Positive Solutions for a System of Fractional q -Difference Equations with Multi-Point Boundary Conditions

: We explore the existence, uniqueness, and multiplicity of positive solutions to a system of fractional q -difference equations that include fractional q -integrals. This investigation is carried out under coupled multi-point boundary conditions featuring q -derivatives and fractional q -derivatives of various orders. The proofs of our principal findings employ a range of fixed-point theorems, including the Guo–Krasnosel’skii fixed-point theorem, the Leggett–Williams fixed-point theorem, the Schauder fixed-point theorem, and the Banach contraction mapping principle.


Introduction
In recent decades, there has been a notable surge in the exploration of nonlocal boundary value problems, including those involving multi-point scenarios, within the realm of ordinary differential or difference equations.This area of research is experiencing rapid growth, spurred not only by theoretical interest but also by the practical applications of modeling various phenomena in engineering, physics, and life sciences.To illustrate, consider systems with feedback controls, such as the steady states of a thermostat.In this context, a second-order ordinary differential equation subject to a three-point boundary condition can capture the dynamics, where a controller at one end adjusts the heat based on the temperature recorded at another point.Another instance is found in the vibrations of a guy wire with a uniform cross-section composed of N parts of varying densities.Such scenarios can be effectively modeled as multi-point boundary value problems (refer to [1]).
The field of q-difference calculus, also known as quantum calculus, traces its origins back to the pioneering work of Jackson ( [17,18]).To explore various applications of this discipline, readers are directed to the research of Ernst ([19]).The fractional q-difference calculus originated in the works of Al-Salam ( [20]) and Agarwal ([21]).For advancements in this branch, encompassing q-analogs of integral and differential fractional operators, including properties like the fractional Leibniz q-formula, q-analogs of Cauchy's formula, q-Laplace transform, q-Taylor's formula, and q-analogs of the Mittag-Leffler function, refer to the papers [22][23][24][25][26].
The novel aspect of our problem (1),(2), compared to (3),(4) from [2], lies in the inclusion of generalized coupled boundary conditions (2) for the system of q-fractional difference equations in Equation (1).In this formulation, the q-fractional derivative of order ς for the unknown function u at the point 1 is contingent upon the q-fractional derivatives of various orders for both functions u and v at different points within the interval (0, 1).Similarly, the q-fractional derivative of order ϑ for the unknown function v at the point 1 is linked to the q-fractional derivatives of distinct orders for functions u and v at diverse points within the interval (0, 1).Furthermore, unlike the approach in the paper [2], we have explored the presence of positive solutions to our specific problem.
Our paper is organized as follows: Section 2 introduces key definitions and properties from q-calculus and fractional q-calculus, along with an existence result for the associated linear problem, the relevant Green functions, and their properties.Section 3 will then present the primary existence results for problem (1),(2), while Section 4 will provide illustrative examples to demonstrate the applicability of our theorems.Finally, Section 5 concludes the paper by summarizing the findings and presenting the overall conclusions.

Preliminary Results
In this section, we will introduce certain definitions and properties derived from q-calculus and fractional q-calculus.Additionally, we will outline some auxiliary findings that will play a pivotal role in the subsequent section.
Definition 1.The q-derivative of a real function f is defined by Definition 2. The q-derivatives of higher order of a real function f are defined by Definition 3. The q-integral of a function f defined in the interval [0, b] is defined by Definition 5.The q-integrals of a function f of higher order are defined by The fundamental theorem of q-calculus says that (D q I q f)(t) = f(t), and if f is continuous at t = 0, then (I q D q f)(t) = f(t) − f(0).The properties of the operators D q and I q are presented in [6,27].Below, we present some properties that will be used later.
Lemma 1.For a, t, s, α ∈ R, and f : where t D q denotes the q-derivative with respect to variable t.Definition 6 ([21]).Let f be a function defined on [0, 1].The fractional q-integral of the Riemann-Liouville type of order α ≥ 0 is defined by (I 0 q f)(t) = f(t) and Definition 7 ([24]).The fractional q-derivative of the Riemann-Liouville type of order α ≥ 0 is defined by (D 0 q f)(t) = f(t) and where m is the smallest integer greater than or equal to α.
Proof.By Definition 6 and Lemma 2 (e) (with α = κ and λ = 0), we obtain In what follows, we will study the linear problem associated with our problem (1),(2).We consider the system of fractional q-difference equations subject to boundary conditions (2), where h, k ∈ C[0, 1].
In a similar manner, we find where G 3 , G 4 , and g 2 are given by ( 37) and (38).So, we deduce the formulas in (36) for the solution (u(t), With a similar proof to that of Lemma 12 from [4], we obtain the next result.

