Existence, Stability, and Circular Interactions in m-Cyclic Coupled Systems of Sequential (k,ψ)-Hilfer and (k,ψ)-Caputo Type with Boundary Conditions

Abstract

This paper examines an m-cyclic coupled system of sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional differential equations with boundary conditions. The nonlinearities follow a cyclic pattern: for $j=1,\dots ,m-1$, $f_j$ depends on $x_j$ and $x_{j+1}$ and $f_m$ depends on $x_m$ and $x_1$, forming a closed loop of interactions. We convert the system into an equivalent integral equation and establish existence and uniqueness results using four fixed-point theorems: the Banach contraction principle, Schaefer’s theorem, Krasnosel’skiĭ’s theorem, and the Leray–Schauder alternative. A thorough Ulam–Hyers stability analysis is presented with explicit stability constants. Numerical examples illustrate the applicability of the theoretical findings.

1. Introduction

Fractional calculus has become an essential tool for modeling systems with memory and hereditary properties [1,2,3]. Among the various fractional operators, the Hilfer derivative [4] unifies the Riemann–Liouville and Caputo formulations through a parameter $\beta \in [0,1]$. Sousa and Oliveira [5] introduced the $\psi$-Hilfer derivative, and Kucche and Mali [6] further extended this to the $(k,\psi )$-framework, allowing kernel adaptation through the parameter k and the function $\psi$.

This paper investigates an m-cyclic coupled system of sequential fractional differential equations. The term sequential refers to the composition of two fractional derivatives: a Caputo derivative of order ${\zeta }_j$ acting on $x_j$ followed by a Hilfer derivative of order ${\theta }_j$ and type ${\beta }_j$. The study of sequential fractional differential equations is motivated by the fundamental limitations of non-sequential formulations. As demonstrated by Vatsala et al. [7], linear Caputo fractional differential equations containing lower-order fractional derivatives cannot be solved in general, whereas sequential formulations admit explicit solutions via Laplace transform methods and Mittag-Leffler functions. This theoretical advantage, combined with the physical interpretation of sequential derivatives as hierarchical memory mechanisms, justifies the sequential structure adopted in the present work.

The combination of two fractional derivatives is not just a mathematical idea; it happens naturally in physical processes that involve two different memory mechanisms working at different scales. For example, in fractional viscoelasticity, the relationship between stress and strain can be represented by a Hilfer derivative acting on a Caputo derivative of the strain. This approach captures both the material’s historical dependence and the rate at which deformation occurs. Syam et al. [8] recently showed how to use sequential fractional derivatives in wave equations. This showed how well they can model complicated physical phenomena.

In recent years, the theory of sequential fractional differential equations has grown a lot. Ahmad and Nieto [9] commenced the investigation of sequential Caputo boundary value problems with three-point conditions, establishing fundamental existence results. Bai [10] expanded upon this research by investigating fractional multi-point boundary value problems at resonance. Zhang and Liu [11,12] further extended the resonance analysis to infinite-point and multi-point settings.

Erkan et al. [13,14] recently established comprehensive existence and uniqueness results for sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo equations with non-separated boundary conditions, thereby providing a rigorous framework for single-equation analysis. Furthermore, Salem and Almaghamsi [15] progressed this study by examining resonance phenomena in sequential fractional differential equations through coincidence degree theory, resulting in the identification of solvability conditions for three-point boundary value problems.

In parallel, the study of coupled fractional systems has attracted considerable attention. Henderson and Luca [16] investigated systems of Riemann–Liouville fractional differential equations with coupled boundary conditions. Su [17] extended such analysis to Caputo-type systems with integral boundary conditions. More recently, researchers have looked at coupled systems with different fractional operators: Baleanu et al. [18] examined coupled systems utilizing Caputo–Fabrizio derivatives, whereas Khan et al. [19] investigated sequential hybrid systems. Neama et al. [20] investigated m-cyclic coupled systems with Caputo–Fabrizio derivatives, underscoring the significance of cyclic coupling in multi-component systems, which is particularly pertinent to this study. Nevertheless, thus far, no research has integrated the adaptability of the $(k,\psi )$-operators—facilitating kernel adjustment via the parameters k and $\psi$—with the cyclic coupling architecture within a sequential context.

In a related direction, discrete fractional difference operators—particularly nabla fractional differences—have gained prominence for modeling memory effects in discrete-time systems. Abdeljawad [21] established the foundational properties of Riemann–Liouville and Caputo differences, while Goodrich [22] demonstrated their utility in boundary value problems exhibiting nonlocal memory. This discrete perspective complements our continuous $(k,\psi )$-framework, which offers tunable memory kernels through the parameters k and $\psi$, adaptable to both short-range and long-range memory in cyclic continuous systems.

This work fills an important gap by combining two important features: hierarchical memory through sequential $(k,\psi )$-operators and closed-loop interactions through cyclic coupling. This combination is very important because it shows how memory can move through a closed loop. Some examples are ring oscillators with fractional-order feedback, cyclic gene regulatory networks with multiple time scales, and circular supply chains with delayed information flow. Unlike the resonance problems studied in [15], our boundary conditions satisfy ${\varsigma }_{j1}+{\varsigma }_{j2}\ne 0$ and ${\varsigma }_{j3}+{\varsigma }_{j4}\ne 0$, ensuring that the linear operator is invertible. The integral representation is well-defined without requiring coincidence degree theory. This setting allows us to apply standard fixed-point theorems directly.

The combination of sequential fractional derivatives with cyclic coupling is purposeful and not arbitrary. In physical systems where memory effects circulate through a closed loop, the appropriate mathematical description often involves hierarchical memory. In such systems, the current state of each component depends not only on its own past but also on the past rate of change of the next component. For example, consider a ring of m viscoelastic elements. In this setup, the stress in element j depends on the strain history of that same element (described by a Caputo derivative) and on the strain rate history of element $j+1$ (which involves a Hilfer derivative acting on the Caputo derivative of element j). This yields precisely the sequential structure ${}_k{}^{H}D^{\theta ,\beta ;\psi }\left({}_k{}^{C}D^{\zeta ;\psi }{x}_j\right)$ with cyclic coupling $f_j(x_j,x_{j+1})$.

Although these systems have not been extensively studied in the literature, the mathematical structure outlined above offers a natural framework for modeling closed-loop systems with hierarchical memory. This work proposes this unified framework as a basis for future applications in cyclic physical systems.

