Boundary value problems for fractional difference equations with three-point fractional sum boundary conditions
© Sitthiwirattham et al.; licensee Springer. 2013
Received: 8 May 2013
Accepted: 17 September 2013
Published: 7 November 2013
In this paper, we consider a discrete fractional boundary value problem of the form
where , , and is a continuous function. Existence and uniqueness of the solutions are proved by using the contraction mapping theorem, the nonlinear contraction theorem and Schaefer’s fixed point theorem. Some illustrative examples are also presented.
Keywordsfractional difference equations boundary value problem existence uniqueness fixed point theorems
Fractional calculus is an emerging field recently drawing attention from both theoretical and applied disciplines. Fractional order differential equations play a vital role in describing many phenomena related to physics, chemistry, mechanics, control systems, flow in porous media, electrical networks, mathematical biology and viscoelasticity. For a reader interested in the systematic development of the topic, we refer to the books [1–3]. A variety of results on initial and boundary value problems of fractional differential equations and inclusions can easily be found in the literature on the topic. For some recent results, we can refer to [4–12] and references cited therein.
Discrete fractional calculus and fractional difference equations represent a very new area for researchers. Some real-world phenomena are being studied with the help of discrete fractional operators. A good account of papers dealing with discrete fractional boundary value problems can be found in [13–26] and references cited therein.
where , is a continuous function and is a given function. Existence and uniqueness of solutions are obtained by the contraction mapping theorem, the Brouwer theorem and the Guo-Krasnoselskii theorem.
where , so that , , so that , is continuous and nonnegative for , and is a given function. Existence and uniqueness of solutions are obtained by the contraction mapping theorem and the Brouwer theorem.
where , , and is a continuous function.
The plan of this paper is as follows. In Section 2 we recall some definitions and basic lemmas. Also, we derive a representation for the solution to (1.3) by converting the problem to an equivalent summation equation. In Section 3, using this representation, we prove existence and uniqueness of the solutions of boundary value problem (1.3) by the help of the contraction mapping theorem, the nonlinear contraction theorem and Schaefer’s fixed point theorem. Some illustrative examples are presented in Section 4.
In this section, we introduce notations, definitions and lemmas which are used in the main results.
Definition 2.1 We define the generalized falling function by , for any t and α, for which the right-hand side is defined. If is a pole of the gamma function and is not a pole, then .
for and . We also define the α th fractional difference for by , where and is chosen so that .
Lemma 2.1 Let t and α be any numbers for whichandare defined. Then.
for some, with.
To define the solution of boundary value problem (1.3), we need the following lemma which deals with linear variant of boundary value problem (1.3) and gives a representation of the solution.
Substituting a constant into (2.6), we obtain (2.3). □
3 Main results
for , where is defined by (2.4). It is easy to see that problem (1.3) has solutions if and only if the operator F has fixed points.
then problem (1.3) has a unique solution in.
Therefore, F is a contraction. Hence, by the Banach fixed point theorem, we get that F has a fixed point which is a unique solution of problem (1.3) on . □
Next, we can still deduce the existence of a solution to (1.3). We shall use nonlinear contraction to accomplish this.
Lemma 3.1 (Boyd and Wong )
Let E be a Banach space and letbe a nonlinear contraction. Then F has a unique fixed point in E.
and Λ is defined in (2.4).
Then boundary value problem (1.3) has a unique solution.
such that and for all .
for . From (3.4), it follows that . Hence F is a nonlinear contraction. Therefore, by Lemma 3.1, the operator F has a unique fixed point in , which is a unique solution of problem (1.3). □
The following result is based on Schaefer’s fixed point theorem.
Theorem 3.3 Suppose that there exists a constantsuch thatfor eachand all.
Then problem (1.3) has at least one solution on.
Proof We shall use Schaefer’s fixed point theorem to prove that the operator F defined by (3.1) has a fixed point. We divide the proof into four steps.
Since f is a continuous function, we have as . This means that F is continuous.
where Ω is defined by (3.3).
This means that the set is an equicontinuous set. As a consequence of Steps I to III together with the Arzelá-Ascoli theorem, we get that is completely continuous.
This shows that the set E is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that F has a fixed point which is a solution of problem (1.3). □
4 Some examples
In this section, in order to illustrate our result, we consider some examples.
Hence, by Theorem 3.1, boundary value problem (4.1)-(4.3) has a unique solution.
Here , , , , , , . It is clear that for . Thus, we conclude from Theorem 3.3 that (4.4)-(4.6) has at least one solution.
The third author is a member of Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group at King Abdulaziz University, Jeddah, Saudi Arabia.
The present paper was done while J Tariboon and T Sitthiwirattham visited the Department of Mathematics of the University of Ioannina, Greece. It is a pleasure for them to thank Professor SK Ntouyas for his warm hospitality. This research of J Tariboon and T Sitthiwirattham is supported by King Mongkut’s University of Technology North Bangkok, Thailand.
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