- Research Article
- Open Access

# Existence Results for Nonlinear Fractional Difference Equation

- Fulai Chen
^{1}, - Xiannan Luo
^{2}and - Yong Zhou
^{2}Email author

**2011**:713201

https://doi.org/10.1155/2011/713201

© Fulai Chen et al. 2011

**Received:**27 September 2010**Accepted:**12 December 2010**Published:**20 December 2010

## Abstract

This paper is concerned with the initial value problem to a nonlinear fractional difference equation with the Caputo like difference operator. By means of some fixed point theorems, global and local existence results of solutions are obtained. An example is also provided to illustrate our main result.

## Keywords

- Difference Operator
- Fractional Differential Equation
- Local Existence
- Fractional Difference
- Finite Difference Equation

## 1. Introduction

where is a Caputo like discrete fractional difference, is continuous in and . is a real Banach space with the norm , .

Fractional differential equation has 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 [1]. However, there are few literature to develop the theory of the analogues fractional finite difference equation [2–6]. Atici and Eloe [2] developed the commutativity properties of the fractional sum and the fractional difference operators, and discussed the uniqueness of a solution for a nonlinear fractional difference equation with the Riemann-Liouville like discrete fractional difference operator. To the best of our knowledge, this is a pioneering work on discussing initial value problems (IVP for short) in discrete fractional calculus. Anastassiou [4] defined a Caputo like discrete fractional difference and compared it to the Riemann-Liouville fractional discrete analog.

For convenience of numerical calculations, the fractional differential equation is generally discretized to corresponding difference one which makes that the research about fractional difference equations becomes important. Following the definition of Caputo like difference operator defined in [4], here we investigate the existence and uniqueness of solutions for the IVP (1.1). A merit of this IVP with Caputo like difference operator is that its initial condition is the same form as one of the integer-order difference equation.

## 2. Preliminaries and Lemmas

We start with some necessary definitions from discrete fractional calculus theory and preliminary results so that this paper is self-contained.

Here is defined for mod (1) and is defined for mod (1); in particular, maps functions defined on to functions defined on , where . In addition, . Atici and Eloe [2] pointed out that this definition of the th fractional sum is the development of the theory of the fractional calculus on time scales [7].

Definition 2.2 (see [4]).

Theorem 2.3 (see [4]).

where is defined on with .

Lemma 2.4.

Proof.

Suppose that for is a solution of (1.1), that is for , then we can obtain (2.6) according to Theorem 2.3.

Let , respectively, we have that for , which implies that is a solution of (1.1).

Lemma 2.5.

Proof.

Lemma 2.6 (see [2]).

In particular, , where is a constant.

The following fixed point theorems will be used in the text.

Theorem 2.7 (Leray-Schauder alternative theorem [8]).

Let be a Banach space with closed and convex. Assume is a relatively open subset of with and is a continuous, compact map. Then either

(1) has a fixed point in ; or

(2)there exist and with .

Theorem 2.8 (Schauder fixed point theorem [9]).

If is a closed, bounded convex subset of a Banach space and is completely continuous, then has a fixed point in .

Theorem 2.9 (Ascoli-Arzela theorem [10]).

Let be a Banach space, and is a function family of continuous mappings . If is uniformly bounded and equicontinuous, and for any , the set is relatively compact, then there exists a uniformly convergent function sequence in .

Lemma 2.10 (Mazur Lemma [11]).

If is a compact subset of Banach space , then its convex closure is compact.

## 3. Local Existence and Uniqueness

Set , where .

Theorem 3.1.

Proof.

By applying Banach contraction principle, has a fixed point which is a unique solution of the IVP (1.1).

Theorem 3.2.

Proof.

which implies that .

which means that the set is an equicontinuous set.

Since is convex and compact, we know that . Hence, for any , the set ( ) is relatively compact. From Theorem 2.9, every contains a uniformly convergent subsequence ( ) on which means that the set is relatively compact. Since is a bounded, equicontinuous and relatively compact set, we have that is completely continuous.

Therefore, the Leray-Schauder fixed point theorem guarantees that has a fixed point, which means that there exists at least one solution of the IVP (1.1) on .

Corollary 3.3.

Assume that there exist such that for any and , and the set is relatively compact for every , then there exists at least one solution of the IVP (1.1) on .

Proof.

Let , , we directly obtain the result by applying Theorem 3.2.

Corollary 3.4.

Assume that the function satisfies , and the set is relatively compact for every , then there exists at least one solution of the IVP (1.1) on .

Proof.

According to , for any , there exists such that for any . Let , then Corollary 3.4 holds by Corollary 3.3.

Corollary 3.5.

and the set is relatively compact for every , then there exists at least one solution of the IVP (1.1) on .

Proof.

By inequity (3.16), there exist positive constants , such that , for all . Let . Then we have , for all . Let , then Corollary 3.5 holds by Corollary 3.3.

## 4. Global Uniqueness

Theorem 4.1.

Assume is globally Lipschitz continuous (with constant ) on , then the IVP (1.1) has a unique solution provided that .

Proof.

Since , by applying Banach contraction principle, has a fixed point which is a unique solution of the IVP (1.1) on .

Since , by applying Banach contraction principle, has a fixed point which is a unique solution of the IVP (1.1) on .

for . Similar to the deduction of (4.3), we may obtain that the IVP (1.1) has a unique solution on , then exists.

then is the unique solution of (1.1) on .

## 5. Example

Example 5.1.

According to Theorem 4.1, the IVP (5.1) has a unique solution provided that . In fact, we can employ the method of successive approximations to obtain the solution of (5.1).

Equation (5.7) implies that, when , the result of the IVP (5.5) is the same as one of the corresponding integer-order IVP (5.6).

Remark 5.2.

Example 5.1 is similar to Example 3.1 in [2] in which the difference operator is in the Riemann-Liouville like discrete sense. Compared with the solution of Example 3.1 in [2] defined on , where , the solution of Example 5.1 in this paper is defined on . This difference makes that fractional difference equation with the Caputo like difference operator is more similar to classical integer-order difference equation.

## Declarations

### Acknowledgments

This work was supported by the Natural Science Foundation of China (10971173), the Scientific Research Foundation of Hunan Provincial Education Department (09B096), the Aid Program for Science and Technology Innovative Research Team in Higher Educational Institutions of Hunan Province, and the Construct Program of the Key Discipline in Hunan Province.

## Authors’ Affiliations

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