024

(38)

Moreover we can verify that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [

1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816 le

3

211990512

1

402|V minus 119906|

le3

2(5

2)

12 1

402|V minus 119906|

=9radic120587

10240|V minus 119906|

asymp 00016 |V minus 119906| lt 119896 |V minus 119906|

(39)

for 119905 isin [12 52]N12

V 119906 isin RTherefore byTheorem 8 problem (37) has a unique solu-

tion

Acknowledgment

This work was supported by the Longdong University GrantXYZK-1010 and XYZK-1007

References

[1] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[2] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[3] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

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

[6] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[7] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[8] J Sabatier O P Agrawal and J A T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer Heidelberg Germany2007

[9] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[10] Z Bai ldquoOn positive solutions of a nonlocal fractional boundaryvalue problemrdquoNonlinear AnalysisTheory Methods amp Applica-tions A vol 72 no 2 pp 916ndash924 2010

[11] Z Bai and Y Zhang ldquoSolvability of fractional three-pointboundary value problems with nonlinear growthrdquo AppliedMathematics and Computation vol 218 no 5 pp 1719ndash17252011

[12] C Bai ldquoTriple positive solutions for a boundary value problemof nonlinear fractional differential equationrdquo Electronic Journalof Qualitative Theory of Differential Equations vol 2008 article24 2008

[13] X Xu D Jiang and C Yuan ldquoMultiple positive solutions for theboundary value problem of a nonlinear fractional differentialequationrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 71 no 10 pp 4676ndash4688 2009

[14] B Ahmad ldquoExistence of solutions for irregular boundary valueproblems of nonlinear fractional differential equationsrdquoAppliedMathematics Letters of Rapid Publication vol 23 no 4 pp 390ndash394 2010

[15] S Zhang ldquoPositive solutions to singular boundary value prob-lem for nonlinear fractional differential equationrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1300ndash13092010

[16] W Jiang ldquoThe existence of solutions to boundary value prob-lems of fractional differential equations at resonancerdquoNonlinearAnalysis Theory Methods amp Applications A vol 74 no 5 pp1987ndash1994 2011

[17] Y Chen and X Tang ldquoSolvability of sequential fractional ordermulti-point boundary value problems at resonancerdquo AppliedMathematics and Computation vol 218 no 14 pp 7638ndash76482012

[18] A Guezane-Lakoud and R Khaldi ldquoSolvability of a fractionalboundary value problem with fractional integral conditionrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 4 pp 2692ndash2700 2012

[19] R Hilscher and V Zeidan ldquoNonnegativity and positivity ofquadratic functionals in discrete calculus of variations surveyrdquoJournal of Difference Equations and Applications vol 11 no 9pp 857ndash875 2005

[20] W G Kelley and A C Peterson Difference Equations AnIntroduction with Applications Academic Press New York NYUSA 1991

[21] F M Atici and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[22] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[23] F M Atıcı and P W Eloe ldquoDiscrete fractional calculus withthe nabla operatorrdquo Electronic Journal of Qualitative Theory ofDifferential Equations vol 2009 article 3 12 pages 2009

[24] F M Atıcı and S Sengul ldquoModeling with fractional differenceequationsrdquo Journal of Mathematical Analysis and Applicationsvol 369 no 1 pp 1ndash9 2010

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 6: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

6 Discrete Dynamics in Nature and Society

= ((119896 [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

times (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)minus1

)V minus 119906 = 119871 V minus 119906

(33)

where 119871 = (119896[|1 minus 1205731|(119886120572 + 120572 + 119887) + |120573

1|(120572 + 119887)](119902 minus

1)(119887 + 1)119872119902minus2

prod119887minus1

119894=1(120572 + 119894))(|1 minus 120573

1|119887) In view of (30) F

is a contraction Thus the conclusion of the theorem followsby the contraction mapping principle This completes theproof

4 Examples

In this section we will illustrate the possible application ofthe above established analytical results with the following twoconcrete examples

Example 1 Consider the discrete fractional boundary valueproblem

Δ [1206013(Δ32

119862119906)] (119905)

= 3(119905 +1

2)

12

times [1

2+ sin2 ( 119906 (119905 + 12)

10radic3+ 120579)

