Research Article | Open Access
Existence and Uniqueness of Solutions for a Discrete Fractional Mixed Type Sum-Difference Equation Boundary Value Problem
By means of Schauder’s fixed point theorem and contraction mapping principle, we establish the existence and uniqueness of solutions to a boundary value problem for a discrete fractional mixed type sum-difference equation with the nonlinear term dependent on a fractional difference of lower order. Moreover, a suitable choice of a Banach space allows the solutions to be unbounded and two representative examples are presented to illustrate the effectiveness of the main results.
For , such that is a nonnegative integer, we denote and throughout this paper. It is also worth noting that, in what follows, for any function defined on , we appeal to the convention , when with .
In this paper, we will consider the following discrete fractional mixed type sum-difference equation boundary value problem: where , and denote the discrete Riemann-Liouville fractional differences of order and , respectively, , , and where , , and .
Continuous fractional calculus is a generalization of ordinary differentiation and integration on an arbitrary order that can be noninteger. This subject, as old as the problem of ordinary differential calculus, can go back to the times when Leibniz and Newton invented differential calculus. The theory of fractional differential equations has received a lot of attention and now constitutes a new important mathematical branch due to its extensive applications in various fields of science and engineer. For more details, see [1–13] and references therein.
It is well known that discrete analogues of differential equations can be very useful in applications , in particular for using computer to simulate the behavior of solutions for certain dynamic equations. However, compared to the long and rich history of continuous fractional calculus, discrete fractional calculus attracted mathematicians and scientists into its fairly new research area in a short period of time. In this time period, the theory of discrete calculus has been developed in many directions parallel to the theory in continuous fractional calculus such as initial value problems and boundary value problems for fractional difference equations, discrete Mittag-Leffler functions, and inequalities with discrete fractional operators; see [15–39] and the references therein. At the same time, in , Atıcı and Şengül have shown the usefulness of discrete Gompertz fractional difference equation for tumor growth model, which implies that discrete fractional difference calculus will provide a new excellent tool to model real world phenomena in the future.
Although, among all recently research topics, the branch of discrete finite fractional difference boundary value problems is currently undergoing active investigation [16, 31–38], significantly less is known about discrete infinite fractional difference boundary value problems with the nonlinear term dependent on a fractional difference operator. Here, we should point out that in , Lv and Feng, by simple analogy with the ordinary case, introduced some basic definitions of discrete fractional calculus for Banach-valued functions and initially studied a class of discrete infinite fractional mixed type sum-difference equation boundary value problems in abstract spaces by using contracting mapping principle. Furthermore, as far as we know, the theory of discrete fractional mixed type sum-difference equations boundary value problems is still a new research area. So in this paper, we continue to focus on this topic for real-valued functions and provide some sufficient conditions for the existence and uniqueness of solutions to problem (1). Particularly note that problem (1) is not like the problem in  and the biggest difference is the nonlinear term in (1) explicitly dependent on the discrete fractional difference operator of lower order. Hence, these differences that cause the main difficulties that we have to deal with in this paper are those of constructing a special Banach space and establishing an appropriate compactness criterion in it.
The outline for the remainder of this paper is as follows. In Section 2, we recall some useful preliminaries for discrete fractional calculus and present the basic space and its compactness criterion for studying problem (1). In Section 3, by employing Schauder’s fixed point theorem and contraction mapping principle, we establish the existence and uniqueness results of problem (1). In Section 4, two concrete examples are provided to illustrate the possible applications of the obtained analytical results.
In this section, we firstly present here some necessary definitions and basic results about discrete fractional calculus.
Definition 1 (see ). For any and , the falling factorial function is defined as provided that the right-hand side is well defined. We appeal to the convention that if is a pole of the Gamma function and is not a pole, then .
Definition 2 (see ). The th discrete fractional sum of a function , for , is defined by Also, we define the trivial sum .
Definition 3 (see ). The th discrete Riemann-Liouville fractional difference of a function , for , is defined by where is the smallest integer greater than or equal to and is the th order forward difference operator. If , then .
Remark 4. From Definitions 2 and 3, it is easy to see that maps functions defined on to functions defined on and maps functions defined on to functions defined on , where is the smallest integer greater than or equal to . For ease of notation, we throughout this paper omit the subscript in and when it is not to lead to domains confusion and general ambiguity.
Lemma 5 (see ). Let and . Then
Lemma 6 (see ). Let and with . Then for , , .
Lemma 7 (see ). Let and be given. Then, for , ,
Lemma 8 (see ). Let , and . Then
Next, we define the space, equiped with the norm
Furthermore, using the linear functional analysis theory, we can easily verify that is a Banach space, and then we present the following compactness criterion in it.
