Structural Analysis of Coupled ψ-Hilfer Pantograph Langevin Systems via Measure of Noncompactness

Abstract

This paper investigates a class of coupled $\psi$-Hilfer fractional pantograph–Langevin equations with nonlocal integral boundary conditions. By reformulating the problem as an equivalent fixed point equation and employing Mönch’s fixed point theorem together with the Kuratowski measure of noncompactness, we establish sufficient conditions for the existence of at least one solution. Under additional Lipschitz-type assumptions, we prove Ulam–Hyers stability on a suitable closed ball and derive explicit, computable stability constants. A concrete numerical example is presented in which all hypotheses are verified and the stability constants are explicitly computed (e.g., $K_1\approx 3.811$, $K_2\approx 2.761$), illustrating the applicability of the theoretical results. The study contributes additional qualitative results to the analysis of fractional pantograph–Langevin systems within the unified framework of $\psi$-Hilfer fractional derivatives.

1. Introduction

In recent times, fractional calculus has become a popular and significant tool. It involves an expansion of the differentiation and integration concepts to non-integer orders, which allows for the modeling of the behavior and dynamics of complex systems that ordinary calculus cannot fully describe [1,2,3]. Fractional-order differential equations have proven to be highly beneficial, especially in fields like finance, biology, engineering, economics, and physics. Several research fields, including control theory, signal processing, and image processing, have utilized fractional calculus. It has also been employed to solve problems related to heat conduction, diffusion, and wave propagation. Furthermore, fractional calculus has shown potential in modeling and analyzing phenomena such as anomalous diffusion, viscoelasticity, and fractional Brownian motion [4,5]. Recently, pantograph equations with proportional delay have gained attention in modeling electrodynamics and population dynamics, with Sun et al. [6] investigating $\psi$-Hilfer pantograph equations at resonance, and Louakar et al. [7] analyzing Hilfer pantograph Langevin equations with nonlocal conditions.

Fractional-order differential equations are proving to be more accurate and useful than their ordinary counterparts, as they can capture complex behavior and dynamics that are often hard to model using traditional equations. Over the past few years, this approach to analysis has produced a wealth of results, as demonstrated by studies such as [8,9,10,11,12]. The theoretical analysis is complemented by numerical investigations, as demonstrated by Goedegebure and Marynets [13], who studied Hilfer fractional problems with fractional-periodic conditions and provided numerical approximations.

For example, in [11], the authors employed Darbo’s fixed point theorem to establish both the existence and uniqueness of a solution to a fractional differential equation (FDE) that incorporates the Hadamard fractional derivative (H-FD) with variable order. The FDE consists of a system of equations in which the H-FD and a continuous function act on a variable function within a designated interval, subject to specific boundary conditions. The Hadamard fractional derivative and fractional integral of variable order play crucial roles in the formulation of the FDE:

$$\left\{\begin{array}{c}{}^H{D}^{\rho \left(\kappa \right)}u\left(\kappa \right)={\mathcal{F}}_1(\kappa ,u\left(\kappa \right)),\kappa \in [1,T],\hfill \\ u\left(1\right)=0,u\left(T\right)=\sum _{j=1}^m{\delta }_{1j}u\left({\eta }_j\right),\hfill \end{array}\right.$$

where $\rho \in (1,2],{\mathcal{F}}_1:[1,T]\times \mathbb{R}\to \mathbb{R}$ is continuous, and ${\delta }_{1j}\in \mathbb{R}$.

The investigation carried out in [8] involves the theory of existence for a particular class of coupled nonlinear systems of sequential FDEs. This class of equations involves coupled, non-conjugate, Riemann–Stieltjes, and non-conjugate integral-multipoint boundary conditions:

$$\left\{\begin{array}{c}{}^c{D}^{\rho }u\left(\kappa \right)={\mathcal{I}}_1(\kappa ,u\left(\kappa \right),v\left(\kappa \right)),\hfill \\ {}^c{D}^{{\sigma }_i}v\left(\kappa \right)={\mathcal{I}}_2(\kappa ,u\left(\kappa \right),v\left(\kappa \right)),\hfill \end{array}\right.$$

with coupled boundary conditions

$$\left\{\begin{array}{c}u\left(0\right)=0,u\left(1\right)={\int }_0^{1}u\left(s\right)dA\left(s\right)+\sum _{j=1}^m{b}_{j}v\left({\xi }_j\right),\hfill \\ v\left(0\right)=0,v\left(1\right)={\int }_0^{1}v\left(s\right)dA\left(s\right)+\sum _{j=1}^m{c}_{j}u\left({\eta }_j\right),\hfill \end{array}\right.$$

where ${}^c{D}^{\rho }$ denotes the Caputo fractional derivative of order $p, 0<\rho <{\sigma }_i<v<1$, ${\mathcal{I}}_1,{\mathcal{I}}_2:[0,1]\times {\mathbb{R}}^2\to \mathbb{R}$ are continuous, and A is a function of bounded variation.

The article referenced as [9] investigates the existence and uniqueness of a multi-point boundary value problem with H-FD using a sequential approach:

$$\left\{\begin{array}{c}{}^H{D}^{\rho }\left({}^H{D}^{\omega }u\left(\kappa \right)\right)={\mathcal{F}}_1(\kappa ,u\left(\kappa \right)),\kappa \in [1,T],\hfill \\ u\left(1\right)=0,u\left(T\right)=\sum _{j=1}^m{\delta }_{1j}u\left({\eta }_j\right),\hfill \end{array}\right.$$

where ${}^H{D}^{\rho }$ is the Hadamard fractional derivative of order $\rho ,\omega >0$, and ${\delta }_{1j}\in \mathbb{R}$.

In [12], the authors utilized the theorems established by Banach and Schaefer to establish adequate criteria for ensuring the existence of solutions to the subsequent fractional differential equation with H-FD, and additionally, they ensured the stability of the problem

$$\left\{\begin{array}{c}{}^H{D}^{\rho }u\left(\kappa \right)={\mathcal{F}}_1(\kappa ,u\left(\kappa \right),v\left(\kappa \right)),\hfill \\ {}^H{D}^{\beta }v\left(\kappa \right)={\mathcal{F}}_2(\kappa ,u\left(\kappa \right),v\left(\kappa \right)),\kappa \in [1,T],\hfill \end{array}\right.$$

with coupled boundary conditions

$$\left\{\begin{array}{c}u\left(1\right)={\delta }_{1}v\left(1\right),u\left(T\right)={\delta }_{2}v\left(T\right),\hfill \end{array}\right.$$

where ${}^H{D}^{\theta }$ denotes the H-FD of order $\theta \in \{\rho ,\beta \}$, ${\mathcal{F}}_1,{\mathcal{F}}_2:[1,T]\times {\mathbb{R}}^2\to \mathbb{R}$ are given functions, and ${\delta }_1{\delta }_2\ne 1$.

In [14], using fixed point theory, the authors presented novel results on the existence and uniqueness of a generalized Caputo-type initial value problem with a delay. These results were based on the characteristics of a space of continuous and measurable functions and were very helpful in demonstrating the existence of a single solution to various systems defined with a generalized Caputo fractional derivative and a delay term. Authors in [15] dealt with a new category of the sequential hybrid inclusion boundary value problem with three-point integral-derivative boundary conditions.

The $\psi$-Hilfer fractional derivative represents a significant generalization of the above notions, unifying various fractional operators within a single framework. By choosing the function $\psi$ appropriately, one recovers classical derivatives: Riemann–Liouville ($\psi \left(t\right)=t$), Hadamard ($\psi \left(t\right)=lnt$), and Caputo-type variants. The $\psi$-Hilfer framework has been extensively developed in recent years. Choden et al. [16] established a generalized Laplace transform theory for $\psi$-Hilfer derivatives in weighted spaces, providing analytical tools for solving hybrid fractional problems. Verma and Sumelka [17] extended this framework to complex-order $\psi$-Hilfer equations, exploring stability in systems with memory and hereditary properties. Zerbib et al. [18] studied hybrid $\psi$-Hilfer equations with generalized proportional derivatives using topological degree theory. Furthermore, Erkan et al. [19] investigated coupled systems combining $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo sequential derivatives with non-separated boundary conditions.

Recently, an increasing number of physical phenomena, such as those related to plasma physics, fluid mechanics, electromagnetic theory, geophysics, and fluid motions, have been modeled by nonlinear fractional differential equations (NFDEs) [20,21,22,23]. Due to the precise results provided by these equations, many researchers have been drawn to explore them across several scientific fields and from different angles. Consequently, a range of methods and approaches have been created and suggested by scholars over time [10,24,25,26].

Further, stability concepts have been a subject of research for a considerable time and have found use in a diverse array of dynamic systems, including the Mittag–Leffler function, Lyapunov stability, and Ulam–Hyers stability. It is important to distinguish between these concepts: while Lyapunov stability concerns the long-term behavior of solutions near equilibrium points, Ulam–Hyers stability addresses a fundamentally different question—if we have an approximate solution, how close is it to an exact solution? This notion is particularly crucial for numerical approximations and real-world modeling, where exact solutions are often unattainable. These forms of stability have been extensively examined and enhanced in multiple research efforts, as evidenced by works like [27,28,29,30,31]. For coupled systems with delay, Benzarouala and Tunç [32] established Ulam–Hyers–Rassias stability for nonlinear Caputo integro-fractional delay coupled systems. Bettayeb et al. [33] established existence, uniqueness, and Ulam–Hyers stability for implicit neutral tempered $\psi$-Caputo fractional coupled systems with delay in generalized Banach spaces.

Boundary value problems (BVPs) are essential for understanding the behavior of solutions to differential equations within specific regions. Such problems usually require solving a differential equation while adhering to given boundary conditions. The study of BVPs is particularly significant in the realm of nonlinear fractional differential equations (NFDEs) within the field of fractional calculus (FC). Extending traditional differential equations to non-integer orders necessitates new problem-solving methods, and BVPs for NFDEs are no exception. In the case of physical phenomena modeled by NFDEs, the physical interpretation of boundary conditions (BCs) is crucial. For example, in a heat conduction problem, the temperature at the system’s boundaries plays a crucial role in determining the overall temperature profile of the material. Likewise, in a viscoelastic system, the stress or strain at the boundaries can impact the system’s overall response. Therefore, the boundary conditions provide a physical understanding of the differential equation solution, and the solution must fulfill these conditions to hold any physical significance. The study of BVPs for NFDEs has far-reaching implications across several fields, including physics, engineering, and biology, among others [34,35,36,37]. The physical interpretation of boundary conditions is particularly critical in these fields, and developing new techniques and methods for solving BVPs for NFDEs is an ongoing area of research within the field of fractional calculus [38,39,40,41].

In [42], the authors discussed the existence and stability of a nonlinear coupled system of sequential fractional differential equations that are subject to a new type of coupled boundary conditions. The authors used Schaefer’s fixed point theorem and Banach’s contraction mapping principle to prove the existence and uniqueness of the solutions to the problem. To assess the stability of the solutions, they employed the Hyers–Ulam type. Other studies cited include [8,28].

To the best of our knowledge, no study has investigated coupled $\psi$-Hilfer fractional pantograph Langevin equations with nonlocal integral boundary conditions. This gap in the literature motivates the present work.

In this article, we investigate the following coupled system of $\psi$-Hilfer fractional pantograph–Langevin equations with nonlocal integral boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_1,{\beta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_2,{\beta }_2;\psi }+{\nu }_1\right)\theta \left(\kappa \right)={\mathcal{J}}_1(\kappa ,\theta \left(\kappa \right),\theta \left({\lambda }_1\kappa \right),\varphi \left(\kappa \right)),\hfill \\ {}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_1,{\delta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_2,{\delta }_2;\psi }+{\nu }_2\right)\varphi \left(\kappa \right)={\mathcal{J}}_2(\kappa ,\varphi \left(\kappa \right),\varphi \left({\lambda }_2\kappa \right),\theta \left(\kappa \right)),\kappa \in [a,b],\hfill \\ \theta \left(a\right)=0,\theta \left(b\right)=\sum _{i=1}^n{\omega }_i\left({\mathcal{I}}^{{\sigma }_i;\psi }\theta \right)\left({\eta }_i\right),\hfill \\ \varphi \left(a\right)=0,\varphi \left(b\right)=\sum _{i=1}^n{\vartheta }_i\left({\mathcal{I}}^{{\pi }_i;\psi }\varphi \right)\left({\xi }_i\right),\hfill \end{array}\right.\end{array}$$

(1)

where ${}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\psi }$ denotes the $\psi$-Hilfer fractional derivative of order $\alpha \in (0,1)$ and type $\beta \in [0,1]$, ${\mathcal{I}}^{·;\psi }$ is the $\psi$-Riemann–Liouville fractional integral, ${\nu }_1,{\nu }_2\in \mathbb{R}$ are constants, ${\lambda }_1,{\lambda }_2\in (0,1)$ are pantograph parameters, ${\sigma }_i,{\pi }_i>0$ are the orders of the fractional integrals appearing in the boundary conditions, and ${\omega }_i,{\vartheta }_i\in \mathbb{R}$ are given coefficients. The functions ${\mathcal{J}}_1,{\mathcal{J}}_2:[a,b]\times {\mathbb{R}}^3\to \mathbb{R}$ are given nonlinearities.