+1

390

1003816100381610038161003816100381610038161003816119906 (119905 +

1

2)1003816100381610038161003816100381610038161003816] 119905 isin [0 2]N

0

10119906 (minus1

2) = 119906 (

9

2)

10Δ119906 (minus1

2) = 10Δ119906 (

1

2) = Δ119906(

7

2)

(34)

here 120579 is a real number

Conclusion Problem (34) has a unique nonnegative solution

Proof Corresponding to problem (1) 119901 = 3 gt 2 119902 = 32 120572 =

32 1205731= 110 120573

2= 110 119887 = 2 and 119891(119905 119906) = 3119905

12

[12 +

sin2((11990610radic3) + 120579) + (1390)|119906|] (119905 119906) isin [12 52]N12

timesRChoosing 120582 = 1 and 119896 = 3100 by direct calculation we

can verify that

119896 =3

100lt (

10038161003816100381610038161 minus 1205731

1003816100381610038161003816 119887)

times ( (119902 minus 1) (119887 + 1)

times [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times [120582Γ (120572 + 1)]119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=18radic3120587

14

325asymp 01277

(35)

It is easy to verify that

120582120572119905120572minus1

=3

211990512

le 311990512

[1

2+ sin2 ( 119906

10radic3+ 120579) +

1

390|119906|]

= 119891 (119905 119906) (119905 119906) isin [1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816

= 311990512

10038161003816100381610038161003816100381610038161003816sin2 ( V

10radic3+ 120579)

minus sin2 ( 119906

10radic3+ 120579)

+1

390|V| minus

1

390|119906|

10038161003816100381610038161003816100381610038161003816

le 3(5

2)

12

(1

300+

1

390) |V minus 119906|

=69

4160radic120587 |V minus 119906|

asymp 00294 |V minus 119906| lt 119896 |V minus 119906|

(36)

for 119905 isin [12 52]N12

119906 V isin R Therefore by Theorem 6the boundary value problem (34) has a unique solutionFurthermore from the nonnegativeness of 119891 and the expres-sion of F we also get that the unique solution of (34) isnonnegative

Example 2 Consider the nonlinear discrete fractionalboundary value problem

Δ [12060132

(Δ32

119862119906)] (119905)

=3

2(119905 +

1

2)

12

times sin2 (119906 (119905 + 12)

40+ 120596) 119905 isin [0 2]N

0

2119906 (minus1

2) = minus119906(

9

2)

2Δ119906 (minus1

2) = 2Δ119906(

1

2) = Δ119906(

7

2)

(37)

where 120596 is a real number

Conclusion Problem (37) has a unique solution

Proof The problem (37) can be regarded as problem (1)where 119901 = 32 lt 2 119902 = 3 gt 2 120572 = 32 120573

1= minus12

1205732= 12 119887 = 2 and119891(119905 119906) = (32)119905

12sin2((11990640)+120596) (119905 119906)

Discrete Dynamics in Nature and Society 7

isin [12 52]N12

timesR Taking 119892(119905) = (32)11990512 119905 isin [12 52]N

12

then119872 = (105Γ(12))32 Let 119896 = 1600 asymp 00017 we have

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=64

15225radic120587asymp 00024

(38)

Moreover we can verify that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [

1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816 le

3

211990512

1

402|V minus 119906|

le3

2(5

2)

12 1

402|V minus 119906|

=9radic120587

10240|V minus 119906|

asymp 00016 |V minus 119906| lt 119896 |V minus 119906|

(39)

for 119905 isin [12 52]N12

V 119906 isin RTherefore byTheorem 8 problem (37) has a unique solu-

tion

Acknowledgment

This work was supported by the Longdong University GrantXYZK-1010 and XYZK-1007

References

[1] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[2] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[3] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

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

[6] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[7] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[8] J Sabatier O P Agrawal and J A T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer Heidelberg Germany2007

[9] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[10] Z Bai ldquoOn positive solutions of a nonlocal fractional boundaryvalue problemrdquoNonlinear AnalysisTheory Methods amp Applica-tions A vol 72 no 2 pp 916ndash924 2010

[11] Z Bai and Y Zhang ldquoSolvability of fractional three-pointboundary value problems with nonlinear growthrdquo AppliedMathematics and Computation vol 218 no 5 pp 1719ndash17252011

