Research Article | Open Access
On Nonlinear Fractional Sum-Difference Equations via Fractional Sum Boundary Conditions Involving Different Orders
We study existence and uniqueness results for Caputo fractional sum-difference equations with fractional sum boundary value conditions, by using the Banach contraction principle and Schaefer’s fixed point theorem. Our problem contains different numbers of order in fractional difference and fractional sums. Finally, we present some examples to show the importance of these results.
In this paper we consider a Caputo fractional sum-difference equation with nonlocal fractional sum boundary value conditions of the formwhere , , , and is the Caputo fractional difference operator of order . For , and are given functions and is a given functional, and for ,
Mathematicians have employed this fractional calculus in recent years to model and solve various applied problems. In particular, fractional calculus is a powerful tool for the processes which appear in nature, for example, biology, ecology, and other areas, and can be found in [1, 2] and the references therein. The continuous fractional calculus has received increasing attention within the last ten years or so, and the theory of fractional differential equations has been a new important mathematical branch due to its extensive applications in various fields of science, such as physics, mechanics, chemistry, and engineering. Although the discrete fractional calculus has seen slower progress, within the recent several years, a lot of papers have appeared, which has helped to build up some of the basic theory of this area; see [3–17] and references cited therein.
At present, there is a development of boundary value problems for fractional difference equations which shows an operation of the investigative function. The study may also have another function which is related to the one we are interested in. These creations are incorporating with nonlocal conditions which are both extensive and more complex, for instance.
Agarwal et al.  investigated the existence of solutions for two fractional boundary value problems:where and is a given function, and where , , and is a given function.
Kang et al.  obtained sufficient conditions for the existence of positive solutions for a nonlocal boundary value problemwhere , , are given functions and are given functionals.
Sitthiwirattham  examined a Caputo fractional sum boundary value problem with a -Laplacian of the formwhere , , , , , is a constant, is a continuous function, and is the -Laplacian operator.
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) 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) by the help of the Banach fixed point theorem and Schaefer’s fixed point theorem. Some illustrative examples are presented in Section 4.
In the following, there are notations, definitions, and lemmas which are used in the main results.
Definition 1. One defines the generalized falling function by , for any and for which the right-hand side is defined. If is a pole of Gamma function and is not a pole, then .
Lemma 2 (see ). Assume that the following factorial functions are well defined: (i), where .(ii)If , then for any .(iii).
Definition 3. For and defined on , the -order fractional sum of is defined by where and .
Definition 4. For and defined on , the -order Caputo fractional difference of is defined by where and is chosen so that . If , then .
Lemma 5 (see ). Assume that and . Then for some , .
The following lemma deals with linear variant of boundary value problem (1) and gives a representation of the solution.
Lemma 6. Let , , , and be given. Then the problemhas the unique solutionwhere
Proof. Using Lemma 5, a general solution for (10) can be written in the formfor . Applying the first boundary condition of (10) implies So,The second condition of (10) implies A constant can be obtained by solving the above equation, so Substituting a constant into (15), we getLet . Thenwe simplify (19) becomes (12).
Substituting into (18), we obtain (11).
3. Main Results
Now we are in a position to establish the main results. First, we transform boundary value problem (1) into a fixed point problem.
For , let be a Banach space and let denote the Banach space of all continuous functions from endowed with a topology of uniform convergence with the norm denoted by . For this purpose, we consider the operator bywhere
It is easy to see that problem (1) has solutions if and only if operator has fixed points.
Theorem 7. Assume that is continuous and maps bounded subsets of into relatively compact subsets of , is continuous with , and is a given functional. In addition, suppose the following: There exist constants such that for each and There exists a constant such that for each For each Consider , whereThen problem (1) has a unique solution on
Proof. We will show that is a contraction. For any and for each , we have Consequently, is a contraction. Therefore, by the Banach fixed point theorem, we get that has a fixed point which is a unique solution of problem (1) on .
The following result is based on Schaefer’s fixed point theorem.
Theorem 8 (Arzelá-Ascoli Theorem (see )). A set of function in with the sup norm is relatively compact if and only if it is uniformly bounded and equicontinuous on .
Theorem 9 (see ). If a set is closed and relatively compact then it is compact.
Theorem 10 (Schaefer’s fixed point theorem (see )). Assume that X is a Banach space and that is continuous compact mapping. Moreover assume that the set is bounded. Then has a fixed point.
Theorem 11. Assume that is continuous and maps bounded subsets of into relatively compact subsets of and is a given functional. In addition, suppose that holds, and suppose the following: There exists a constant such that for each and There exists a constant such that for each Then problem (1) has at least one solution on .
Proof. We will use Schaefer’s fixed point theorem to prove this result. Let be the operator defined in (20). It is clear that is completely continuous. So, it remains to show that the set is bounded.
Let ; then for some . Thus, for each , we have which implies that, for each , we have where and are defined on (25). This shows that set is bounded. As a consequence of Schaefer’s fixed point theorem, we conclude that has a fixed point which is a solution of problem (1).
4. Some Examples
In this section, in order to illustrate our results, we consider some examples.
Example 1. Consider the following fractional sum boundary value problem:Here , , , , , , , and Let ; we haveso holds with , , and we have , andso holds with .
Since , we have then is satisfied.
Also, we have We can show that Hence, by Theorem 7, boundary value problem (33) has a unique solution.
Example 2. Consider the following fractional sum boundary value problem:Here , , , , and , , and Clearly for , we have Hence, conditions , , and of Theorem 11 are satisfied, and consequently boundary value problem (40) has at least one solution.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
The authors would like to thank the editor and the referees for their useful comments. This research was funded by King Mongkut’s University of Technology North Bangkok (Contract no. KMUTNB-GOV-58-50).
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