Due to the lack of compactness of the domain and the presence of nonlocal boundary conditions, standard fixed point theorems like Banach’s or Schauder’s are not directly applicable. Therefore, we employ Mönch’s fixed point theorem in conjunction with the Kuratowski measure of noncompactness to establish the existence of a solution. Furthermore, we analyze the Ulam–Hyers stability of the system, providing explicit stability constants.

Our main contributions are:

(i)

Establishing the existence of at least one solution to (1) via Mönch’s fixed point theorem combined with the Kuratowski measure of noncompactness;

(ii)

Deriving sufficient conditions for Ulam–Hyers stability of the system with explicitly computable stability constants;

(iii)

Providing a concrete numerical example that verifies all theoretical assumptions.

The paper is organized as follows: Section 2 recalls essential preliminaries on $\psi$-fractional calculus and the Kuratowski measure of noncompactness. Section 3 presents the functional setting, operator formulation, and the main existence result using Mönch’s theorem. Section 4 develops the Ulam–Hyers stability analysis. Section 5 provides a numerical example illustrating the applicability of our results. Finally, Section 6 offers concluding remarks and directions for future research.

2. Preliminaries

In this section, we recall essential definitions and properties from fractional calculus and the theory of measure of noncompactness. Throughout this paper, let $\psi \in C^1([a,b],\mathbb{R})$ be a strictly increasing function with ${\psi }^{\prime }\left(\kappa \right)>0$ for all $\kappa \in [a,b]$.

2.1. Fractional Calculus

Definition 1

([5]). The Riemann–Liouville fractional integral of order $\alpha >0$ for a function f is defined by

$${\mathcal{I}}_{0^{+}}^{\alpha }f\left(\kappa \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\kappa }{(\kappa -v)}^{\alpha -1}f\left(v\right)dv,\kappa >0.$$

(2)

Definition 2

([5]). For $n-1<\alpha <n$ with $n\in \mathbb{N}$, the Riemann–Liouville fractional derivative of order α is

$$D_{0^{+}}^{\alpha }f\left(\kappa \right)={\displaystyle \frac{1}{\Gamma (n-\alpha )}}{\left({\displaystyle \frac{d}{d\kappa }}\right)}^n{\int }_0^{\kappa }{(\kappa -v)}^{n-\alpha -1}f\left(v\right)dv,\kappa >0.$$

(3)

Definition 3

([5]). The Caputo fractional derivative of order $r\in [n-1,n)$ for $f\in A{C}^n([0,\infty ),\mathbb{R})$ is defined by

$${}^c{D}_{0^{+}}^{r}f\left(\kappa \right)=D_{0^{+}}^r\left(f\left(\kappa \right)-\sum _{k=0}^{n-1}{\displaystyle \frac{{\kappa }^k}{k!}}{f}^{\left(k\right)}\left(0\right)\right),\kappa >0.$$

(4)

The $\psi$-fractional operators generalize the preceding notions and provide a flexible framework that includes several classical operators as special cases.

Definition 4

([5]). The ψ-Riemann–Liouville fractional integral of order $\alpha >0$ for a function f is defined by

$${\mathcal{I}}_{a^{+}}^{\alpha ;\psi }f\left(\kappa \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{\kappa }{\psi }^{\prime }\left(s\right){\left(\psi \left(\kappa \right)-\psi \left(s\right)\right)}^{\alpha -1}f\left(s\right)ds,\kappa >a.$$

(5)

Definition 5

([5]). The ψ-Hilfer fractional derivative of order $\alpha \in (n-1,n)$ and type $\beta \in [0,1]$ is denoted by ${}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\psi }$ and defined as

$${}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\psi }f\left(\kappa \right)={\mathcal{I}}_{a^{+}}^{\beta (n-\alpha );\psi }{\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\kappa \right)}}{\displaystyle \frac{d}{d\kappa }}\right)}^n{\mathcal{I}}_{a^{+}}^{(1-\beta )(n-\alpha );\psi }f\left(\kappa \right).$$

(6)

For convenience, we set

$$\gamma =\alpha +\beta (n-\alpha ),f_{\psi }^{\left[k\right]}\left(\kappa \right)={\left({\displaystyle \frac{1}{{\psi }^{\prime }\left(\kappa \right)}}{\displaystyle \frac{d}{d\kappa }}\right)}^{k}f\left(\kappa \right).$$

(7)

Lemma 1

([5]). Let $f\in C^n([a,b],\mathbb{R})$, $n-1<\alpha <n$, and $0\le \beta \le 1$. Then

$${\mathcal{I}}_{a^{+}}^{\alpha ;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\psi }f\right)\left(\kappa \right)=f\left(\kappa \right)-\sum _{k=1}^n{\displaystyle \frac{{\left(\psi \left(\kappa \right)-\psi \left(a\right)\right)}^{\gamma -k}}{\Gamma (\gamma -k+1)}}\left[{\left({\mathcal{I}}_{a^{+}}^{(1-\beta )(n-\alpha );\psi }f\right)}_{\psi }^{[n-k]}\right]\left(a\right).$$

(8)

Lemma 2

([5]). For $\alpha >0$, $\nu >0$, and $\kappa >a$, the following identities hold:

$$\begin{array}{cc}\hfill {\mathcal{I}}_{a^{+}}^{\alpha ;\psi }\left({(\psi (·)-\psi \left(a\right))}^{\nu -1}\right)\left(\kappa \right)& ={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu +\alpha )}}{\left(\psi \left(\kappa \right)-\psi \left(a\right)\right)}^{\nu +\alpha -1},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill {}^H{\mathcal{D}}_{a^{+}}^{\alpha ,\beta ;\psi }\left({(\psi (·)-\psi \left(a\right))}^{\nu -1}\right)\left(\kappa \right)& ={\displaystyle \frac{\Gamma \left(\nu \right)}{\Gamma (\nu -\alpha )}}{\left(\psi \left(\kappa \right)-\psi \left(a\right)\right)}^{\nu -\alpha -1},\nu >n.\hfill \end{array}$$

(10)

Remark 1.

For brevity, throughout this paper we denote the ψ-Riemann–Liouville fractional integral operator ${\mathcal{I}}_{a^{+}}^{\alpha ;\psi }$ simply by ${\mathcal{I}}^{\alpha ;\psi }$ whenever the lower limit a is clear from the context.

2.2. Measure of Noncompactness and Fixed Point Theory

We now recall the Kuratowski measure of noncompactness.

Definition 6

([43]). Let X be a Banach space and $\Omega \subset X$ be bounded. The Kuratowski measure of noncompactness $\mu (\Omega )$ is defined by

$$\mu (\Omega )=inf\left\{r>0:\Omega \mathit{can}\mathit{be}\mathit{covered}\mathit{by}\mathit{finitely}\mathit{many}\mathit{sets}\mathit{of}\mathit{diameter}\le r\right\}.$$

(11)

Lemma 3

([43]). Let $\mathcal{W}\subset C\left(\right[a,b],X)$ be bounded and equicontinuous. Then, the mapping $\kappa \mapsto \mu \left(\mathcal{W}\right(\kappa \left)\right)$ is continuous on $[a,b]$ and

$$\mu \left({\int }_a^b\mathcal{W}\left(s\right)ds\right)\le {\int }_a^b\mu \left(\mathcal{W}\left(s\right)\right)ds.$$

(12)

The following fixed point theorem due to Mönch will be used in the sequel.

Theorem 1

([40] (Mönch fixed point theorem)). Let D be a closed, bounded, and convex subset of a Banach space X with $0\in D$. Let $T:D\to D$ be continuous and satisfy: for any countable subset $U\subset D$,

$$U\subset \overline{conv}\left(T\left(U\right)\cup \left\{0\right\}\right)⟹U\mathit{is}\mathit{relatively}\mathit{compact}.$$

(13)

Then T has a fixed point in D.

3. Existence of Solutions via Mönch’s Fixed Point Theorem

In this section, we establish the existence of at least one solution to the nonlinear boundary value problem (1) by means of Mönch’s fixed point theorem combined with the Kuratowski measure of noncompactness. For the existence analysis, we additionally assume that the delayed arguments remain inside the interval of definition, namely,

$${\lambda }_1[a,b]\subseteq [a,b],{\lambda }_2[a,b]\subseteq [a,b].$$

(14)

This condition is automatically satisfied in the important case $a=0$. We begin by reformulating (1) as a fixed point problem on a suitable Banach space.

3.1. Functional Setting and Operator Formulation

Let

$$\mathcal{E}=C\left(\right[a,b],\mathbb{R})$$

be the Banach space of continuous real-valued functions on $[a,b]$, equipped with the supremum norm

$${\parallel u\parallel }_{\infty }=\underset{\kappa \in [a,b]}{sup}\left|u\left(\kappa \right)\right|.$$

Define the product space

$$\tilde{\mathcal{E}}=\mathcal{E}\times \mathcal{E},$$

endowed with the norm

$${\parallel (\theta ,\varphi )\parallel }_{\tilde{\mathcal{E}}}={\parallel \theta \parallel }_{\infty }+{\parallel \varphi \parallel }_{\infty }.$$

To derive the operator associated with (1), we first consider the linear auxiliary problem

$$\left\{\begin{array}{c}{}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_1,{\beta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_2,{\beta }_2;\psi }+{\nu }_1\right)\theta \left(\kappa \right)={\mathfrak{h}}_1\left(\kappa \right),\hfill \\ {}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_1,{\delta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_2,{\delta }_2;\psi }+{\nu }_2\right)\varphi \left(\kappa \right)={\mathfrak{h}}_2\left(\kappa \right),\hfill \\ {\displaystyle \theta \left(a\right)=0,\theta \left(b\right)=\sum _{i=1}^n{\omega }_i\left({\mathcal{I}}^{{\sigma }_i;\psi }\theta \right)\left({\eta }_i\right),}\hfill \\ {\displaystyle \varphi \left(a\right)=0,\varphi \left(b\right)=\sum _{i=1}^n{\vartheta }_i\left({\mathcal{I}}^{{\pi }_i;\psi }\varphi \right)\left({\xi }_i\right),}\hfill \end{array}\right.\kappa \in [a,b],$$

(15)

where ${\mathfrak{h}}_1,{\mathfrak{h}}_2\in C([a,b],\mathbb{R})$.

Set

$${\gamma }_1={\zeta }_1+{\beta }_1(1-{\zeta }_1),{\gamma }_2={\mathcal{P}}_1+{\delta }_1(1-{\mathcal{P}}_1),$$

and define

$${\Delta }_1=\sum _{i=1}^n{\omega }_i{\displaystyle \frac{{\left(\psi \left({\eta }_i\right)-\psi \left(a\right)\right)}^{{\gamma }_1+{\zeta }_2+{\sigma }_i-1}}{\Gamma ({\gamma }_1+{\zeta }_2+{\sigma }_i)}}-{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\gamma }_1+{\zeta }_2-1}}{\Gamma ({\gamma }_1+{\zeta }_2)}},$$

(16)

$${\Delta }_2=\sum _{i=1}^n{\vartheta }_i{\displaystyle \frac{{\left(\psi \left({\xi }_i\right)-\psi \left(a\right)\right)}^{{\gamma }_2+{\mathcal{P}}_2+{\pi }_i-1}}{\Gamma ({\gamma }_2+{\mathcal{P}}_2+{\pi }_i)}}-{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\gamma }_2+{\mathcal{P}}_2-1}}{\Gamma ({\gamma }_2+{\mathcal{P}}_2)}}.$$

(17)

Throughout the paper, we assume

$${\Delta }_1\ne 0,{\Delta }_2\ne 0.$$

For $(\theta ,\varphi )\in \tilde{\mathcal{E}}$, introduce the auxiliary nonlinear terms

$${\mathfrak{J}}_1^{\theta ,\varphi }\left(\tau \right)={\mathcal{J}}_1\left(\tau ,\theta \left(\tau \right),\theta \left({\lambda }_1\tau \right),\varphi \left(\tau \right)\right),{\mathfrak{J}}_2^{\theta ,\varphi }\left(\tau \right)={\mathcal{J}}_2\left(\tau ,\varphi \left(\tau \right),\varphi \left({\lambda }_2\tau \right),\theta \left(\tau \right)\right).$$

