Existence results of nonlocal Robin mixed Hahn and q-difference boundary value problems

In this paper, we aim to study a nonlocal Robin boundary value problem for fractional sequential fractional Hahn-q-equation. The existence and uniqueness results for this problem are revealed by using the Banach fixed point theorem. In addition, the existence of at least one solution is studied by using Schauder’s fixed point theorem. The theorems for existence results are obtained.


Introduction
The quantum difference operator has been applied in many mathematical areas such as orthogonal polynomials, combinatorics, and the calculus of variations [1][2][3][4]. The research works related to the quantum difference operator have been published continuously.
We observe that the study of the boundary value problem of mixed difference operators had not been studied until the work of Dumrongpokaphan et al. [57]. They studied sequential fractional q-Hahn-difference equation. In this paper, we propose a sequential fractional Hahn-q-difference equation where the difference operators are reverse. Our problem is a nonlocal Robin boundary value problem for sequential fractional Hahn-qdifference equation of the form We organize the paper as follows. In Sect. 2, we provide some basic knowledge. In Sect. 3, we prove the existence and uniqueness of a solution to problem (1.1) by using the Banach fixed point theorem. In Sect. 4, we prove the existence of at least one solution to problem (1.1) by using Schauder's fixed point theorem. Finally, in the last section, we provide an example to show applications of our results.

Preliminaries
In this section, we recall some notations, definitions, and lemmas used in the main results.

Definition 2.1
For q ∈ (0, 1), the q-derivative of a real function f is defined by The q-integral of a function f defined on the interval [0, T] is defined by where the infinite series is convergent.

Definition 2.3
Let I be any closed interval of R containing a, b, and ω 0 , and f : I → R is a given function. We define q, x ∈ I, and the series converges at x = a and x = b. The sum to the right-hand side of the above equation is called the Jackson-Nörlund sum.
We note that the actual domain of function f is [a, b] q,ω ⊂ I.
In what follows, we define fractional q-integral, fractional Hahn integral, fractional qdifference, and fractional Hahn difference of Riemann-Liouville type.

Definition 2.5
For α, ω > 0, q ∈ (0, 1), and f defined on [ω 0 , T] q,ω , the fractional Hahn integral is defined by , the fractional q-derivative of the Riemann-Liouville type of order α is defined by , ω > 0, and f defined on [ω 0 , T] q,ω , the fractional Hahn difference of Riemann-Liouville type of order α is defined by The q-gamma and q-beta functions are defined by Next, we aim to find a solution of the linear variant of mixed problem (1.1) where the following auxiliary lemmas will be used for simplifying calculations.

4)
and the constants A η , A T , B η , B T , and Ω are defined by

5)
(2.7) Proof We first take fractional Hahn integral of order α for (2.1). Then the problem becomes fractional q-difference equation as follows: After taking fractional q-integral of order β for (2.10), we get the solution which is in the form In order to find the unknown constants C 1 and C 0 that appeared in (2.11), we first take fractional q-difference and fractional Hahn difference of order γ for (2.11) to get for t ∈ [ω 0 , T], respectively. Substitute t = η into (2.11) and (2.12) and use the first condition of (2.1). Then we get (2.14) Similarly, substitute t = T into (2.11) and (2.13) and employ the second condition of (2.1).

Existence and uniqueness result
In this section, we employ the Banach fixed point theorems to consider the existence and uniqueness result for problem (1.1). Let C = C([0, T], R) be a Banach space of all function u with the norm defined by We define an operator F : C → C by and the constants A η , A T , B η , B T , Ω are defined by (2.5)-(2.9), respectively. If one can prove that F has a fixed point, we can conclude that problem (1.1) has a solution.
Similarly, we see from (3.3) that Taking fractional fractional q-difference of order θ and fractional Hahn difference of order ν for (3.1), we get for t ∈ [0, T], and Similarly, we have From (3.9), (3.12), and (3.13), it implies that Therefore, F is a contraction by (H 3 ). Thus, F has a fixed point, which is a unique solution of problem (1.1) by using the Banach fixed point theorem.

Existence of at least one solution
In this section, we also prove the existence of at least one solution of (1.1) by using Schauder's fixed point theorem. Firstly, we provide some basic knowledge that is used in this section as follows.   Based on the above lemmas, we prove the existence of at least one solution of (1.1) as shown in the following theorem. Proof The proof is divided into three steps as follows.
Step I. We verify that F maps bounded sets into bounded sets in B R = {u ∈ C : u C ≤ R}. We let max t∈[0,T] |F(t, 0, 0, 0)| = M, sup u∈C |φ 1 (u)| = N 1 , sup u∈C |φ 2 (u)| = N 2 and choose a constant We denote that For each t ∈ [0, T] and u ∈ B R , we have Similarly, we find that From (4.2)-(4.3), we find that Hence, which implies that F is uniformly bounded.
Step II. It is obvious that the operator F is continuous on B R due to the continuity of F.
Step III. We prove that F is equicontinuous on B R . For any t 1 , t 2 ∈ I T q,ω with t 1 < t 2 , we find that and D ν q,ω Fu (t 2 ) -D ν q,ω Fu (t 1 ) When |t 2t 1 | → 0, we find that the right-hand side of (4.6)-(4.8) tends to be zero. Hence, F is relatively compact on B R and the set F(B R ) is an equicontinuous set. From Steps I to III and the Arzelá-Ascoli theorem, we can conclude that F : C → C is completely continuous. Therefore, problem (1.1) has at least one solution by Schauder's fixed point theorem.

Conclusion
A nonlocal Robin boundary value problem for fractional sequential fractional Hahn-qequation (1.1) is studied. Our problem contains both fractional Hahn and fractional qdifference operators, which is a new idea. We establish the conditions for the existence and uniqueness of solution for problem (1.1) by using the Banach fixed point theorem, and the conditions of at least one solution by using Schauder's fixed point theorem.