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Discrete Dynamics in Nature and Society
Volume 2013 (2013), Article ID 104276, 9 pages
Existence Results for Fractional Difference Equations with Three-Point Fractional Sum Boundary Conditions
1Department of Mathematics, Faculty of Applied Science, King Mongkut’s University of Technology, North Bangkok, Bangkok, Thailand
2Department of Mathematics, University of Ioannina, 451 10 Ioannina, Greece
Received 3 June 2013; Accepted 7 August 2013
Academic Editor: Eric R. Kaufmann
Copyright © 2013 Thanin Sitthiwirattham et al. This 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.
We consider a discrete fractional boundary value problem of the form where , , , and is a continuous function. The existence of at least one solution is proved by using Krasnoselskii's fixed point theorem and Leray-Schauder's nonlinear alternative. Some illustrative examples are also presented.
Fractional differential and difference equations have received increasing attention during recent years since fractional derivatives provide an excellent instrument for the description of memory and hereditary properties of various materials and processes. For a reader interested in the systematic development of the topic, a good account of papers dealing with fractional or discrete fractional boundary value problems can be found in [1–25] and references cited therein.
In this paper we consider a nonlinear discrete fractional boundary value problem of the form where,,, andis a continuous function. Letbe the nonnegative integers. Denote.
Problem (1) was studied recently by authors in  where existence and uniqueness results are obtained by using Banach’s contraction mapping principle, the nonlinear contraction, and Schaefer’s fixed point theorem. Here we continue this study by proving some new existence results by using Krasnoselskii’s fixed point theorem and Leray-Schauder’s nonlinear alternative. Thus the results of this paper complement the results of .
The plan of this paper is as follows. In Section 2 we recall some definitions and basic lemmas. Also we give a representation for the solution to (1) by converting the problem to an equivalent summation equation. In Section 3, using this representation, we prove our main results. Some illustrative examples are presented in Section 4.
In this section, we introduce notations, definitions, and lemmas needed in the following.
Definition 1. One defines the generalized falling function by, for anyandfor which the right-hand side is defined. Ifis a pole of the Gamma function andis not a pole, then.
Definition 2 (see ). Theth fractional sum of a function, for, is defined by forand. One also defines theth fractional difference forby, whereandare chosen so that.
Lemma 3. Letandbe any numbers for whichandare defined. Then.
Lemma 4 (see ). LetThen for some, with
Theorem 5 (Arzelá-Ascoli theorem, see ). Letbe a Banach space anda function family of continuous mappings. If S is uniformly bounded and equicontinuous, and, for any, the setis relatively compact, then there exists a uniformly convergent function sequenceforin.
Lemma 6 (Mazur Lemma, see ). Ifis a compact subset of Banach space, then its convex closure is compact.
To define the solution of the boundary value problem (1) we need the following lemma, proved in  which deals with linear variant of the boundary value problem (1) and gives a representation of the solution.
Lemma 7. Let and let be given. Then the problem has a unique solution
3. Main Results
Letbe the Banach space of all continuous functionswith the normDefine an operatorby for, whereis defined by (4). It is easy to see that the problem (1) has solutions if and only if the operatorhas fixed points.
In the following, for the sake of convenience, we set a constant
Our first result is based on the Krasnoselskii’s fixed point theorem .
Theorem 8. Letbe a bounded closed convex and nonempty subset of a Banach space. Letbe operators such that: (i)whenever(ii)is compact and continuous(iii)is a contraction mapping. Then there existssuch that.
Theorem 9. Assume that(H1) there exists a constantsuch thatfor eachand(H2), for all, with;(H3) the set,is relatively compact for any.
If whereis given by (8), then the boundary value problem (1) has at least one solution on.
Proof. By settingand choosing a constant
we now consider a suitable ball.
In view of Lemma 7, we define the operatorsandon the ballas
For, we have
This implies that. Therefore,. Now, we will prove thatis a contraction mapping. For any, we have By (9), we get thatis a contraction mapping.
Next, we will show thatis compact and continuous. Continuity ofcoupled with the assumptionimplies that the operatoris continuous and uniformly bounded onas
From, there exists a constantsuch that. For any, there exists a neighborhoodof,, such that, for,
Then, we have This means that the set is an equicontinuous set.
In view of Lemma 6 and the condition thatis relatively compact, we have thatis compact. For any, where
Since is convex and compact, we have that. Therefore, for any, the set, for, is relatively compact. From Theorem 5, everycontains a uniformly convergent subsequence, for, onwhich means that the set is relatively compact. Sinceis a bounded, equicontinuous, and relatively compact set, we have thatis compact on. Therefore, all the assumptions of Theorem 8 are satisfied, and the conclusion of Theorem 8 implies that boundary value problem (1) has at least one solution on. This completes the proof.
Corollary 10. Assume that there exists a constantsuch thatfor anyand , and the set , and is relatively compact for every; then there exists at least one solution of boundary value problem (1) on .
Proof. Letting for, we get the result by using Theorem 9.
The second result is based on Leray-Schauder nonlinear alternative .
Theorem 11. Letbe a Banach space,a closed, convex subset of,an open subset of , and. Suppose thatis a continuous, compact (i.e.,is a relatively compact subset of) map. Then either(i)has a fixed point in or (ii)there are(the boundary ofin) andwith.
Theorem 12. Letbe a continuous function. Assume that(H4) there exist a function and a continuous nondecreasing functionsuch that, for all(H5) the setis relatively compact for any, whereis defined as inand,;(H6) there exists a constantsuch that
Then the boundary value problem (1) has at least one solution on.
Proof. We consider the operator equation, wheredefined by (7).
Now, we will show thatmaps bounded sets into bounded sets in. For a positive number, letbe a bounded set in. Then
Next we show thatmaps bounded sets into equicontinuous sets of. Let, whereis a bounded set of. For any, there exists a neighborhoodof,, such that, for,
Then, we have
This means that the setis an equicontinuous set. By, Lemma 6, and Theorem 5, it follows thatis completely continuous.
Let. Then, for and using the computations as in the first step, we get
Consequently, In view of, there existssuch that. Let us set
Note that the operatoris continuous and completely continuous. From the choice of, there is nosuch thatfor some. Consequently, by the nonlinear alternative of Leray-Schauder type (Theorem 11), we deduce thathas a fixed pointwhich is a solution of the problem (1). This completes the proof.
4. Some Examples
In this section, in order to illustrate our result, we consider two examples.
Example 1. Consider the following three-point fractional sum boundary value problem:
Here,,,, and Also we find
Sinceis satisfied with, we can show that
Example 2. Consider the following three-point fractional sum boundary value problem:
Here ,,,,,, and . It is clear that .
The authors would like to thank the reviewers for their valuable comments and suggestions on the paper. This research of T. Sitthiwirattham and J. Tariboon is supported by King Mongkut’s University of Technology North Bangkok, Thailand. Sotiris K. Ntouyas is Member of Nonlinear Analysis and Applied Mathematics (NAAM) Research Group, King Abdulaziz University, Jeddah, Saudi Arabia.
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