To improve readability, we also define the boundary correction functionals

$$\begin{array}{cc}\hfill {\mathfrak{B}}_1(\theta ,\varphi )& =\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }{\mathfrak{J}}_1^{\theta ,\varphi }\right)\left(b\right)-{\nu }_1\left({\mathcal{I}}^{{\zeta }_2;\psi }\theta \right)\left(b\right)\hfill \\ & -\sum _{i=1}^n{\omega }_i\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2+{\sigma }_i;\psi }{\mathfrak{J}}_1^{\theta ,\varphi }\right)\left({\eta }_i\right)+{\nu }_1\sum _{i=1}^n{\omega }_i\left({\mathcal{I}}^{{\zeta }_2+{\sigma }_i;\psi }\theta \right)\left({\eta }_i\right),\hfill \end{array}$$

(18)

$$\begin{array}{cc}\hfill {\mathfrak{B}}_2(\theta ,\varphi )& =\left({\mathcal{I}}^{{\mathcal{P}}_1+{\mathcal{P}}_2;\psi }{\mathfrak{J}}_2^{\theta ,\varphi }\right)\left(b\right)-{\nu }_2\left({\mathcal{I}}^{{\mathcal{P}}_2;\psi }\varphi \right)\left(b\right)\hfill \\ & -\sum _{i=1}^n{\vartheta }_i\left({\mathcal{I}}^{{\mathcal{P}}_1+{\mathcal{P}}_2+{\pi }_i;\psi }{\mathfrak{J}}_2^{\theta ,\varphi }\right)\left({\xi }_i\right)+{\nu }_2\sum _{i=1}^n{\vartheta }_i\left({\mathcal{I}}^{{\mathcal{P}}_2+{\pi }_i;\psi }\varphi \right)\left({\xi }_i\right).\hfill \end{array}$$

(19)

We now define the operator $\mathcal{T}:\tilde{\mathcal{E}}\to \tilde{\mathcal{E}}$ by

$$\mathcal{T}(\theta ,\varphi )\left(\kappa \right)=\left({\mathcal{T}}_1(\theta ,\varphi )\left(\kappa \right),{\mathcal{T}}_2(\theta ,\varphi )\left(\kappa \right)\right),\kappa \in [a,b],$$

where

$${\mathcal{T}}_1(\theta ,\varphi )\left(\kappa \right)=\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }{\mathfrak{J}}_1^{\theta ,\varphi }\right)\left(\kappa \right)-{\nu }_1\left({\mathcal{I}}^{{\zeta }_2;\psi }\theta \right)\left(\kappa \right)+{\displaystyle \frac{{\left(\psi \left(\kappa \right)-\psi \left(a\right)\right)}^{{\gamma }_1+{\zeta }_2-1}}{{\Delta }_1\Gamma ({\gamma }_1+{\zeta }_2)}}{\mathfrak{B}}_1(\theta ,\varphi ),$$

(20)

and

$${\mathcal{T}}_2(\theta ,\varphi )\left(\kappa \right)=\left({\mathcal{I}}^{{\mathcal{P}}_1+{\mathcal{P}}_2;\psi }{\mathfrak{J}}_2^{\theta ,\varphi }\right)\left(\kappa \right)-{\nu }_2\left({\mathcal{I}}^{{\mathcal{P}}_2;\psi }\varphi \right)\left(\kappa \right)+{\displaystyle \frac{{\left(\psi \left(\kappa \right)-\psi \left(a\right)\right)}^{{\gamma }_2+{\mathcal{P}}_2-1}}{{\Delta }_2\Gamma ({\gamma }_2+{\mathcal{P}}_2)}}{\mathfrak{B}}_2(\theta ,\varphi ).$$

(21)

Lemma 4

(Equivalence). A pair $(\theta ,\varphi )\in \tilde{\mathcal{E}}$ is a solution of the boundary value problem (1) if and only if it is a fixed point of the operator $\mathcal{T}$.

Proof. 

Suppose first that $(\theta ,\varphi )$ is a solution of (1). Define

$${\mathfrak{h}}_1\left(\kappa \right)={\mathcal{J}}_1\left(\kappa ,\theta \left(\kappa \right),\theta \left({\lambda }_1\kappa \right),\varphi \left(\kappa \right)\right),{\mathfrak{h}}_2\left(\kappa \right)={\mathcal{J}}_2\left(\kappa ,\varphi \left(\kappa \right),\varphi \left({\lambda }_2\kappa \right),\theta \left(\kappa \right)\right).$$

Then, $(\theta ,\varphi )$ satisfies the linear system (15) with these choices of ${\mathfrak{h}}_1,{\mathfrak{h}}_2$. Applying the $\psi$-fractional integral operators and using the composition property established in Lemma 1, together with the boundary conditions to determine the integration constants, yields exactly the integral representation (20) and (21). Hence,

$$(\theta ,\varphi )=\mathcal{T}(\theta ,\varphi ).$$

Conversely, assume that $(\theta ,\varphi )=\mathcal{T}(\theta ,\varphi )$. Applying the operator

$${}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_1,{\beta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_2,{\beta }_2;\psi }+{\nu }_1\right)$$

to the identity $\theta ={\mathcal{T}}_1(\theta ,\varphi )$, and similarly

$${}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_1,{\delta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_2,{\delta }_2;\psi }+{\nu }_2\right)$$

to $\varphi ={\mathcal{T}}_2(\theta ,\varphi )$, we recover the two equations in (1). The boundary conditions are encoded in the correction terms ${\mathfrak{B}}_1(\theta ,\varphi )$ and ${\mathfrak{B}}_2(\theta ,\varphi )$, together with the nonvanishing assumptions ${\Delta }_1\ne 0$ and ${\Delta }_2\ne 0$. Therefore, $(\theta ,\varphi )$ solves (1). □

3.2. Assumptions

We impose the following hypotheses on the nonlinearities ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$.

(H${}_1$)

The functions ${\mathcal{J}}_1,{\mathcal{J}}_2:[a,b]\times {\mathbb{R}}^3\to \mathbb{R}$ satisfy the Carathéodory conditions: for each $(u_1,u_2,u_3)\in {\mathbb{R}}^3$, the mapping $\kappa \mapsto {\mathcal{J}}_j(\kappa ,u_1,u_2,u_3)$ is measurable; for almost every $\kappa \in [a,b]$, the mapping $(u_1,u_2,u_3)\mapsto {\mathcal{J}}_j(\kappa ,u_1,u_2,u_3)$ is continuous; and for each $r>0$, there exists ${\phi }_j^r\in L^1([a,b],{\mathbb{R}}_{+})$ such that

$$\left|{\mathcal{J}}_j(\kappa ,u_1,u_2,u_3)\right|\le {\phi }_j^r\left(\kappa \right)$$

whenever $|u_1|,|u_2|,|u_3|\le r$, for almost every $\kappa \in [a,b]$.

(H${}_2$)

There exist functions ${\eta }_1,{\eta }_2\in L^1([a,b],{\mathbb{R}}_{+})$ and nondecreasing continuous functions ${\Theta }_1,{\Theta }_2:{\mathbb{R}}_{+}\to {\mathbb{R}}_{+}$ such that

$$\begin{gathered} |{\mathcal{J}}_1(\kappa ,u_1,u_2,u_3)|\le {\eta }_1\left(\kappa \right){\Theta }_1\left(\right|u_1|+|u_2|+|u_3\left|\right), \\ |{\mathcal{J}}_2(\kappa ,u_1,u_2,u_3)|\le {\eta }_2\left(\kappa \right){\Theta }_2\left(\right|u_1|+|u_2|+|u_3\left|\right), \end{gathered}$$

for almost every $\kappa \in [a,b]$ and all $u_1,u_2,u_3\in \mathbb{R}$.

(H${}_3$)

There exist functions ${\xi }_1,{\xi }_2\in L^1([a,b],{\mathbb{R}}_{+})$ such that for every bounded set $S\subset \tilde{\mathcal{E}}$ and almost every $\kappa \in [a,b]$,

$$\begin{gathered} {\mu }_{\mathbb{R}}\left({\mathcal{J}}_1(\kappa ,{\widehat{S}}_1\left(\kappa \right))\right)\le {\xi }_1\left(\kappa \right){\mu }_{\tilde{\mathcal{E}}}\left(S\right), \\ {\mu }_{\mathbb{R}}\left({\mathcal{J}}_2(\kappa ,{\widehat{S}}_2\left(\kappa \right))\right)\le {\xi }_2\left(\kappa \right){\mu }_{\tilde{\mathcal{E}}}\left(S\right), \end{gathered}$$

where

$$\begin{gathered} {\widehat{S}}_1\left(\kappa \right)=\left\{(u\left(\kappa \right),u\left({\lambda }_1\kappa \right),v\left(\kappa \right)):(u,v)\in S\right\}, \\ {\widehat{S}}_2\left(\kappa \right)=\left\{(v\left(\kappa \right),v\left({\lambda }_2\kappa \right),u\left(\kappa \right)):(u,v)\in S\right\}, \end{gathered}$$

and ${\mu }_{\mathbb{R}}$, ${\mu }_{\tilde{\mathcal{E}}}$ denote the Kuratowski measures of noncompactness in $\mathbb{R}$ and $\tilde{\mathcal{E}}$, respectively.

3.3. Main Existence Result

For later use, define

$$\begin{array}{cc}\hfill {\Lambda }_1^{\left(1\right)}=& {\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\zeta }_1+{\zeta }_2}}{\Gamma ({\zeta }_1+{\zeta }_2+1)}}+{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\gamma }_1+{\zeta }_2-1}}{\left|{\Delta }_1\right|\Gamma ({\gamma }_1+{\zeta }_2)}}[{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\zeta }_1+{\zeta }_2}}{\Gamma ({\zeta }_1+{\zeta }_2+1)}}\hfill \\ & +\sum _{i=1}^n\left|{\omega }_i\right|{\displaystyle \frac{{\left(\psi \left({\eta }_i\right)-\psi \left(a\right)\right)}^{{\zeta }_1+{\zeta }_2+{\sigma }_i}}{\Gamma ({\zeta }_1+{\zeta }_2+{\sigma }_i+1)}}],\hfill \end{array}$$

(22)

$${\Lambda }_1^{\left(2\right)}={\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\zeta }_2}}{\Gamma ({\zeta }_2+1)}}+{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\gamma }_1+{\zeta }_2-1}}{\left|{\Delta }_1\right|\Gamma ({\gamma }_1+{\zeta }_2)}}\left[{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\zeta }_2}}{\Gamma ({\zeta }_2+1)}}+\sum _{i=1}^n\left|{\omega }_i\right|{\displaystyle \frac{{\left(\psi \left({\eta }_i\right)-\psi \left(a\right)\right)}^{{\zeta }_2+{\sigma }_i}}{\Gamma ({\zeta }_2+{\sigma }_i+1)}}\right],$$

(23)

$$\begin{array}{cc}\hfill {\Lambda }_2^{\left(1\right)}=& {\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\mathcal{P}}_1+{\mathcal{P}}_2}}{\Gamma ({\mathcal{P}}_1+{\mathcal{P}}_2+1)}}+{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\gamma }_2+{\mathcal{P}}_2-1}}{\left|{\Delta }_2\right|\Gamma ({\gamma }_2+{\mathcal{P}}_2)}}[{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\mathcal{P}}_1+{\mathcal{P}}_2}}{\Gamma ({\mathcal{P}}_1+{\mathcal{P}}_2+1)}}\hfill \\ & +\sum _{i=1}^n\left|{\vartheta }_i\right|{\displaystyle \frac{{\left(\psi \left({\xi }_i\right)-\psi \left(a\right)\right)}^{{\mathcal{P}}_1+{\mathcal{P}}_2+{\pi }_i}}{\Gamma ({\mathcal{P}}_1+{\mathcal{P}}_2+{\pi }_i+1)}}],\hfill \end{array}$$

(24)

$${\Lambda }_2^{\left(2\right)}={\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\mathcal{P}}_2}}{\Gamma ({\mathcal{P}}_2+1)}}+{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\gamma }_2+{\mathcal{P}}_2-1}}{\left|{\Delta }_2\right|\Gamma ({\gamma }_2+{\mathcal{P}}_2)}}\left[{\displaystyle \frac{{\left(\psi \left(b\right)-\psi \left(a\right)\right)}^{{\mathcal{P}}_2}}{\Gamma ({\mathcal{P}}_2+1)}}+\sum _{i=1}^n\left|{\vartheta }_i\right|{\displaystyle \frac{{\left(\psi \left({\xi }_i\right)-\psi \left(a\right)\right)}^{{\mathcal{P}}_2+{\pi }_i}}{\Gamma ({\mathcal{P}}_2+{\pi }_i+1)}}\right].$$

(25)

Also set

$${\eta }_1^{\ast }=\parallel {\eta }_1{\parallel }_{L^1},{\eta }_2^{\ast }=\parallel {\eta }_2{\parallel }_{L^1},{\xi }_1^{\ast }=\parallel {\xi }_1{\parallel }_{L^1},{\xi }_2^{\ast }={\parallel {\xi }_2\parallel }_{L^1}.$$

The choice of Mönch’s fixed point theorem is natural here: the nonlocal boundary conditions and the delay structure do not guarantee compactness of the associated operator in a form suitable for direct application of the Banach contraction principle or the classical Schauder theorem. The combination of Mönch’s theorem with the Kuratowski measure of noncompactness allows one to treat this lack of compactness in a quantitative way.

Theorem 2

(Existence via Mönch’s theorem). Assume that (H${}_1$)–(H${}_3$) hold and that (14) is satisfied. If there exists $R>0$ such that

$${\eta }_1^{\ast }{\Theta }_1\left(3R\right){\Lambda }_1^{\left(1\right)}+{\eta }_2^{\ast }{\Theta }_2\left(3R\right){\Lambda }_2^{\left(1\right)}+\left(|{\nu }_1|{\Lambda }_1^{\left(2\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)R\le R,$$

(26)

and

$$q:={\xi }_1^{\ast }{\Lambda }_1^{\left(1\right)}+{\xi }_2^{\ast }{\Lambda }_2^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}<1,$$

(27)

then the boundary value problem (1) has at least one solution $(\theta ,\varphi )\in \tilde{\mathcal{E}}$ satisfying

$${\parallel (\theta ,\varphi )\parallel }_{\tilde{\mathcal{E}}}\le R.$$

Proof. 

We divide the proof into five steps.

Step 1: Construction of an invariant ball. Define

$$B_R=\left\{(\theta ,\varphi )\in \tilde{\mathcal{E}}:{\parallel (\theta ,\varphi )\parallel }_{\tilde{\mathcal{E}}}\le R\right\}.$$

Then, $B_R$ is closed, bounded, convex, and contains the origin.

Let $(\theta ,\varphi )\in B_R$. By (14), the delayed evaluations $\theta \left({\lambda }_1\kappa \right)$ and $\varphi \left({\lambda }_2\kappa \right)$ are well-defined for $\kappa \in [a,b]$, and

$$\left|\theta \left(\kappa \right)\right|\le R,|\theta \left({\lambda }_1\kappa \right)|\le R,|\varphi \left(\kappa \right)|\le R.$$

Hence, hypothesis (H${}_2$) gives

$$\left|{\mathfrak{J}}_1^{\theta ,\varphi }\left(\kappa \right)\right|\le {\eta }_1\left(\kappa \right){\Theta }_1\left(3R\right),\kappa \in [a,b].$$

Similarly,

$$\left|{\mathfrak{J}}_2^{\theta ,\varphi }\left(\kappa \right)\right|\le {\eta }_2\left(\kappa \right){\Theta }_2\left(3R\right),\kappa \in [a,b].$$

Applying the positivity of the $\psi$-fractional kernels, together with the estimates encoded in (22)–(25), we obtain

$$\parallel {\mathcal{T}}_1{(\theta ,\varphi )\parallel }_{\infty }\le {\eta }_1^{\ast }{\Theta }_1\left(3R\right){\Lambda }_1^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}R,$$

and

$$\parallel {\mathcal{T}}_2{(\theta ,\varphi )\parallel }_{\infty }\le {\eta }_2^{\ast }{\Theta }_2\left(3R\right){\Lambda }_2^{\left(1\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}R.$$

Therefore,

$${\parallel \mathcal{T}(\theta ,\varphi )\parallel }_{\tilde{\mathcal{E}}}\le {\eta }_1^{\ast }{\Theta }_1\left(3R\right){\Lambda }_1^{\left(1\right)}+{\eta }_2^{\ast }{\Theta }_2\left(3R\right){\Lambda }_2^{\left(1\right)}+\left(|{\nu }_1|{\Lambda }_1^{\left(2\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)R.$$

By (26), the right-hand side is bounded by R. Hence,

$$\mathcal{T}\left(B_R\right)\subseteq B_R.$$

Step 2: Continuity of $\mathcal{T}$. Let ${\left\{({\theta }_m,{\varphi }_m)\right\}}_{m\ge 1}\subset B_R$ converge to $(\theta ,\varphi )$ in $\tilde{\mathcal{E}}$. By (H${}_1$), for almost every $\kappa \in [a,b]$,

$${\mathfrak{J}}_1^{{\theta }_m,{\varphi }_m}\left(\kappa \right)\to {\mathfrak{J}}_1^{\theta ,\varphi }\left(\kappa \right),{\mathfrak{J}}_2^{{\theta }_m,{\varphi }_m}\left(\kappa \right)\to {\mathfrak{J}}_2^{\theta ,\varphi }\left(\kappa \right).$$

Moreover, by (H${}_2$),

$$\left|{\mathfrak{J}}_1^{{\theta }_m,{\varphi }_m}\left(\kappa \right)\right|\le {\eta }_1\left(\kappa \right){\Theta }_1\left(3R\right),\left|{\mathfrak{J}}_2^{{\theta }_m,{\varphi }_m}\left(\kappa \right)\right|\le {\eta }_2\left(\kappa \right){\Theta }_2\left(3R\right),$$

and the dominating functions belong to $L^1\left([a,b]\right)$. By dominated convergence and continuity of the $\psi$-fractional integral operators from $L^1\left([a,b]\right)$ into $C\left(\right[a,b\left]\right)$, it follows that

$${\mathcal{T}}_1({\theta }_m,{\varphi }_m)\to {\mathcal{T}}_1(\theta ,\varphi ),{\mathcal{T}}_2({\theta }_m,{\varphi }_m)\to {\mathcal{T}}_2(\theta ,\varphi )$$

uniformly on $[a,b]$. Thus, $\mathcal{T}$ is continuous on $B_R$.

Step 3: Equicontinuity of $\mathcal{T}\left(B_R\right)$. Let $(\theta ,\varphi )\in B_R$ and $a\le {\kappa }_1<{\kappa }_2\le b$. Since the functions ${\mathfrak{J}}_1^{\theta ,\varphi }$ and ${\mathfrak{J}}_2^{\theta ,\varphi }$ are uniformly $L^1$-bounded on $B_R$, and the kernels

$$(\tau ,\kappa )\mapsto {\psi }^{\prime }\left(\tau \right){\left(\psi \left(\kappa \right)-\psi \left(\tau \right)\right)}^{\alpha -1}$$

depend continuously on $\kappa$ for $\tau \le \kappa$, standard estimates for fractional Volterra operators imply that there exist constants $C_1,C_2>0$, independent of $(\theta ,\varphi )\in B_R$, such that

$$\begin{gathered} |{\mathcal{T}}_1(\theta ,\varphi )\left({\kappa }_2\right)-{\mathcal{T}}_1(\theta ,\varphi )\left({\kappa }_1\right)|\le C_1|\psi \left({\kappa }_2\right)-\psi \left({\kappa }_1\right)|, \\ |{\mathcal{T}}_2(\theta ,\varphi )\left({\kappa }_2\right)-{\mathcal{T}}_2(\theta ,\varphi )\left({\kappa }_1\right)|\le C_2|\psi \left({\kappa }_2\right)-\psi \left({\kappa }_1\right)|. \end{gathered}$$

Because $\psi$ is continuous on the compact interval $[a,b]$, it is uniformly continuous. Hence, $\mathcal{T}\left(B_R\right)$ is equicontinuous.

Step 4: Verification of Mönch’s condition. Let $S\subset B_R$ be countable and satisfy

$$S\subset \overline{conv}\left(\mathcal{T}\left(S\right)\cup \left\{\right(0,0\left)\right\}\right).$$

For each $\kappa \in [a,b]$, define

$$m\left(\kappa \right)={\mu }_{{\mathbb{R}}^2}\left(S\left(\kappa \right)\right),S\left(\kappa \right)=\{(\theta \left(\kappa \right),\varphi \left(\kappa \right)):(\theta ,\varphi )\in S\}.$$

Since the Kuratowski measure is invariant under closure and convex hull,

$$m\left(\kappa \right)\le {\mu }_{{\mathbb{R}}^2}\left(\mathcal{T}\left(S\right)\left(\kappa \right)\right)\le {\mu }_{\mathbb{R}}\left({\mathcal{T}}_1\left(S\right)\left(\kappa \right)\right)+{\mu }_{\mathbb{R}}\left({\mathcal{T}}_2\left(S\right)\left(\kappa \right)\right).$$

Using the measure estimate for integral operators established in Lemma 3, hypothesis (H${}_3$), and the definitions of ${\Lambda }_1^{\left(1\right)}$, ${\Lambda }_1^{\left(2\right)}$, ${\Lambda }_2^{\left(1\right)}$, and ${\Lambda }_2^{\left(2\right)}$, we obtain

$${\mu }_{\mathbb{R}}\left({\mathcal{T}}_1\left(S\right)\left(\kappa \right)\right)\le \left({\xi }_1^{\ast }{\Lambda }_1^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}\right)\underset{\tau \in [a,b]}{sup}m\left(\tau \right),$$

and

$${\mu }_{\mathbb{R}}\left({\mathcal{T}}_2\left(S\right)\left(\kappa \right)\right)\le \left({\xi }_2^{\ast }{\Lambda }_2^{\left(1\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)\underset{\tau \in [a,b]}{sup}m\left(\tau \right).$$

Consequently,

$$m\left(\kappa \right)\le \left({\xi }_1^{\ast }{\Lambda }_1^{\left(1\right)}+{\xi }_2^{\ast }{\Lambda }_2^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)\underset{\tau \in [a,b]}{sup}m\left(\tau \right)=q\underset{\tau \in [a,b]}{sup}m\left(\tau \right).$$

Taking the supremum over $\kappa \in [a,b]$ gives

$$\underset{\kappa \in [a,b]}{sup}m\left(\kappa \right)\le q\underset{\kappa \in [a,b]}{sup}m\left(\kappa \right).$$

Since $q<1$ by (27), it follows that

$$\underset{\kappa \in [a,b]}{sup}m\left(\kappa \right)=0.$$

Hence, $m\left(\kappa \right)=0$ for all $\kappa \in [a,b]$, so $S\left(\kappa \right)$ is relatively compact in ${\mathbb{R}}^2$ for every $\kappa \in [a,b]$.

Together with the equicontinuity established in Step 3, the Arzelà–Ascoli theorem implies that S is relatively compact in $\tilde{\mathcal{E}}$.

Step 5: Application of Mönch’s theorem. We have shown that:

• $B_R$ is closed, bounded, convex, and $0\in B_R$;
• $\mathcal{T}:B_R\to B_R$ is continuous;
• Every countable set $S\subset B_R$ satisfying

  $$S\subset \overline{conv}\left(\mathcal{T}\left(S\right)\cup \left\{\right(0,0\left)\right\}\right)$$

  is relatively compact.

Therefore, Mönch’s fixed point theorem (Theorem 1) guarantees that $\mathcal{T}$ has a fixed point in $B_R$. By Lemma 4, that fixed point is a solution of (1). This completes the proof. □

Remark 2.

Condition (26) is automatically satisfied for sufficiently large R whenever ${\Theta }_1$ and ${\Theta }_2$ have sublinear growth. Indeed, if there exist constants $K_1,K_2>0$ and exponents ${\alpha }_1,{\alpha }_2\in (0,1)$ such that

$${\Theta }_1\left(r\right)\le K_1{r}^{{\alpha }_1},{\Theta }_2\left(r\right)\le K_2{r}^{{\alpha }_2},r\ge 0,$$

then the nonlinear terms on the left-hand side of (26) grow like $R^{max\{{\alpha }_1,{\alpha }_2\}}$, whereas the right-hand side grows linearly in R.

Remark 3.

Condition (27) provides a quantitative smallness requirement ensuring that the noncompactness generated by the nonlinear and delay terms is sufficiently controlled. In particular, this condition can be enforced when the interval length $\psi \left(b\right)-\psi \left(a\right)$ is small enough or when the coefficients ${\nu }_1,{\nu }_2$ and the $L^1$-norms ${\xi }_1^{\ast },{\xi }_2^{\ast }$ are suitably bounded.

4. Ulam–Hyers Stability Analysis

This section establishes the Ulam–Hyers stability of the boundary value problem (1). Stability analysis requires stronger conditions than existence theory, particularly Lipschitz continuity of the nonlinearities and quantitative control over perturbations. We systematically develop the stability theory within the framework of $\psi$-fractional calculus, ensuring consistency with the notation and results established in Section 2 and Section 3.

4.1. Definitions and Preliminary Results

We begin with the definition of Ulam–Hyers stability tailored to boundary value problems, where approximate solutions must satisfy the boundary conditions exactly. It is important to distinguish between different stability concepts: while Lyapunov stability concerns the long-term behavior of solutions near equilibrium points, Ulam–Hyers stability addresses a fundamentally different question—if we have an approximate solution, how close is it to an exact solution? This notion is particularly crucial for numerical approximations and real-world modeling, where exact solutions are often unattainable.

Definition 7

(Ulam–Hyers stability on $B_R$ for BVPs). The boundary value problem (1) is said to be Ulam–Hyers stable on $B_R$, where

$$B_R=\left\{(\theta ,\varphi )\in \tilde{\mathcal{E}}:{\parallel (\theta ,\varphi )\parallel }_{\tilde{\mathcal{E}}}\le R\right\},$$

if there exist constants $K_1,K_2>0$ such that for every ${ϵ}_1,{ϵ}_2>0$ and for every pair $(\theta ,\varphi )\in B_R$ satisfying the inequalities

$$\left\{\begin{array}{c}\left|{}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_1,{\beta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_2,{\beta }_2;\psi }+{\nu }_1\right)\theta \left(\kappa \right)-{\mathcal{J}}_1(\kappa ,\theta \left(\kappa \right),\theta \left({\lambda }_1\kappa \right),\varphi \left(\kappa \right))\right|\le {ϵ}_1,\hfill \\ \left|{}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_1,{\delta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_2,{\delta }_2;\psi }+{\nu }_2\right)\varphi \left(\kappa \right)-{\mathcal{J}}_2(\kappa ,\varphi \left(\kappa \right),\varphi \left({\lambda }_2\kappa \right),\theta \left(\kappa \right))\right|\le {ϵ}_2,\hfill \end{array}\right.\kappa \in [a,b],$$

(28)

and satisfying exactly the boundary conditions of (1), there exists a solution $({\theta }^{\ast },{\varphi }^{\ast })$ of (1) such that

$$\parallel (\theta ,\varphi )-({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}\le K_1{ϵ}_1+K_2{ϵ}_2.$$

Remark 4.

The requirement that approximate solutions satisfy the boundary conditions exactly is standard in stability analysis for nonlocal boundary value problems. This ensures that the integral representation from Lemma 4 can be applied without introducing additional boundary perturbation terms. Restricting the approximate solutions to the ball $B_R$ is natural since all exact solutions obtained via Theorem 2 lie in this ball, and the contraction property established in Lemma 8 holds on $B_R$.

The following lemma converts the pointwise inequality (28) into an equation with explicit bounded perturbation functions.

Lemma 5

(Perturbed differential equation). Let $(\theta ,\varphi )\in B_R$ satisfy (28) with ${ϵ}_1,{ϵ}_2>0$ and the boundary conditions of (1). Then, there exist measurable functions ${\mathcal{Z}}_1,{\mathcal{Z}}_2:[a,b]\to \mathbb{R}$ such that

$$|{\mathcal{Z}}_1\left(\kappa \right)|\le {ϵ}_1,\left|{\mathcal{Z}}_2\left(\kappa \right)\right|\le {ϵ}_2\mathit{for}\mathit{all}\kappa \in [a,b],$$

and

$$\left\{\begin{array}{c}{}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_1,{\beta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_2,{\beta }_2;\psi }+{\nu }_1\right)\theta \left(\kappa \right)={\mathcal{J}}_1(\kappa ,\theta \left(\kappa \right),\theta \left({\lambda }_1\kappa \right),\varphi \left(\kappa \right))+{\mathcal{Z}}_1\left(\kappa \right),\hfill \\ {}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_1,{\delta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_2,{\delta }_2;\psi }+{\nu }_2\right)\varphi \left(\kappa \right)={\mathcal{J}}_2(\kappa ,\varphi \left(\kappa \right),\varphi \left({\lambda }_2\kappa \right),\theta \left(\kappa \right))+{\mathcal{Z}}_2\left(\kappa \right),\hfill \end{array}\right.\kappa \in [a,b].$$

(29)

Proof. 

From (28), we can take ${\mathcal{Z}}_1\left(\kappa \right)$ and ${\mathcal{Z}}_2\left(\kappa \right)$ to be the respective differences between the left and right sides of the inequalities. Specifically, define

$$\begin{gathered} {\mathcal{Z}}_1\left(\kappa \right)={}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_1,{\beta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\zeta }_2,{\beta }_2;\psi }+{\nu }_1\right)\theta \left(\kappa \right)-{\mathcal{J}}_1(\kappa ,\theta \left(\kappa \right),\theta \left({\lambda }_1\kappa \right),\varphi \left(\kappa \right)), \\ {\mathcal{Z}}_2\left(\kappa \right)={}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_1,{\delta }_1;\psi }\left({}^H{\mathcal{D}}_{a^{+}}^{{\mathcal{P}}_2,{\delta }_2;\psi }+{\nu }_2\right)\varphi \left(\kappa \right)-{\mathcal{J}}_2(\kappa ,\varphi \left(\kappa \right),\varphi \left({\lambda }_2\kappa \right),\theta \left(\kappa \right)). \end{gathered}$$

The bounds $\left|{\mathcal{Z}}_1\left(\kappa \right)\right|\le {ϵ}_1$ and $\left|{\mathcal{Z}}_2\left(\kappa \right)\right|\le {ϵ}_2$ follow directly from (28), and (29) is satisfied by construction. □

Using Lemma 4, we now derive an integral equation satisfied by approximate solutions.

Lemma 6

(Perturbed fixed point equation). Under the conditions of Lemma 5, the pair $(\theta ,\varphi )$ satisfies

$$(\theta ,\varphi )=\mathcal{T}(\theta ,\varphi )+\mathcal{P}({\mathcal{Z}}_1,{\mathcal{Z}}_2),$$

(30)

where $\mathcal{T}$ is the operator defined in (20) and (21), and $\mathcal{P}:L^{\infty }([a,b],\mathbb{R})\times L^{\infty }([a,b],\mathbb{R})\to \tilde{\mathcal{E}}$ is the linear perturbation operator defined component-wise by

$$\begin{array}{cc}\hfill {\mathcal{P}}_1({\mathcal{Z}}_1,{\mathcal{Z}}_2)\left(\kappa \right)& =\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }{\mathcal{Z}}_1\right)\left(\kappa \right)\hfill \\ \hfill & +{\displaystyle \frac{{(\psi \left(\kappa \right)-\psi \left(a\right))}^{{\gamma }_1+{\zeta }_2-1}}{{\Delta }_1\Gamma ({\gamma }_1+{\zeta }_2)}}[\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }{\mathcal{Z}}_1\right)\left(b\right)\hfill \\ \hfill & -\sum _{i=1}^n{\omega }_i\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2+{\sigma }_i;\psi }{\mathcal{Z}}_1\right)\left({\eta }_i\right)],\hfill \end{array}$$

(31)

$$\begin{array}{cc}\hfill {\mathcal{P}}_2({\mathcal{Z}}_1,{\mathcal{Z}}_2)\left(\kappa \right)& =\left({\mathcal{I}}^{{\mathcal{P}}_1+{\mathcal{P}}_2;\psi }{\mathcal{Z}}_2\right)\left(\kappa \right)\hfill \\ \hfill & +{\displaystyle \frac{{(\psi \left(\kappa \right)-\psi \left(a\right))}^{{\gamma }_2+{\mathcal{P}}_2-1}}{{\Delta }_2\Gamma ({\gamma }_2+{\mathcal{P}}_2)}}[\left({\mathcal{I}}^{{\mathcal{P}}_1+{\mathcal{P}}_2;\psi }{\mathcal{Z}}_2\right)\left(b\right)\hfill \\ \hfill & -\sum _{i=1}^n{\vartheta }_i\left({\mathcal{I}}^{{\mathcal{P}}_1+{\mathcal{P}}_2+{\pi }_i;\psi }{\mathcal{Z}}_2\right)\left({\xi }_i\right)],\hfill \end{array}$$

(32)

with $\mathcal{P}({\mathcal{Z}}_1,{\mathcal{Z}}_2)=({\mathcal{P}}_1({\mathcal{Z}}_1,{\mathcal{Z}}_2),{\mathcal{P}}_2({\mathcal{Z}}_1,{\mathcal{Z}}_2))$.

Proof. 

Apply Lemma 4 to (29) with

$$\begin{gathered} {\mathfrak{h}}_1\left(\kappa \right)={\mathcal{J}}_1(\kappa ,\theta \left(\kappa \right),\theta \left({\lambda }_1\kappa \right),\varphi \left(\kappa \right))+{\mathcal{Z}}_1\left(\kappa \right), \\ {\mathfrak{h}}_2\left(\kappa \right)={\mathcal{J}}_2(\kappa ,\varphi \left(\kappa \right),\varphi \left({\lambda }_2\kappa \right),\theta \left(\kappa \right))+{\mathcal{Z}}_2\left(\kappa \right). \end{gathered}$$

The linearity of the integral representation with respect to the right-hand side yields (30), with $\mathcal{P}$ collecting all terms involving ${\mathcal{Z}}_1$ and ${\mathcal{Z}}_2$. □

To quantify the effect of perturbations, we introduce explicit stability constants derived from the problem parameters.

Definition 8

(Stability constants). Define

$$\begin{array}{cc}\hfill {\Pi }_1& ={\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\zeta }_1+{\zeta }_2}}{\Gamma ({\zeta }_1+{\zeta }_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\gamma }_1+{\zeta }_2-1}}{|{\Delta }_1|\Gamma ({\gamma }_1+{\zeta }_2)}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\zeta }_1+{\zeta }_2}}{\Gamma ({\zeta }_1+{\zeta }_2+1)}}+\sum _{i=1}^n\left|{\omega }_i\right|{\displaystyle \frac{{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{{\zeta }_1+{\zeta }_2+{\sigma }_i}}{\Gamma ({\zeta }_1+{\zeta }_2+{\sigma }_i+1)}}\right],\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill {\Pi }_2& ={\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\mathcal{P}}_1+{\mathcal{P}}_2}}{\Gamma ({\mathcal{P}}_1+{\mathcal{P}}_2+1)}}\hfill \\ \hfill & +{\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\gamma }_2+{\mathcal{P}}_2-1}}{|{\Delta }_2|\Gamma ({\gamma }_2+{\mathcal{P}}_2)}}\hfill \\ \hfill & \times \left[{\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\mathcal{P}}_1+{\mathcal{P}}_2}}{\Gamma ({\mathcal{P}}_1+{\mathcal{P}}_2+1)}}+\sum _{i=1}^n\left|{\vartheta }_i\right|{\displaystyle \frac{{(\psi \left({\xi }_i\right)-\psi \left(a\right))}^{{\mathcal{P}}_1+{\mathcal{P}}_2+{\pi }_i}}{\Gamma ({\mathcal{P}}_1+{\mathcal{P}}_2+{\pi }_i+1)}}\right].\hfill \end{array}$$

(34)

Remark 5.

Comparing (33) and (34) with (22) and (24), we observe that ${\Pi }_1={\Lambda }_1^{\left(1\right)}$ and ${\Pi }_2={\Lambda }_2^{\left(1\right)}$. This is expected, as the perturbation operator $\mathcal{P}$ has the same structure as the principal part of $\mathcal{T}$ without the nonlinear terms.

Lemma 7

(Perturbation bound). For any bounded measurable functions ${\mathcal{Z}}_1,{\mathcal{Z}}_2:[a,b]\to \mathbb{R}$,

$$\parallel \mathcal{P}({\mathcal{Z}}_1,{\mathcal{Z}}_2){\parallel }_{\tilde{\mathcal{E}}}\le {\Pi }_1\parallel {\mathcal{Z}}_1{\parallel }_{\infty }+{\Pi }_2{\parallel {\mathcal{Z}}_2\parallel }_{\infty }.$$

In particular, if $|{\mathcal{Z}}_1\left(\kappa \right)|\le {ϵ}_1$ and $|{\mathcal{Z}}_2\left(\kappa \right)|\le {ϵ}_2$ for all $\kappa \in [a,b]$, then

$$\parallel \mathcal{P}({\mathcal{Z}}_1,{\mathcal{Z}}_2){\parallel }_{\tilde{\mathcal{E}}}\le {\Pi }_1{ϵ}_1+{\Pi }_2{ϵ}_2.$$

Proof. 

