Existence and Compactness of the Solution Set for a Coupled Caputo Fractional System with ϕ-Laplacian Operators and Nonlocal Boundary Conditions

Abstract

In this paper, we investigate a class of coupled fractional differential systems involving Caputo derivatives and nonlinear $\varphi$-Laplacian operators subject to nonlocal boundary conditions. By transforming the problem into an equivalent integral system via appropriate Green’s functions, the existence of solutions is studied within a generalized Banach space framework. Using a Leray–Schauder type fixed point theorem and suitable growth conditions on the nonlinear terms, we establish the existence of at least one bounded solution. Furthermore, we prove that the solution set is compact. An illustrative example involving the p-Laplacian operator is provided to demonstrate the applicability of the obtained theoretical results.

1. Introduction

Fractional calculus has emerged as a powerful mathematical framework for modeling processes with memory and hereditary effects. Such phenomena naturally arise in a wide range of applications including viscoelasticity, control theory, anomalous diffusion, biology, and fluid dynamics [1,2,3,4,5]. Compared to classical integer-order models, fractional differential equations provide greater flexibility and often yield more accurate descriptions of complex dynamical systems. As a result, the qualitative analysis of fractional differential equations has attracted significant attention in recent years [6,7].

Among the various definitions of fractional derivatives, the Caputo derivative is particularly suitable for applications since it allows the incorporation of initial and boundary conditions in terms of integer-order derivatives. This feature has led to extensive studies on boundary value problems (BVPs) involving Caputo derivatives. In parallel, nonlinear fractional differential equations have been widely investigated due to their ability to model complex nonlinear phenomena.

A prominent class of nonlinear operators arising in differential equations is the p-Laplacian operator and its generalization, the $\varphi$-Laplacian operator. These operators appear naturally in nonlinear diffusion, non-Newtonian fluid mechanics, and reaction–diffusion systems. The $\varphi$-Laplacian framework extends the classical p-Laplacian and allows the treatment of a broader class of nonlinearities. Consequently, fractional differential equations involving $\varphi$-Laplacian operators have been extensively studied in the literature [8,9,10,11,12,13].

Another important aspect in the study of fractional BVPs is the nature of boundary conditions. In particular, nonlocal boundary conditions, where the value of the solution or its derivatives at a point depends on values at other points, arise naturally in many real-world applications such as heat transfer, population dynamics, and control systems. However, the presence of nonlocal conditions significantly increases the mathematical complexity of the problem, especially when combined with nonlinear operators and fractional derivatives.

In recent years, several works have addressed fractional boundary value problems with nonlinear operators and nonlocal conditions. For instance, Zibar et al. [14] studied existence and stability results for coupled systems involving mixed fractional derivatives. Jiang and Sun [15] investigated positive solutions for p-Laplacian fractional differential equations with nonlocal boundary conditions. Alghanmi [16] analyzed nonlocal BVPs involving generalized fractional derivatives in Banach spaces, while Batit Özen [17] considered ℵ-Caputo fractional problems with p-Laplacian operators. Furthermore, numerical aspects of nonlinear fractional problems have been explored in [18].

Despite these contributions, most existing results focus either on single equations, specific types of fractional derivatives, or standard functional settings. Moreover, the analysis of coupled systems involving $\varphi$-Laplacian operators, Caputo derivatives, and nonlocal boundary conditions within a generalized Banach space framework remains relatively limited.

Motivation and contribution.

The main objective of this paper is to fill this gap by studying a class of coupled fractional systems that simultaneously involve:

• Caputo fractional derivatives of order in $(1,2]$,
• nonlinear $\varphi$-Laplacian operators,
• nonlocal boundary conditions of both function and derivative type,
• and a generalized Banach space setting.

The novelty of our work can be summarized as follows:

• We consider a coupled system rather than a single equation, which introduces additional analytical difficulties due to the interaction between components.
• We treat a general $\varphi$-Laplacian operator, which extends many existing results restricted to the classical p-Laplacian case.
• We combine Caputo derivatives with nonlocal boundary conditions, providing a framework that captures both memory effects and spatial interactions.
• We establish both existence and compactness of the solution set in a generalized Banach space, which strengthens the qualitative analysis of the problem.

Drawing from these considerations, we study the following coupled system:

$$\left\{\begin{array}{c}{}^{c}D_{0^{+}}^{\alpha 1}\left({\varphi }_1\left({}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)\right)\right)=f(\xi ,x\left(\xi \right),y\left(\xi \right)),\xi \in [0,1],\\ {}^{c}D_{0^{+}}^{\alpha 2}\left({\varphi }_2\left({}^{c}D_{0^{+}}^{\beta 2}y\left(\xi \right)\right)\right)=g(\xi ,x\left(\xi \right),y\left(\xi \right)),\xi \in [0,1],\\ x\left(0\right)=0,x\left(1\right)=a_{1}x\left({\lambda }_1\right),{}^{c}D_{0^{+}}^{\beta 1}x\left(0\right)=0,{}^{c}D_{0^{+}}^{\beta 1}x\left(1\right)=b_1{}^{c}D_{0^{+}}^{\beta 1}x\left({\mu }_1\right),\\ y\left(0\right)=0,y\left(1\right)=a_{2}y\left({\lambda }_2\right),{}^{c}D_{0^{+}}^{\beta 2}y\left(0\right)=0,{}^{c}D_{0^{+}}^{\beta 2}y\left(1\right)=b_2{}^{c}D_{0^{+}}^{\beta 2}y\left({\mu }_2\right).\end{array}\right.$$

(1)

Here, ${\alpha }_i,{\beta }_i\in (1,2]$, ${\lambda }_i,{\mu }_i\in (0,1)$, $a_i\in [0,1]$, and $b_i\in \left(0,{\varphi }_i^{-1}\left({\mu }_i^{-1}\right)\right)$ for $i=1,2$.

The presence of coupling, nonlinearity, and nonlocal constraints makes problem (1) highly nontrivial and requires refined analytical techniques.

To solve this problem, we first construct suitable Green’s functions and transform the system into an equivalent system of integral equations. This reformulation allows us to apply a Leray–Schauder type fixed point theorem in a generalized Banach space framework. Under appropriate growth conditions, we establish the existence of at least one bounded solution and prove that the solution set is compact.

The paper is organized as follows. Section 2 presents the necessary preliminaries. Section 3 is devoted to the construction of Green’s functions and the derivation of the equivalent integral formulation. Section 4 contains the main existence and compactness results. Finally, Section 5 provides an illustrative example.

2. Preliminaries

This section is devoted to introducing some fundamental concepts that will play a crucial role in the analysis carried out in the following sections.

Definition 1

([19]). Let $0\le a<b<+\infty$ and let $\alpha >0$. The left-sided Caputo fractional derivative of order α of a function h is defined by

$${}^{c}D_{\mathrm{a}+}^{\alpha }h\left(\zeta \right)={\int }_a^{\zeta }{\displaystyle \frac{{(\zeta -\xi )}^{n-\alpha -1}}{\mathsf{\Gamma }(n-\alpha )}}{h}^{\left(n\right)}\left(\xi \right)d\xi ,\zeta >a,$$

where $n=\left[\alpha \right]+1$.

Definition 2

([20]). Let X be a nonempty set. A mapping

$$d:X\times X⟶{\mathbb{R}}_{+}^m$$

is called a generalized metric in the sense of Perov if it satisfies the usual metric axioms componentwise. The pair $(X,d)$ is then referred to as a generalized metric space.

Within a Perov-type generalized metric space, fundamental concepts such as sequence convergence, Cauchy criteria, completeness, and the characterization of open and closed sets are formulated in a manner similar to their counterparts in standard metric space theory.