Motivated by [13,14,20], we consider the following m-cycle coupled systems of sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo types:

$$\left\{\begin{array}{c}{}_{k_1}{}^{H}D_{0^{+}}^{{\theta }_1,{\beta }_1;{\psi }_1}\left({}_{k_1}{}^{C}D_{0^{+}}^{{\zeta }_1;{\psi }_1}{x}_1\left(\varrho \right)\right)=f_1(\varrho ,x_1\left(\varrho \right),x_2\left(\varrho \right)),\varrho \in [0,b],\hfill \\ {}_{k_2}{}^{H}D_{0^{+}}^{{\theta }_2,{\beta }_2;{\psi }_2}\left({}_{k_2}{}^{C}D_{0^{+}}^{{\zeta }_2;{\psi }_2}{x}_2\left(\varrho \right)\right)=f_2(\varrho ,x_2\left(\varrho \right),x_3\left(\varrho \right)),\varrho \in [0,b],\hfill \\ ⋮\hfill \\ {}_{k_m}{}^{H}D_{0^{+}}^{{\theta }_m,{\beta }_m;{\psi }_m}\left({}_{k_m}{}^{C}D_{0^{+}}^{{\zeta }_m;{\psi }_m}{x}_m\left(\varrho \right)\right)=f_m(\varrho ,x_m\left(\varrho \right),x_1\left(\varrho \right)),\varrho \in [0,b],\hfill \end{array}\right.$$

(1)

These are subject to the following boundary conditions:

$$\left\{\begin{array}{c}{\varsigma }_{11}{x}_1\left(0\right)+{\varsigma }_{12}{x}_1\left(b\right)=0,\hfill \\ {\varsigma }_{21}{x}_2\left(0\right)+{\varsigma }_{22}{x}_2\left(b\right)=0,\hfill \\ ⋮\hfill \\ {\varsigma }_{m1}{x}_m\left(0\right)+{\varsigma }_{m2}{x}_m\left(b\right)=0,\hfill \end{array}\right.$$

(2)

and

$$\left\{\begin{array}{c}{\varsigma }_{1,3}{}_{k_1}{}^{C}D_{0^{+}}^{{\delta }_{k_1}+{\zeta }_1-k_1;{\psi }_1}{x}_1\left(0\right)+{\varsigma }_{1,4}{}_{k_1}{}^{C}D_{0^{+}}^{{\delta }_{k_1}+{\zeta }_1-k_1;{\psi }_1}{x}_1\left(b\right)=0,\hfill \\ {\varsigma }_{2,3}{}_{k_2}{}^{C}D_{0^{+}}^{{\delta }_{k_2}+{\zeta }_2-k_2;{\psi }_2}{x}_2\left(0\right)+{\varsigma }_{2,4}{}_{k_2}{}^{C}D_{0^{+}}^{{\delta }_{k_2}+{\zeta }_2-k_2;{\psi }_2}{x}_2\left(b\right)=0,\hfill \\ ⋮\hfill \\ {\varsigma }_{m,3}{}_{k_m}{}^{C}D_{0^{+}}^{{\delta }_{k_m}+{\zeta }_m-k_m;{\psi }_m}{x}_m\left(0\right)+{\varsigma }_{m,4}{}_{k_m}{}^{C}D_{0^{+}}^{{\delta }_{k_m}+{\zeta }_m-k_m;{\psi }_m}{x}_m\left(b\right)=0.\hfill \end{array}\right.$$

(3)

For each $j=1,\dots ,m$, the parameters satisfy $0<{\theta }_j,{\zeta }_j<1$, $0\le {\beta }_j\le 1$, $k_j>0$, with ${\delta }_{k_j}={\theta }_j+{\beta }_j(k_j-{\theta }_j)$ and ${\delta }_{k_j}+{\zeta }_j-k_j>0$. The functions ${\psi }_j\in C^1([0,b],\mathbb{R})$ are increasing with ${\psi }_j^{\prime }\left(\varrho \right)\ne 0$, and ${\varsigma }_{j,1},{\varsigma }_{j,2},{\varsigma }_{j,3},{\varsigma }_{j,4}\in \mathbb{R}$ are constants. The nonlinearities $f_j:[0,b]\times {\mathbb{R}}^2\to \mathbb{R}$ are continuous, and the cyclic convention $x_{m+1}=x_1$ closes the loop.

The specific structure of this system—sequential $(k,\psi )$-fractional operators combined with cyclic coupling—presents three interconnected challenges that distinguish it from previously studied systems.

First, the integral representation must account for both the sequential nature and the cyclic coupling. The composition of fractional derivatives creates two layers of memory: the Caputo derivative keeps track of the history of $x_j$, and the Hilfer derivative keeps track of the history of how quickly $x_j$ changes. Each component $x_j$ depends on $x_j$ and $x_{j+1}$ through the nonlinearity $f_j$ when cyclic coupling is used. The boundary conditions connect the values at 0 and b. Deriving an equivalent integral formulation requires carefully solving for the constants arising from the inversion formulas while respecting the cyclic structure.

Second, the contraction analysis reveals a factor of 2 that arises purely from the cyclic coupling. The contraction constant for standard coupled systems without cyclic coupling is $\sum L_j{\Omega }_j$. Because of the closed loop, each term $|x_j-y_j|$ shows up twice in the sum, which gives us $2\sum L_j{\Omega }_j$. This factor shows how perturbations spread around the cycle, and it is a direct result of the cyclic geometry.

Third, the stability analysis must take into account how changes move through the cycle. The Ulam–Hyers stability constants from Section 5 explicitly include the cyclic coupling by using constants ${\Omega }_j^{\left(g\right)}$, which give more weight to the effects of perturbations from each component and its neighbor.

The main contributions of this work are:

• Derivation of an equivalent integral representation for the m-cyclic coupled sequential system.
• Application of four fixed-point theorems (Banach contraction, Schaefer, Krasnosel’skiĭ, Leray–Schauder) to establish existence and uniqueness under different conditions on $f_j$.
• Explicit contraction conditions incorporating the cyclic coupling factor 2.
• Explicit Ulam–Hyers stability constants that quantify perturbation propagation through the cycle.

The organization of the remaining sections is as follows: In Section 2, we recall essential definitions and establish the standing assumptions. Section 3 presents the derivation of the integral representation. The existence and uniqueness results, utilizing four fixed-point theorems, are discussed in Section 4. Section 5 analyzes Ulam–Hyers stability, providing explicit constants. Numerical examples are offered in Section 6. Finally, Section 7 concludes the discussion.

2. Preliminaries

2.1. Standing Assumptions

Throughout this paper, we assume the following:

(A1)

For each $j=1,\dots ,m$, $0<{\theta }_j,{\zeta }_j<1$, $0\le {\beta }_j\le 1$, $k_j>0$, ${\delta }_{k_j}={\theta }_j+{\beta }_j(k_j-{\theta }_j)$, and ${\delta }_{k_j}+{\zeta }_j-k_j>0$.

(A2)

${\psi }_j\in C^1([0,b],\mathbb{R})$ is strictly increasing with ${\psi }_j^{\prime }\left(\varrho \right)>0$ for all $\varrho \in [0,b]$.

(A3)

${\varsigma }_{j1}+{\varsigma }_{j2}\ne 0$ and ${\varsigma }_{j3}+{\varsigma }_{j4}\ne 0$ for each j.

(A4)

$f_j:[0,b]\times {\mathbb{R}}^2\to \mathbb{R}$ are continuous.

2.2. Fractional Operators

Definition 1

([23]). For $\theta \in \mathbb{C}$ with $\Re \left(\theta \right)>0$ and $k>0$, the k-gamma function is

$${\Gamma }_k\left(\theta \right)={\int }_0^{\infty }{\tau }^{\theta -1}{e}^{-{\displaystyle \frac{{\tau }^k}{k}}}d\tau ,$$

with ${\Gamma }_k(\theta +k)=\theta {\Gamma }_k\left(\theta \right)$ and ${\Gamma }_k\left(\theta \right)=k^{{\displaystyle \frac{\theta }{k}}-1}\Gamma \left({\displaystyle \frac{\theta }{k}}\right)$.

Definition 2

([24]). Let $x:[a,b]\to \mathbb{R}$ be integrable. For increasing ψ with ${\psi }^{\prime }\left(t\right)>0$, the $(k,\psi )$-Riemann–Liouville fractional integral of order $\theta >0$ is

$${}_{k}I_{a^{+}}^{\theta ,\psi }x\left(t\right)={\displaystyle \frac{1}{k{\Gamma }_k\left(\theta \right)}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{{\displaystyle \frac{\theta }{k}}-1}x\left(\tau \right)d\tau ,t>a.$$

Definition 3

([6]). Let $\theta >0$, $k>0$, $n=\lceil \theta /k\rceil$, and $x\in C^n([a,b],\mathbb{R})$. The $(k,\psi )$-Caputo fractional derivative is

$${}_k{}^{C}D_{a^{+}}^{\theta ,\psi }x\left(t\right)={\displaystyle \frac{1}{k{\Gamma }_k(nk-\theta )}}{\int }_a^t{\psi }^{\prime }\left(\tau \right){(\psi \left(t\right)-\psi \left(\tau \right))}^{n-{\displaystyle \frac{\theta }{k}}-1}{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(\tau \right)}}{\displaystyle \frac{d}{d\tau }}\right)}^{n}x\left(\tau \right)d\tau .$$

Definition 4

([6]). Let $\theta >0$, $k>0$, $0\le \beta \le 1$, $n=\lceil \theta /k\rceil$, and $x\in C^n([a,b],\mathbb{R})$. The $(k,\psi )$-Hilfer fractional derivative is

$${}_k{}^{H}D_{a^{+}}^{\theta ,\beta ;\psi }x\left(t\right)={}_{k}I_{a^{+}}^{\beta (nk-\theta );\psi }{\left({\displaystyle \frac{k}{{\psi }^{\prime }\left(t\right)}}{\displaystyle \frac{d}{dt}}\right)}^n{}_{k}I_{a^{+}}^{(1-\beta )(nk-\theta );\psi }x\left(t\right).$$

Definition 5

(Solution). A vector $\mathbf{x}=(x_1,\dots ,x_m)$ with $x_j\in C^1([0,b],\mathbb{R})$ is called a solution of system (1) and (2) if it satisfies the equations for all $\varrho \in [0,b]$ and the boundary conditions. The $C^1$ regularity ensures that the inner Caputo derivative is continuous, making the sequential derivative well-defined.

Remark 1.

While we assume $x_j\in C^1([0,b],\mathbb{R})$ for simplicity, the results extend naturally to functions whose $(k,\psi )$-Caputo derivative exists almost everywhere and is integrable. This is the natural solution space for fractional differential equations of order $0<{\zeta }_j<1$ (see, e.g., Kilbas et al. [1]). The $C^1$ assumption is convenient but not essential for the fixed-point analysis, which relies only on the integral representation (4) and the continuity of the Nemytskii operator.

Lemma 1

(Existence of Fractional Integrals and Derivatives). Let $x\in C\left(\right[a,b],\mathbb{R})$ and $\alpha >0$. Then, the $(k,\psi )$-Riemann–Liouville fractional integral ${}_{k}I_{a^{+}}^{\alpha ;\psi }x\left(t\right)$ exists for all $t\in [a,b]$ and defines a continuous function on $[a,b]$. Moreover, if $x\in C^1([a,b],\mathbb{R})$, then the $(k,\psi )$-Caputo derivative ${}_k{}^{C}D_{a^{+}}^{\alpha ;\psi }x\left(t\right)$ exists and is continuous.

Lemma 2

(Inversion for Caputo [6]). For $x\in C^n([a,b],\mathbb{R})$ with $n=\lceil \theta /k\rceil$,

$${}_{k}I_{a^{+}}^{\theta ;\psi }\left({}_k{}^{C}D_{a^{+}}^{\theta ,\psi }x\left(t\right)\right)=x\left(t\right)-{\displaystyle \sum _{j=0}^{n-1}}{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^j}{{\Gamma }_k(jk+k)}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}\right)}^{j}x\right]}_{t=a}.$$

Lemma 3

(Inversion for Hilfer [6]). For $x\in C^n([a,b],\mathbb{R})$ with $n=\lceil \theta /k\rceil$ and ${\delta }_k=\theta +\beta (nk-\theta )$,

$${}_{k}I_{a^{+}}^{\theta ;\psi }\left({}_k{}^{H}D_{a^{+}}^{\theta ,\beta ;\psi }x\left(t\right)\right)=x\left(t\right)-{\displaystyle \sum _{j=1}^n}{\displaystyle \frac{{(\psi \left(t\right)-\psi \left(a\right))}^{\frac{{\delta }_k}{k}-j}}{{\Gamma }_k({\delta }_k-jk+k)}}{\left[{\left({\displaystyle \frac{k}{{\psi }^{\prime }}}{\displaystyle \frac{d}{dt}}\right)}^{n-j}\left({}_{k}I_{a^{+}}^{nk-{\delta }_k;\psi }x\right)\right]}_{t=a}.$$

Lemma 4

(Composition [6]). For ${\theta }_1,{\theta }_2,k>0$ with ${\theta }_2>{\theta }_1$,

$${}_k{}^{C}D_{a^{+}}^{{\theta }_1;\psi }\left({}_{k}I_{a^{+}}^{{\theta }_2;\psi }x\left(t\right)\right)={}_{k}I_{a^{+}}^{{\theta }_2-{\theta }_1;\psi }x\left(t\right).$$

Lemma 5

(Power Function [6]). For $\theta ,k>0$ and $\mu \in \mathbb{R}$ with $\mu /k>-1$,

$${}_k{}^{C}D_{a^{+}}^{\theta ;\psi }{(\psi \left(t\right)-\psi \left(a\right))}^{{\displaystyle \frac{\mu }{k}}}={\displaystyle \frac{{\Gamma }_k(\mu +k)}{{\Gamma }_k(\mu +k-\theta )}}{(\psi \left(t\right)-\psi \left(a\right))}^{{\displaystyle \frac{\mu -\theta }{k}}}.$$

3. Integral Equation Representation

Let $X=C\left(\right[0,b],\mathbb{R})$ be the Banach space of continuous functions on $[0,b]$ equipped with the norm ${\parallel x\parallel }_{\infty }={sup}_{\varrho \in [0,b]}\left|x\left(\varrho \right)\right|$. Let $\mathcal{X}=X^m$ be the product Banach space with the norm ${\parallel \mathbf{x}\parallel }_{\mathcal{X}}={\sum }_{j=1}^m{\parallel x_j\parallel }_{\infty }$.

Lemma 6

(Integral Representation). Under the standing assumptions (Section 2.1), a vector $(x_1,\dots ,x_m)$ with $x_j\in C^1([0,b],\mathbb{R})$ is a solution of system (1) and (2) if and only if, for each $j=1,\dots ,m$ and all $\varrho \in [0,b]$,

$$\begin{array}{cc}\hfill x_j\left(\varrho \right)& =\left({\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\mathcal{G}}_j\left(b\right)-{\mathcal{G}}_j\left(\varrho \right)\right){}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)+{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right),\hfill \end{array}$$

(4)

with $x_{m+1}=x_1$ and

$${\mathcal{G}}_j\left(\varrho \right)={\displaystyle \frac{{\varsigma }_{j4}}{{\varsigma }_{j3}+{\varsigma }_{j4}}}{\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}.$$

(5)

Proof. 

Let $(x_1,\dots ,x_m)$ be a solution of system (1) and (2) in the sense of Definition 5. For each $j=1,2,\dots ,m$, define auxiliary functions

$$u_j\left(\varrho \right)={}_{k_j}{}^{C}D_{0^{+}}^{{\zeta }_j;{\psi }_j}{x}_j\left(\varrho \right),\varrho \in [0,b].$$

(6)

Since $x_j\in C^1([0,b],\mathbb{R})$, by Lemma 1, $u_j$ is continuous. Moreover, because $f_j$ is continuous, the mapping $\varrho \mapsto f_j(\varrho ,x_j\left(\varrho \right),x_{j+1}\left(\varrho \right))$ is a superposition (Nemytskii) operator [25]; such operators preserve continuity under Carathéodory conditions. Hence, all fractional integrals in the following derivation are well-defined.

Then, the differential equations in system (1) become

$${}_{k_j}{}^{H}D_{0^{+}}^{{\theta }_j,{\beta }_j;{\psi }_j}{u}_j\left(\varrho \right)=f_j(\varrho ,x_j\left(\varrho \right),x_{j+1}\left(\varrho \right)),j=1,2,\dots ,m.$$

(7)

Applying the $(k_j,{\psi }_j)$-Riemann–Liouville fractional integral of order ${\theta }_j$ to both sides of (7) and using Lemma 3 with $n=1$ (since $0<{\theta }_j/k_j<1$), we obtain

$$u_j\left(\varrho \right)={}_{k_j}I_{0^{+}}^{{\theta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)+c_{j1}{\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}}{k_j}-1}}{{\Gamma }_{k_j}\left({\delta }_{k_j}\right)}},j=1,2,\dots ,m,$$

(8)

where $c_{j1}$ are constants given by

$$c_{j1}={\left[{}_{k_j}I_{0^{+}}^{k_j-{\delta }_{k_j};{\psi }_j}{u}_j\right]}_{\varrho =0},j=1,2,\dots ,m.$$

(9)

Note that Lemma 3 is applicable here because $u_j$ is continuous and the inversion formula holds for continuous functions under the given assumptions.

Since $u_j\left(\varrho \right)={}_{k_j}{}^{C}D_{0^{+}}^{{\zeta }_j;{\psi }_j}{x}_j\left(\varrho \right)$, applying the $(k_j,{\psi }_j)$-Riemann–Liouville fractional integral of order ${\zeta }_j$ to both sides and using Lemma 2 with $n=1$ (since $0<{\zeta }_j/k_j<1$) yields

$$x_j\left(\varrho \right)={}_{k_j}I_{0^{+}}^{{\zeta }_j;{\psi }_j}{u}_j\left(\varrho \right)+d_{j0},j=1,2,\dots ,m,$$

(10)

where $d_{j0}=x_j\left(0\right)$ are constants.

Substituting (8) into (10) and using Lemma 4 (the semigroup property), we obtain

$$x_j\left(\varrho \right)={}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)+c_{j1}{}_{k_j}I_{0^{+}}^{{\zeta }_j;{\psi }_j}\left({\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}}{k_j}-1}}{{\Gamma }_{k_j}\left({\delta }_{k_j}\right)}}\right)+d_{j0}.$$

(11)

Using Lemma 5 to compute the fractional integral of the power function, we have

$${}_{k_j}I_{0^{+}}^{{\zeta }_j;{\psi }_j}\left({({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{{\displaystyle \frac{{\delta }_{k_j}}{k_j}}-1}\right)={\displaystyle \frac{{\Gamma }_{k_j}\left({\delta }_{k_j}\right)}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{{\displaystyle \frac{{\delta }_{k_j}+{\zeta }_j}{k_j}}-1}.$$

(12)

Substituting (12) into (11) gives the general form

$$x_j\left(\varrho \right)={}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)+c_{j1}{\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}+d_{j0}.$$

(13)

Applying Lemmas 4 and 5 to (13), we compute

$$\begin{array}{cc}\hfill {}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{x}_j\left(\varrho \right)& ={}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)\hfill \\ \hfill & +c_{j1}{\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1-\frac{{\delta }_{k_j}+{\zeta }_j-k_j}{k_j}}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j-({\delta }_{k_j}+{\zeta }_j-k_j))}}.\hfill \end{array}$$

(14)

Simplifying the exponent explicitly can be described as follows:

$$\begin{array}{cc}\hfill {\displaystyle \frac{{\delta }_{k_j}+{\zeta }_j}{k_j}}-1-{\displaystyle \frac{{\delta }_{k_j}+{\zeta }_j-k_j}{k_j}}& ={\displaystyle \frac{{\delta }_{k_j}+{\zeta }_j-k_j-({\delta }_{k_j}+{\zeta }_j-k_j)}{k_j}}=0.\hfill \end{array}$$

Using ${\Gamma }_{k_j}\left(k_j\right)=1$, we obtain the simplified form

$${}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{x}_j\left(\varrho \right)={}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)+c_{j1}.$$

(15)

In particular, at the endpoints, we have

$${}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{x}_j\left(0\right)=c_{j1},$$

and

$${}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{x}_j\left(b\right)={}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)+c_{j1}.$$

(16)

From (13), we evaluate $x_j$ at the endpoints:

$$x_j\left(0\right)=d_{j0},$$

and

$$x_j\left(b\right)={}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)+c_{j1}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}+d_{j0}.$$

(17)

Now we substitute (17) and (16) into the boundary conditions of system (1). For the first boundary condition of the j-th component,

$${\varsigma }_{j1}{d}_{j0}+{\varsigma }_{j2}\left({}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j\left(b\right)+c_{j1}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}+d_{j0}\right)=0.$$

(18)

For the second boundary condition of the j-th component,

$${\varsigma }_{j3}{c}_{j1}+{\varsigma }_{j4}\left({}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j\left(b\right)+c_{j1}\right)=0.$$

(19)

From (19), we solve for $c_{j1}$:

$$({\varsigma }_{j3}+{\varsigma }_{j4})c_{j1}=-{\varsigma }_{j4}{}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j\left(b\right),$$

which gives

$$c_{j1}=-{\displaystyle \frac{{\varsigma }_{j4}}{{\varsigma }_{j3}+{\varsigma }_{j4}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j\left(b\right).$$

(20)

Now substitute these into (18) to solve for $d_{j0}$. First, rearrange (18):

$$({\varsigma }_{j1}+{\varsigma }_{j2})d_{j0}+{\varsigma }_{j2}{c}_{j1}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}=-{\varsigma }_{j2}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j\left(b\right).$$

(21)

Solving for $d_{j0}$ can be described as follows:

$$d_{j0}=-{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j\left(b\right)-{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{c}_{j1}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}.$$

(22)

Substituting (20) into (22) yields

$$\begin{array}{cc}\hfill d_{j0}=& -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j\left(b\right)\hfill \\ \hfill & +{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\displaystyle \frac{{\varsigma }_{j4}}{{\varsigma }_{j3}+{\varsigma }_{j4}}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}{}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j\left(b\right).\hfill \end{array}$$

(23)

Now substitute the expressions for $c_{j1}$ and $d_{j0}$ back into (13). For $x_j\left(\varrho \right)$, we have

$$x_j\left(\varrho \right)={}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j\left(\varrho \right)+c_{j1}{\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}+d_{j0}.$$

(24)

Using (20) and (23) and introducing the notation ${\mathcal{G}}_j\left(\varrho \right)$ as defined in (5), we obtain

$$\begin{array}{cc}\hfill x_j\left(\varrho \right)=& {}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j\left(\varrho \right)-{\displaystyle \frac{{\varsigma }_{j4}}{{\varsigma }_{j3}+{\varsigma }_{j4}}}{\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}{}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j\left(b\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j\left(b\right)\hfill \\ \hfill & +{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\displaystyle \frac{{\varsigma }_{j4}}{{\varsigma }_{j3}+{\varsigma }_{j4}}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}{}_{k_j}I_{0^{+}}^{{\theta }_j+k_j-{\delta }_{k_j};{\psi }_j}{f}_j\left(b\right).\hfill \end{array}$$

(25)

Rearranging terms and using ${\mathcal{G}}_j\left(\varrho \right)$ and ${\mathcal{G}}_j\left(b\right)$, we can write

$$\begin{array}{cc}\hfill x_j\left(\varrho \right)=& \left({\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\mathcal{G}}_j\left(b\right)-{\mathcal{G}}_j\left(\varrho \right)\right){}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)+{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right),\hfill \end{array}$$

(26)

with the cyclic convention $x_{m+1}=x_1$. This completes the proof. □

Remark 2.

In the special case where $k_j=1$, ${\psi }_j\left(\varrho \right)=\varrho$, ${\beta }_j=1$ for all j, the Hilfer derivative reduces to the Caputo derivative, and the integral representation simplifies to the form studied in [26]. Note that the order ${\theta }_j-{\delta }_{k_j}+k_j$ in the fractional integral simplifies to ${\theta }_j-({\theta }_j+{\beta }_j(k_j-{\theta }_j))+k_j=k_j-{\beta }_j(k_j-{\theta }_j)$, which is positive under assumption (A1) since ${\beta }_j(k_j-{\theta }_j)<k_j$.

4. Existence and Uniqueness Results

Based on Lemma 6, we define the operator $T:\mathcal{X}\to \mathcal{X}$ by

$$T(x_1,x_2,\dots ,x_m)\left(\varrho \right)=\left(\begin{array}{c}{T}_1(x_1,x_2,\dots ,x_m)\left(\varrho \right)\\ T_2(x_1,x_2,\dots ,x_m)\left(\varrho \right)\\ ⋮\\ T_m(x_1,x_2,\dots ,x_m)\left(\varrho \right)\end{array}\right),$$

(27)

where, for each $j=1,2,\dots ,m$,

$$\begin{array}{cc}\hfill T_j(x_1,\dots ,x_m)\left(\varrho \right)& =\left({\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\mathcal{G}}_j\left(b\right)-{\mathcal{G}}_j\left(\varrho \right)\right){}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\hfill \\ \hfill & +{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right),\hfill \end{array}$$

(28)

with $x_{m+1}=x_1$ and ${\mathcal{G}}_j\left(\varrho \right)$ defined as in (5).

We first establish some essential estimates that will be used throughout the fixed-point arguments.

Lemma 7

(Estimates for Fractional Integrals). For $j=1,2,\dots ,m$, let $f_j:[0,b]\times {\mathbb{R}}^2\to \mathbb{R}$ be continuous functions. Then, for any $(x_1,\dots ,x_m)\in \mathcal{X}$, the following estimates hold:

$$\left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)\right|\le {\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}{\parallel f_j(·,x_j,x_{j+1})\parallel }_{\infty },$$

(29)

and

$$\left|{}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\right|\le {\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j-{\delta }_{k_j}+k_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j-{\delta }_{k_j}+2k_j)}}{\parallel f_j(·,x_j,x_{j+1})\parallel }_{\infty },$$

(30)

where $\parallel f_j(·,x_j,x_{j+1}){\parallel }_{\infty }={sup}_{\varrho \in [0,b]}\left|f_j(\varrho ,x_j\left(\varrho \right),x_{j+1}\left(\varrho \right))\right|$.

Proof. 

From the definition of the $(k,\psi )$-Riemann–Liouville fractional integral, we have

$$\begin{array}{cc}\hfill \left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)\right|& \le {\displaystyle \frac{1}{k_j{\Gamma }_{k_j}({\theta }_j+{\zeta }_j)}}{\int }_0^{\varrho }{\psi }_j^{\prime }\left(\tau \right){({\psi }_j\left(\varrho \right)-{\psi }_j\left(\tau \right))}^{{\displaystyle \frac{{\theta }_j+{\zeta }_j}{k_j}}-1}\left|f_j(\tau ,x_j\left(\tau \right),x_{j+1}\left(\tau \right))\right|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{\parallel f_j(·,x_j,x_{j+1}){\parallel }_{\infty }}{k_j{\Gamma }_{k_j}({\theta }_j+{\zeta }_j)}}{\int }_0^{\varrho }{\psi }_j^{\prime }\left(\tau \right){({\psi }_j\left(\varrho \right)-{\psi }_j\left(\tau \right))}^{{\displaystyle \frac{{\theta }_j+{\zeta }_j}{k_j}}-1}d\tau \hfill \\ \hfill & ={\displaystyle \frac{\parallel f_j(·,x_j,x_{j+1}){\parallel }_{\infty }}{k_j{\Gamma }_{k_j}({\theta }_j+{\zeta }_j)}}·{\displaystyle \frac{k_j}{{\theta }_j+{\zeta }_j}}{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{{\displaystyle \frac{{\theta }_j+{\zeta }_j}{k_j}}}\hfill \\ \hfill & ={\displaystyle \frac{{({\psi }_j\left(\varrho \right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}{\parallel f_j(·,x_j,x_{j+1})\parallel }_{\infty },\hfill \end{array}$$

since ${\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)=({\theta }_j+{\zeta }_j){\Gamma }_{k_j}({\theta }_j+{\zeta }_j)$. The estimate (30) follows similarly by noting that the integral is evaluated at $\varrho =b$. □

Now we define the following constants that will appear in the contraction conditions for each $j=1,2,\dots ,m$:

$$\begin{array}{cc}\hfill {\Omega }_j& =\left({\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}·{\displaystyle \frac{|{\varsigma }_{j4}|}{|{\varsigma }_{j3}+{\varsigma }_{j4}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{|{\varsigma }_{j4}|}{|{\varsigma }_{j3}+{\varsigma }_{j4}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}\right){\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j-{\delta }_{k_j}+k_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j-{\delta }_{k_j}+2k_j)}}\hfill \\ \hfill & +{\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}+{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}.\hfill \end{array}$$

(31)

Remark 3

(Comparison of Fixed-Point Theorems). The four fixed-point theorems employed in Section 4 are not redundant; each applies under distinct conditions on the nonlinearities $f_j$. The table below summarizes the key assumptions and scenarios where each theorem is uniquely applicable.

Theorem	Key Assumption	When Others Fail
Banach (Theorem 1)	Global Lipschitz with  $2\sum L_j{\Omega }_j<1$	Nonlinearities not Lipschitz; no uniqueness needed
Schaefer (Theorem 3)	Uniform boundedness:  $|f_j|\le M_j$	Lipschitz fails (e.g., oscillatory nonlinearities)
Krasnosel’skiĭ (Theorem 5)	Split: contraction + compact	Full T not a contraction, but part is
Leray–Schauder (Theorem 7)	Sublinear growth:  $|f_j|\le {\omega }_j\left(\varrho \right){\sigma }_j\left(\right|x_j|+|x_{j+1}\left|\right)$	Boundedness fails (e.g., linear or power growth)

Examples 1–4 in Section 6 illustrate these distinctions explicitly, showing for each case why the chosen theorem is necessary and alternative theorems do not apply.

4.1. Uniqueness via the Banach Contraction Mapping Principle

Theorem 1

(Banach Contraction Principle). Assume that $f_j:[0,b]\times {\mathbb{R}}^2\to \mathbb{R}$ for $j=1,2,\dots ,m$ are continuous functions satisfying the Lipschitz condition

$$|f_j(\varrho ,u_j,u_{j+1})-f_j(\varrho ,v_j,v_{j+1})|\le L_j\left(\right|u_j-v_j|+|u_{j+1}-v_{j+1}\left|\right),j=1,2,\dots ,m,$$

(32)

for all $\varrho \in [0,b]$ and $u_j,u_{j+1},v_j,v_{j+1}\in \mathbb{R}$, where $L_j>0$ are constants, with the cyclic convention $u_{m+1}=u_1$, $v_{m+1}=v_1$. If

$$2{\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j<1,$$

(33)

where ${\Omega }_j$ are defined by (31), then the m-cyclic coupled system (1) and (2) has a unique solution in $\mathcal{X}={\left(C([0,b],\mathbb{R})\right)}^m$.

Proof. 

We show that the operator T defined by (28) is a contraction mapping on $\mathcal{X}$. Let $\mathbf{x}=(x_1,\dots ,x_m),\mathbf{y}=(y_1,\dots ,y_m)\in \mathcal{X}$. For any $\varrho \in [0,b]$, we estimate for each j

$$\begin{array}{cc}\hfill & |T_j\left(\mathbf{x}\right)\left(\varrho \right)-T_j\left(\mathbf{y}\right)\left(\varrho \right)|\hfill \\ \hfill & \le \left({\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}|{\mathcal{G}}_j\left(b\right)|+|{\mathcal{G}}_j\left(\varrho \right)|\right)\hfill \\ \hfill & \times \left|{}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}\left(f_j(·,x_j,x_{j+1})-f_j(·,y_j,y_{j+1})\right)\left(b\right)\right|\hfill \\ \hfill & +{\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}\left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}\left(f_j(·,x_j,x_{j+1})-f_j(·,y_j,y_{j+1})\right)\left(b\right)\right|\hfill \\ \hfill & +\left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}\left(f_j(·,x_j,x_{j+1})-f_j(·,y_j,y_{j+1})\right)\left(\varrho \right)\right|.\hfill \end{array}$$

(34)

Note that $|{\mathcal{G}}_j\left(\varrho \right)|\le {\displaystyle \frac{|{\varsigma }_{j4}|}{|{\varsigma }_{j3}+{\varsigma }_{j4}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}$ for all $\varrho \in [0,b]$, and the same bound holds for $|{\mathcal{G}}_j\left(b\right)|$.