We estimate ${\mathcal{P}}_1({\mathcal{Z}}_1,{\mathcal{Z}}_2)\left(\kappa \right)$ for a fixed $\kappa \in [a,b]$. For the first term,

$$\begin{array}{cc}\hfill \left|\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }{\mathcal{Z}}_1\right)\left(\kappa \right)\right|& =\left|{\displaystyle \frac{1}{\Gamma ({\zeta }_1+{\zeta }_2)}}{\int }_a^{\kappa }{\psi }^{\prime }\left(s\right){(\psi \left(\kappa \right)-\psi \left(s\right))}^{{\zeta }_1+{\zeta }_2-1}{\mathcal{Z}}_1\left(s\right)ds\right|\hfill \\ & \le {\displaystyle \frac{\parallel {\mathcal{Z}}_1{\parallel }_{\infty }}{\Gamma ({\zeta }_1+{\zeta }_2)}}{\int }_a^{\kappa }{\psi }^{\prime }\left(s\right){(\psi \left(\kappa \right)-\psi \left(s\right))}^{{\zeta }_1+{\zeta }_2-1}ds\hfill \\ & ={\displaystyle \frac{\parallel {\mathcal{Z}}_1{\parallel }_{\infty }}{\Gamma ({\zeta }_1+{\zeta }_2+1)}}{(\psi \left(\kappa \right)-\psi \left(a\right))}^{{\zeta }_1+{\zeta }_2}\hfill \\ & \le {\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\zeta }_1+{\zeta }_2}}{\Gamma ({\zeta }_1+{\zeta }_2+1)}}{\parallel {\mathcal{Z}}_1\parallel }_{\infty }.\hfill \end{array}$$

For the second term, we have similarly

$$\begin{array}{cc}& \left|{\displaystyle \frac{{(\psi \left(\kappa \right)-\psi \left(a\right))}^{{\gamma }_1+{\zeta }_2-1}}{{\Delta }_1\Gamma ({\gamma }_1+{\zeta }_2)}}\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }{\mathcal{Z}}_1\right)\left(b\right)\right|\hfill \\ & \le {\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\gamma }_1+{\zeta }_2-1}}{|{\Delta }_1|\Gamma ({\gamma }_1+{\zeta }_2)}}·{\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\zeta }_1+{\zeta }_2}}{\Gamma ({\zeta }_1+{\zeta }_2+1)}}{\parallel {\mathcal{Z}}_1\parallel }_{\infty }.\hfill \end{array}$$

For each $i=1,\dots ,n$,

$$\begin{array}{cc}& \left|{\displaystyle \frac{{(\psi \left(\kappa \right)-\psi \left(a\right))}^{{\gamma }_1+{\zeta }_2-1}}{{\Delta }_1\Gamma ({\gamma }_1+{\zeta }_2)}}{\omega }_i\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2+{\sigma }_i;\psi }{\mathcal{Z}}_1\right)\left({\eta }_i\right)\right|\hfill \\ & \le {\displaystyle \frac{{(\psi \left(b\right)-\psi \left(a\right))}^{{\gamma }_1+{\zeta }_2-1}}{|{\Delta }_1|\Gamma ({\gamma }_1+{\zeta }_2)}}·|{\omega }_i|{\displaystyle \frac{{(\psi \left({\eta }_i\right)-\psi \left(a\right))}^{{\zeta }_1+{\zeta }_2+{\sigma }_i}}{\Gamma ({\zeta }_1+{\zeta }_2+{\sigma }_i+1)}}{\parallel {\mathcal{Z}}_1\parallel }_{\infty }.\hfill \end{array}$$

Summing these estimates gives

$$|{\mathcal{P}}_1({\mathcal{Z}}_1,{\mathcal{Z}}_2)\left(\kappa \right)|\le {\Pi }_1{\parallel {\mathcal{Z}}_1\parallel }_{\infty }.$$

Taking the supremum over $\kappa \in [a,b]$ yields $\parallel {\mathcal{P}}_1({\mathcal{Z}}_1,{\mathcal{Z}}_2){\parallel }_{\infty }\le {\Pi }_1{\parallel {\mathcal{Z}}_1\parallel }_{\infty }$. A parallel argument gives $\parallel {\mathcal{P}}_2({\mathcal{Z}}_1,{\mathcal{Z}}_2){\parallel }_{\infty }\le {\Pi }_2{\parallel {\mathcal{Z}}_2\parallel }_{\infty }$. Combining these completes the proof. □

4.2. Lipschitz Conditions and Contraction Property

To establish Ulam–Hyers stability, we require Lipschitz conditions on the nonlinearities and a contraction property for the operator $\mathcal{T}$.

(H${}_4$)

(Lipschitz conditions) There exist constants $L_{11},L_{12},L_{13}\ge 0$ such that for all $\kappa \in [a,b]$ and all $x_i,y_i\in \mathbb{R}$$(i=1,2,3)$,

$$|{\mathcal{J}}_1(\kappa ,x_1,x_2,x_3)-{\mathcal{J}}_1(\kappa ,y_1,y_2,y_3)|\le L_{11}|x_1-y_1|+L_{12}|x_2-y_2|+L_{13}|x_3-y_3|.$$

Similarly, there exist $L_{21},L_{22},L_{23}\ge 0$ such that

$$|{\mathcal{J}}_2(\kappa ,x_1,x_2,x_3)-{\mathcal{J}}_2(\kappa ,y_1,y_2,y_3)|\le L_{21}|x_1-y_1|+L_{22}|x_2-y_2|+L_{23}|x_3-y_3|.$$

(H${}_5$)

(Contraction condition) Define the combined Lipschitz constants

$$L_1=L_{11}+L_{12}+L_{13},L_2=L_{21}+L_{22}+L_{23},$$

and the contraction constant

$$\rho :=\left(L_1{\Lambda }_1^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}\right)+\left(L_2{\Lambda }_2^{\left(1\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right),$$

where ${\Lambda }_1^{\left(1\right)},{\Lambda }_1^{\left(2\right)},{\Lambda }_2^{\left(1\right)},{\Lambda }_2^{\left(2\right)}$ are the constants defined in (22)–(25). Assume $\rho <1$.

(H${}_6$)

(Invariance condition) There exists $R>0$ such that $\mathcal{T}\left(B_R\right)\subseteq B_R$, where

$$B_R=\{(\theta ,\varphi )\in \tilde{\mathcal{E}}{:\parallel (\theta ,\varphi )\parallel }_{\tilde{\mathcal{E}}}\le R\}.$$

This condition ensures that the fixed point argument takes place within a closed ball.

Lemma 8

(Contraction on $B_R$). Assume that hypotheses (H${}_4$) and (H${}_5$) hold. Then, the operator $\mathcal{T}$ is a contraction on $B_R$ with constant $\rho <1$.

Proof. 

Let $({\theta }_1,{\varphi }_1),({\theta }_2,{\varphi }_2)\in B_R$. For $\kappa \in [a,b]$, using (H${}_4$) and the fact that ${\lambda }_1,{\lambda }_2\in (0,1)$ together with (14), we have

$$\begin{array}{cc}& |{\mathcal{J}}_1(\kappa ,{\theta }_1\left(\kappa \right),{\theta }_1\left({\lambda }_1\kappa \right),{\varphi }_1\left(\kappa \right))-{\mathcal{J}}_1(\kappa ,{\theta }_2\left(\kappa \right),{\theta }_2\left({\lambda }_1\kappa \right),{\varphi }_2\left(\kappa \right))|\hfill \\ & \le L_{11}|{\theta }_1\left(\kappa \right)-{\theta }_2\left(\kappa \right)|+L_{12}|{\theta }_1\left({\lambda }_1\kappa \right)-{\theta }_2\left({\lambda }_1\kappa \right)|+L_{13}|{\varphi }_1\left(\kappa \right)-{\varphi }_2\left(\kappa \right)|\hfill \\ & \le L_1{\parallel ({\theta }_1,{\varphi }_1)-({\theta }_2,{\varphi }_2)\parallel }_{\tilde{\mathcal{E}}}.\hfill \end{array}$$

Now consider ${\mathcal{T}}_1$. From (20),

$$\begin{array}{cc}& |{\mathcal{T}}_1({\theta }_1,{\varphi }_1)\left(\kappa \right)-{\mathcal{T}}_1({\theta }_2,{\varphi }_2)\left(\kappa \right)|\hfill \\ & \le \left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }|{\mathfrak{J}}_1^{{\theta }_1,{\varphi }_1}-{\mathfrak{J}}_1^{{\theta }_2,{\varphi }_2}|\right)\left(\kappa \right)+|{\nu }_1|\left({\mathcal{I}}^{{\zeta }_2;\psi }|{\theta }_1-{\theta }_2|\right)\left(\kappa \right)\hfill \\ & +{\displaystyle \frac{{(\psi \left(\kappa \right)-\psi \left(a\right))}^{{\gamma }_1+{\zeta }_2-1}}{|{\Delta }_1|\Gamma ({\gamma }_1+{\zeta }_2)}}[\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2;\psi }|{\mathfrak{J}}_1^{{\theta }_1,{\varphi }_1}-{\mathfrak{J}}_1^{{\theta }_2,{\varphi }_2}|\right)\left(b\right)\hfill \\ & +|{\nu }_1|\left({\mathcal{I}}^{{\zeta }_2;\psi }|{\theta }_1-{\theta }_2|\right)\left(b\right)\hfill \\ & +\sum _{i=1}^n|{\omega }_i|\left({\mathcal{I}}^{{\zeta }_1+{\zeta }_2+{\sigma }_i;\psi }|{\mathfrak{J}}_1^{{\theta }_1,{\varphi }_1}-{\mathfrak{J}}_1^{{\theta }_2,{\varphi }_2}|\right)\left({\eta }_i\right)\hfill \\ & +|{\nu }_1|\sum _{i=1}^n|{\omega }_i|\left({\mathcal{I}}^{{\zeta }_2+{\sigma }_i;\psi }|{\theta }_1-{\theta }_2|\right)\left({\eta }_i\right)].\hfill \end{array}$$

Applying the Lipschitz estimate and evaluating the fractional integrals using the constants defined in (22) and (23) yields

$$|{\mathcal{T}}_1({\theta }_1,{\varphi }_1)\left(\kappa \right)-{\mathcal{T}}_1({\theta }_2,{\varphi }_2)\left(\kappa \right)|\le \left(L_1{\Lambda }_1^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}\right){\parallel ({\theta }_1,{\varphi }_1)-({\theta }_2,{\varphi }_2)\parallel }_{\tilde{\mathcal{E}}}.$$

Taking the supremum over $\kappa$ gives

$$\parallel {\mathcal{T}}_1({\theta }_1,{\varphi }_1)-{\mathcal{T}}_1({\theta }_2,{\varphi }_2){\parallel }_{\infty }\le \left(L_1{\Lambda }_1^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}\right){\parallel ({\theta }_1,{\varphi }_1)-({\theta }_2,{\varphi }_2)\parallel }_{\tilde{\mathcal{E}}}.$$

A similar computation for ${\mathcal{T}}_2$ yields

$$\parallel {\mathcal{T}}_2({\theta }_1,{\varphi }_1)-{\mathcal{T}}_2({\theta }_2,{\varphi }_2){\parallel }_{\infty }\le \left(L_2{\Lambda }_2^{\left(1\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right){\parallel ({\theta }_1,{\varphi }_1)-({\theta }_2,{\varphi }_2)\parallel }_{\tilde{\mathcal{E}}}.$$

Therefore,

$$\begin{gathered} \parallel \mathcal{T}({\theta }_1,{\varphi }_1)-\mathcal{T}({\theta }_2,{\varphi }_2){\parallel }_{\tilde{\mathcal{E}}}=\parallel {\mathcal{T}}_1({\theta }_1,{\varphi }_1)-{\mathcal{T}}_1({\theta }_2,{\varphi }_2){\parallel }_{\infty }+{\parallel {\mathcal{T}}_2({\theta }_1,{\varphi }_1)-{\mathcal{T}}_2({\theta }_2,{\varphi }_2)\parallel }_{\infty } \\ \le \left[\left(L_1{\Lambda }_1^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}\right)+\left(L_2{\Lambda }_2^{\left(1\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)\right]\parallel ({\theta }_1,{\varphi }_1)-({\theta }_2,{\varphi }_2){\parallel }_{\tilde{\mathcal{E}}}=\rho {\parallel ({\theta }_1,{\varphi }_1)-({\theta }_2,{\varphi }_2)\parallel }_{\tilde{\mathcal{E}}}, \end{gathered}$$

with $\rho <1$ by (H${}_5$). □

4.3. Main Stability Theorem

Theorem 3

(Ulam–Hyers stability). Assume that hypotheses (H${}_1$), (H${}_2$), (H${}_4$), (H${}_5$), and (H${}_6$) hold, together with the domain condition (14). Then the boundary value problem (1) is Ulam–Hyers-stable on the ball $B_R$. Specifically, for any ${ϵ}_1,{ϵ}_2>0$ and any pair $(\theta ,\varphi )\in B_R$ satisfying (28) with the boundary conditions of (1), there exists a unique solution $({\theta }^{\ast },{\varphi }^{\ast })\in B_R$ of (1) such that

$$\parallel (\theta ,\varphi )-({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}\le K_1{ϵ}_1+K_2{ϵ}_2,$$

where the stability constants are given explicitly by

$$K_1={\displaystyle \frac{{\Pi }_1}{1-\rho }},K_2={\displaystyle \frac{{\Pi }_2}{1-\rho }}.$$

Proof. 

