Uniqueness of continuous solution to q-Hilfer fractional hybrid integro-difference equation of variable order

In this paper, the authors introduced a novel definition based on Hilfer fractional derivative, which name $q$-Hilfer fractional derivative of variable order. And the uniqueness of solution to $q$-Hilfer fractional hybrid integro-difference equation of variable order of the form \eqref{eq:varorderfrac} with $0 < \alpha(t) < 1$, $0 \leq \beta \leq 1$, and $0 < q < 1$ is studied. Moreover, an example is provided to demonstrate the result.


Introduction
Fractional calculus caught much attention towards mathematical worlds (see [1,2,3,4,5,6,7,8,13,14,15,22]). In fact, fractional calculus is a branch of mathematical analysis, which separate itself from normal calculus, with non-integers order of derivatives and integrals as special characteristics. The development of fractional calculus started from the first-order derivative such that In this case, it is said that the discrete version of such operator is called h-derivative, which is Fractional calculus is developed towards time, and various experts propose many definitions of fractional derivatives. The two famous senses that caught the most attention in the differential equation are Caputo fractional derivative and Riemann-Liouville fractional derivative. Subsequently, Hilfer developed the general definition of fractional derivative by interpolating such operators motivated by these two derivatives. Determine n − 1 < α < n and β ∈ [0, 1], the three visualizations of Caputo, Riemann-Liouville and Hilfer derivatives are given as follows. Firstly, The left Riemann-Liouville fractional derivative of order α for the function f(t) is defined by Secondly, The left Caputo fractional derivative of order α is defined by Lastly, The Hilfer fractional derivative [18] is defined by As the consequences, these common definitions lead to further enormous generalization of fractional derivatives such as fractional derivatives of a function with respect to another function [10,27], fractional proportional derivative, variable-order fractional derivatives [9,21,29], etc. Also, there are several methods used to illustrate the existence and uniqueness of solution such as Banach fixed point theorem, Schaefer fixed point theorem, Schauder fixed point theorem, etc. (see [16,17,26,12,31,20]) In 1909, Jackson [19] introduced the new branch of calculus by defining q-derivative with 0 < q < 1 as and q-integral operator such that Moreover, the definition of q-derivative and q-integral is studied and gradually developed by many researchers (see [11,23,24,28]). The definitions of q-derivative and q-integral are developed, which are based on the q-Riemann-Liouville fractional integral. In this work, motivated by [17,26,29,30,21], and the Hilfer operator in [18], the authors will introduce a novel definition based on Hilfer fractional derivative, which name q-Hilfer fractional derivative of variable order. Also, the main purpose of this paper is to study q-Fractional Hybrid Integro-Difference Equation of Variable Order (q-FHIDEVO) of the form where 0 < α(t) < 1, 0 β 1 and 0 < q < 1. Our result illustrates the uniqueness of the solution. This paper is constructed as follows. In section 2, the notation and concept of qfractional calculus will be introduced. In section 3, the concept of variable order and essential conditions to display the uniqueness and stability of the solution to q-FHIDEVO will be displayed. In sections 4 and 5, the uniqueness of solution in subinterval and the uniqueness of continuous solution will be presented, respectively. Lastly, in section 6, the example will be illustrated.

Preliminaries and Framework
The preliminaries section will introduce the necessary definition of operator, space, and concept of q-difference equation.

Definition 2.2. [11] For any
For p = 1 it can be denoted the space as L q (a, b). Definition 2.3. [28] Let, q ∈ (0, 1) and α > 0, then the q-Riemann-Liouville fractional integral is defined as Also, let α, β 0 and f(t) is a function on [0, T ], then there are following properties Definition 2.5. [11] Let n − 1 < α < n, the q-Caputo fractional derivative of the function Motivated by definition 2.4 and definition 2.5, based on Hilfer fractional derivative, authors shall introduce the operator of the q-Hilfer fractional derivative as follows.

Variable approach and mild solution
For q = 1, it can be noted the space as AC[a, b].
Proof. The proof is trivial. By property (1) pursuant to the definition 2.3 and the definition 2.6, we obtain q I α t q D α,β t f(t) = q I γ t q D γ t f(t). Subsequently, applies theorem 3.4 with n = 1 , we will obtain the illustrated result.
Moving into the variable concept, the authors define the q-Hilfer derivative with order 0 < α(t) < 1 and 0 β 1, and the q-fractional integral of variable order as follows.
Definition 3.6. Let, q ∈ (0, 1) and α(t) > 0, then the q-Riemann-Liouville fractional integral of variable order is defined as Definition 3.7. Let 0 < α(t) < 1, 0 β 1 and 0 < q < 1 then, the q-Hilfer variable order fractional derivative of the function f(t) is defined by It is obvious that when α(t) = α, the operator is the same as definition 2.6.
In this work, the fractional order hybrid integro-difference equation with initial condition given by (1.1), where f : [0, T ] × R → R, g : [0, T ] × R × R → R and initial data x 0 , f 0 ∈ R, will be analysed.
Secondly, we define the α-approximation functionα(t) : (0, T ] → (0, 1) as piecewise continuous function respect to P. The functionα is written bỹ where I k is the indicator on P k . In other words, I k (t) = 1 for t ∈ P k . Otherwise, I k (t) = 0. Consequently, the function α(t) = lim N→∞α (t), as |α k − α k−1 | → 0 for any |t k − t k−1 | → 0. Hence (1.1) can be represented by Now, we present the definition of solution to problem (1.1) , which is fundamental to this article. From the theorem 3.5 and the equation (3.1), the integral represent solution x k (t) in subinterval P k is written by Moreover, the continuous mild solution x(t) = ∞ k=1 I k (t)x k (t) is written by

Uniqueness of solution in subinterval
In this part, the authors will illustrate the uniqueness of solution according to the k th -subinterval.

Theorem 4.2. The equation (1.1) has a solution in
To display the uniqueness of solution, we state the essential assumptions as follows: (A0) There exists positive constant M f such that (A1) There exist positive constants L 1 , L 2 , M g such that Proof. For each k = 1, 2, . . ., we define the contraction mapping Q : Then, The proof is completed.

uniqueness of continuous solution
In this part, the uniqueness of (3.4) will be displayed.

From this point, it is obvious that
Next, suppose z = α(t) and s = 1 − α(t) into inequality of theorem 5.1, we get Now, combining inequalities together, the new inequality holds Rearrange the inequality, the inequality holds .
The proof is completed.
Proof. Generating the approximation contraction functionφ on (0, T ] by aggregate the contraction constant in each subinterval P k we get Thus, by take limit N → ∞, the fundamental contraction function ϕ * (t) is displayed as According to the theorem 5.2, it is obvious that Since there exists the contraction function ϕ(t) < 1, the continuous solution x(t) is a unique solution on L q [0, T ]. The proof is completed.
This mean the contraction function ϕ(t) is written as

Conclusion
In this work, the authors introduce novel operators in quantum calculus, which are q-Hilfer fractional derivative and q-Hilfer fractional derivative of variable order. Also, we present the novel proof of the uniqueness of continuous solutions to q-FHIDEVO. The uniqueness of solution is proved by using Banach fixed point theorem under Lipschitz conditions for nonlinear terms.