Using Lemma 7 and the Lipschitz condition (32), we obtain

$$\begin{array}{cc}\hfill & \left|{}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}\left(f_j(·,x_j,x_{j+1})-f_j(·,y_j,y_{j+1})\right)\left(b\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j-{\delta }_{k_j}+k_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j-{\delta }_{k_j}+2k_j)}}{L}_j\left(\parallel x_j-y_j{\parallel }_{\infty }+{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\right).\hfill \end{array}$$

(35)

Similarly,

$$\begin{array}{cc}\hfill & \left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}\left(f_j(·,x_j,x_{j+1})-f_j(·,y_j,y_{j+1})\right)\left(b\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}{L}_j\left(\parallel x_j-y_j{\parallel }_{\infty }+{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\right),\hfill \end{array}$$

(36)

and

$$\begin{array}{cc}\hfill & \left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}\left(f_j(·,x_j,x_{j+1})-f_j(·,y_j,y_{j+1})\right)\left(\varrho \right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}{L}_j\left(\parallel x_j-y_j{\parallel }_{\infty }+{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\right).\hfill \end{array}$$

(37)

Substituting these estimates into (34) yields

$$\begin{array}{cc}\hfill \parallel T_j\left(\mathbf{x}\right)-T_j{\left(\mathbf{y}\right)\parallel }_{\infty }& \le L_j\left(\parallel x_j-y_j{\parallel }_{\infty }+{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\right){\Omega }_j.\hfill \end{array}$$

(38)

Summing over all $j=1,\dots ,m$ and using the cyclic convention $x_{m+1}=x_1$, $y_{m+1}=y_1$, we obtain

$$\begin{array}{cc}\hfill {\parallel T\left(\mathbf{x}\right)-T\left(\mathbf{y}\right)\parallel }_{\mathcal{X}}& ={\displaystyle \sum _{j=1}^m}{\parallel T_j\left(\mathbf{x}\right)-T_j\left(\mathbf{y}\right)\parallel }_{\infty }\hfill \\ \hfill & \le {\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j\left(\parallel x_j-y_j{\parallel }_{\infty }+{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\right)\hfill \\ \hfill & ={\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j\parallel x_j-y_j{\parallel }_{\infty }+{\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\hfill \\ \hfill & ={\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j\parallel x_j-y_j{\parallel }_{\infty }+{\displaystyle \sum _{j=1}^m}{L}_{j-1}{\Omega }_{j-1}{\parallel x_j-y_j\parallel }_{\infty }\hfill \\ \hfill & \le 2\left({\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j\right){\parallel \mathbf{x}-\mathbf{y}\parallel }_{\mathcal{X}},\hfill \end{array}$$

(39)

where we define $L_0=L_m$ and ${\Omega }_0={\Omega }_m$ for the cyclic indexing. Note that the factor 2 arises from the cyclic coupling; since each index j appears both as j and $j-1$ in the sum, we obtain the worst-case bound $2\sum L_j{\Omega }_j$. Condition (33) with $2{\sum }_{j=1}^m{L}_j{\Omega }_j<1$ implies that T is a contraction mapping on $\mathcal{X}$. By the Banach contraction principle, T has a unique fixed point, which is the unique solution of system (1) and (2). □

4.2. Existence via Schaefer’s Fixed-Point Theorem

We now present an existence result using Schaefer’s fixed-point theorem, which requires a boundedness condition on the nonlinearities.

Theorem 2

(Schaefer’s Fixed-Point Theorem [27]). Let X be a Banach space and $T:X\to X$ a completely continuous operator. If the set

$$E=\{x\in X:x=\varsigma T(x)\mathit{for}\mathit{some}\varsigma \in [0,1\left]\right\}$$

is bounded, then T has a fixed point.

First, we establish the complete continuity of the operator T in $C\left(\right[0,b],\mathbb{R})$.

Lemma 8

(Complete Continuity). Under the standing assumptions, the operator $T:\mathcal{X}\to \mathcal{X}$ defined by (28) is completely continuous.

Proof. 

Let ${\left\{{\mathbf{x}}_n\right\}}_{n\in \mathbb{N}}=\left\{(x_{1n},x_{2n},\dots ,x_{mn})\right\}$ be a sequence in $\mathcal{X}$ such that ${\mathbf{x}}_n\to \mathbf{x}$ in $\mathcal{X}$. Since each $f_j$ is continuous, we have $f_j(\varrho ,x_{jn}\left(\varrho \right),x_{j+1,n}\left(\varrho \right))\to f_j(\varrho ,x_j\left(\varrho \right),x_{j+1}\left(\varrho \right))$ pointwise for each $\varrho \in [0,b]$. Moreover, the convergence is uniform due to the continuity of $f_j$ on compact sets. Then, using the estimates from Lemma 7, we obtain

$$\parallel T_j\left({\mathbf{x}}_n\right)-T_j\left(\mathbf{x}\right){\parallel }_{\infty }\le {\Omega }_j{\parallel f_j(·,x_{jn},x_{j+1,n})-f_j(·,x_j,x_{j+1})\parallel }_{\infty }\to 0\mathrm{as}n\to \infty .$$

Thus, T is continuous.

Now let $B_R={\{\mathbf{x}\in \mathcal{X}:\parallel \mathbf{x}\parallel }_{\mathcal{X}}\le R\}$ be a bounded set. For any $\mathbf{x}\in B_R$, by the continuity of $f_j$, there exists a constant $M_j\left(R\right)>0$ such that $\parallel f_j(·,x_j,x_{j+1}){\parallel }_{\infty }\le M_j\left(R\right)$. Then, using Lemma 7, we have

$$\parallel T_j{\left(\mathbf{x}\right)\parallel }_{\infty }\le M_j\left(R\right){\Omega }_j.$$

Hence, ${\parallel T\left(\mathbf{x}\right)\parallel }_{\mathcal{X}}\le {\sum }_{j=1}^m{M}_j\left(R\right){\Omega }_j$, so $T\left(B_R\right)$ is bounded.