Let $\vartheta ,r\in {\mathbb{R}}^m$ with $\vartheta =({\vartheta }_1,{\vartheta }_2,\dots ,{\vartheta }_m)$ and $r=(r_1,r_2,\dots ,r_m)$. We write $\vartheta \le r$ if ${\vartheta }_i\le r_i$ for all $i\in \{1,\dots ,m\}$, and $\vartheta <r$ if ${\vartheta }_i<r_i$ for every i. Moreover, we denote

$$\left|\vartheta \right|:=\left(\right|{\vartheta }_1|,\dots ,|{\vartheta }_m\left|\right),max(u,\vartheta ):=(max(u_1,{\vartheta }_1),\dots ,max(u_m,{\vartheta }_m)).$$

If $c\in \mathbb{R}$, the notation $\vartheta \le c$ means ${\vartheta }_i\le c$ for all i.

For $x_0\in X$ and $r\in {\mathbb{R}}_{+}^m$, we define the open ball

$$B(x_0,r):=\{x\in X:d(x_0,x)<r\},$$

and the closed ball

$$\overline{B}(x_0,r):=\{x\in X:d(x_0,x)\le r\}.$$

We now recall a Leray–Schauder type alternative adapted to generalized Banach spaces.

Theorem 1

([21,22]). Let X be a generalized Banach space and let $N:X\to X$ be a completely continuous operator. Then one of the following assertions holds:

(i) 

the fixed-point equation $N\left(x\right)=x$ admits at least one solution in X;

(ii) 

the set

$$\mathcal{M}=\{x\in X:\mu N(x)=x,\mu \in (0,1\left)\right\}$$

is unbounded.

3. Auxiliary Results

To rigorously introduce the concept of a solution for problem (1), we first present several lemmas that will serve as the foundation for the subsequent analysis.

Lemma 1.

Let $\vartheta :[0,1]⟶\mathbb{R}$; ${\beta }_1\in (1,2]$; $a_1\in [0,1]$ and ${\lambda }_1\in (0,1)$. The unique solution of the problem

$${}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)=\vartheta \left(\xi \right),\xi \in [0,1]$$

(2)

$$x\left(0\right)=0,x\left(1\right)=a_{1}x\left({\lambda }_1\right)$$

(3)

is given by

$${\displaystyle x\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta )\vartheta \left(\zeta \right)d\zeta ,}$$

where $G_1$ is the green function given by

$${\displaystyle G_1(\xi ,\zeta )=\frac{1}{(a_1{\lambda }_1-1)\mathsf{\Gamma }\left({\beta }_1\right)}\left\{\begin{array}{c}\xi {(1-\zeta )}^{{\beta }_1-1}-a_1\xi {({\lambda }_1-\zeta )}^{{\beta }_1-1}+(a_1{\lambda }_1-1){(\xi -\zeta )}^{{\beta }_1-1},\hfill \\ 0\le \zeta \le \xi \le 1,\zeta \le {\lambda }_1,\hfill \\ \xi {(1-\zeta )}^{{\beta }_1-1}-(a_1{\lambda }_1-1){(\xi -\zeta )}^{{\beta }_1-1},0<{\lambda }_1\le \zeta \le \xi \le 1,\hfill \\ \xi {(1-\zeta )}^{{\beta }_1-1}-a_1\xi {({\lambda }_1-\zeta )}^{{\beta }_1-1},0\le \xi \le \zeta \le {\lambda }_1<1,\hfill \\ \xi {(1-\zeta )}^{{\beta }_1-1},0\le \xi \le \zeta \le 1,{\lambda }_1\le \zeta .\hfill \end{array}\right.}$$

Proof. 

Let ${\beta }_1\in (1,2]$ and $\vartheta \in C\left(\right[0,1],\mathbb{R})$. We consider the fractional differential equation

$${}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)=\vartheta \left(\xi \right),\xi \in [0,1],$$

(4)

subject to the boundary conditions

$$x\left(0\right)=0,x\left(1\right)=a_{1}x\left({\lambda }_1\right).$$

(5)

Step 1: General solution of the fractional equation.

Applying the fractional integral operator $I_{0^{+}}^{{\beta }_1}$ to both sides of (4), and using the well-known property of Caputo derivatives, we obtain

$$x\left(\xi \right)=c_0+c_1\xi +I_{0^{+}}^{{\beta }_1}\vartheta \left(\xi \right),$$

where $c_0,c_1\in \mathbb{R}$ are constants and

$$I_{0^{+}}^{{\beta }_1}\vartheta \left(\xi \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left({\beta }_1\right)}}{\int }_0^{\xi }{(\xi -\zeta )}^{{\beta }_1-1}\vartheta \left(\zeta \right)d\zeta .$$

Step 2: Determination of the constants.

Using the condition $x\left(0\right)=0$, we immediately obtain

$$c_0=0.$$

Next, we evaluate $x\left(1\right)$ and $x\left({\lambda }_1\right)$:

$$x\left(1\right)=c_1+I_{0^{+}}^{{\beta }_1}\vartheta \left(1\right),x\left({\lambda }_1\right)=c_1{\lambda }_1+I_{0^{+}}^{{\beta }_1}\vartheta \left({\lambda }_1\right).$$

Substituting into the boundary condition $x\left(1\right)=a_{1}x\left({\lambda }_1\right)$, we get

$$c_1+I_{0^{+}}^{{\beta }_1}\vartheta \left(1\right)=a_1{\lambda }_1{c}_1+a_1{I}_{0^{+}}^{{\beta }_1}\vartheta \left({\lambda }_1\right).$$

Solving for $c_1$, we obtain

$$c_1={\displaystyle \frac{I_{0^{+}}^{{\beta }_1}\vartheta \left(1\right)-a_1{I}_{0^{+}}^{{\beta }_1}\vartheta \left({\lambda }_1\right)}{a_1{\lambda }_1-1}}.$$

Step 3: Integral representation of the solution.

Substituting $c_0=0$ and the expression of $c_1$ into the general solution, we obtain

$$x\left(\xi \right)={\displaystyle \frac{\xi }{a_1{\lambda }_1-1}}\left(I_{0^{+}}^{{\beta }_1}\vartheta \left(1\right)-a_1{I}_{0^{+}}^{{\beta }_1}\vartheta \left({\lambda }_1\right)\right)+I_{0^{+}}^{{\beta }_1}\vartheta \left(\xi \right).$$

Using the integral representation of the fractional integral, we write

$$I_{0^{+}}^{{\beta }_1}\vartheta \left(1\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left({\beta }_1\right)}}{\int }_0^1{(1-\zeta )}^{{\beta }_1-1}\vartheta \left(\zeta \right)d\zeta ,$$

$$I_{0^{+}}^{{\beta }_1}\vartheta \left({\lambda }_1\right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left({\beta }_1\right)}}{\int }_0^{{\lambda }_1}{({\lambda }_1-\zeta )}^{{\beta }_1-1}\vartheta \left(\zeta \right)d\zeta .$$

Hence,

$$x\left(\xi \right)\begin{array}{c}={\int }_0^1{\displaystyle \frac{\xi {(1-\zeta )}^{{\beta }_1-1}}{(a_1{\lambda }_1-1)\mathsf{\Gamma }\left({\beta }_1\right)}}\vartheta \left(\zeta \right)d\zeta \hfill \\ -{\int }_0^{{\lambda }_1}{\displaystyle \frac{a_1\xi {({\lambda }_1-\zeta )}^{{\beta }_1-1}}{(a_1{\lambda }_1-1)\mathsf{\Gamma }\left({\beta }_1\right)}}\vartheta \left(\zeta \right)d\zeta \hfill \\ +{\int }_0^{\xi }{\displaystyle \frac{{(\xi -\zeta )}^{{\beta }_1-1}}{\mathsf{\Gamma }\left({\beta }_1\right)}}\vartheta \left(\zeta \right)d\zeta .\hfill \end{array}$$

Step 4: Construction of the Green’s function.

To express the solution in the form

$$x\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta )\vartheta \left(\zeta \right)d\zeta ,$$

we split the domain according to the relative positions of $\zeta$, $\xi$, and ${\lambda }_1$.

• Case 1: $0\le \zeta \le \xi \le 1$, $\zeta \le {\lambda }_1$.

All three integrals contribute, yielding

$$G_1(\xi ,\zeta )={\displaystyle \frac{1}{(a_1{\lambda }_1-1)\mathsf{\Gamma }\left({\beta }_1\right)}}\left[\xi {(1-\zeta )}^{{\beta }_1-1}-a_1\xi {({\lambda }_1-\zeta )}^{{\beta }_1-1}+(a_1{\lambda }_1-1){(\xi -\zeta )}^{{\beta }_1-1}\right].$$

Case 2: $0<{\lambda }_1\le \zeta \le \xi \le 1$.

The second integral vanishes, hence

$$G_1(\xi ,\zeta )={\displaystyle \frac{1}{(a_1{\lambda }_1-1)\mathsf{\Gamma }\left({\beta }_1\right)}}\left[\xi {(1-\zeta )}^{{\beta }_1-1}-(a_1{\lambda }_1-1){(\xi -\zeta )}^{{\beta }_1-1}\right].$$

Case 3: $0\le \xi \le \zeta \le {\lambda }_1$.

The third integral vanishes, giving

$$G_1(\xi ,\zeta )={\displaystyle \frac{1}{(a_1{\lambda }_1-1)\mathsf{\Gamma }\left({\beta }_1\right)}}\left[\xi {(1-\zeta )}^{{\beta }_1-1}-a_1\xi {({\lambda }_1-\zeta )}^{{\beta }_1-1}\right].$$

Case 4: $0\le \xi \le \zeta \le 1$, ${\lambda }_1\le \zeta$.

Only the first integral contributes, hence

$$G_1(\xi ,\zeta )={\displaystyle \frac{\xi {(1-\zeta )}^{{\beta }_1-1}}{(a_1{\lambda }_1-1)\mathsf{\Gamma }\left({\beta }_1\right)}}.$$

Combining all cases, we conclude that

$$x\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta )\vartheta \left(\zeta \right)d\zeta ,$$

where $G_1(\xi ,\zeta )$ is the Green’s function defined above.

• This completes the proof.  □

Lemma 2.

Let $\vartheta :[0,1]⟶\mathbb{R}$; ${\alpha }_1,{\beta }_1\in (1,2]$; $a_1,b_1\in [0,1]$ and ${\lambda }_1,{\mu }_1\in (0,1)$. The fractional BVP

$${}^{c}D_{0^{+}}^{\alpha 1}\left({\varphi }_1\left({}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)\right)\right)=\vartheta \left(\xi \right),\xi \in [0,1]$$

(6)

$$x\left(0\right)=0,x\left(1\right)=a_{1}x\left({\lambda }_1\right),{}^{c}D_{0^{+}}^{\beta 1}x\left(0\right)=0,{}^{c}D_{0^{+}}^{\beta 1}x\left(1\right)=b_1{}^{c}D_{0^{+}}^{\beta 1}x\left({\mu }_1\right)$$

(7)

has a unique solution given by

$${\displaystyle x\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta ){\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )\vartheta \left(\eta \right)d\eta }\right)d\zeta ,}$$

where $H_1$ is the green function given by

$${\displaystyle H_1(\xi ,\zeta )=\frac{1}{({\varphi }_1\left(b_1\right){\mu }_1-1)\mathsf{\Gamma }\left({\alpha }_1\right)}\left\{\begin{array}{c}\xi {(1-\zeta )}^{{\alpha }_1-1}-{\varphi }_1\left(b_1\right)\xi {({\mu }_1-\zeta )}^{{\alpha }_1-1}+({\varphi }_1\left(b_1\right){\mu }_1-1){(\xi -\zeta )}^{{\alpha }_1-1},\hfill \\ 0\le \zeta \le \xi \le 1,\zeta \le {\mu }_1,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_1-1}-({\varphi }_1\left(b_1\right){\mu }_1-1){(\xi -\zeta )}^{{\alpha }_1-1},\hfill \\ 0<{\mu }_1\le \zeta \le \xi \le 1,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_1-1}-{\varphi }_1\left(b_1\right)\xi {({\mu }_1-\zeta )}^{{\alpha }_1-1},0\le \xi \le \zeta \le {\mu }_1<1,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_1-1},0\le \xi \le \zeta \le 1,{\mu }_1\le \zeta .\hfill \end{array}\right.}$$

Proof. 

Let ${\alpha }_1,{\beta }_1\in (1,2]$ and $\vartheta \in C\left(\right[0,1],\mathbb{R})$. We consider the fractional boundary value problem

$${}^{c}D_{0^{+}}^{\alpha 1}\left({\varphi }_1\left({}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)\right)\right)=\vartheta \left(\xi \right),\xi \in [0,1],$$

(8)

subject to

$$x\left(0\right)=0,x\left(1\right)=a_{1}x\left({\lambda }_1\right),{}^{c}D_{0^{+}}^{\beta 1}x\left(0\right)=0,{}^{c}D_{0^{+}}^{\beta 1}x\left(1\right)=b_1{}^{c}D_{0^{+}}^{\beta 1}x\left({\mu }_1\right).$$

(9)

Step 1: Reduction of order.

Set

$$y\left(\xi \right)={\varphi }_1\left({}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)\right).$$

Then problem (8) reduces to

$${}^{c}D_{0^{+}}^{\alpha 1}y\left(\xi \right)=\vartheta \left(\xi \right).$$

Applying the fractional integral operator $I_{0^{+}}^{{\alpha }_1}$, we obtain

$$y\left(\xi \right)=c_0+c_1\xi +I_{0^{+}}^{{\alpha }_1}\vartheta \left(\xi \right),$$

where

$$I_{0^{+}}^{{\alpha }_1}\vartheta \left(\xi \right)={\displaystyle \frac{1}{\mathsf{\Gamma }\left({\alpha }_1\right)}}{\int }_0^{\xi }{(\xi -\zeta )}^{{\alpha }_1-1}\vartheta \left(\zeta \right)d\zeta .$$

Step 2: Determination of constants.

Using ${}^{c}D_{0^{+}}^{\beta 1}x\left(0\right)=0$, we get

$$y\left(0\right)={\varphi }_1\left(0\right)=0\Rightarrow c_0=0.$$

Next, evaluate $y\left(1\right)$ and $y\left({\mu }_1\right)$:

$$y\left(1\right)=c_1+I_{0^{+}}^{{\alpha }_1}\vartheta \left(1\right),y\left({\mu }_1\right)=c_1{\mu }_1+I_{0^{+}}^{{\alpha }_1}\vartheta \left({\mu }_1\right).$$

From the boundary condition

$${}^{c}D_{0^{+}}^{\beta 1}x\left(1\right)=b_1{}^{c}D_{0^{+}}^{\beta 1}x\left({\mu }_1\right),$$

and using the multiplicative property of ${\varphi }_1$, we obtain

$$y\left(1\right)={\varphi }_1\left(b_1\right)y\left({\mu }_1\right).$$

Thus,

$$c_1+I_{0^{+}}^{{\alpha }_1}\vartheta \left(1\right)={\varphi }_1\left(b_1\right)\left(c_1{\mu }_1+I_{0^{+}}^{{\alpha }_1}\vartheta \left({\mu }_1\right)\right).$$

Solving for $c_1$, we obtain

$$c_1={\displaystyle \frac{I_{0^{+}}^{{\alpha }_1}\vartheta \left(1\right)-{\varphi }_1\left(b_1\right)I_{0^{+}}^{{\alpha }_1}\vartheta \left({\mu }_1\right)}{{\varphi }_1\left(b_1\right){\mu }_1-1}}.$$

Step 3: Integral representation of $y\left(\xi \right)$.

Substituting $c_0=0$ and $c_1$ into $y\left(\xi \right)$ gives

$$y\left(\xi \right)={\displaystyle \frac{\xi }{{\varphi }_1\left(b_1\right){\mu }_1-1}}\left(I_{0^{+}}^{{\alpha }_1}\vartheta \left(1\right)-{\varphi }_1\left(b_1\right)I_{0^{+}}^{{\alpha }_1}\vartheta \left({\mu }_1\right)\right)+I_{0^{+}}^{{\alpha }_1}\vartheta \left(\xi \right).$$

Using the integral form, we obtain

$$y\left(\xi \right)\begin{array}{c}={\int }_0^1{\displaystyle \frac{\xi {(1-\zeta )}^{{\alpha }_1-1}}{({\varphi }_1\left(b_1\right){\mu }_1-1)\mathsf{\Gamma }\left({\alpha }_1\right)}}\vartheta \left(\zeta \right)d\zeta \hfill \\ -{\int }_0^{{\mu }_1}{\displaystyle \frac{{\varphi }_1\left(b_1\right)\xi {({\mu }_1-\zeta )}^{{\alpha }_1-1}}{({\varphi }_1\left(b_1\right){\mu }_1-1)\mathsf{\Gamma }\left({\alpha }_1\right)}}\vartheta \left(\zeta \right)d\zeta \hfill \\ +{\int }_0^{\xi }{\displaystyle \frac{{(\xi -\zeta )}^{{\alpha }_1-1}}{\mathsf{\Gamma }\left({\alpha }_1\right)}}\vartheta \left(\zeta \right)d\zeta .\hfill \end{array}$$

Step 4: Construction of the Green’s function $H_1$.

We express

$$y\left(\xi \right)={\int }_0^1{H}_1(\xi ,\zeta )\vartheta \left(\zeta \right)d\zeta .$$

Splitting the domain:

• Case 1: $0\le \zeta \le \xi \le 1$, $\zeta \le {\mu }_1$

$$H_1(\xi ,\zeta )={\displaystyle \frac{1}{({\varphi }_1\left(b_1\right){\mu }_1-1)\mathsf{\Gamma }\left({\alpha }_1\right)}}\left[\xi {(1-\zeta )}^{{\alpha }_1-1}-{\varphi }_1\left(b_1\right)\xi {({\mu }_1-\zeta )}^{{\alpha }_1-1}+({\varphi }_1\left(b_1\right){\mu }_1-1){(\xi -\zeta )}^{{\alpha }_1-1}\right].$$

Case 2: ${\mu }_1\le \zeta \le \xi \le 1$

$$H_1(\xi ,\zeta )={\displaystyle \frac{1}{({\varphi }_1\left(b_1\right){\mu }_1-1)\mathsf{\Gamma }\left({\alpha }_1\right)}}\left[\xi {(1-\zeta )}^{{\alpha }_1-1}-({\varphi }_1\left(b_1\right){\mu }_1-1){(\xi -\zeta )}^{{\alpha }_1-1}\right].$$

Case 3: $0\le \xi \le \zeta \le {\mu }_1$

$$H_1(\xi ,\zeta )={\displaystyle \frac{1}{({\varphi }_1\left(b_1\right){\mu }_1-1)\mathsf{\Gamma }\left({\alpha }_1\right)}}\left[\xi {(1-\zeta )}^{{\alpha }_1-1}-{\varphi }_1\left(b_1\right)\xi {({\mu }_1-\zeta )}^{{\alpha }_1-1}\right].$$

Case 4: $0\le \xi \le \zeta \le 1$, ${\mu }_1\le \zeta$

$$H_1(\xi ,\zeta )={\displaystyle \frac{\xi {(1-\zeta )}^{{\alpha }_1-1}}{({\varphi }_1\left(b_1\right){\mu }_1-1)\mathsf{\Gamma }\left({\alpha }_1\right)}}.$$

Thus,

$$y\left(\xi \right)={\int }_0^1{H}_1(\xi ,\zeta )\vartheta \left(\zeta \right)d\zeta .$$

Step 5: Recovery of $x\left(\xi \right)$.

Since $y\left(\xi \right)={\varphi }_1\left({}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)\right)$, we obtain

$${}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)={\varphi }_1^{-1}\left({\int }_0^1{H}_1(\xi ,\zeta )\vartheta \left(\zeta \right)d\zeta \right).$$

Thus, the problem reduces to a fractional equation of the form

$${}^{c}D_{0^{+}}^{\beta 1}x\left(\xi \right)=\tilde{\vartheta }\left(\xi \right),$$

where

$$\tilde{\vartheta }\left(\xi \right)={\varphi }_1^{-1}\left({\int }_0^1{H}_1(\xi ,\zeta )\vartheta \left(\zeta \right)d\zeta \right).$$

Applying Lemma 1, we conclude that

$$x\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta ){\varphi }_1^{-1}\left({\int }_0^1{H}_1(\zeta ,\eta )\vartheta \left(\eta \right)d\eta \right)d\zeta .$$

This completes the proof.  □

For ${\alpha }_2,{\beta }_2\in (1,2]$; $a_2,b_2\in [0,1]$ and ${\lambda }_2,{\mu }_2\in (0,1)$, similar results are obtained for the following BVP

$${}^{c}D_{0^{+}}^{\alpha 2}\left({\varphi }_2\left({}^{c}D_{0^{+}}^{\beta 2}y\left(\xi \right)\right)\right)=\vartheta \left(\xi \right),\xi \in [0,1]$$

(10)

$$y\left(0\right)=0,y\left(1\right)=a_{2}x\left({\lambda }_2\right),{}^{c}D_{0^{+}}^{\beta 2}y\left(0\right)=0,{}^{c}D_{0^{+}}^{\beta 2}y\left(1\right)=b_2{}^{c}D_{0^{+}}^{\beta 2}x\left({\mu }_2\right)$$

(11)

its unique solution is given by

$${\displaystyle y\left(\xi \right)={\int }_0^1{G}_2(\xi ,\zeta ){\varphi }_2^{-1}\left({\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )\vartheta \left(\eta \right)d\eta }\right)d\zeta ,}$$

where

$${\displaystyle G_2(\xi ,\zeta )=\frac{1}{(a_2{\lambda }_2-1)\mathsf{\Gamma }\left({\beta }_2\right)}\left\{\begin{array}{c}\xi {(1-\zeta )}^{{\beta }_2-1}-a_2\xi {({\lambda }_2-\zeta )}^{{\beta }_2-1}+(a_2{\lambda }_2-1){(\xi -\zeta )}^{{\beta }_2-1},\hfill \\ 0\le \zeta \le \xi \le 1,\zeta \le {\lambda }_2,\hfill \\ \xi {(1-\zeta )}^{{\beta }_2-1}-(a_2{\lambda }_2-1){(\xi -\zeta )}^{{\beta }_2-1},0<{\lambda }_2\le \zeta \le \xi \le 1,\hfill \\ \xi {(1-\zeta )}^{{\beta }_2-1}-a_2\xi {({\lambda }_2-\zeta )}^{{\beta }_2-1},0\le \xi \le \zeta \le {\lambda }_2<1,\hfill \\ \xi {(1-\zeta )}^{{\beta }_2-1},0\le \xi \le \zeta \le 1,{\lambda }_2\le \zeta .\hfill \end{array}\right.}$$

and

$${\displaystyle H_2(\xi ,\zeta )=\frac{1}{({\varphi }_2\left(b_2\right){\mu }_2-1)\mathsf{\Gamma }\left({\alpha }_2\right)}\left\{\begin{array}{c}\xi {(1-\zeta )}^{{\alpha }_2-1}-{\varphi }_2\left(b_2\right)\xi {({\mu }_2-\zeta )}^{{\alpha }_2-1}+({\varphi }_2\left(b_2\right){\mu }_2-1){(\xi -\zeta )}^{{\alpha }_2-1},\hfill \\ 0\le \zeta \le \xi \le 1,\zeta \le {\mu }_2,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_2-1}-({\varphi }_2\left(b_2\right){\mu }_2-1){(\xi -\zeta )}^{{\alpha }_2-1},\hfill \\ 0<{\mu }_2\le \zeta \le \xi \le 1,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_2-1}-{\varphi }_2\left(b_2\right)\xi {({\mu }_2-\zeta )}^{{\alpha }_2-1},0\le \xi \le \zeta \le {\mu }_2<1,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_2-1},0\le \xi \le \zeta \le 1,{\mu }_2\le \zeta .\hfill \end{array}\right.}$$

Discussion on the Growth Conditions

In this work, the assumptions imposed on the nonlinear functions f and g are of growth-type and are commonly used in the study of fractional boundary value problems. These conditions play a crucial role in ensuring that the associated operator is well-defined, continuous, and compact in the underlying functional space, which allows the application of the Leray–Schauder fixed-point theorem.

More precisely, the imposed growth conditions guarantee that the nonlinear terms do not grow too rapidly with respect to the unknown functions. This ensures that the corresponding integral operator maps bounded sets into bounded and equicontinuous sets, which is essential for obtaining compactness via the Arzelà–Ascoli theorem.

We emphasize that these assumptions are sufficiently general to include a broad class of nonlinearities frequently encountered in applications, including polynomial-type and sublinear growth behaviors. Moreover, they are consistent with the hypotheses adopted in many related works on fractional differential equations involving p-Laplacian and $\varphi$-Laplacian operators.

It is also worth noting that, although these conditions may not be minimal, they represent a natural and effective framework for applying topological fixed-point techniques such as the Leray–Schauder theorem. Weakening these assumptions would generally require the use of alternative analytical methods, such as monotone operator theory, upper and lower solution techniques, or measures of noncompactness, which are beyond the scope of the present study.

Therefore, the chosen hypotheses strike a balance between generality and mathematical tractability, allowing us to establish the existence and compactness of solutions in a rigorous and unified framework.

4. Main Results

This section examines the existence of solutions for problem (1). We begin by defining the appropriate function spaces. Consider the interval $J=[0,1]$, and let $C(J,\mathbb{R})$ represent the set of continuous functions from J to $\mathbb{R}$. From this space, we select the Banach space

$$C_b:=\{\chi \in C(J,\mathbb{R})\mid \chi \mathrm{is}\mathrm{bounded}\},$$

equipped with the norm

$${\parallel \chi \parallel }_{C_b}:=\underset{\xi \in J}{sup}\left|\chi \left(\xi \right)\right|.$$

Next, we define the generalized Banach space $\mathsf{\Gamma }:=C_b\times C_b,$ with the norm

$${\displaystyle \parallel }({\chi }_1,{\chi }_2)\parallel :=max\{\parallel {\chi }_1{\parallel }_{C_b},\parallel {\chi }_2{\parallel }_{C_b}\}.$$

With these spaces established, the following result can be deduced directly from Lemmas 1 and 2.

Lemma 3.

Consider the continuous functions $f,g:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$. The coupled system (1) then admits a unique solution $(x,y)\in \mathsf{\Gamma }$ given by

$$\left\{\begin{array}{c}{\displaystyle x\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta ){\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)d\zeta }\\ {\displaystyle y\left(\xi \right)={\int }_0^1{G}_2(\xi ,\zeta ){\varphi }_2^{-1}\left({\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)d\zeta ,}\end{array}\right.$$

(12)

where $G_i$ and $H_i$ for $i=1,2$ are the Green functions given by

$${\displaystyle G_i(\xi ,\zeta )=\frac{1}{(a_i{\lambda }_i-1)\mathsf{\Gamma }\left({\beta }_i\right)}\left\{\begin{array}{c}\xi {(1-\zeta )}^{{\beta }_i-1}-a_i\xi {({\lambda }_i-\zeta )}^{{\beta }_i-1}+(a_i{\lambda }_i-1){(\xi -\zeta )}^{{\beta }_i-1},\hfill \\ 0\le \zeta \le \xi \le 1,\zeta \le {\lambda }_i,\hfill \\ \xi {(1-\zeta )}^{{\beta }_i-1}-(a_i{\lambda }_i-1){(\xi -\zeta )}^{{\beta }_i-1},0<{\lambda }_i\le \zeta \le \xi \le 1,\hfill \\ \xi {(1-\zeta )}^{{\beta }_i-1}-a_i\xi {({\lambda }_i-\zeta )}^{{\beta }_i-1},0\le \xi \le \zeta \le {\lambda }_i<1,\hfill \\ \xi {(1-\zeta )}^{{\beta }_i-1},0\le \xi \le \zeta \le 1,{\lambda }_i\le \zeta .\hfill \end{array}\right.}$$

and

$${\displaystyle H_i(\xi ,\zeta )=\frac{1}{({\varphi }_i\left(b_i\right){\mu }_i-1)\mathsf{\Gamma }\left({\alpha }_i\right)}\left\{\begin{array}{c}\xi {(1-\zeta )}^{{\alpha }_i-1}-{\varphi }_i\left(b_i\right)\xi {({\mu }_i-\zeta )}^{{\alpha }_i-1}+({\varphi }_i\left(b_i\right){\mu }_i-1){(\xi -\zeta )}^{{\alpha }_i-1},\hfill \\ 0\le \zeta \le \xi \le 1,\zeta \le {\mu }_i,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_i-1}-({\varphi }_i\left(b_i\right){\mu }_i-1){(\xi -\zeta )}^{{\alpha }_i-1},\hfill \\ 0<{\mu }_i\le \zeta \le \xi \le 1,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_i-1}-{\varphi }_i\left(b_i\right)\xi {({\mu }_i-\zeta )}^{{\alpha }_i-1},0\le \xi \le \zeta \le {\mu }_i<1,\hfill \\ \xi {(1-\zeta )}^{{\alpha }_i-1},0\le \xi \le \zeta \le 1,{\mu }_i\le \zeta .\hfill \end{array}\right.}$$

Definition 3.

A pair of functions $(x,y)\in \mathsf{\Gamma }$ is called a solution of problem (1.1) if:

• x and y are continuous and bounded on $[0,1]$,
• the system of fractional differential equations in (1.1) is satisfied for all $\xi \in [0,1]$,
• all the prescribed boundary conditions associated with problem (1.1) are fulfilled.

Now we assume the following hypothesis which will be used in the sequel.

${\mathcal{H}}_1$

There exist continuous nondecreasing functions ${\psi }_i,{\phi }_i:[0,\infty ]⟶[0,\infty )$ and nonnegative functions $h_i,k_i,l_i\in L^1(J,\mathbb{R})$, i = 1, 2 such that

$$\left|f(\xi ,x,y)\right|\le h_1\left(\xi \right){\psi }_1\left(\right|x\left|\right)+k_1\left(\xi \right){\phi }_1\left(\right|y\left|\right)+l_1\left(\xi \right),$$

$$\left|g(\xi ,x,y)\right|\le h_2\left(\xi \right){\psi }_2\left(\right|x\left|\right)+k_2\left(\xi \right){\phi }_2\left(\right|y\left|\right)+l_2\left(\xi \right),$$

for $\xi \in J$ and $x,y\in \mathbb{R}$.

${\mathcal{H}}_2$

Growth condition under $\varphi$: the functions ${\psi }_i$ and ${\phi }_i$ for $i=1,2$ from (${\mathcal{H}}_1$) satisfy

$${\displaystyle \underset{\varrho \to \infty }{lim}\frac{{\psi }_i\left(\varrho \right)+{\phi }_i\left(\varrho \right)}{{\varphi }_i\left(\varrho \right)}=0.}$$

(13)

Set

$${\displaystyle G_i^{*}:=\underset{0\le \xi \le 1}{sup}{\int }_0^1|G_i(\xi ,\zeta )|d\zeta and{H}_i^{*}:=\underset{0\le \xi \le 1}{sup}{\int }_0^1\left|H_i(\xi ,\zeta )\right|d\zeta fori=1,2.}$$

Theorem 2.

Suppose hypotheses $\left({\mathcal{H}}_1\right)$ and $\left({\mathcal{H}}_2\right)$ hold. Then problem (1) admits at least one bounded solution. Furthermore, the solution set

$$S=\left\{\right(x,y)\in \mathsf{\Gamma }:(x,y\left)solves\right(\left)\right\}$$

is compact.

Proof. 

Reformulate problem (1) as an equivalent fixed-point equation. Define the operator $N:\mathsf{\Gamma }\to \mathsf{\Gamma }$ by

$$N(x,y)=\left(N_1(x,y),N_2(x,y)\right).$$

where

$${\displaystyle N_1(x,y)={\int }_0^1{G}_1(\xi ,\zeta ){\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)d\zeta }$$

and

$${\displaystyle N_2(x,y)={\int }_0^1{G}_2(\xi ,\zeta ){\varphi }_2^{-1}\left({\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)d\zeta .}$$

Initially, we verify that the operator N is well-defined. Take $(x,y)\in \mathsf{\Gamma }$ and $\xi \in J$. In this case,

$${\displaystyle {\displaystyle |}{N}_1(x,y){\displaystyle |}\le {\int }_0^1\left|G_1(\xi ,\zeta )\right|\left|{\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)\right|d\zeta }$$

Since

$$\begin{array}{ccc}\left|{\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right|d\eta \hfill & \le \hfill & {\displaystyle H_1^{*}{\int }_0^1\left(h_1\left(\eta \right){\psi }_1\left(\right|x\left|\right)+k_1\left(\eta \right){\phi }_1\left(\right|y\left|\right)+l_1\left(\eta \right)\right)d\eta }\hfill \\ \hfill & \le \hfill & H_1^{*}\left(\parallel h_1{\parallel }_{L^1}{\psi }_1\left({\parallel x\parallel }_{C_b}\right)+\parallel k_1{\parallel }_{L^1}{\phi }_1\left({\parallel y\parallel }_{C_b}\right)+{\parallel l_1\parallel }_{L^1}\right):={\ell }_1\hfill \end{array}$$

(14)

Similarly we prove that

$$\left|{\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right|d\eta \le H_2^{*}\left(\parallel h_2{\parallel }_{L^1}{\psi }_2\left({\parallel x\parallel }_{C_b}\right)+\parallel k_2{\parallel }_{L^1}{\phi }_2\left({\parallel y\parallel }_{C_b}\right)+{\parallel l_2\parallel }_{L^1}\right):={\ell }_2,$$

(15)

it follows that

$${\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta \in \overline{B}(0,{\ell }_1),}$$

and

$${\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta \in \overline{B}(0,{\ell }_2).}$$

Since ${\varphi }_i^{-1}$, $i=1,2$ are continuous, then

$${\displaystyle \underset{\chi \in \overline{B}(0,{\ell }_i)}{sup}\left|{\varphi }_i^{-1}\left(\chi \right)\right|<\infty ,}$$

thus

$${\displaystyle \parallel N_1{(x,y)\parallel }_{C_b}\le G_1^{*}\underset{\chi \in \overline{B}(0,{\ell }_1)}{sup}\left|{\varphi }_1^{-1}\left(\chi \right)\right|,}$$

$${\displaystyle \parallel N_2{(x,y)\parallel }_{C_b}\le G_2^{*}\underset{\chi \in \overline{B}(0,{\ell }_2)}{sup}\left|{\varphi }_2^{-1}\left(\chi \right)\right|.}$$

Thus, the operator N is well-defined.

Evidently, fixed points of N correspond to solutions of problem (1). Now we aim to verify that N meet the criteria of Theorem 1 which we will accomplish in four stages.

Step 1:

Invariance under the operators.

We will show that N maps bounded sets into bounded sets in $\mathsf{\Gamma }$, it suffices to show that for any $r=(r_1,r_2)$ there exists a positive constant vector $L=(L_1,L_2)$ such that for each $(x,y)\in B_r=\{(x,y)\in \mathsf{\Gamma }:\parallel (x,y)\parallel \le r\}$ we have $\parallel N(x,y)\parallel \le L$.

For each $(x,y)\in B_r$ we obtain by (14) and (15)

$$\left|{\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right|d\eta \le H_1^{*}\left(\parallel h_1{\parallel }_{L^1}{\psi }_1\left(r_1\right)+\parallel k_1{\parallel }_{L^1}{\phi }_2\left(r_2\right)+{\parallel l_2\parallel }_{L^1}\right):={\tilde{\ell }}_1,$$

and

$$\left|{\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right|d\eta \le H_2^{*}\left(\parallel h_2{\parallel }_{L^1}{\psi }_2\left(r_1\right)+\parallel k_2{\parallel }_{L^1}{\phi }_2\left(r_2\right)+{\parallel l_2\parallel }_{L^1}\right):={\tilde{\ell }}_2,$$

it follows that

$${\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta \in \overline{B}(0,{\tilde{\ell }}_1),}$$

and

$${\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta \in \overline{B}(0,{\tilde{\ell }}_2).}$$

Since ${\varphi }_i^{-1}$, $i=1,2$ are continuous, then we obtain

$${\displaystyle \parallel N_1{(x,y)\parallel }_{C_b}\le G_1^{*}\underset{\chi \in \overline{B}(0,{\tilde{\ell }}_1)}{sup}\left|{\varphi }_1^{-1}\left(\chi \right)\right|:=L_1,}$$

and

$${\displaystyle \parallel N_2{(x,y)\parallel }_{C_b}\le G_2^{*}\underset{\chi \in \overline{B}(0,{\tilde{\ell }}_2)}{sup}\left|{\varphi }_2^{-1}\left(\chi \right)\right|:=L_2.}$$

Hence

$$\left(\parallel \left(N_1(x,y)\parallel ,\parallel N_2(x,y)\right)\parallel \right)\le (L_1,L_2)$$

Step 2:

Continuity of N.

