Existence Theory for a Fractional q-Integro-Difference Equation with q-Integral Boundary Conditions of Different Orders

In this paper, we study the existence of solutions for a new class of fractional q-integro-difference equations involving Riemann-Liouville q-derivatives and a q-integral of different orders, supplemented with boundary conditions containing q-integrals of different orders. The first existence result is obtained by means of Krasnoselskii’s fixed point theorem, while the second one relies on a Leray-Schauder nonlinear alternative. The uniqueness result is derived via the Banach contraction mapping principle. Finally, illustrative examples are presented to show the validity of the obtained results. The paper concludes with some interesting observations.


Introduction and Preliminaries
Fractional calculus, dealing with differential and integral operators of arbitrary order, serves as a powerful modelling tool for many real-world phenomena. An interesting feature of such operators is their nonlocal nature that accounts for the history of the phenomena involved in the fractional models. Motivated by the extensive applications of fractional calculus, many researchers turned to the theoretical development of fractional-order initial and boundary value problems. Now, the literature on the topic contains many interesting and important results on the existence and uniqueness of solutions, and other properties of solutions for fractional-order problems. The available material includes different types of derivatives such as Riemann-Liouville, Caputo, Hadamard, etc. and a variety of boundary conditions. For some recent works on the topic, for instance, see [1][2][3][4][5][6][7][8] and the references therein.
Fractional q-difference equations (fractional analogue of q-difference equations) also received significant attention. One can find preliminary work on the topic in [9], while some interesting details about initial and boundary value problems of q-difference and fractional q-difference equations can be found in the book [10].
We make use of Krasnoselskii's fixed point theorem and a Leray-Schauder nonlinear alternative to prove the existence results, while the uniqueness result is proved via Banach contraction mapping principle for the given problem.
Let us first recall some necessary concepts and definitions about q-fractional calculus and fixed point theory.
Recall that I β q I α q u(t) = I β+α q u(t) for α, β ∈ R + [9] and The Riemann-Liouville fractional q-derivative of order α > 0 for a function u : (0, ∞) → R is defined by [9] Next, we state some fixed point theorems related to our work. Then, there exists z ∈ M such that z = Az + Bz (Krasnoselskii's fixed point theorem [32]). Lemma 2. Let X be a closed and convex subset of a Banach space E and let Y be an open subset of X with 0 ∈ Y. Then, a continuous compact map H : Y → X has a fixed point in Y or there is a y ∈ ∂Y and σ ∈ (0, 1) such that y = σH(y), where ∂Y is the boundary of Y in X (Nonlinear alternative for single-valued maps [33]).

Main Results
Now, we prove the following lemma which characterizes the structure of solutions for boundary value problems (1) and (2).
The function u is a solution for the fractional q-difference boundary value problem if and only if u is a solution for the fractional q-integral equation Proof. Let u be a solution of the q-fractional boundary value problem (4). Then, we have Taking the Riemann-Liouville fractional q-integral of order α to both sides of the above equation, we get where c 1 , c 2 ∈ R are arbitrary constants. Since 1 < α < 2, it follows from the first boundary condition that c 2 = 0. Thus, On the other hand, if σ ∈ {γ 1 , γ 2 }, then we have Now, by using the second boundary value condition and substituting the values σ ∈ {γ 1 , γ 2 } into the above expression, we obtain Solving the above equation for c 1 , we find that where ∆ is defined in (3). Substituting the value of c 1 in (6), we get the solution (5). Conversely, it is clear that u is a solution for the fractional q-difference Equation (4) whenever u is a solution for the fractional q-integral Equation (5). This completes the proof.
In relation to the problems (1) and (2), we introduce an operator T : where u ∈ E and t ∈ [0, 1]. In the sequel, we set Now, we are ready to present our main results. The first existence result is based on Krasnoselskii's fixed point theorem. (i) there exists a positive constant L such that for each u, v ∈ R, (ii) For each u ∈ R, there exists a continuous function m on [0, 1] such that If Λ 0 + LΛ 1 < 1, then the fractional q-integro-difference Equation (1) with q-integral boundary conditions (2) has at least one solution on [0, 1], where Λ 0 and Λ 1 are defined by (8).
Since g is continuous, we get T 2 u n − T 2 u → 0 as u n → u. In consequence, it follows that the operator T 2 is continuous on B r .
In the next step, we show that the operator T 2 is compact. Let us first show that T 2 is uniformly bounded. For each u ∈ B r and t ∈ [0, 1], we have In order to establish the equicontinuity of the operator T 2 , we assume that t 1 , t 2 ∈ [0, 1] such that t 2 > t 1 . We will show that T 2 maps bounded sets into equicontinuous sets. For each u ∈ B r , we have Observe that the right-hand side of the above inequality is independent of u ∈ B r and tends to zero as t 1 → t 2 . This shows that T 2 is equicontinuous. Therefore, the operator T 2 is relatively compact on B r and the Arzelá-Ascoli theorem implies that T 2 is completely continuous and so T 2 is compact operator on B r .
Finally, we prove that the operator T 1 is a contraction. For any u, v ∈ B r and t ∈ [0, 1], we obtain Since Λ 0 + LΛ 1 < 1, T 1 is a contraction. Thus, all the assumptions of Lemma 1 are satisfied. Therefore, the fractional q-integro-difference Equation (1) with q-integral boundary conditions (2) has at least one solution on [0, 1] and the proof is completed.
In the following result, we prove the existence of solutions for the problem (1) and (2) by means of a Leray-Schauder nonlinear alternative.
Then, the fractional q-integro-difference Equation (1) Secondly, we show that T maps bounded sets into equicontinuous sets of E. Let t 1 , t 2 ∈ [0, 1] with t 1 < t 2 and u ∈ B R . Then, we have Thus, the Arzelá-Ascoli theorem applies and hence T : E → E is completely continuous. In the last step, we show that all solutions to the equation u = θT u are bounded for θ ∈ [0, 1]. For that, let u be a solution of u = θT u for θ ∈ [0, 1]. Then, for t ∈ [0, 1], we apply the strategy used in the first step to obtain By the condition (H 4 ), we can find a positive number Ξ such that u = Ξ. Introduce a set and observe that the operator T : U → E is continuous and completely continuous. With this choice of U, we cannot find u ∈ ∂U satisfying the relation u = θT x for some θ ∈ (0, 1). Therefore, it follows by a nonlinear alternative of the Leray-Schauder type (Lemma 2) that the operator T has a fixed point in U. Thus, there exists a solution of problems (1) and (2)  In our final result, the uniqueness of solutions for the given problem is shown with the aid of a Banach contraction mapping principle [34].

Conclusions
We have derived some new existence and uniqueness results for a nonlinear fractional q-integro-difference equation equipped with q-integral boundary conditions. The obtained results significantly contribute to the literature on boundary value problems of fractional q-integro-difference equations and yield several new results as special cases. Some of these results are listed below.
(c) Our results with a = 0 and b = 0 correspond to the ones with purely integral nonlinearity and purely non-integral nonlinearity, respectively.
Funding: This research received no external funding.