Abstract
In this manuscript we investigate the existence of the fractional finite difference equation (FFDE) via the boundary condition and the sum boundary condition for order , where , , and . Along the same lines, we discuss the existence of the solutions for the following FFDE: via the boundary conditions and and the sum boundary condition for order , where , , , and with .
MSC:34A08.
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1 Introduction
By the late 19th century, combined efforts made by several mathematicians led to a fairly solid understanding of fractional calculus in the continuous setting but significantly less is still known about discrete fractional calculus (see for example [1, 2] and [3] and the references therein). Recently, there has been a strong interest in this subject but still little progress was made in developing the theory of fractional finite difference equations (see [4–11] and [12] and the references therein).
Discrete fractional calculus is a powerful tool for the processes which appears in nature, e.g. biology, ecology and other areas (see for example [13] and [14] and the references therein), where the discrete models have to be considered in order to describe properly the complexity of the dynamical processes with memory effect. We notice that the existence of solutions for fractional finite difference equations is a hot topic of the fractional calculus with direct implications in modeling of some real world phenomena which have only discrete behaviors.
Motivated by the above mentioned results, in this paper we investigate the fractional finite difference equation
via the boundary conditions and , where , , , , and .
Moreover, we investigate the FFDE given by
via the boundary conditions , , and , where , , , and with .
In the following we present the basic definitions and theorems used in this manuscript. In Section 3 we present the main result. The manuscript ends with our conclusions.
2 Preliminaries
As you know, the gamma function is defined by
which converges in the right half of the complex plane . It is well known that and for all . Now, we define
for all [15]. If is a pole of the gamma function and is not a pole, then we define [16]. For example, we have . Also, one can verify that and .
In this paper, we use the notations for all and for all real numbers p and q whenever is a natural number.
Let with for some natural number m. The μ th fractional sum of f based at a is defined as [3]
for all , where is the forward jump operator. Similarly, we define
for all [17]. Note that the domain of is for and for . Also, for the natural number , we have the known formula [16]
We define for all , too.
Lemma 2.1 [16]
Let be a mapping and m a natural number. Then the general solution of the equation is given by
for all , where are arbitrary constants.
Let be a mapping and m 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 . The next theorem plays an important role in our main results.
Theorem 2.2 [18]
Every continuous function from a compact, convex, nonempty subset of a Banach space to itself has a fixed point.
3 Main results
In the following, we are ready to provide the main results. First, we investigate the FFDE
via the boundary conditions and , where , , , , and .
Lemma 3.1 Let , , , , and . Then is a solution of the problem
via the boundary conditions and if and only if is a solution of the fractional sum equation
where
whenever or ,
whenever ,
whenever and
whenever or .
Proof Let be a solution of the problem
via the boundary conditions and . By using Lemma 2.1, we get
Since , we have
Since and
. On the other hand, we have . Thus,
Hence,
and so
To calculate , taking the summation on both sides of the above relation gives us
Hence,
and so by interchanging the order of summations, we have
Since
by replacing (3.3) in (3.1), we get
Now, let be a solution of the fractional sum equation
Then is a solution of the equation
It is easy to check that . Also, we have
Moreover, we have . This completes the proof. □
Some authors tried to find the maximum or exact value of in some papers (see for example [16, 19] and [17]). Now, we show that is bounded, where is the Green function of the last result.
Lemma 3.2 For each and , we have
for some positive number .
Proof Since for all , we have
for all and . Thus, is a (finite) real number, for all , and . Consequently, both sums in the statement are finite, because is finite. □
Theorem 3.3 Let be bounded and continuous in its second and third variables. Then the fractional finite difference equation via the boundary conditions and has a solution with , for all admissible t.
Proof Since g is bounded, there exists a constant C such that for all and . Let be the Banach space of real valued functions defined on via the norm
and . One can check easily that is a compact, convex, and nonempty subset of . Now, define the map T on by
for all . First, we show that . Let and . Then
Since was arbitrary, and so . Now, we show that T is continuous. Let be given. Since g is continuous in its second and third variables, it is uniformly continuous in its second and third variables on and so there exists such that for all and with and . Thus, we get
for all . Hence, and so T is continuous on . By using Theorem 2.2, T has a fixed point and so, by using Lemma 3.1, the fractional finite difference equation
via the boundary conditions and has a solution in . □
Now, we consider the fractional finite difference equation via the boundary conditions , and , where and with .
Lemma 3.4 Let , , , , and with . Then is a solution of the problem via the boundary conditions , , and if and only if is a solution of the fractional sum equation
where
whenever , ,
whenever , ,
whenever , , ,
whenever , and
whenever , .
Proof Let be a solution of the problem via the boundary conditions , , and . By using Lemma 2.1, we get
Similar to the proof of Lemma 3.1, by using the boundary value conditions we obtain ,
and
Thus,
To calculate , by taking the summation on both sides of the above relation gives us
and so
Since and
by replacing (3.5) in (3.4), we get
Now, let be a solution of the fractional sum equation
Similar to proof of the Lemma 3.1, we conclude that is a solution to the problem
via the boundary conditions , , and . This completes the proof. □
By using similar proofs of Lemma 3.2 and Theorem 3.3, we obtain the next results.
Lemma 3.5 For each and , we have
for some positive number .
Theorem 3.6 Assume that is continuous and bounded in its second variable. Then the fractional finite difference equation via the boundary conditions , , and has a solution with , for all admissible t.
4 An example
Now, we provide an example for the first investigated problem.
Example 4.1 Consider the equation
via the boundary value conditions and . We show that this equation has a solution with for all admissible t. Let , , , and
in the first problem. Thus, we should investigate the fractional finite difference equation
via the boundary value conditions and . Note that the map
is continuous and bounded in its second and third variables. Now, we show that . Also, the Green function is given by
whenever , ,
whenever ,
whenever , , and
for . Thus, for each , one of the values of the Green function G satisfies
Similar calculations give us the values of G summarized in Table 1.
Thus, . Hence by using Theorem 3.6, (4.1) has a solution with for all admissible t.
5 Conclusions
In this manuscript based on a fixed point theorem we provided the existence results for two fractional finite difference equations in the presence of the sum boundary conditions. One example illustrates our results.
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Acknowledgements
Research of the third and fourth authors was supported by Azarbaijan Shahid Madani University.
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Agarwal, R.P., Baleanu, D., Rezapour, S. et al. The existence of solutions for some fractional finite difference equations via sum boundary conditions. Adv Differ Equ 2014, 282 (2014). https://doi.org/10.1186/1687-1847-2014-282
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DOI: https://doi.org/10.1186/1687-1847-2014-282