Abstract
We investigate the existence of solutions for a -dimensional system of fractional finite difference equations by using the Kranoselskii’s fixed point theorem. We present an example in order to illustrate our results.
1. Introduction
The fractional calculus revealed during the last decade its huge potential applications in many branches of science and engineering (see, e.g., [1–9]). A new and promising direction within fractional calculus is the discrete fractional calculus (see [6, 7, 10–14]). The advantages of this type of calculus are that it treats better phenomena with memory effect (see [10, 11, 14]). We recall that some researchers have been investigating discrete fractional calculus for special equations via very definite boundary conditions (see, e.g., [12, 13, 15–24] and the references therein). Many researchers could focus on this field by considering natural potential of fractional finite difference equations. In this paper, we investigate the existence of solutions for -dimensional system of fractional finite difference equations: where , , and are continuous functions for . One-dimensional version of the problem has been studied by Goodrich [18]. Also, Pan et al. studied two-dimensional version of the problem [24]. We show that the problem (1) is equivalent to a summation equation and by using Krasnoselskii's fixed point theorem we investigate solutions of the problem. In this way, we present an example to illustrate our result.
2. Preliminaries
It is known that the finite fractional difference theory is important in many branches of science and engineering (see, e.g., [13, 16, 18, 19, 21, 25, 26] and the references therein). The Gamma function is defined by for the complex numbers in which the real part of is positive (see [8]). Note that the domain of the Gamma function is (see [8]). Now, we recall for all whenever the right-hand side is defined (see [16]). If is a pole of the Gamma function and is not a pole, then (see [16]). We recall that (see [16]). One can verify that and .
In this paper, we use the standard notations for all and for all real numbers and whenever is a natural number. Let with for some natural number . Then, the th fractional sum of based at is defined by for all , where is the forward jump operator (see [16]). Similarly, we define for all . Note that the domain of is for and for (see [16]). Also, for the natural number , we have to recall the formula We define the trivial sum for all .
Lemma 1 (see [13]). Let be a mapping and a natural number. Then, the general solution of the equation is given by for all , where are arbitrary constants.
Let be a mapping and a natural number. By using a similar proof, one can check that the general solution of the equation is given by for all . In particular, the general solution has the following representation: for all . By considering the details, note that whenever . Also for with and , the domain of the operator is given by , , , , , and (for more details see [13, 21, 22]). One can find next result about composing a difference with a sum in [12].
Lemma 2. Let be a map, , and with . Then for all and for all .
By using Lemma 1 and last lemma for , we get We are going to use this in our main results. A nonempty, closed subset of a topological vector space is called a cone whenever and for all and nonnegative real numbers (for more details and examples see [27] and references therein).
Lemma 3 (see [28]). Let be a Banach space and a cone in . Assume that and are open subsets of such that and . Suppose that is a completely continuous operator. If either for all and for all or for all and for all , then has at least one fixed point in .
3. Main Result
In this section we provide the main results. For next result, consider the problem (1).
Lemma 4. The fractional finite difference equation via the boundary conditions has a solution if and only if is a solution of the summation equation , where the Green function is given by for all . Here, and is one of equations of the system.
Proof. Let and letbe a solution of the fractional finite difference equation , . By using Lemma 1, we get . By using the boundary condition , we obtain and so . Now by using the boundary condition , we get Hence, and so Now, let be a solution of the fractional sum equation Then, . Since , we get . Also, Hence, we get Moreover, . Since and , we get This completes the proof.
Hereafter, for simplicity we use the notations and for all .
Lemma 5 (see [18]). The Green function (7) satisfies for all and and for all and there exist such that for all .
Goodrich showed that (see [18]), where and Note that can be written in the simple form , becauseNote that hold because . Suppose that is the Banach space of the maps via the usual maximum norm . Consider the space via the norm . It is clear that is a Banach space (see [29]). Now, define the map by where for . Also, consider the cone defined by where . First, for the operator we show that whenever the functions are nonnegative for . Let . Then, we have where . Hence, and so . For providing our main result, we use similar conditions which have been given by Goodrich in [18] and Henderson et al. in [30].
Theorem 6. Suppose that for all : such that and for some where is the Green function (7) and . Then the -dimensional system of fractional finite difference equations (1) has at least one solution.
Proof. Consider the operator defined by (17) and the cone . It is clear that is completely continuous because it is a summation operator on a finite set. Choose such that for all . Put . Then, and for all . Also, we have for all . Hence, for all . Now, choose such that and for all . Also, choose such that . Now, put . Then, and for all . Thus, by using (25) we get for all . Hence, for all . By using Lemma 3, has at least one fixed point in and so by using Lemma 4, the -dimensional system of fractional finite difference equations (1) has at least one solution.
4. Example
Here, we provide an example to illustrate our last result.
Example 1. Consider the 5-dimensional fractional finite difference equation system: We show that the problem has at least one solution, where Let , , , , , , and . Thus, the system (29) is a special case of the system (1). It is easy to check that for . Put , and for . Then, by a calculation we get , , , , and . Thus, . On the other hand by calculation of some limits, one can get that , , , , , , , , , and . Moreover, we have Similarly, we obtain Now, let . Then, and we have Thus by using Theorem 6, the 5-dimensional system of fractional finite difference equations (29) has at least one solution.
5. Conclusions
In this paper, based on main idea of Goodrich we review the existence of solutions for a -dimensional system of fractional finite difference equations. In fact we are going to extend the work of Goodrich in a sense. We give an example to illustrate our last result.
Conflict of Interests
The authors declare that there is no conflict of interests regarding the publication of this paper.
Acknowledgments
Research of the second and third authors was supported by Azarbaijan Shahid Madani University. Also, the authors express their gratitude to the referees for their helpful suggestions which improved the final version of this paper.