Positive Solutions for a Class of Integral Boundary Value Problem of Fractional q -Difference Equations

: This paper studies a class of integral boundary value problem of fractional q -difference equations. We ﬁrst give an explicit expression for the associated Green’s function and obtain an important property of the function. The new property allows us to prove sufﬁcient conditions for the existence of positive solutions based on the associated parameter. The results are derived from the application of a ﬁxed point theorem on order intervals


Introduction
It is known that fractional differential equations and their discrete analogues, fractional difference equations, have broad applications in many areas such as neural computing [1], transportation modelling [2], and dynamical systems [3,4]. Compared to the traditional integer ordered calculus, fractional ordered derivatives have less requirements for the smoothness of the functions [5]. This property provides some advantages in modeling computational algorithms and other applications [6,7]. From the perspectives of numerical analysis, fractional integral equations can be solved via various neural networks [8].
The concept of quantum calculus (q-calculus) was first developed by Jackson [9,10]. The theory has advanced applications in mathematical physics, impulsive waves, and signal analysis [11][12][13][14][15]. Fractional q-difference calculus is the counterpart of fractional integral and difference operators [16,17]. For a comprehensive introduction to this subject, we refer to the book [12].
The study of positive solutions for Boundary Value Problems (BVPs) is paramount due to its significant applications. For example, positive solutions of a BVP arising from chemical reactor theory [18] represent the temperature during the reaction. In the literature, existence of positive solutions for differential equations (DE) and fractional differential equations (FDE) have been extensively studied [19,20]. However, results for q-difference fractional boundary value problems are relatively rare [21,22].
The following nonlinear q-fractional BVP was considered in [21]: A common approach to prove the existence of positive solutions for BVPs is to convert the problem to an integral equation and then apply fixed point theorems. Results of [21] on sufficient conditions for the existence of positive solutions were obtained following this idea. The well-known Guo-Krasnoselskii fixed point theorem was applied.
Later, as a special case of the boundary conditions for BVP (1), when β = 0, the following parameter dependent fractional-difference BVP was examined in [23]: BVP (2) is equivalent to a class of time-independent fractional-difference Schrödinger equations that simulate the evolution of the slowly varying amplitude of a nonlinear wave [23]. Again, applying the Guo-Krasnoselskii fixed point theorem, the existence of positive solutions depending on the values of the parameter λ were obtained.
Recently, the uniqueness of solutions for BVP involving a parameter and the integral operator was obtained in [24]: The approach of [24] relies on the iterative technique and a fixed point theorem that ensures uniqueness [25].
Motivated by the above studies, in this paper, we consider the fractional q-difference equation associated with a parameter and subject to integral boundary conditions: where the parameters λ > 0, 2 < α ≤ 3, and f : [0, +∞) → [0, +∞) is a continuous function. D α q denotes the q-fractional differential operator of order α. The same as the work of [21,23,24], we will use the Riemann-Liouville type fractional derivative [26]. Different from methods applied in [21,23,24], we utilize a relatively new fixed point theorem on order intervals [27] that provides more information on the solutions.
In Section 2, we present some definitions and lemmas for fractional q-derivative and fractional q-integral. The Green's function of the fractional integral BVP is obtained. Its properties are proved. In Section 3, new sufficient conditions for the existence of positive solutions are proved.

Preliminaries on q-Calculus
The following notations and definitions can be found in [21,28]. Suppose q ∈ (0, 1) and define The q-gamma function is defined as follows: The q-derivative of a function f is defined by The following four formulas will be used in the sequel: where t D q denotes the derivative with respect to variable t.

Definition 1 ([17]
). Assume that α ≥ 0 and f is a function defined on [0, 1]. The fractional q-integral of the Riemann-Liouville type is defined by

Definition 2 ([14]
). The fractional q-derivative of the Riemann-Liouville type of f is defined by where m is the smallest integer greater than or equal to α. 11,14]). Let α, β ≥ 0 and f be a function defined on [0, 1]. We have the following formulas:

Lemma 2 ([22]
). Assuming that α > 0 and p is a positive integer, then Then, the solution of integral boundary value problem is equivalent to the solution of Proof. In view of Definition 2 and Lemma 1, we obtain that It follows from Lemma 2 that the solution u of (9) is given by where Taking the derivative of both sides of Equation (10), with the help of (7) and (8), we obtain Using the boundary conditions of D q u(0) = 0 and D q u (1) Thus, u can be calculated as Let C = 1 0 u(s)d q s. Integrating (11) with respect to t from 0 to 1, we obtain By using equations In view of (6), substituting C into the formula (11), we deduce that  Proof. We first define two functions In view of Remark 1, we have Notice that [α] q − 1 > 0, thus Therefore, G(t, qs) ≥ 0. In addition, for fixed t ∈ [0, 1], i.e., g 1 (t, qs) and g 2 (t, qs) are increasing functions of t, so G(t, qs) is an increasing function to t for fixed s ∈ [0, 1]. If t ≥ qs, then If t ≤ qs, then Thus, the proof is complete.

Main Results
For convenience, we denote where h(s) = s α−1 .
If there exist positive numbers 0 < a < b such that T : That is, H(P) ⊂ P.
Let D ⊂ P be bounded. There exists a positive constant M > 0 such that u ≤ M for each u ∈ D. Assume L = max u ≤M | f (u)| + 1; then, in view of Lemma 4(ii), we have This shows that, for each parameter λ > 0, T(D) is bounded.
It is well-known that u ∈ X is a solution of (4) if and only if u is a fixed point of H.
Since λ < 1 MF 0 , we can choose a small enough number of ε 1 > 0, such that Let a = δ 2 . Then, for u ∈ [at α−1 , a], we can obtain On the other hand, using the condition λ > 1 N f ∞ , there exist c > 0 and ε 2 > 0 such that By Theorem 1, H has a fixed point u λ ∈ [at α−1 , b]. It is a positive solution of (4).

Theorem 3.
Assuming that MF ∞ < N f 0 holds, then BVP (4) has at least one positive solution for Proof. From the definition of the cone P, we have u 0 = h(t), ϕ = 1. For u ∈ X + , ||u|| = u(1). Therefore, u 0 and ϕ satisfy the conditions of Theorem 1.