Let ${\left((x_n,y_n)\right)}_n$ be a sequence that converges to $(x,y)$ in $B_r$, then for each $\xi \in J$, we have

$$|N_1(x_n,y_n)-N_1(x,y)| \le \begin{array}{c}{\displaystyle G_1^{*}{\int }_0^1\left({\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x_n\left(\eta \right),y_n\left(\eta \right))d\eta }\right)\right.}\hfill \\ -\left.{\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)\right)d\zeta .\hfill \end{array}$$

By the dominated convergence theorem, we have

$${\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x_n\left(\eta \right),y_n\left(\eta \right))d\eta -{\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta ⟶0asn⟶\infty .}$$

Since ${\varphi }_1^{-1}$ is continuous, we obtain

$$|N_1(x_n,y_n)-N_1(x,y)| \le \begin{array}{c}{\displaystyle G_1^{*}{\int }_0^1\left({\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x_n\left(\eta \right),y_n\left(\eta \right))d\eta }\right)\right.}\hfill \\ -\left.{\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)\right)d\zeta ⟶0asn⟶\infty .\hfill \end{array}$$

Thus

$${\displaystyle \parallel }{N}_1(x_n,y_n)-N_1(x,y){{\displaystyle \parallel }}_{C_b}⟶0asn⟶\infty$$

Similarly we obtain

$${\displaystyle \parallel }{N}_2(x_n,y_n)-N_2(x,y){{\displaystyle \parallel }}_{C_b}⟶0asn⟶\infty$$

Step 3:

Equicontinuity of N.

We will show that N maps bounded sets into equicontinuous sets in $\mathsf{\Gamma }$. Let ${\xi }_1,{\xi }_2\in [0,1]$ be such that ${\xi }_1<{\xi }_2$ and let $(x,y)\in B_r$. Then we have

$$\left|N_1(x\left({\xi }_2\right),y\left({\xi }_2\right))-N_1(x\left({\xi }_1\right)-y\left({\xi }_1\right))\right|\begin{array}{c} \le {\displaystyle {\int }_0^1\left|{\varphi }^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)\right|\left|G_1({\xi }_2,\zeta )-G_1({\xi }_1,\zeta )\right|d\zeta }\hfill \\ \le {\displaystyle \underset{\chi \in \overline{B}(0,{\tilde{\ell }}_1)}{sup}\left|{\varphi }_1^{-1}\left(\chi \right)\right|{\int }_0^1\left|G_1({\xi }_2,\zeta )-G_1({\xi }_1,\zeta )\right|d\zeta }\hfill \end{array}$$

As ${\xi }_1\to {\xi }_2$, the right-hand side of the above inequality tends to zero. Similarly we obtain

$$\left|N_2(x\left({\xi }_2\right),y\left({\xi }_2\right))-N_2(x\left({\xi }_1\right)-y\left({\xi }_1\right))\right|⟶0as{\xi }_1\to {\xi }_2$$

Step 4:

We will show that the set $\mathcal{M}$ defined by

$$\mathcal{M}=\left\{\right(x,y)\in \mathsf{\Gamma }:(x,y)=\mu N(x,y),\mu \in (0,1\left)\right\}$$

is bounded.

Let $(x,y)\in \mathcal{M}$, then there exists $\mu \in (0,1)$ such that

$$x=\mu N_1(x,y)y=\mu N_2(x,y).$$

We have

$${\parallel x\parallel }_{C_b}\le G_1^{*}{\varphi }_1^{-1}\left(P_1{\psi }_1{(\parallel x\parallel }_{C_b})+Q_1{\phi }_1{(\parallel y\parallel }_{C_b})+K_1\right)$$

and

$${\parallel y\parallel }_{C_b}\le G_2^{*}{\varphi }_2^{-1}\left(P_2{\psi }_2{(\parallel x\parallel }_{C_b})+Q_2{\phi }_2{(\parallel y\parallel }_{C_b})+K_2\right),$$

where for $i=1,2$

$$P_i=H_i^{*}\parallel h_i{\parallel }_{L^1},Q_i=H_i^{*}\parallel k_i{\parallel }_{L^1},K_i=H_i^{*}{\parallel l_i\parallel }_{L^1}.$$

Suppose that $\mathcal{M}$ is unbounded, then there exists a sequence $(x_n,y_n)\in \mathcal{M}$ such that

$${\varrho }_n:=\parallel (x_n,y_n)\parallel ⟶\infty asn\to \infty .$$

From the inequalities above we obtain

$${\varrho }_n\le G^{*}{\varphi }^{-1}\left(P\psi \left({\varrho }_n\right)+Q\phi \left({\varrho }_n\right)+K\right)$$

where

$$G^{*}=max(G_1^{*},G_2^{*})$$

and

$${\varphi }^{-1}\left({P\psi (\parallel x\parallel }_{C_b}{)+Q\phi (\parallel y\parallel }_{C_b})+K\right)=max\left({\varphi }_i^{-1}\left(P_i{\psi }_i{(\parallel x\parallel }_{C_b})+Q_i{\phi }_i{(\parallel y\parallel }_{C_b})+K_i\right)\right),i=1,2.$$

Thus we obtain

$${\displaystyle 1\le \varphi \left(G^{*}\right)\frac{P\psi \left({\varrho }_n\right)+Q\phi \left({\varrho }_n\right)+K}{\varphi \left({\varrho }_n\right)},}$$

as $n\to \infty$ and by hypothesis (${\mathcal{H}}_2$), the right-hand side of the previous inequality tends to zero which is absurd. As a consequence of the alternative of Leray-Schauder type we deduce that the operator N has at least one bounded solution.

Now we show that S is compact. It is clear that $S\subset N\left(S\right)$, from step 4 we deduce that S is bounded set in $\mathsf{\Gamma }$. Since N is compact, S is compact if and only if S is closed.

Let ${\left\{(x_n,y_n)\right\}}_n$ be a sequence in S converges to $(x,y)$. Thus

$${\displaystyle x_n\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta ){\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x_n\left(\eta \right),y_n\left(\eta \right))d\eta }\right)d\zeta }$$

and

$${\displaystyle y_n\left(\xi \right)={\int }_0^1{G}_2(\xi ,\zeta ){\varphi }_2^{-1}\left({\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x_n\left(\eta \right),y_n\left(\eta \right))d\eta }\right)d\zeta .}$$

Following the approach of Step 2, the proof proceeds analogously.

$${\displaystyle x\left(\xi \right)={\int }_0^1{G}_1(\xi ,\zeta ){\varphi }_1^{-1}\left({\displaystyle {\int }_0^1{H}_1(\zeta ,\eta )f(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)d\zeta ,\xi \in J}$$

and

$${\displaystyle y\left(\xi \right)={\int }_0^1{G}_2(\xi ,\zeta ){\varphi }_2^{-1}\left({\displaystyle {\int }_0^1{H}_2(\zeta ,\eta )g(\eta ,x\left(\eta \right),y\left(\eta \right))d\eta }\right)d\zeta ,\xi \in J}$$

This imlies that S is compact.  □

5. Example

In this section, we present a concrete example to illustrate the applicability of our main theoretical results and to provide further insight into the qualitative behavior of the considered system.