[12] C Bai ldquoTriple positive solutions for a boundary value problemof nonlinear fractional differential equationrdquo Electronic Journalof Qualitative Theory of Differential Equations vol 2008 article24 2008

[13] X Xu D Jiang and C Yuan ldquoMultiple positive solutions for theboundary value problem of a nonlinear fractional differentialequationrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 71 no 10 pp 4676ndash4688 2009

[14] B Ahmad ldquoExistence of solutions for irregular boundary valueproblems of nonlinear fractional differential equationsrdquoAppliedMathematics Letters of Rapid Publication vol 23 no 4 pp 390ndash394 2010

[15] S Zhang ldquoPositive solutions to singular boundary value prob-lem for nonlinear fractional differential equationrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1300ndash13092010

[16] W Jiang ldquoThe existence of solutions to boundary value prob-lems of fractional differential equations at resonancerdquoNonlinearAnalysis Theory Methods amp Applications A vol 74 no 5 pp1987ndash1994 2011

[17] Y Chen and X Tang ldquoSolvability of sequential fractional ordermulti-point boundary value problems at resonancerdquo AppliedMathematics and Computation vol 218 no 14 pp 7638ndash76482012

[18] A Guezane-Lakoud and R Khaldi ldquoSolvability of a fractionalboundary value problem with fractional integral conditionrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 4 pp 2692ndash2700 2012

[19] R Hilscher and V Zeidan ldquoNonnegativity and positivity ofquadratic functionals in discrete calculus of variations surveyrdquoJournal of Difference Equations and Applications vol 11 no 9pp 857ndash875 2005

[20] W G Kelley and A C Peterson Difference Equations AnIntroduction with Applications Academic Press New York NYUSA 1991

[21] F M Atici and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[22] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[23] F M Atıcı and P W Eloe ldquoDiscrete fractional calculus withthe nabla operatorrdquo Electronic Journal of Qualitative Theory ofDifferential Equations vol 2009 article 3 12 pages 2009

[24] F M Atıcı and S Sengul ldquoModeling with fractional differenceequationsrdquo Journal of Mathematical Analysis and Applicationsvol 369 no 1 pp 1ndash9 2010

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 7: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

Discrete Dynamics in Nature and Society 7

isin [12 52]N12

timesR Taking 119892(119905) = (32)11990512 119905 isin [12 52]N

12

then119872 = (105Γ(12))32 Let 119896 = 1600 asymp 00017 we have

119896 lt (10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 119887)

times ( [10038161003816100381610038161 minus 120573

1

1003816100381610038161003816 (119886120572 + 120572 + 119887) +10038161003816100381610038161205731

1003816100381610038161003816 (120572 + 119887)]

times (119902 minus 1) (119887 + 1)119872119902minus2

119887minus1

prod

119894=1

(120572 + 119894))

minus1

=64

15225radic120587asymp 00024

(38)

Moreover we can verify that

1003816100381610038161003816119891 (119905 119906)1003816100381610038161003816 le 119892 (119905) (119905 119906) isin [

1

25

2]N12

timesR

1003816100381610038161003816119891 (119905 V) minus 119891 (119905 119906)1003816100381610038161003816 le

3

211990512

1

402|V minus 119906|

le3

2(5

2)

12 1

402|V minus 119906|

=9radic120587

10240|V minus 119906|

asymp 00016 |V minus 119906| lt 119896 |V minus 119906|

(39)

for 119905 isin [12 52]N12

V 119906 isin RTherefore byTheorem 8 problem (37) has a unique solu-

tion

Acknowledgment

This work was supported by the Longdong University GrantXYZK-1010 and XYZK-1007

References

[1] S G Samko A A Kilbas and O I Marichev FractionalIntegrals and Derivatives Theory and Applications Gordon andBreach Science Yverdon Switzerland 1993

[2] W G Glockle and T F Nonnenmacher ldquoA fractional calculusapproach to self-similar protein dynamicsrdquo Biophysical Journalvol 68 no 1 pp 46ndash53 1995

[3] R Metzler W Schick H Kilian and T F NonnenmacherldquoRelaxation in filled polymers a fractional calculus approachrdquoThe Journal of Chemical Physics vol 103 no 16 pp 7180ndash71861995

