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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.
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]).
In this paper, we will investigate the system of fractional q-difference equations
supplemented with the multi-point boundary conditions
Here, , , , , , ; , , , , , , , , , , ; , ; , and for , , , ; is the fractional q-derivative of order for , , , , ; is the q-derivative of order i for and ; and is the fractional q-integral of order for .
We will establish conditions on the functions f and g under which problem (1),(2) possesses at least one positive solution. Our proofs will leverage several 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. Subsequently, we will provide references to pertinent papers that are closely related to our investigated problem. In [2], the authors studied the solvability for the system of nonlinear fractional q-difference equations
subject to the coupled nonlocal boundary conditions
where , , , , , , , , , , , and the integrals in (4) are Riemann–Stieltjes integrals with - and -bounded variation functions. By using varied fixed-point theorems, they obtained the existence and uniqueness results for the solutions of problem (3),(4). In [3], the authors examined the existence of solutions for the fractional q-difference equation with nonlinear integral conditions
where , , , and is the Caputo fractional q-derivative of order . In the proof of the main result, they utilized measures of noncompactness and the Mönch fixed-point theorem. In [4], the authors investigated the existence, uniqueness, and multiplicity of positive solutions to the fractional q-difference equation with the nonlocal boundary conditions
where , , , , , , and satisfies Caratheodory-type conditions. In the proof of the main theorems, they applied several fixed-point theorems. In [5], by using the Guo–Krasnosel’skii fixed-point theorem, the author analyzed the existence of positive solutions to the fractional q-difference equation with the boundary conditions
where , , , and is a nonnegative continuous function. In [6], the author explored the existence of nontrivial solutions to the nonlinear q-fractional boundary value problem
where , , and is a nonnegative continuous function. To prove the main finding, he also used the Guo–Krasnosel’skii fixed-point theorem. In [7], the authors proved the existence of solutions to the second-order q-difference equation with the boundary conditions
where , , , and is a fixed real number. In [8], the authors studied the existence of solutions to the second-order q-difference equation subject to the boundary conditions
where , , is a fixed constant, is a fixed real number, and . Regarding additional papers that investigate systems of fractional q-difference equations with either coupled or uncoupled boundary conditions or that focus on fractional q-difference equations specifically, we refer to the following papers: [9,10,11,12,13,14,15,16].
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 for the unknown function at the point 1 is contingent upon the q-fractional derivatives of various orders for both functions and at different points within the interval . Similarly, the q-fractional derivative of order for the unknown function at the point 1 is linked to the q-fractional derivatives of distinct orders for functions and at diverse points within the interval . 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.
2. 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.
Let . Define the number
Next, to introduce the q-gamma function, we present the q-analog of the power function with :
For , define
If , then .
The q-gamma function is defined by
This function satisfies the relation .
Definition1.
The q-derivative of a real function is defined by
Definition2.
The q-derivatives of higher order of a real function are defined by
Definition3.
The q-integral of a function defined in the interval is defined by
Definition4.
If and is defined in the interval , then its q-integral from a to b is given by
Definition5.
The q-integrals of a function of higher order are defined by
The fundamental theorem of q-calculus says that , and if is continuous at , then . The properties of the operators and are presented in [6,27]. Below, we present some properties that will be used later.
Lemma1.
For , and , we have
(i) ;
(ii) ;
(iii) If , then ;
(iv) If , then ;
(v) If and , then ;
(vi) ,
where denotes the q-derivative with respect to variable .
Definition6
([21]). Let be a function defined on . The fractional q-integral of the Riemann–Liouville type of order is defined by and
Definition7
([24]). The fractional q-derivative of the Riemann–Liouville type of order is defined by and
where m is the smallest integer greater than or equal to α.
Lemma2
([21,24]). Let , , , and , and let be a function defined on . Then, the following relations are satisfied:
(a) ;
(b) ;
(c) If , then ;
(d) ;
(e) .
Lemma3
([6]). Let and p be a positive integer. Then, the following relation holds:
Lemma4.
If , then for , we have
where .
Proof.
By Definition 6 and Lemma 2 (e) (with and ), 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
Problem (1),(2) can be expressed equivalently to the following system of fractional q-integral equations:
Let be the Banach space endowed with the norm , and let be the Banach space with the norm . We also introduce the cone by
We now define the operator , , for , where are given by
for all and .
We observe that constitutes a positive solution to problem (49) (or equivalently, (1),(2)) if and only if it serves as a fixed point for the operator . Consequently, our subsequent analysis will focus on examining the existence of fixed points for .
Under assumptions and , using standard arguments, we deduce that operator is completely continuous. In addition, by Lemma 9, we obtain
that is, .
For , with , we introduce the constants
We remark that and and and .
Our initial existence result for positive solutions to problem (1),(2) relies on the Guo–Krasnosel’skii fixed-point theorem (refer to [28]).
Theorem1.
Let with . Assume that and are satisfied. In addition, we suppose that there exist two positive constants and the constants , , , and such that
(H3)
(H4)
Then, problem (1),(2) has at least one positive solution, such thatand .
Proof.
We introduce the set . Then, for , we have , so for all . Because and , by Lemma 4, we find , and for all , . We define and for .
Then, by Lemma 8 and , we obtain
and
Then, we deduce
that is,
Now, we define the set . Then, for , we have , so for all .
Therefore, by Lemma 8 and , we obtain
and
Then, we conclude
and so
As we mentioned before, the operator is completely continuous. Then, by (57), (61), and the Guo–Krasnosel’skii fixed-point theorem, we deduce that the operator has a fixed point , which is a solution of problem (1),(2). This solution satisfies , , for all , , for all ; because , we obtain or , that is, for all or for all . □
Subsequently, we will establish the existence of at least three positive solutions to problem (1),(2) using the Leggett–Williams fixed-point theorem (refer to Theorem 3.3 in [29] or Theorem 2.3 in [30]).
Theorem2.
Letwith . Assume thatandare satisfied. In addition, we suppose that there exist positive constantssuch that
(H5)
(H6)
(H7)
Then, problem (1),(2) has at least three positive solutions