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Existence Results for Nonlinear Fractional Difference Equation
Advances in Difference Equations volume 2011, Article number: 713201 (2011)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ1_HTML.gif)
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.
Let . The
th fractional sum
is defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ2_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ3_HTML.gif)
Here is the
th order forward difference operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ4_HTML.gif)
Theorem 2.3 (see [4]).
For ,
noninteger,
,
, it holds
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ5_HTML.gif)
where is defined on
with
.
In particular, when and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ6_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ8_HTML.gif)
On the other hand, Theorem 2.3 yields that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ9_HTML.gif)
Comparing with the above two equations, it is obtained that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ10_HTML.gif)
Let , respectively, we have that
for
, which implies that
is a solution of (1.1).
Lemma 2.5.
One has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ11_HTML.gif)
Proof.
For ,
,
,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ12_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ13_HTML.gif)
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ14_HTML.gif)
Lemma 2.6 (see [2]).
Let and assume
is not a nonpositive integer. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ16_HTML.gif)
Proof.
Define a mapping by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ17_HTML.gif)
for . Now we show that
is contraction. For any
it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ19_HTML.gif)
Proof.
Let be the operator defined by (3.2), we define the set
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ20_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ21_HTML.gif)
Assume that there exist and
such that
. We claim that
. In fact,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ22_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ23_HTML.gif)
We have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ24_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ26_HTML.gif)
then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ27_HTML.gif)
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
,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ28_HTML.gif)
where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ29_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ30_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ32_HTML.gif)
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
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ33_HTML.gif)
For any ,
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ36_HTML.gif)
then is the unique solution of (1.1) on
.
5. Example
Example 5.1.
Consider the fractional difference equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ37_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ38_HTML.gif)
Applying Lemma 2.6, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ39_HTML.gif)
By induction, it follows that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ40_HTML.gif)
Taking the limit , we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ41_HTML.gif)
which is the unique solution of (5.1). In particular, when , the IVP (5.1) becomes the following integer-order IVP
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ42_HTML.gif)
which has the unique solution . At the same time, (5.5) becomes that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2011%2F713201/MediaObjects/13662_2010_Article_67_Equ43_HTML.gif)
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.
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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.
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Chen, F., Luo, X. & Zhou, Y. Existence Results for Nonlinear Fractional Difference Equation. Adv Differ Equ 2011, 713201 (2011). https://doi.org/10.1155/2011/713201
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DOI: https://doi.org/10.1155/2011/713201