[4] I Podlubny Fractional Differential Equations vol 198 ofMath-ematics in Science and Engineering Academic Press San DiegoCalif USA 1999

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

[6] A A Kilbas and J J Trujillo ldquoDifferential equations offractional order methods results and problems Irdquo ApplicableAnalysis vol 78 no 1-2 pp 153ndash192 2001

[7] A A Kilbas and J J Trujillo ldquoDifferential equations of frac-tional order methods results and problems IIrdquo ApplicableAnalysis vol 81 no 2 pp 435ndash493 2002

[8] J Sabatier O P Agrawal and J A T Machado Advances inFractional Calculus Theoretical Developments and Applicationsin Physics and Engineering Springer Heidelberg Germany2007

[9] Z Bai and H Lu ldquoPositive solutions for boundary valueproblem of nonlinear fractional differential equationrdquo Journalof Mathematical Analysis and Applications vol 311 no 2 pp495ndash505 2005

[10] Z Bai ldquoOn positive solutions of a nonlocal fractional boundaryvalue problemrdquoNonlinear AnalysisTheory Methods amp Applica-tions A vol 72 no 2 pp 916ndash924 2010

[11] Z Bai and Y Zhang ldquoSolvability of fractional three-pointboundary value problems with nonlinear growthrdquo AppliedMathematics and Computation vol 218 no 5 pp 1719ndash17252011

[12] C Bai ldquoTriple positive solutions for a boundary value problemof nonlinear fractional differential equationrdquo Electronic Journalof Qualitative Theory of Differential Equations vol 2008 article24 2008

[13] X Xu D Jiang and C Yuan ldquoMultiple positive solutions for theboundary value problem of a nonlinear fractional differentialequationrdquo Nonlinear Analysis Theory Methods amp ApplicationsA vol 71 no 10 pp 4676ndash4688 2009

[14] B Ahmad ldquoExistence of solutions for irregular boundary valueproblems of nonlinear fractional differential equationsrdquoAppliedMathematics Letters of Rapid Publication vol 23 no 4 pp 390ndash394 2010

[15] S Zhang ldquoPositive solutions to singular boundary value prob-lem for nonlinear fractional differential equationrdquo Computersamp Mathematics with Applications vol 59 no 3 pp 1300ndash13092010

[16] W Jiang ldquoThe existence of solutions to boundary value prob-lems of fractional differential equations at resonancerdquoNonlinearAnalysis Theory Methods amp Applications A vol 74 no 5 pp1987ndash1994 2011

[17] Y Chen and X Tang ldquoSolvability of sequential fractional ordermulti-point boundary value problems at resonancerdquo AppliedMathematics and Computation vol 218 no 14 pp 7638ndash76482012

[18] A Guezane-Lakoud and R Khaldi ldquoSolvability of a fractionalboundary value problem with fractional integral conditionrdquoNonlinear Analysis Theory Methods amp Applications A vol 75no 4 pp 2692ndash2700 2012

[19] R Hilscher and V Zeidan ldquoNonnegativity and positivity ofquadratic functionals in discrete calculus of variations surveyrdquoJournal of Difference Equations and Applications vol 11 no 9pp 857ndash875 2005

[20] W G Kelley and A C Peterson Difference Equations AnIntroduction with Applications Academic Press New York NYUSA 1991

[21] F M Atici and PW Eloe ldquoA transformmethod in discrete frac-tional calculusrdquo International Journal of Difference Equationsvol 2 no 2 pp 165ndash176 2007

[22] F M Atici and P W Eloe ldquoInitial value problems in discretefractional calculusrdquo Proceedings of the American MathematicalSociety vol 137 no 3 pp 981ndash989 2009

[23] F M Atıcı and P W Eloe ldquoDiscrete fractional calculus withthe nabla operatorrdquo Electronic Journal of Qualitative Theory ofDifferential Equations vol 2009 article 3 12 pages 2009

[24] F M Atıcı and S Sengul ldquoModeling with fractional differenceequationsrdquo Journal of Mathematical Analysis and Applicationsvol 369 no 1 pp 1ndash9 2010

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 8: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

8 Discrete Dynamics in Nature and Society

[25] FM Atıcı and PW Eloe ldquoTwo-point boundary value problemsfor finite fractional difference equationsrdquo Journal of DifferenceEquations and Applications vol 17 no 4 pp 445ndash456 2011

