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.

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][2][3][4][5][6][7][8][9][10][11][12][13] and references therein.
It is well known that discrete analogues of differential equations can be very useful in applications [14], 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  and the references therein.At the same time, in [27], Atıcı and S ¸engül 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][32][33][34][35][36][37][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 [39], 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 [39] 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.

Preliminaries
In this section, we firstly present here some necessary definitions and basic results about discrete fractional calculus.
Definition 1 (see [30]).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 +1−] is a pole of the Gamma function and  + 1 is not a pole, then  ] = 0.
Definition 3 (see [30]).The ]th discrete Riemann-Liouville fractional difference of a function  : N  → R, for ] > 0, is defined by where  is the smallest integer greater than or equal to ] and Δ  is the th order forward difference operator.If ] =  ∈ N 1 , then Δ   () = Δ  ().
Remark 4. From Definitions 2 and 3, it is easy to see that Δ −]  maps functions defined on N  to functions defined on N +] and Δ ]  maps functions defined on N  to functions defined on N +−] , where  is the smallest integer greater than or equal to ].For ease of notation, we throughout this paper omit the subscript  in Δ ]  () and Δ −]  () when it is not to lead to domains confusion and general ambiguity.

Lemma 9.
Let  ⊆  be a bounded set.If for any given  > 0, there exists a positive integer  = () such that , 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  ∈ ), is a finite dimension Banach space.So we know that  N + −2 is relatively compact from the boundness of ; hence  N + −2 is totally bounded; namely, for any  > 0, there exist finitely many ball where Similarly, denote  −1 is also a Banach space with the norm | and it can be covered by finitely many balls   (Δ −1 V  ); that is, where Step 2. Define   = { ∈  : Let us consider the case  ∈ N + −2 .It is obvious that . Now, let us take   ∈   ; then  can be covered by the balls  4 (  ),  ∈ N  1 ,  ∈ N  1 , where In fact, for any  ∈ , the argument in Step 1 implies that there exist  and  such that For arbitrary  ∈ N + , ( 12) and ( 17) yield that           () and for any  ∈ N +1 , ( 13) and (18) ensure that Relations ( 17)- (20) show that ‖ −   ‖  < 4.Therefore,  is totally bounded and this lemma is proved.

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: (C 2 ) There exist functions   : N −1 → [0, ∞),  ∈ N 5  1 , with and for , and there exist nonnegative numbers   ,  ∈ N 4  1 , and a function  : for  ∈ N −1 , , V, , , , V, ,  ∈ R.
Remark 12. From the expression of (, ), we can easily find that (, ) ≥ 0 and (, )/(1 +  −1 ) < 1/Γ() for (, ) ∈ For any  ∈ , define an operator F by (34) 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  ∈ N 0 .So ( 35) and (37) imply that F :  →  is well defined and bounded.Furthermore, from Lemma 11, we can transform problem (1) into an operator equation  = F and it is clear to see that  is a solution of problem (1) which is equivalent to a fixed point of F. for ) is stronger than (C 2 ).So under assumptions (C 1 ) and (C  2 ), the operator F :  →  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 𝑓 : N
is continuous, and suppose that conditions ( 1 ) and ( 2 ) hold.Then problem (1) has at least one solution  ∈ .

Proof.
In what follows, we divide this proof into three steps.
Step 2. Let  be s subset of .We employ Lemma 9 to verify that F is relatively compact.
In view of Lemma 10 and the boundness of , there exists By ( 34) and ( 36), we have Observing (42), together with lim  → +∞ ( −1 /(1 +  −1 )) = 1 and the conditions of Lemma 9, we only need to show that, for any  > 0, there exists sufficiently large positive integer  such that, for any  1 ,  2 ∈ N + , and for any Then, by virtue of (51) and (52), we conclude that ‖F  − F‖  ≤  as  > , which asserts the continuity of F. 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 ( 1 ) and (  2 ) hold.Then problem (1) has a unique solution  ∈ .
Proof.For any , V ∈ , in view of (C  2 ) and Remark 12, we have On the other hand, by (36) So, from (53), (54) and the facts that  * < Γ() and Γ() ∈ (0, 1] when  ∈ (1, 2], we know that F is a contraction mapping.By means of Banach contraction mapping principle, we get that F has a unique fixed point in ; that is, problem (1) has a unique solution.This completes the proof.

Examples
In this section, we will illustrate the possible applications of the above established analytical results with the following two concrete examples.
From the expression of , it is easy to see that  is continuous.Furthermore, we can verify that  * = sup (57) So condition (C 1 ) is satisfied.