Lemma 9. Let be a bounded set. If for any given , there exists a positive integer such that whenever , , and ; then is relatively compact in .
Proof. Evidently, it is sufficient to prove that is totally bounded. In what follows we divide this proof into two steps.
Step 1. Let us consider the case .
Denote by the restriction of on . Then , equipped with the norm , is a finite dimension Banach space. So we know that is relatively compact from the boundness of ; hence is totally bounded; namely, for any , there exist finitely many ball , , , such that where .
Similarly, denote . Then is also a Banach space with the norm and it can be covered by finitely many balls ; that is, where .
Step 2. Define
Let us consider the case . It is obvious that . Now, let us take ; then can be covered by the balls , where In fact, for any , the argument in Step 1 implies that there exist and such that , . Hence, for and , we have For arbitrary , (12) and (17) yield that and for any , (13) and (18) ensure that Relations (17)–(20) show that . Therefore, is totally bounded and this lemma is proved.
3. Main Result
In this section, we will establish the existence and uniqueness of solutions for problem (1) by using Schauder’s fixed point theorem and contraction mapping principle. For the sake of convenience and to abbreviate our presentation, for any function , we denote in the sequel discussion and list here the following conditions: () and < .()There exist functions , , with and such that for ., and there exist nonnegative numbers , and a function with such that for .
Lemma 10. If and hold, then, for any ,
Lemma 11. If and hold, then the unique solution of problem (1) is where
Proof. If satisfies the equation of problem (1), then Lemma 6 implies that for some , . By , we get .
Therefore, By virtue of Lemmas 5, 7, and 8, we have Using the condition in (31), we obtain Now, substitution of into (30) gives where is defined by (28). The proof is completed.
Remark 12. From the expression of , we can easily find that and for .
For any , define an operator by and due to Lemma 10 and Remark 12, we have On the other hand, by virtue of Lemmas 5, 7, 8, and 10, we get which hold for . So (35) and (37) imply that is well defined and bounded. Furthermore, from Lemma 11, we can transform problem (1) into an operator equation and it is clear to see that is a solution of problem (1) which is equivalent to a fixed point of .
Remark 13. Setting in , we have for , which implies that condition is stronger than . So under assumptions and , the operator defined by (34) is also well defined.
Now, we are in the position to give the main results of this work.
Theorem 14. Assume that is continuous, and suppose that conditions and hold. Then problem (1) has at least one solution .
Proof. In what follows, we divide this proof into three steps.
Step 1. Choose and let Then, for any , by (35), (37), and the fact , we can verify that , which implies .
Step 2. Let be s subset of . We employ Lemma 9 to verify that is relatively compact.
In view of Lemma 10 and the boundness of , there exists such that By (34) and (36), we have Observing (42), together with and the conditions of Lemma 9, we only need to show that, for any , there exists sufficiently large positive integer such that, for any , and for any with , Relation (41) yields that there exists a positive number such that On the other hand, from the monotonicity of and , there exist such that, for any and , Now taking , by virtue of (41), (45), and (46), we getMoreover, from (45), we have which holds for any with and arbitrary . Moreover, it follows from (47) and (48) that (43) and (44) hold. Consequently, by Lemma 9, is relatively compact.
Step 3. is a continuous operator.
Let such that as . Then by , for any there exists a positive integer such that On the other hand, from the continuity of , we know that there exists such that, for any and , Therefore, for and , by (49)-(50) and Remark 12, we can obtain that Meanwhile, for and , applying (49)-(50) again, we can easily verify that Then, by virtue of (51) and (52), we conclude that as , which asserts the continuity of .
Therefore, by Schauder’s fixed point theorem, we obtain that problem (1) has at least one solution in and the proof is finished.
Theorem 15. Suppose that conditions and hold. Then problem (1) has a unique solution .
Proof. For any , in view of and Remark 12, we have On the other hand, by (36) and using again, we have So, from (53), (54) and the facts that and when , we know that is a contraction mapping. By means of Banach contraction mapping principle, we get that has a unique fixed point in ; that is, problem (1) has a unique solution. This completes the proof.
In this section, we will illustrate the possible applications of the above established analytical results with the following two concrete examples.
Example 1. Consider the discrete fractional difference boundary value problem: Conclusion. Problem (55) has at least one solution .
Proof. Corresponding to problem (1), we have , .
From the expression of , it is easy to see that is continuous. Furthermore, we can verify that So condition is satisfied.
On the other hand, by using a simple inequality we have and therefore where By directly calculation, we have Thus, condition holds. So, by Theorem 14, our conclusion follows.
Example 2. Consider the following problem: