On the Nonlocal Fractional Delta-Nabla Sum Boundary Value Problem for Sequential Fractional Delta-Nabla Sum-Difference Equations

: In this paper, we propose sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions. The Banach contraction principle and the Schauder’s ﬁxed point theorem are used to prove the existence and uniqueness results of the problem. The different orders in one fractional delta differences, one fractional nabla differences, two fractional delta sum, and two fractional nabla sum are considered. Finally, we present an illustrative example.

We note that there are a few papers using the delta-nabla calculus as a tool. For example, Malinowska and Torres [38] presented the delta-nabla calculus of variations. Dryl and Torres [39,40] studied the delta-nabla calculus of variations for composition functionals on time scales, and a general delta-nabla calculus of variations on time scales with application to economics. Ghorbanian and Rezapour [41] proposed a two-dimensional system of delta-nabla fractional difference inclusions. Liu, Jin and Hou [42] investigated existence of positive solutions for discrete delta-nabla fractional boundary value problems with p-Laplacian.
In this paper, we aim to extend the study of delta-nabla calculus that has appeared in discrete fractional boundary value problems. We have found that the research works related to delta-nabla calculus were presented as above. However, the boundary value problem for sequential fractional delta-nabla difference equation has not been studied before. Our problem is sequential fractional delta-nabla sum-difference equations with nonlocal fractional delta-nabla sum boundary conditions as given by where t ∈ N 0,T := {0, 1, . . . , T}; α ∈ (1, 2]; β, θ, φ, ω, γ ∈ (0, 1]; α + β ∈ (2, 3]; T > 0; κ, λ are given constants; F ∈ C N α−2,T+α × R 3 , R ; and for ϕ, The objective of this research is to investigate the solution of the boundary value problem (1). The basic knowledge is disscussed in Section 2, the existence results are presented in Section 3, and an example is provided in Section 4.

Preliminaries
We give the notations, definitions, and lemmas as follows. The forward operator and the backward operator are defined as σ(t) := t + 1, and ρ(t) := t − 1, respectively.
For t, α ∈ R, we define the generalized falling and rising functions as follows: • The generalized falling function for any t + 1 − α is not a pole of the Gamma function. If t + 1 − α is a pole and t + 1 is not a pole, then t α = 0.
• The generalized rising function for any t is not a pole of the Gamma function. If t is a pole and t + α is not a pole, then t α = 0.

Definition 1.
For α > 0 and f defined on N a := {a, a + 1, . . .}, the α-order fractional delta sum of f is defined by The α-order fractional nabla sum of f is defined by

Definition 2.
For α > 0, N ∈ N where 0 ≤ N − 1 < α < N and f defined on N a , the α-order Riemann-Liouville fractional delta difference of f is defined by The α-order Riemann-Liouville fractional nabla difference of f is defined by Lemma 1 ([15]). Let 0 ≤ N − 1 < α ≤ N, N ∈ N and y : N a → R. Then, We next provide a linear variant of our problem (1).

Existence and Uniqueness Result
Define C = C(N α−2,T+α , R) is the Banach space of all function u and define the norm as In addition, we define the operator F : C → C by where Λ = 0 is defined by (7) Obviously, the operator F has the fixed points if and only if the boundary value problem (1) has solutions. Firstly, we show the existence and uniqueness result of the boundary value problem (1) by using the Banach contraction principle.
Suppose that the following conditions hold: Then the problem (1) has a unique solution on N α−2,T+α provided that where Proof. We shall show that F is a contraction. For any u, v ∈ C and for each t ∈ N α−2,T+α , we have and Next, we consider the following (∇ −θ F u) and (∆ −φ F u) as Similarly to the above, we have From (20), (23) and (24), we get By (14), Consequently, F is a contraction. Therefore, by the Banach fixed point theorem, we get that F has a fixed point which is a unique solution of the problem (1) on t ∈ N α−2,T+α .

Existence of at Least One Solution
Next, we provide Arzelá-Ascoli theorem and Schauder's fixed point theorem that will be used to prove the existence of at least one solution of (1). Proof. The proof is organized as follows.

Lemma 5 ([43]). A set is compact if it is closed and relatively compact.
Step I. Verify F map bounded sets into bounded sets in B R = {u ∈ C(N α−2,T+α ) : u C ≤ R}.
Step II. Since F and H are continuous, the operator F is continuous on B R .
Step III. Prove that F is equicontinuous on B R . For any > 0, there exists a positive constant ρ * = max{δ 1 , δ 2 , δ 3 , δ 4 , δ 5 , , δ 6 } such that for t 1 , Then we have Similarly to the above, we have It implies that the set F (B R ) is equicontinuous. By the results of Steps I to III and the Arzelá-Ascoli theorem, F : C → C is completely continuous. By Schauder fixed point theorem, the boundary value problem (1) has at least one solution.
Hence, by Theorem 1, the problem (36) has a unique solution.

Conclusions
We estabish the conditions for the existence and unique results of the solution for a fractional delta-nabla difference equations with fractional delta-nabla sum-difference boundary value conditions by using Banach contraction principle and the conditions for result of at least one solution by using the Schauder's fixed point theorem.