Consider the fractional boundary value problem

$$\left\{\begin{array}{c}{}^{c}D_{0^{+}}^{{\displaystyle \frac{5}{3}}}\left({\varphi }_1\left({}^{c}D_{0^{+}}^{{\displaystyle \frac{7}{5}}}x\left(\xi \right)\right)\right)={\displaystyle \frac{(1+{\xi }^2)x^q}{ln(1+x)}}+(2+e^{\xi })y^q+\xi +1,\hfill \\ {}^{c}D_{0^{+}}^{{\displaystyle \frac{6}{5}}}\left({\varphi }_2\left({}^{c}D_{0^{+}}^{{\displaystyle \frac{3}{2}}}y\left(\xi \right)\right)\right)=(2+\xi )(x^q+1)+(1+{\xi }^3)(y^q+1)+e^{\xi },\hfill \\ x\left(0\right)=0,x\left(1\right)={\displaystyle \frac{1}{4}}x\left({\displaystyle \frac{1}{2}}\right),{}^{c}D_{0^{+}}^{{\displaystyle \frac{7}{5}}}x\left(0\right)=0,{}^{c}D_{0^{+}}^{{\displaystyle \frac{7}{5}}}x\left(1\right)=7^{{\displaystyle \frac{1}{2p}}}{}^{c}D_{0^{+}}^{{\displaystyle \frac{7}{5}}}x\left({\displaystyle \frac{1}{7}}\right),\\ y\left(0\right)=0,y\left(1\right)={\displaystyle \frac{1}{6}}y\left({\displaystyle \frac{1}{3}}\right),{}^{c}D_{0^{+}}^{{\displaystyle \frac{3}{2}}}y\left(0\right)=0,{}^{c}D_{0^{+}}^{{\displaystyle \frac{3}{2}}}y\left(1\right)=5^{{\displaystyle \frac{1}{2p}}}{}^{c}D_{0^{+}}^{{\displaystyle \frac{3}{2}}}y\left({\displaystyle \frac{1}{5}}\right),\end{array}\right.$$

(16)

where

$$f(\xi ,x,y)={\displaystyle \frac{(1+{\xi }^2)x^q}{ln(1+x)}}+(2+e^{\xi })y^q+\xi +1,$$

$$g(\xi ,x,y)=(2+\xi )(x^q+1)+(1+{\xi }^3)(y^q+1)+e^{\xi }.$$

Step 1: Verification of the operator assumptions.

Let ${\varphi }_1\left(x\right)={\varphi }_2\left(x\right)=x^p$ with $p\ge 1$. Then ${\varphi }_i$$(i=1,2)$ are continuous, strictly increasing homeomorphisms satisfying ${\varphi }_i\left(0\right)=0$ and the multiplicative property. Hence, the structural assumptions on the $\varphi$-Laplacian operators are fulfilled.

• Step 2: Decomposition of the nonlinearities.

Define

$${\psi }_1\left(x\right)={\displaystyle \frac{x^q}{ln(1+x)}},{\psi }_2\left(x\right)=x^q+1,$$

$${\phi }_1\left(y\right)=y^q,{\phi }_2\left(y\right)=y^q+1,$$

and

$$\begin{array}{ccc}{h}_1\left(\xi \right)=1+{\xi }^2,& k_1\left(\xi \right)=2+e^{\xi },& l_1\left(\xi \right)=\xi +1,\\ h_2\left(\xi \right)=2+\xi ,& k_2\left(\xi \right)=1+{\xi }^3,& l_2\left(\xi \right)=e^{\xi }.\end{array}$$

Thus, the nonlinear functions can be written in the form required by hypothesis $\left({\mathcal{H}}_1\right)$, which ensures that f and g satisfy appropriate growth conditions with respect to x and y.

• Step 3: Verification of the growth conditions.

Since $q<p$, we have

$$\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\psi }_1\left(\varrho \right)}{{\varphi }_1\left(\varrho \right)}}=\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\varrho }^q}{ln(1+\varrho ){\varrho }^p}}=0,\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\phi }_1\left(\varrho \right)}{{\varphi }_1\left(\varrho \right)}}=\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\varrho }^q}{{\varrho }^p}}=0.$$

Similarly,

$$\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\psi }_2\left(\varrho \right)}{{\varphi }_2\left(\varrho \right)}}=\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\varrho }^q+1}{{\varrho }^p}}=0,\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\phi }_2\left(\varrho \right)}{{\varphi }_2\left(\varrho \right)}}=\underset{\varrho \to \infty }{lim}{\displaystyle \frac{{\varrho }^q+1}{{\varrho }^p}}=0.$$

Therefore, hypothesis $\left({\mathcal{H}}_2\right)$ is satisfied.

• Step 4: Interpretation of the model.

The nonlinear terms in (16) combine polynomial-type growth with logarithmic damping (through $ln(1+x)$) and exponential dependence on $\xi$. This reflects a balance between nonlinear amplification and saturation effects. In particular:

• The term ${\displaystyle \frac{x^q}{ln(1+x)}}$ grows slower than $x^q$, ensuring subcritical growth.
• The coupling terms $y^q$ and $x^q$ describe mutual interaction between the two state variables.
• The exponential terms $e^{\xi }$ introduce non-uniform forcing along the domain.

These features illustrate the flexibility of the theoretical framework in handling complex nonlinear behaviors.

• Step 5: Conclusion.

All the assumptions of Theorem 2 are satisfied. Hence, problem (16) admits at least one bounded solution on $[0,1]$.

Remark 1.

Although the present work is mainly theoretical, the above example provides a suitable framework for numerical investigation. In particular, iterative schemes based on the associated integral formulation could be used to approximate solutions. Numerical simulations and graphical representations would provide additional insight into the qualitative behavior of the system and will be considered in future work.

Physical interpretation. Although nonlocal boundary conditions are often associated with Riemann–Liouville fractional derivatives, their use in the Caputo framework is also meaningful from an applied perspective.

The Caputo derivative is particularly suitable for modeling systems where initial conditions are given in terms of integer-order derivatives, which is common in physical applications such as velocity and displacement in mechanics.

The nonlocal boundary conditions considered in this work, for example,

$$x\left(1\right)=a_{1}x\left({\lambda }_1\right),{}^{c}D_{0^{+}}^{\beta 1}x\left(1\right)=b_1{}^{c}D_{0^{+}}^{\beta 1}x\left({\mu }_1\right),$$

represent global constraints linking the state of the system at the boundary to its values at interior points.

Such conditions naturally arise in several applications, including:

• heat transfer processes with distributed measurements,
• population dynamics with spatial interactions,
• control systems where terminal states depend on intermediate states,
• anomalous diffusion processes involving memory effects.

Therefore, the combination of Caputo fractional derivatives with nonlocal boundary conditions allows the modeling of systems that simultaneously incorporate memory effects and spatial interactions, thereby extending the applicability of the model.

6. Conclusions

In this paper, we investigated a coupled system of fractional differential equations involving Caputo derivatives, nonlinear $\varphi$-Laplacian operators, and nonlocal boundary conditions. By constructing appropriate Green’s functions, the original BVP was transformed into an equivalent system of integral equations, which allowed the problem to be studied within a suitable generalized Banach space framework.

Under appropriate growth assumptions on the nonlinear terms, we applied a Leray–Schauder type fixed-point theorem to establish the existence of at least one bounded solution for the considered system. In addition, we proved that the set of solutions is compact, providing further insight into the topological structure of the solution space. These results extend and generalize several existing works on fractional differential equations by allowing coupled systems, general $\varphi$-Laplacian operators, and nonlocal boundary conditions involving fractional derivatives.

An illustrative example involving the p-Laplacian operator was presented to demonstrate the applicability of the theoretical results and to verify the validity of the imposed hypotheses. The example confirms that the developed approach can handle a wide class of nonlinear fractional models.

The results obtained in this work contribute to the growing literature on nonlinear fractional BVPs and open the door to further investigations. Possible directions for future research include the study of uniqueness and stability of solutions, the extension to variable-order or impulsive fractional systems, and the analysis of similar problems under more general boundary conditions or in different functional settings.