[26] C S Goodrich ldquoSolutions to a discrete right-focal fractionalboundary value problemrdquo International Journal of DifferenceEquations vol 5 no 2 pp 195ndash216 2010

[27] C S Goodrich ldquoContinuity of solutions to discrete fractionalinitial value problemsrdquo Computers amp Mathematics with Appli-cations vol 59 no 11 pp 3489ndash3499 2010

[28] C S Goodrich ldquoExistence and uniqueness of solutions toa fractional difference equation with nonlocal conditionsrdquoComputers amp Mathematics with Applications vol 61 no 2 pp191ndash202 2011

[29] C S Goodrich ldquoExistence of a positive solution to a system ofdiscrete fractional boundary value problemsrdquo Applied Mathe-matics and Computation vol 217 no 9 pp 4740ndash4753 2011

[30] C S Goodrich ldquoOn a discrete fractional three-point boundaryvalue problemrdquo Journal of Difference Equations and Applica-tions vol 18 no 3 pp 397ndash415 2012

[31] C S Goodrich ldquoOn discrete sequential fractional boundaryvalue problemsrdquo Journal of Mathematical Analysis and Applica-tions vol 385 no 1 pp 111ndash124 2012

[32] F Chen X Luo and Y Zhou ldquoExistence results for nonlinearfractional difference equationrdquo Advances in Difference Equa-tions vol 2011 Article ID 713201 12 pages 2011

[33] N R O Bastos R A C Ferreira andD FM Torres ldquoDiscrete-time fractional variational problemsrdquo Signal Process vol 91 no3 pp 513ndash524 2011

[34] N RO Bastos RAC Ferreira andD FMTorres ldquoNecessaryoptimality conditions for fractional difference problems of thecalculus of variationsrdquo Discrete and Continuous DynamicalSystems vol 29 no 2 pp 417ndash437 2011

[35] T Abdeljawad ldquoOn Riemann and Caputo fractional differ-encesrdquo Computers ampMathematics with Applications vol 62 no3 pp 1602ndash1611 2011

[36] M Holm ldquoSum and difference compositions in discrete frac-tional calculusrdquo Cubo vol 13 no 3 pp 153ndash184 2011

[37] R A C Ferreira ldquoExistence and uniqueness of solution to somediscrete fractional boundary value problems of order less thanonerdquo Journal of Difference Equations and Applications vol 19no 5 pp 712ndash718 2013

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

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

[40] Z Han H Lu S Sun and D Yang ldquoPositive solutions toboundary-value problems of p-Laplacian fractional differentialequations with a parameter in the boundaryrdquo Electronic Journalof Differential Equations vol 2012 article 213 14 pages 2012

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

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

[43] X Liu and M Jia ldquoOn the solvability of a fractional differentialequationmodel involving the p-Laplacian operatorrdquo ComputersampMathematics with Applications vol 64 no 10 pp 3267ndash32752012

[44] H Lu ZHan S Sun and J Liu ldquoExistence on positive solutionsfor boundary value problems of nonlinear fractional differentialequations with p-Laplacianrdquo Advances in Difference Equationsvol 2013 article 30 2013

[45] W Lv ldquoExistence of solutions for discrete fractional boundaryvalue problems with a p-Laplacian operatorrdquo Advances inDifference Equations vol 2012 article 163 2012

[46] K S Miller and B Ross ldquoFractional difference calculusrdquo inProceedings of the International Symposium on Univalent Func-tions Fractional Calculus andTheir Applications Ellis HorwoodSeries in Mathematics amp Its Applications pp 139ndash152 NihonUniversity Koriyama Japan May 1988 Horwood ChichesterUK 1989

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

Page 9: Research Article Solvability for Discrete Fractional Boundary ...ied discrete fractional boundary value problems involving the Riemann-Liouville fractional di erence intensively and

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


MATRIX APPROACH TO DISCRETE FRACTIONAL CALCULUS Igor Podlubnypeople.tuke.sk/igor.podlubny/pspdf/ma2dfc.pdf · MATRIX APPROACH TO DISCRETE FRACTIONAL CALCULUS Igor Podlubny ∗ DedicatedtoProf

MATRIX APPROACH TO DISCRETE FRACTIONAL CALCULUS Igor Podlubnypeople.tuke.sk/igor.podlubny/pspdf/ma2dfc.pdf · MATRIX APPROACH TO DISCRETE FRACTIONAL CALCULUS Igor Podlubny ∗ DedicatedtoProf

Documents
Boundary-Value Problems Ordinary Differential Equations: Discrete

Boundary-Value Problems Ordinary Differential Equations: Discrete

Documents
Discrete Fractional Fourier Transform as a Fast Algorithm for … · 2014-02-03 · 1 Discrete Fractional Fourier Transform as a Fast Algorithm for Evaluating the Diffraction Pattern

Discrete Fractional Fourier Transform as a Fast Algorithm for … · 2014-02-03 · 1 Discrete Fractional Fourier Transform as a Fast Algorithm for Evaluating the Diffraction Pattern

Documents
Discrete Surfaces Segmentation into Discrete Planes ...Discrete Surfaces Segmentation into Discrete Planes 461 Definition 3. Let O be a 3D discrete object. The inner boundary of O

Discrete Surfaces Segmentation into Discrete Planes ...Discrete Surfaces Segmentation into Discrete Planes 461 Definition 3. Let O be a 3D discrete object. The inner boundary of O

Documents
Backstepping-based boundary control design for a ...mechatronics.ucmerced.edu/sites/mechatronics.ucmerced.edu/files/… · Backstepping-based boundary control design for a fractional

Backstepping-based boundary control design for a ...mechatronics.ucmerced.edu/sites/mechatronics.ucmerced.edu/files/… · Backstepping-based boundary control design for a fractional

Documents
Fractional boundary value problems and Lyapunov-type inequalities … · 2016-06-21 · lems. We derive Lyapunov-type inequalities for linear fractional differential equations and

Fractional boundary value problems and Lyapunov-type inequalities … · 2016-06-21 · lems. We derive Lyapunov-type inequalities for linear fractional differential equations and

Documents
On the Solution of Fractional Order Nonlinear Boundary Value

On the Solution of Fractional Order Nonlinear Boundary Value

Documents
JSTAT Fractional diffusion equation, boundary conditions ...hvribeiro.org/wp-content/uploads/2014/08/P08019.pdf · J. Stat. Mech. (2014) P08019 Fractional diffusion equation, boundary

JSTAT Fractional diffusion equation, boundary conditions ...hvribeiro.org/wp-content/uploads/2014/08/P08019.pdf · J. Stat. Mech. (2014) P08019 Fractional diffusion equation, boundary

Documents
Outer boundary of the expanding cosmos: Discrete fields and

Outer boundary of the expanding cosmos: Discrete fields and

Documents
Kinetic boundary layers for the Boltzmann equation on discrete

Kinetic boundary layers for the Boltzmann equation on discrete

Documents
Fractional generalizations of - Reed College · 2000-04-19 · 2 Fractional Leibniz’formulæ which in its most frequently encountered formulation might be abbreviated fDg= boundary

Fractional generalizations of - Reed College · 2000-04-19 · 2 Fractional Leibniz’formulæ which in its most frequently encountered formulation might be abbreviated fDg= boundary

Documents
Discrete Fractional Calculus and Its Applications to Tumor

Discrete Fractional Calculus and Its Applications to Tumor

Documents
The discrete fractional Fourier transform based on the DFT ... · The discrete fractional Fourier transform based on the DFT matrix Ahmet Serbes , Lutfiye Durak-Ata Yildiz Technical

The discrete fractional Fourier transform based on the DFT ... · The discrete fractional Fourier transform based on the DFT matrix Ahmet Serbes , Lutfiye Durak-Ata Yildiz Technical

Documents
Non linear problems with Fractional Diffusions · Non linear problems involving fractional diffusions appear in several areas of applied mathematics: •Boundary diffusion (see for

Non linear problems with Fractional Diffusions · Non linear problems involving fractional diffusions appear in several areas of applied mathematics: •Boundary diffusion (see for

Documents
JSTAT Fractional diffusion equation, boundary conditions ...old.pfi.uem.br/complex/paper/2014/P08019.pdf · J. Stat. Mech. (2014) P08019 Fractional diffusion equation, boundary conditions

