Some results on fractional Hahn di ﬀ erence boundary value problems

: Fractional Hahn boundary value problems are signi ﬁ cant tools to describe mathematical and physical phenomena depending on non - di ﬀ erentiable functions. In this work, we develop certain aspects of the theory of fractional Hahn boundary value problems involving fractional Hahn derivatives of the Caputo type. First, we construct the Green function for an α th - order fractional boundary value problem, with α 1 2 < < , and discuss some important properties of the Green function. The solutions to the proposed problems are obtained in terms of the Green function. The uniqueness of the solutions is proved by various ﬁ xed point theorems. The Banach ’ s contraction mapping theorem, the Schauder ’ s theorem, and the Browder ’ s theorem are used.


Introduction
Recently, the Hahn calculus and fractional Hahn difference equations have gained much attention. The Hahn calculus (also called q ω , -calculus) can be dated back to 1949, Hahn's work [1]. Based on the fractional Hahn calculus, the fractional Hahn difference equations were established that can describe some physical processes appearing in quantum dynamics, discrete dynamical systems, discrete stochastic processes, and many others. Here, one should point out that the Hahn difference equations are usually defined on a time scale set I q ω , , with σ t qt ω q ω , ( ) = + being the scale index. The Hahn difference operator is defined by [1]: We note that this operator is combined from the well-known operators: the forward difference operator and the Jackson q-difference operator.
With the development of the Hahn calculus theory, some related concepts and results have also been introduced and studied, such as the theory of linear Hahn difference equations, Leibniz's rule and Fubini's theorem associated with Hahn difference operator, and q ω , -Taylor expansion [2][3][4][5][6][7][8] (see [9][10][11][12][13][14] for more details on Hahn and fractional Hahn difference equations). Up to now, compared with the classical fractional differential equations, the study of the fractional Hahn difference equations is still immature. At present, the literature have many studies on the existence and uniqueness of solutions of fractional Hahn difference boundary value problems. In [10][11][12][13], the Banach's fixed point theorem and the Schauder's fixedpoint theorem are used to prove the existence and uniqueness results of Caputo fractional Hahn difference boundary value problems for fractional Hahn integro-difference equations. In [11], the authors studied a nonlocal Robin boundary value problem for the fractional Hahn integro-difference equation. The existence and uniqueness results were proved by using the Banach's fixed point theorem and the Schauder's fixed point theorem. More recently, nonlocal fractional symmetric Hahn integral boundary value problems for the fractional symmetric Hahn integro-difference equation were studied in [12].
In this work, we are going to gain further insight into the theorem of fractional Hahn difference boundary value problems. Mainly, we consider the following Caputo fractional Hahn difference boundary value problem: )→ being given functionals. We aim to study the existence and uniqueness of the solution to Problem (1.1) by using the Banach's fixed point theorem and the existence of at least one solution by using the Browder's and the Schauder's fixed point theorems. This work is organized as follows: Section 2 provides some basic definitions and relevant results on the Hahn and fractional Hahn calculi. In Section 3, we study the existence of solutions of Caputo fractional Hahn difference boundary value problems. Section 4 is devoted to the uniqueness of the solutions by using various fixed point theorems. Finally, illustrative examples are given in Section 5.
q ω For a b , ∈ , the q-analogue of the power function a b q The q ω , -analogue of the power function a b q ω n , Note that a a q α α = and a ω a ω .
q ω The q-gamma function is defined as [16]: , the sequence σ t q ω n n , 0 Also, we consider provided that f is differentiable at ω 0 in the usual sense. We call D f q ω , the q ω , -derivative of f and say that f is q ω , -differentiable on I.
The n q ω th , -derivative, n ∈ of a function f I : → is given by: Lemma 2.1.
q ω q ω q ω 8,9] Let I be a closed interval of containing a b , , and ω 0 .
provided that the series converges at x a = and x b = .
[8] Let f g I , : → be q ω , -integrable functions on I , k ∈ , and a b c I , , ∈ with a c b < < . Then,

( ) exists for every x I ∈ and
D F x f x x I , .
∈ , and f I : , q ω T , → the fractional Hahn difference operator of Caputo type of order α is defined as: Thus, the following results can be proved.
∈ , and f be a function on I q ω T , . Then, the function is a solution of the following Caputo-type fractional Hahn initial value problem By using Definition 2.3 and Lemma 2.7, we obtain , , satisfies the following FBVP: , .
q ω From the first boundary condition y ω ϕ y , The second boundary condition Now, we define the Green function as follows.
Lemma 2.13. The Green function for FBVP is given by: Lemma 2.14. Let α t I 1, 2 , where G t s , q ω , ( ) is defined by (2.12) and z t ( ) is a solution to FBVP Proof. Applying Lemmas 2.11 and 2.12 to obtain the desired result. □

Properties of a Green's function
In this section, we study some properties of the Green function G t s , q ω , ( ) defined by (2.12).
T ω T ω s ω q T ω s ω q By [9, Lemma 3.1], we obtain and α t σ s d s t ω α α t ω α In the following result, we find the maximum of G t s d s , .

Existence of at least one solution
In this section, we prove the existence of at least one solution to Problem (1.1) by applying the Schauder's fixed point theorem and a special case of Browder's fixed point theorem.  Therefore, T is bounded on S. Then, T has a fixed point. □ In the next section, we prove the uniqueness result for Problem (1.1) by applying the Banach's fixed point theorem. ( ) ‖ ‖ be a Banach space and T X X : → be a contraction mapping. Then, T has a unique fixed point in X.

Existence of a unique solution
Applying the aforementioned lemma helps to prove the following theorems.

Example 2
For the following fractional Hahn boundary value problem we have α ω q ω T , 2, , 4

Conclusion
In this study, we have considered a boundary value problem for a fractional Hahn difference equation subject to two boundary conditions. Our results extend and generalize the results obtained in [18][19][20]. After proving the existence and uniqueness results concerning linear variants of the main nonlinear problem, it is transformed into a fixed point problem. A Green's function is constructed and used to express the solutions to the considered boundary value problems. Banach's, Schauder's, and Browder's fixed point theorems are used to prove the existence and uniqueness results. The main results are illustrated by numerical examples. Some properties of the fractional Hahn calculus needed in our work are also presented. The results of this article are new and enrich the theory of boundary value problems for Hahn difference equations. In future works, we will study the Laplace transform and Baskakov basis functions associated with the fractional Hahn difference operators. Also, the q ω , -analogs of degenerate derivatives and their applications and the Boson operator can be investigated. These potential results will generalize the recent works [21][22][23][24].