Open Access

Existence Results for Nonlinear Fractional Difference Equation

Advances in Difference Equations20102011:713201

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

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.

1. Introduction

This paper deals with the existence of solutions for nonlinear fractional difference equations
(1.1)

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 [26]. 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.

Definition 2.1 (see [2, 3]).

Let . The th fractional sum is defined by
(2.1)

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]).

Let and , where denotes a positive integer, , ceiling of number. Set . The th fractional Caputo like difference is defined as
(2.2)
Here is the th order forward difference operator
(2.3)

Theorem 2.3 (see [4]).

For , noninteger, , , it holds
(2.4)

where is defined on with .

In particular, when and , we have
(2.5)

Lemma 2.4.

A solution is a solution of the IVP (1.1) if and only if is a solution of the the following fractional Taylor's difference formula:
(2.6)

Proof.

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

Conversely, we assume that is a solution of (2.6), then
(2.7)
On the other hand, Theorem 2.3 yields that
(2.8)
Comparing with the above two equations, it is obtained that
(2.9)

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

Lemma 2.5.

One has
(2.10)

Proof.

For , , , , we have
(2.11)
that is,
(2.12)
Then
(2.13)

Lemma 2.6 (see [2]).

Let and assume is not a nonpositive integer. Then
(2.14)

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.

Assume is locally Lipschitz continuous (with constant ) on , then the IVP (1.1) has a unique solution on provided that
(3.1)

Proof.

Define a mapping by
(3.2)
for . Now we show that is contraction. For any it follows that
(3.3)

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

Theorem 3.2.

Assume that there exist such that for , and the set is relatively compact for every , then there exists at least one solution of the IVP (1.1) on provided that
(3.4)

Proof.

Let be the operator defined by (3.2), we define the set as follows:
(3.5)
where
(3.6)
Assume that there exist and such that . We claim that . In fact,
(3.7)
then
(3.8)
We have
(3.9)
that is,
(3.10)

which implies that .

The operator is continuous because that is continuous. In the following, we prove that the operator is also completely continuous in . For any , there exist such that
(3.11)
then we have
(3.12)

which means that the set is an equicontinuous set.

In view of Lemma 2.10 and the condition that is relatively compact, we know that is compact. For any ,
(3.13)
where
(3.14)

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.

Assume the function is nondecreasing continuous and there exist such that
(3.15)
(3.16)

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.

For , let be the operator defined by (3.2). For any it follows that
(4.1)

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

Since exists, for , we may define the following mapping :
(4.2)
For any , , we have
(4.3)

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

In general, since exists, we may define the operator as follows
(4.4)

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

Define as follows
(4.5)

then is the unique solution of (1.1) on .

5. Example

Example 5.1.

Consider the fractional difference equation
(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).

Set
(5.2)
Applying Lemma 2.6, we have
(5.3)
By induction, it follows that
(5.4)
Taking the limit , we obtain
(5.5)
which is the unique solution of (5.1). In particular, when , the IVP (5.1) becomes the following integer-order IVP
(5.6)
which has the unique solution . At the same time, (5.5) becomes that
(5.7)

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

(1)
Department of Mathematics, Xiangnan University
(2)
School of Mathematics and Computational Science, Xiangtan University

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Copyright

© Fulai Chen et al. 2011

This article is published under license to BioMed Central Ltd. 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.