JSTAT Fractional diffusion equation, boundary conditions ...old.pfi.uem.br/complex/paper/2014/P08019.pdf · J. Stat. Mech. (2014) P08019 Fractional diffusion equation, boundary conditions

Documents
Analysis and numerical solution of a Riemann-Liouville fractional ... · Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem

Analysis and numerical solution of a Riemann-Liouville fractional ... · Analysis and numerical solution of a Riemann-Liouville fractional derivative two-point boundary value problem

Documents
BOUNDARY REGULARITY FOR THE FRACTIONAL HEAT EQUATIONuser.math.uzh.ch/...regularity_for_the_fractional_heat_equation_4.pdf · For the fractional heat equation (1.1), Chen-Kim-Song

BOUNDARY REGULARITY FOR THE FRACTIONAL HEAT EQUATIONuser.math.uzh.ch/...regularity_for_the_fractional_heat_equation_4.pdf · For the fractional heat equation (1.1), Chen-Kim-Song

Documents
M.tech Theis - - VLSI Architecture for Discrete Fractional Fourier Transform

M.tech Theis - - VLSI Architecture for Discrete Fractional Fourier Transform

Documents
Boundary-ValueProblems Ordinary Differential Equations ... · Boundary-ValueProblems Ordinary Differential Equations: Discrete Variable Methods INTRODUCTION Inthis chapterwe discuss

Boundary-ValueProblems Ordinary Differential Equations ... · Boundary-ValueProblems Ordinary Differential Equations: Discrete Variable Methods INTRODUCTION Inthis chapterwe discuss

Documents
Boundary Value Problems for Fractional Differential Equationsprr.hec.gov.pk/jspui/bitstream/123456789/2267/1/2508S.pdf · Boundary Value Problems for Fractional Differential Equations:

Boundary Value Problems for Fractional Differential Equationsprr.hec.gov.pk/jspui/bitstream/123456789/2267/1/2508S.pdf · Boundary Value Problems for Fractional Differential Equations:

Documents
Discrete Artificial Boundary Conditions for Nonlinear Schr ...ehrhardt/papers/dufort.pdfKey words: Nonlinear Schro¨dinger equation, unbounded domains, discrete artificial boundary

Discrete Artificial Boundary Conditions for Nonlinear Schr ...ehrhardt/papers/dufort.pdfKey words: Nonlinear Schro¨dinger equation, unbounded domains, discrete artificial boundary

Documents
BOUNDARY VALUE PROBLEMS FOR FRACTIONAL ORDER …

BOUNDARY VALUE PROBLEMS FOR FRACTIONAL ORDER …

Documents
Discrete Fractional Calculus and Its Applications to Tumor Growth

Discrete Fractional Calculus and Its Applications to Tumor Growth

Documents
The discrete fractional Fourier transform based on the DFT matrix

The discrete fractional Fourier transform based on the DFT matrix

Documents
Discrete Images (Chapter 7) Fourier Transform on discrete and bounded domains. Given an image: 1.Zero boundary condition 2.Periodic boundary condition

Discrete Images (Chapter 7) Fourier Transform on discrete and bounded domains. Given an image: 1.Zero boundary condition 2.Periodic boundary condition

Documents
Performance of Discrete Fractional Fourier Transform Classes in Signal Processing Applications

Performance of Discrete Fractional Fourier Transform Classes in Signal Processing Applications

Documents
Solving fractional two-point boundary value problems using continuous analytic method

Solving fractional two-point boundary value problems using continuous analytic method

Documents
Robust model predictive control for discrete-time fractional-order systems

Robust model predictive control for discrete-time fractional-order systems

Science
Fractional boundary value problems on the half · PDF fileOpuscula Math. 37, no. 2 (2017), ... the reader to [7 9,11,13,14,16] ... Fractional boundary value problems on the half line

Fractional boundary value problems on the half · PDF fileOpuscula Math. 37, no. 2 (2017), ... the reader to [7 9,11,13,14,16] ... Fractional boundary value problems on the half line

Documents
NONLOCAL DISCRETE DIFFUSION EQUATIONS AND THE … · Abstract. The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size

NONLOCAL DISCRETE DIFFUSION EQUATIONS AND THE … · Abstract. The analysis of nonlocal discrete equations driven by fractional powers of the discrete Laplacian on a mesh of size

Documents