We proceed in three logically separated steps.

Step 1: Uniqueness of solutions on $B_R$. Under (H${}_4$) and (H${}_5$), Lemma 8 shows that $\mathcal{T}$ is a contraction on $B_R$ with constant $\rho <1$. Since $(B_R,\parallel ·{\parallel }_{\tilde{\mathcal{E}}})$ is a complete metric space (as a closed subset of the Banach space $\tilde{\mathcal{E}}$) and $\mathcal{T}\left(B_R\right)\subseteq B_R$ by (H${}_6$), the Banach fixed point theorem guarantees the existence of a unique fixed point $({\theta }^{\ast },{\varphi }^{\ast })\in B_R$ of $\mathcal{T}$. By Lemma 4, $({\theta }^{\ast },{\varphi }^{\ast })$ is the unique solution of (1) in $B_R$.

Step 2: Error estimate for approximate solutions. Let $(\theta ,\varphi )\in B_R$ satisfy (28) with ${ϵ}_1,{ϵ}_2>0$ and the boundary conditions of (1). By Lemmas 5 and 6, there exist ${\mathcal{Z}}_1,{\mathcal{Z}}_2$ with $\parallel {\mathcal{Z}}_1{\parallel }_{\infty }\le {ϵ}_1$, $\parallel {\mathcal{Z}}_2{\parallel }_{\infty }\le {ϵ}_2$ such that

$$(\theta ,\varphi )=\mathcal{T}(\theta ,\varphi )+\mathcal{P}({\mathcal{Z}}_1,{\mathcal{Z}}_2).$$

Let $({\theta }^{\ast },{\varphi }^{\ast })$ be the unique fixed point of $\mathcal{T}$ in $B_R$. Then

$$\begin{array}{cc}\hfill {\parallel (\theta ,\varphi )-({\theta }^{\ast },{\varphi }^{\ast })\parallel }_{\tilde{\mathcal{E}}}& =\parallel \mathcal{T}(\theta ,\varphi )+\mathcal{P}({\mathcal{Z}}_1,{\mathcal{Z}}_2)-\mathcal{T}({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}\hfill \\ & \le \parallel \mathcal{T}(\theta ,\varphi )-\mathcal{T}({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}+{\parallel \mathcal{P}({\mathcal{Z}}_1,{\mathcal{Z}}_2)\parallel }_{\tilde{\mathcal{E}}}\hfill \\ & \le \rho \parallel (\theta ,\varphi )-({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}+{\Pi }_1\parallel {\mathcal{Z}}_1{\parallel }_{\infty }+{\Pi }_2{\parallel {\mathcal{Z}}_2\parallel }_{\infty }\hfill \\ & \le \rho \parallel (\theta ,\varphi )-({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}+{\Pi }_1{ϵ}_1+{\Pi }_2{ϵ}_2,\hfill \end{array}$$

where we used the contraction property from Lemmas 7 and 8.

Step 3: Derivation of stability constants. Rearranging the inequality yields

$$(1-\rho )\parallel (\theta ,\varphi )-({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}\le {\Pi }_1{ϵ}_1+{\Pi }_2{ϵ}_2.$$

Since $\rho <1$ by (H${}_5$), we obtain the final error estimate:

$$\parallel (\theta ,\varphi )-({\theta }^{\ast },{\varphi }^{\ast }){\parallel }_{\tilde{\mathcal{E}}}\le {\displaystyle \frac{{\Pi }_1}{1-\rho }}{ϵ}_1+{\displaystyle \frac{{\Pi }_2}{1-\rho }}{ϵ}_2.$$

The inequality above shows that for any ${ϵ}_1,{ϵ}_2>0$ and any approximate solution $(\theta ,\varphi )\in B_R$ satisfying (28) with the correct boundary conditions, there exists a unique solution $({\theta }^{\ast },{\varphi }^{\ast })\in B_R$ of (1) such that the error bound holds with the explicit constants $K_1={\Pi }_1/(1-\rho )$ and $K_2={\Pi }_2/(1-\rho )$. This precisely matches Definition 7, completing the proof. □

4.4. Remarks on the Stability Result

Remark 6

(Interpretation of the stability constants). The stability constants $K_1$ and $K_2$ have clear interpretations:

• ${\Pi }_1$ and ${\Pi }_2$ measure the amplification of pointwise perturbations through the integral operator $\mathcal{P}$. These constants depend on the interval length $\left(\psi \right(b)-\psi (a\left)\right)$ through the fractional integral bounds.
• The denominator $1-\rho$ accounts for the contraction property of $\mathcal{T}$; as ρ approaches 1, the stability constants increase, reflecting decreased stability.
• Unlike formulations based on $L^1$-type norms, our final estimate does not introduce an extra multiplicative factor $\left(\psi \right(b)-\psi (a\left)\right)$ from norm conversions; the dependence on interval length is embedded in ${\Pi }_1$ and ${\Pi }_2$ through the fractional integral evaluations.

Remark 7

(Practical verification of conditions). The conditions of Theorem 3 can be verified in applications:

• (H${}_4$) requires Lipschitz continuity, which holds for many physically motivated nonlinearities.
• (H${}_5$) imposes a smallness condition that can often be satisfied by choosing the interval $[a,b]$ sufficiently short or by restricting the parameters ${\nu }_1,{\nu }_2$.
• (H${}_6$) is an a priori bound that follows from growth conditions on ${\Theta }_1$ and ${\Theta }_2$, similar to those in Theorem 2.

Remark 8

(On the best Ulam–Hyers constant). The constants $K_1$ and $K_2$ derived in Theorem 3 are sufficient conditions but are not necessarily optimal. The search for the "best" (smallest) Ulam–Hyers constant is a more challenging and often open problem for complex boundary value problems with nonlocal conditions. Our results provide explicit, computable bounds that guarantee stability, which is sufficient for most applications.

Remark 9

(Comparison with existence theory). The stability analysis complements the existence theory from Section 3. While Theorem 2 establishes existence under general conditions using Mönch’s fixed point theorem, Theorem 3 provides quantitative control over perturbations under stronger Lipschitz conditions. This hierarchical approach is standard in the analysis of fractional boundary value problems, with existence requiring milder assumptions than stability.

5. Numerical Example

In this section, we present a concrete boundary value problem of the form (1) and verify that all hypotheses of Theorem 2 (existence) and Theorem 3 (Ulam–Hyers stability) are satisfied. The purpose of this example is to demonstrate that the abstract conditions developed in Section 3 and Section 4 are not vacuous and can be explicitly verified for a specific system with chosen parameters.

5.1. Definition of the Example

Let $a=0$, $b=1$, and $n=2$. We take the function $\psi$ to be

$$\psi \left(\kappa \right)={\displaystyle \frac{e^{\kappa }}{3}},\kappa \in [0,1].$$

Note that with $a=0$, the domain condition (14) is automatically satisfied since ${\lambda }_1[0,1],{\lambda }_2[0,1]\subseteq [0,1]$ for ${\lambda }_1,{\lambda }_2\in (0,1)$.

The parameters of the system are chosen as follows:

$$\begin{array}{cc}\hfill {\zeta }_1& ={\displaystyle \frac{2}{5}},{\zeta }_2={\displaystyle \frac{1}{5}},{\mathcal{P}}_1={\displaystyle \frac{3}{5}},{\mathcal{P}}_2={\displaystyle \frac{2}{5}},{\beta }_1={\displaystyle \frac{4}{5}},{\beta }_2={\displaystyle \frac{3}{5}},\hfill \\ \hfill {\delta }_1& ={\displaystyle \frac{2}{5}},{\delta }_2={\displaystyle \frac{3}{5}},{\nu }_1={\displaystyle \frac{1}{9}},{\nu }_2={\displaystyle \frac{1}{10}},{\lambda }_1={\displaystyle \frac{1}{3}},{\lambda }_2={\displaystyle \frac{3}{10}},\hfill \\ \hfill {\sigma }_1& ={\displaystyle \frac{3}{2}},{\sigma }_2={\displaystyle \frac{5}{2}},{\pi }_1={\displaystyle \frac{5}{3}},{\pi }_2={\displaystyle \frac{7}{3}},\hfill \\ \hfill {\omega }_1& ={\displaystyle \frac{5}{8}},{\omega }_2={\displaystyle \frac{7}{8}},{\vartheta }_1={\displaystyle \frac{4}{7}},{\vartheta }_2={\displaystyle \frac{5}{7}},{\eta }_1={\displaystyle \frac{1}{3}},{\eta }_2={\displaystyle \frac{1}{2}},{\xi }_1={\displaystyle \frac{2}{3}},{\xi }_2={\displaystyle \frac{1}{3}}.\hfill \end{array}$$

Define the nonlinearities ${\mathcal{J}}_1,{\mathcal{J}}_2:[0,1]\times {\mathbb{R}}^3\to \mathbb{R}$ by

$$\begin{gathered} {\mathcal{J}}_1(\kappa ,u_1,u_2,u_3)={\displaystyle \frac{1}{10(\kappa +9)}}arctan(u_1+u_2)+{\displaystyle \frac{1}{100}}arctan\left(u_3\right)+{\displaystyle \frac{1}{81}}, \\ {\mathcal{J}}_2(\kappa ,u_1,u_2,u_3)={\displaystyle \frac{1}{20\pi }}sin\left(\pi u_1\right)+{\displaystyle \frac{1}{30}}arctan\left(u_2\right)+{\displaystyle \frac{1}{40\sqrt{\kappa +4}}}arctan\left(u_3\right). \end{gathered}$$

With these choices, the system (1) takes the explicit form

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^H{\mathcal{D}}_{0^{+}}^{2/5,4/5;\psi }\left({}^H{\mathcal{D}}_{0^{+}}^{1/5,3/5;\psi }+{\displaystyle \frac{1}{9}}\right)\theta \left(\kappa \right)={\mathcal{J}}_1(\kappa ,\theta \left(\kappa \right),\theta (\kappa /3),\varphi \left(\kappa \right)),\hfill \\ {}^H{\mathcal{D}}_{0^{+}}^{3/5,2/5;\psi }\left({}^H{\mathcal{D}}_{0^{+}}^{2/5,3/5;\psi }+{\displaystyle \frac{1}{10}}\right)\varphi \left(\kappa \right)={\mathcal{J}}_2(\kappa ,\varphi \left(\kappa \right),\varphi (3\kappa /10),\theta \left(\kappa \right)),\hfill \\ \theta \left(0\right)=0,\theta \left(1\right)={\displaystyle \frac{5}{8}}\left({\mathcal{I}}^{3/2;\psi }\theta \right)\left({\displaystyle \frac{1}{3}}\right)+{\displaystyle \frac{7}{8}}\left({\mathcal{I}}^{5/2;\psi }\theta \right)\left({\displaystyle \frac{1}{2}}\right),\hfill \\ \varphi \left(0\right)=0,\varphi \left(1\right)={\displaystyle \frac{4}{7}}\left({\mathcal{I}}^{5/3;\psi }\varphi \right)\left({\displaystyle \frac{2}{3}}\right)+{\displaystyle \frac{5}{7}}\left({\mathcal{I}}^{7/3;\psi }\varphi \right)\left({\displaystyle \frac{1}{3}}\right),\hfill \end{array}\right.\kappa \in [0,1].\end{array}$$

(35)

5.2. Verification of Hypotheses (H${}_1$)–(H${}_3$)

5.2.1. (H${}_1$) Carathéodory Condition

The functions ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ are continuous in $(u_1,u_2,u_3)$ for each fixed $\kappa \in [0,1]$, and for each fixed $(u_1,u_2,u_3)$ they are continuous (hence measurable) in $\kappa$. Thus, (H${}_1$) is satisfied.

5.2.2. (H${}_2$) Growth Condition

Using the elementary bounds $|arctan(·\left)\right|\le \pi /2$, $|sin(\pi ·\left)\right|\le 1$, and $1/\sqrt{\kappa +4}\le 1/2$ on $[0,1]$, we obtain

$$\begin{gathered} |{\mathcal{J}}_1(\kappa ,u_1,u_2,u_3)|\le {\displaystyle \frac{\pi }{20(\kappa +9)}}+{\displaystyle \frac{\pi }{200}}+{\displaystyle \frac{1}{81}}=:{\eta }_1\left(\kappa \right), \\ |{\mathcal{J}}_2(\kappa ,u_1,u_2,u_3)|\le {\displaystyle \frac{1}{20\pi }}+{\displaystyle \frac{\pi }{60}}+{\displaystyle \frac{\pi }{80\sqrt{\kappa +4}}}=:{\eta }_2\left(\kappa \right). \end{gathered}$$

Both ${\eta }_1$ and ${\eta }_2$ belong to $L^1([0,1],{\mathbb{R}}_{+})$, and we may take ${\Theta }_1\left(r\right)={\Theta }_2\left(r\right)\equiv 1$. Hence, (H${}_2$) holds.