Main Results
In this section, we will outline the results concerning the existence of positive solutions for our given problem (1), (2).
Initially, we state our primary assumptions: 1),(2) can be expressed equivalently to the following system of fractional q-integral equations: Let X = C[0, 1] be the Banach space endowed with the norm ∥u∥ = sup t∈[0,1] |u(t)|, and let Y = X × X be the Banach space with the norm ∥(u, v)∥ Y = ∥u∥ + ∥v∥.We also introduce the cone P ⊂ Y by We now define the operator A : P → Y, A(u, v) = (A 1 (u, v), A 2 (u, v)), for (u, v) ∈ P, where A 1 , A 2 : P → X are given by for all t ∈ [0, 1] and (u, v) ∈ P.
We observe that (u, v) constitutes a positive solution to problem (49) (or equivalently, (1),( 2)) if and only if it serves as a fixed point for the operator A. Consequently, our subsequent analysis will focus on examining the existence of fixed points for A.
Under assumptions (H1) and (H2), using standard arguments, we deduce that operator A is completely continuous.In addition, by Lemma 9, we obtain that is, A(P ) ⊂ P.
For ϖ = q n , with n ∈ N, we introduce the constants We remark that L 2 , L 3 , L 2 , and Our initial existence result for positive solutions to problem (1),( 2) relies on the Guo-Krasnosel'skii fixed-point theorem (refer to [28]).
In the final theorem, we will employ the Banach contraction mapping principle.
Theorem 4. Assume that (H1) and (H2) are satisfied.In addition, we suppose that there exist continuous functions H where then the boundary value problem given by ( 1),(2) possesses a unique positive solution (u Additionally, an error estimate is provided by the following inequality: Proof.By using (74), Lemma 4, and Lemma 8, for any (u 1 , v 1 ), (u 2 , v 2 ) ∈ P, we obtain In a similar manner, we deduce Therefore, by ( 78) and (79), we conclude By satisfying condition (75), we establish that the operator A is a contraction mapping.Consequently, according to the Banach fixed-point theorem, it follows that A possesses a unique fixed point (u * , v * ) ∈ P.This fixed point corresponds to the unique positive solution of problem (1), (2).Furthermore, for any (u 0 , v 0 ) ∈ P, the sequence ((u n , v n )) n≥0 defined by (u n , v n ) = A(u n−1 , v n−1 ) for n ≥ 1 converges to (u * , v * ) as n → ∞.The proof of the Banach theorem yields the error estimate (77).
Example 1.We consider the functions f(t, u, v, x, y) = (u + v) 4 + u 3(1+u) + v 5(1+v) + x 1/3 + 1 6 e −y , ∀ t ∈ [0, 1], u, v ≥ 0, u + v ≤ 1,   with an operator, and the fixed points of this operator correspond to the positive solutions of (49).Our primary results hinge on the application of the Guo-Krasnosel'skii fixed-point theorem (in Theorem 1), the Leggett-Williams fixed-point theorem (for Theorem 2), the Schauder fixed-point theorem (in the case of Theorem 3), and the Banach contraction mapping principle (in the context of Theorem 4).In the second-to-last section of the paper, we provide several examples to elucidate and illustrate the implications of our results.

) Definition 4 .
If a ∈ [0, b] and f is defined in the interval [0, b], then its q-integral from a to b is given by b