Existence results of fractional delta–nabla difference equations via mixed boundary conditions

In this article, we purpose existence results for a fractional delta–nabla difference equations with mixed boundary conditions by using Banach contraction principle and Schauder’s fixed point theorem. Our problem contains a nonlinear function involving fractional delta and nabla differences. Moreover, our problem contains different orders in four fractional delta differences, four fractional nabla differences, one fractional delta sum, and one fractional nabla sum. Finally, we present some illustrative examples.


Introduction
Simultaneously with the development of the theory and application of differential calculus, difference calculus has also received more intense attention. In this article, we study the evolution of fractional difference calculus. Recently, fractional difference calculus became an attractive field to researchers since it can be used in ecology, biology, and other applied sciences [1][2][3][4].

Preliminaries
This section is divided into two parts. The first contains the notations, definitions, and lemmas which are used in the main results. In the second part, we provide a lemma presenting a linear variant of problem (1.1).
The forward jump operator is defined by σ (t) := t + 1, and the backward jump operator is defined by ρ(t) := t -1.
For t, α ∈ R, the generalized falling function is defined by where t + 1α is not a pole of the Gamma function. If t + 1α is a pole and t + 1 is not a pole, then t α = 0. The generalized rising function is defined by where t and t + α are not poles of the Gamma function. If t is a pole and t + α is not a pole, then t α = 0.
For convenience, the notations -α f (t) and α f (t) are used instead of -α a f (t) and α a f (t), respectively.

Definition 2.2 ([29]
) For α > 0 and f defined on N a , the α-order fractional nabla sum of f is defined by and the α-order Riemann-Liouville fractional nabla difference of f is defined by where N ∈ N is such that 0 ≤ N -1 < α < N .
The solution of a linear variant of the boundary value problem (1.1) is given in the following lemma.

Main results
In this section, we show existence results of problem (1.1). Let C = C(N α-3,T+α , R) be the Banach space of functions u with the norm defined by where Λ = 0, A 1 , A 2 , and A 3 are given in Lemma 2.3 and the functional Φ[F(u)] is given by The boundary value problem (1.1) has solutions if and only if operator F has fixed points. Theorem 3.1 Let F : N α-3,T+α × R 3 → R be a continuous function and suppose that the following conditions hold: (H 1 ) There exist constants L 1 , L 2 , L 3 > 0 such that for each t ∈ N α-3,T+α and u i , v i ∈ R, i = 1, 2, 3, Proof Letting u, v ∈ C, for each t ∈ N α-3,T+α , we have and We find that and we get Hence, F is a contraction. By the Banach contraction principle, we conclude that F has a unique fixed point which is a unique solution of the problem (1.1) for t ∈ N α-3,T+α .
We next show that our problem (1.1) has at least one solution as follows.

Lemma 3.2 ([43]) If a set is closed and relatively compact, then it is compact.
Lemma 3.3 (Schauder's fixed point theorem, [44]) Let (D, d) be a complete metric space, U a closed convex subset of D, and T : D → D a map such that the set Tu : u ∈ U is relatively compact in D. Then, the operator T has at least one fixed point u * ∈ U: Tu * = u * . Theorem 3.2 Suppose that (H 1 ) and (H 2 ) hold. Then problem (1.1) has at least one solution on N α-3,T+α .

Proof
Step I. We verify that F maps bounded sets into bounded sets in B R , where we consider B R = {u ∈ C : u C ≤ R}.
Step II. Since F is a continuous function, the operator F is continuous on B R .
Step III. We show that F is equicontinuous on B R . For any > 0, there exists a positive constant ρ * = min{δ 1 , δ 2 , δ 3 , δ 4 , δ 5 , δ 6 } such that for t 1 Then, for |t 2t 1 | < ρ * , we have (3.20) Similarly, we have Hence, the set F(B R ) is equicontinuous. Combining the results of Steps I to III with the Arzelá-Ascoli theorem, we get that F : C → C is completely continuous. By using Schauder fixed point theorem, we can conclude that boundary value problem (1.1) has at least one solution.

Conclusions
We consider a fractional delta-nabla difference equation with fractional delta-nabla sumdifference boundary value conditions. In our studies, we employ the Banach contraction principle to investigate the conditions for the existence and uniqueness of solution for our problem. In addition, the conditions for at least one solution is obtained by using the Schauder's fixed point theorem.