5.2.3. (H${}_3$) Measure Condition

Since ${\mathcal{J}}_1$ and ${\mathcal{J}}_2$ are Lipschitz continuous (see the verification of (H${}_4$) below), for any bounded set $S\subset \tilde{\mathcal{E}}$ we have

$${\mu }_{\mathbb{R}}\left({\mathcal{J}}_1(\kappa ,{\widehat{S}}_1\left(\kappa \right))\right)\le L_1{\mu }_{\tilde{\mathcal{E}}}\left(S\right),{\mu }_{\mathbb{R}}\left({\mathcal{J}}_2(\kappa ,{\widehat{S}}_2\left(\kappa \right))\right)\le L_2{\mu }_{\tilde{\mathcal{E}}}\left(S\right),$$

where $L_1$ and $L_2$ are the Lipschitz constants computed in the next subsection. Thus, we may take ${\xi }_1\left(\kappa \right)=L_1$ and ${\xi }_2\left(\kappa \right)=L_2$ (constant functions), which belong to $L^1\left([0,1]\right)$. Consequently, (H${}_3$) is satisfied with ${\xi }_1^{\ast }=L_1$ and ${\xi }_2^{\ast }=L_2$.

5.3. Verification of Lipschitz Condition (H${}_4$)

For all $\kappa \in [0,1]$ and $x_i,y_i\in \mathbb{R}$, we have

$$\begin{array}{cc}\hfill |{\mathcal{J}}_1(\kappa ,x_1,x_2,x_3)-{\mathcal{J}}_1(\kappa ,y_1,y_2,y_3)|& \le {\displaystyle \frac{1}{10(\kappa +9)}}\left(|x_1-y_1|+|x_2-y_2|\right)+{\displaystyle \frac{1}{100}}|x_3-y_3|.\hfill \end{array}$$

Since $1/\left(10\right(\kappa +9\left)\right)\le 1/90$ on $[0,1]$, we obtain

$$L_{11}={\displaystyle \frac{1}{90}},L_{12}={\displaystyle \frac{1}{90}},L_{13}={\displaystyle \frac{1}{100}},L_1=L_{11}+L_{12}+L_{13}={\displaystyle \frac{145}{4500}}\approx 0.032222.$$

Similarly,

$$\begin{array}{cc}\hfill |{\mathcal{J}}_2(\kappa ,x_1,x_2,x_3)-{\mathcal{J}}_2(\kappa ,y_1,y_2,y_3)|& \le {\displaystyle \frac{1}{20}}|x_1-y_1|+{\displaystyle \frac{1}{30}}|x_2-y_2|+{\displaystyle \frac{1}{40\sqrt{\kappa +4}}}|x_3-y_3|.\hfill \end{array}$$

Using $1/\left(40\sqrt{\kappa +4}\right)\le 1/80$ on $[0,1]$, we obtain

$$L_{21}={\displaystyle \frac{1}{20}},L_{22}={\displaystyle \frac{1}{30}},L_{23}={\displaystyle \frac{1}{80}},L_2=L_{21}+L_{22}+L_{23}\approx 0.095833.$$

Thus, (H${}_4$) holds.

5.4. Computation of Structural Constants

First, compute the necessary values of $\psi$:

$$\psi \left(1\right)-\psi \left(0\right)={\displaystyle \frac{e-1}{3}}\approx 0.57276061,$$

$$\psi (1/3)-\psi \left(0\right)\approx 0.13187081,\psi (1/2)-\psi \left(0\right)\approx 0.21624042,\psi (2/3)-\psi \left(0\right)\approx 0.31591135.$$

The parameters ${\gamma }_1$ and ${\gamma }_2$ are given by

$${\gamma }_1={\zeta }_1+{\beta }_1(1-{\zeta }_1)={\displaystyle \frac{22}{25}}=0.88,{\gamma }_2={\mathcal{P}}_1+{\delta }_1(1-{\mathcal{P}}_1)={\displaystyle \frac{19}{25}}=0.76.$$

The constants ${\Delta }_1$ and ${\Delta }_2$ defined in (16) and (17) are computed as

$$\begin{array}{cc}\hfill {\Delta }_1& =\sum _{i=1}^2{\omega }_i{\displaystyle \frac{{(\psi \left({\eta }_i\right)-\psi \left(0\right))}^{{\gamma }_1+{\zeta }_2+{\sigma }_i-1}}{\Gamma ({\gamma }_1+{\zeta }_2+{\sigma }_i)}}-{\displaystyle \frac{{(\psi \left(1\right)-\psi \left(0\right))}^{{\gamma }_1+{\zeta }_2-1}}{\Gamma ({\gamma }_1+{\zeta }_2)}}\approx -0.97382829,\hfill \\ \hfill {\Delta }_2& =\sum _{i=1}^2{\vartheta }_i{\displaystyle \frac{{(\psi \left({\xi }_i\right)-\psi \left(0\right))}^{{\gamma }_2+{\mathcal{P}}_2+{\pi }_i-1}}{\Gamma ({\gamma }_2+{\mathcal{P}}_2+{\pi }_i)}}-{\displaystyle \frac{{(\psi \left(1\right)-\psi \left(0\right))}^{{\gamma }_2+{\mathcal{P}}_2-1}}{\Gamma ({\gamma }_2+{\mathcal{P}}_2)}}\approx -0.94175410.\hfill \end{array}$$

Hence, ${\Delta }_1\ne 0$ and ${\Delta }_2\ne 0$ as required.

Using the definitions in (22)–(25), we obtain

$${\Lambda }_1^{\left(1\right)}\approx 1.626131,{\Lambda }_1^{\left(2\right)}\approx 1.987879,{\Lambda }_2^{\left(1\right)}\approx 1.178044,{\Lambda }_2^{\left(2\right)}\approx 1.870518.$$

5.5. Verification of Contraction Condition (H${}_5$)

With the Lipschitz constants from (H${}_4$) and the constants computed above, the contraction constant $\rho$ defined in (H${}_5$) is

$$\begin{array}{cc}\hfill \rho & =\left(L_1{\Lambda }_1^{\left(1\right)}+\left|{\nu }_1\right|{\Lambda }_1^{\left(2\right)}\right)+\left(L_2{\Lambda }_2^{\left(1\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)\hfill \\ & =(0.032222\times 1.626131+\frac{1}{9}\times 1.987879)\hfill \\ & +(0.095833\times 1.178044+\frac{1}{10}\times 1.870518)\hfill \\ & =(0.0524+0.2209)+(0.1129+0.1871)=0.2733+0.3000=0.5733.\hfill \end{array}$$

Since $\rho =0.5733<1$, condition (H${}_5$) is satisfied.

5.6. Ulam–Hyers Stability Constants

From Remark 6, we have ${\Pi }_1={\Lambda }_1^{\left(1\right)}$ and ${\Pi }_2={\Lambda }_2^{\left(1\right)}$. Thus,

$${\Pi }_1\approx 1.626131,{\Pi }_2\approx 1.178044.$$

The stability constants $K_1$ and $K_2$ from Theorem 3 are then

$$K_1={\displaystyle \frac{{\Pi }_1}{1-\rho }}={\displaystyle \frac{1.626131}{1-0.5733}}={\displaystyle \frac{1.626131}{0.4267}}\approx 3.811,$$

$$K_2={\displaystyle \frac{{\Pi }_2}{1-\rho }}={\displaystyle \frac{1.178044}{0.4267}}\approx 2.761.$$

Remark 10.

The constants $K_1$ and $K_2$ computed above are sufficient conditions guaranteeing Ulam–Hyers stability, as established in Theorem 3. However, they are not necessarily the optimal (smallest) such constants. Determining the best Ulam–Hyers constants for this system remains an open problem, as noted in Remark 8.

5.7. Verification of Invariance Condition (H${}_6$)

First, compute the $L^1$-norms of ${\eta }_1$ and ${\eta }_2$:

$${\eta }_1^{\ast }={\int }_0^1{\eta }_1\left(\kappa \right)d\kappa \approx 0.044603633,{\eta }_2^{\ast }={\int }_0^1{\eta }_2\left(\kappa \right)d\kappa \approx 0.086816107.$$

The invariance condition $\mathcal{T}\left(B_R\right)\subseteq B_R$ requires that for some $R>0$,

$${\eta }_1^{\ast }{\Theta }_1\left(3R\right){\Lambda }_1^{\left(1\right)}+{\eta }_2^{\ast }{\Theta }_2\left(3R\right){\Lambda }_2^{\left(1\right)}+\left(|{\nu }_1|{\Lambda }_1^{\left(2\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)R\le R.$$

Since ${\Theta }_1\equiv {\Theta }_2\equiv 1$ in this example, this inequality reduces to

$${\eta }_1^{\ast }{\Lambda }_1^{\left(1\right)}+{\eta }_2^{\ast }{\Lambda }_2^{\left(1\right)}\le \left(1-|{\nu }_1|{\Lambda }_1^{\left(2\right)}-\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)R.$$

Solving for R gives

$$R\ge {\displaystyle \frac{{\eta }_1^{\ast }{\Lambda }_1^{\left(1\right)}+{\eta }_2^{\ast }{\Lambda }_2^{\left(1\right)}}{1-\left(|{\nu }_1|{\Lambda }_1^{\left(2\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}\right)}}.$$

Substituting the numerical values:

$$|{\nu }_1|{\Lambda }_1^{\left(2\right)}+\left|{\nu }_2\right|{\Lambda }_2^{\left(2\right)}={\displaystyle \frac{1}{9}}\times 1.987879+{\displaystyle \frac{1}{10}}\times 1.870518\approx 0.2209+0.1871=0.4080,$$

$${\eta }_1^{\ast }{\Lambda }_1^{\left(1\right)}+{\eta }_2^{\ast }{\Lambda }_2^{\left(1\right)}\approx 0.044603633\times 1.626131+0.086816107\times 1.178044\approx 0.0725+0.1023=0.1748.$$

Thus,

$$R\ge {\displaystyle \frac{0.1748}{1-0.4080}}={\displaystyle \frac{0.1748}{0.5920}}\approx 0.295.$$

Choosing $R=0.30$ ensures that $\mathcal{T}\left(B_R\right)\subseteq B_R$. Hence, condition (H${}_6$) is satisfied.

5.8. Conclusion of the Example

We have verified that for the system (35) with the chosen parameters and nonlinearities, all hypotheses (H${}_1$)–(H${}_6$) required by Theorem 2 and Theorem 3 are satisfied. Consequently, by Theorem 2, the system admits at least one solution $(\theta ,\varphi )\in B_R$ with ${\parallel (\theta ,\varphi )\parallel }_{\tilde{\mathcal{E}}}\le 0.30$. Moreover, by Theorem 3, the system is Ulam–Hyers-stable on $B_R$ with stability constants $K_1\approx 3.811$ and $K_2\approx 2.761$.

Remark 11.

The example presented above is analytical rather than computational. Its purpose is to demonstrate that the abstract hypotheses of Theorems 2 and 3 can be verified for a concrete system with explicit parameter values. The computation of all constants, including the contraction constant ρ and the stability constants $K_1,K_2$, shows that the conditions are not only theoretically sound but also practically verifiable. A numerical simulation of the solutions would be an interesting direction for future research, but is beyond the scope of the present work.

6. Conclusions

In this paper, we studied a coupled system of $\psi$-Hilfer fractional pantograph–Langevin equations subject to nonlocal integral boundary conditions. By reformulating the problem as an equivalent fixed point equation and applying Mönch’s fixed point theorem together with the Kuratowski measure of noncompactness, we obtained sufficient conditions for the existence of at least one solution. We then established Ulam–Hyers stability on a suitable closed ball under additional Lipschitz-type assumptions, and derived explicit computable stability constants, which were evaluated numerically in the example as $K_1\approx 3.811$ and $K_2\approx 2.761$. A concrete example was included to verify the applicability of the theoretical assumptions and to illustrate the use of the main results.

The results of this work contribute to the qualitative analysis of coupled fractional boundary value problems with proportional delay and nonlocal conditions in the $\psi$-Hilfer setting. The existence theorem is obtained under relatively general growth and noncompactness assumptions, while the stability result requires stronger contraction-type conditions, which is consistent with the usual hierarchy between existence and stability analyses. Thus, the paper achieves the goals set out in the introduction: establishing existence, proving Ulam–Hyers stability, and providing a verifiable example.

Several directions remain open for future research. One natural extension is to investigate generalized forms of stability, such as Ulam–Hyers–Rassias stability, for the same class of systems. Another is to study related problems involving different fractional operators or more general boundary conditions. Although the stability constants obtained here are sufficient for guaranteeing stability, determining the optimal (smallest) constants remains an open problem. It would also be of interest to develop numerical schemes for approximating solutions and to compare their behavior with the theoretical stability bounds obtained here.