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.
MSC: 34A08, 26A33.
Keywords:fractional 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.
Goodrich in  considered a discrete fractional boundary value problem of the form
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.
Pan et al. in  examined the fractional boundary value problem
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.
In this paper we consider the nonlinear discrete fractional boundary value problem of the form
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 .
Definition 2.2 The αth fractional sum of a function f, for , is defined by
for and . We also define the αth fractional difference for by , where and is chosen so that .
Lemma 2.1Lettandαbe any numbers for which and are defined. Then .
Lemma 2.2Let . 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.
and be given. Then the problem
has a unique solution
Proof From Lemma 2.2, we find that a general solution for (2.2) can be written as
for . Applying the first boundary condition of (2.2) and using , we have . So,
Using the fractional sum of order for (2.6), we obtain
The second condition of (2.2) implies
Solving the above equation for a constant , we get
where Λ is defined by (2.4). From the fact that for , we have
Substituting a constant into (2.6), we obtain (2.3). □
3 Main results
In this section, we wish to establish the existence results for problem (1.3). To accomplish this, we define , the Banach space of all function x with the norm defined by and also define an operator by
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.
Theorem 3.1Suppose that there exists a constant such that for each and . If
then problem (1.3) has a unique solution in .
Proof Firstly, we transform problem (1.3) into a fixed point problem, , where the operator is defined by (3.1). Then, for any , we have
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.
Definition 3.1 Let E be a Banach space and let be a mapping. F is said to be a nonlinear contraction if there exists a continuous nondecreasing function such that and for all with the property
Lemma 3.1 (Boyd and Wong )
LetEbe a Banach space and let be a nonlinear contraction. ThenFhas a unique fixed point inE.
Theorem 3.2Suppose that there exists a continuous function such that for all and , where
and Λ is defined in (2.4).
Then boundary value problem (1.3) has a unique solution.
Proof Let the operator be defined by (3.1). We define a continuous nondecreasing function by
such that and for all .
Let . Then we get
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 3.3Suppose that there exists a constant such that for each and 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.
Step I.Continuity ofF. Let be a sequence such that in . Then, for each , we get
Since f is a continuous function, we have as . This means that F is continuous.
Step II.Fmaps bounded sets into bounded sets in . Let us prove that for any , there exists a positive constant L such that for each , we have . Indeed, for any , we obtain
Hence, we deduce that
where Ω is defined by (3.3).
Step III. is equicontinuous with defined as in Step II. For any , there exist , such that
Then we have
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.
Step IV.A priori bounds. We show that the set
Let . Then for some . Thus, for each , we have
Therefore, for , we get
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.
Example 4.1 Consider the following three-point fractional sum boundary value problem:
Here , , , , . Also, we find
From , we have . We can show that
Hence, by Theorem 3.1, boundary value problem (4.1)-(4.3) has a unique solution.
Example 4.2 Consider the following three-point fractional sum boundary value problem:
Here , , , , , , . It is clear that for . Thus, we conclude from Theorem 3.3 that (4.4)-(4.6) has at least one solution.
The authors declare that they have no competing interests.
All authors contributed equally in this article. They read and approved the final manuscript.
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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