To show equicontinuity, let ${\varrho }_1,{\varrho }_2\in [0,b]$ with ${\varrho }_1<{\varrho }_2$. For any $\mathbf{x}\in B_R$, we have

$$\begin{array}{cc}\hfill & |T_j\left(\mathbf{x}\right)\left({\varrho }_2\right)-T_j\left(\mathbf{x}\right)\left({\varrho }_1\right)|\hfill \\ \hfill & \le |{\mathcal{G}}_j\left({\varrho }_2\right)-{\mathcal{G}}_j\left({\varrho }_1\right)|\left|{}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\right|\hfill \\ \hfill & +\left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left({\varrho }_2\right)-{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left({\varrho }_1\right)\right|.\hfill \end{array}$$

The first term tends to 0 uniformly as ${\varrho }_2\to {\varrho }_1$ by the continuity of ${\mathcal{G}}_j$. For the second term, using the definition of the fractional integral, one can show it is bounded by $M_j\left(R\right)$ times ${({\psi }_j\left({\varrho }_2\right)-{\psi }_j\left({\varrho }_1\right))}^{{\displaystyle \frac{{\theta }_j+{\zeta }_j}{k_j}}}$ up to a constant factor, which tends to 0 uniformly as ${\varrho }_2\to {\varrho }_1$. Hence, $T\left(B_R\right)$ is equicontinuous. By the Arzelà–Ascoli theorem, $T\left(B_R\right)$ is relatively compact. Therefore, T is completely continuous. □

Theorem 3

(Existence via Schaefer’s Theorem). Assume that $f_j:[0,b]\times {\mathbb{R}}^2\to \mathbb{R}$ for $j=1,2,\dots ,m$ are continuous functions satisfying the growth condition

$$|f_j(\varrho ,x_j,x_{j+1})|\le M_j,j=1,2,\dots ,m,$$

(40)

for all $\varrho \in [0,b]$ and $x_j,x_{j+1}\in \mathbb{R}$, where $M_j>0$ are constants. Then, the m-cyclic coupled system (1) and (2) has at least one solution in $\mathcal{X}$.

Proof. 

Consider the set

$$E=\{\mathbf{x}\in \mathcal{X}:\mathbf{x}=\varsigma T(\mathbf{x})\mathrm{for}\mathrm{some}\varsigma \in [0,1\left]\right\}.$$

For any $\mathbf{x}\in E$, we have $\mathbf{x}=\varsigma T\left(\mathbf{x}\right)$ for some $\varsigma \in [0,1]$. Then, for each $\varrho \in [0,b]$ and each j, using Lemma 7 and the boundedness condition (40), we obtain

$$\parallel x_j{\parallel }_{\infty }\le M_j{\Omega }_j.$$

Therefore,

$${\parallel \mathbf{x}\parallel }_{\mathcal{X}}={\displaystyle \sum _{j=1}^m}{\parallel x_j\parallel }_{\infty }\le {\displaystyle \sum _{j=1}^m}{M}_j{\Omega }_j.$$

Thus, the set E is bounded. By Lemma 8, T is completely continuous. Applying Schaefer’s fixed-point theorem, T has a fixed point, which is a solution of system (1) and (2). □

4.3. Existence via Krasnosel’skiĭ’s Fixed-Point Theorem

We now present an existence result using Krasnosel’skiĭ’s fixed-point theorem, which is useful when the operator can be split into a contraction and a compact part.

Theorem 4

(Krasnosel’skiĭ’s Fixed-Point Theorem [28]). Let B be a nonempty, closed, convex, and bounded subset of a Banach space X. Let $T_1,T_2:B\to X$ be two operators such that the following apply:

• $T_1\left(x\right)+T_2\left(y\right)\in B$ for all $x,y\in B$;
• $T_1$ is continuous and compact;
• $T_2$ is a contraction mapping.

Then, there exists $z\in B$ such that $z=T_1\left(z\right)+T_2\left(z\right)$.

We split the operator T into two parts $T=T^{\left(1\right)}+T^{\left(2\right)}$, where, for each $j=1,\dots ,m$,

$$T_j^{\left(1\right)}\left(\mathbf{x}\right)\left(\varrho \right)={}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right),$$

(41)

and

$$\begin{array}{cc}\hfill T_j^{\left(2\right)}\left(\mathbf{x}\right)\left(\varrho \right)& =\left({\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\mathcal{G}}_j\left(b\right)-{\mathcal{G}}_j\left(\varrho \right)\right){}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right).\hfill \end{array}$$

(42)

Define the constants for the contractive part for each j:

$$\begin{array}{cc}\hfill {\Omega }_j^{\left(2\right)}& =\left({\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}·{\displaystyle \frac{|{\varsigma }_{j4}|}{|{\varsigma }_{j3}+{\varsigma }_{j4}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{|{\varsigma }_{j4}|}{|{\varsigma }_{j3}+{\varsigma }_{j4}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}\right){\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j-{\delta }_{k_j}+k_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j-{\delta }_{k_j}+2k_j)}}\hfill \\ \hfill & +{\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}.\hfill \end{array}$$

(43)

Theorem 5

(Existence via Krasnosel’skiĭ). Assume that $f_j:[0,b]\times {\mathbb{R}}^2\to \mathbb{R}$ for $j=1,2,\dots ,m$ are continuous functions satisfying the following:

1. 

There exist constants $L_j>0$ such that condition (32) holds;

2. 

There exist continuous functions $g_j:[0,b]\to {\mathbb{R}}^{+}$ such that

$$|f_j(\varrho ,x_j,x_{j+1})|\le g_j\left(\varrho \right),j=1,2,\dots ,m,$$

(44)

for all $\varrho \in [0,b]$ and $x_j,x_{j+1}\in \mathbb{R}$.

If

$$2{\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j^{\left(2\right)}<1,$$

(45)

where ${\Omega }_j^{\left(2\right)}$ are defined by (43), then the m-cyclic coupled system (1) and (2) has at least one solution in $\mathcal{X}$.

Proof. 

Define the set $B_R={\{\mathbf{x}\in \mathcal{X}:\parallel \mathbf{x}\parallel }_{\mathcal{X}}\le R\}$, with R chosen such that

$$R\ge {\displaystyle \sum _{j=1}^m}\left(\parallel g_j{\parallel }_{\infty }{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}+{\parallel g_j\parallel }_{\infty }{\Omega }_j^{\left(2\right)}\right).$$

Step 1: ${\mathit{T}}^{\left(\mathbf{1}\right)}\left(\mathbf{x}\right)+{\mathit{T}}^{\left(\mathbf{2}\right)}\left(\mathbf{y}\right)\in {\mathbf{B}}_{\mathbf{R}}$ for all $\mathbf{x},\mathbf{y}\in {\mathbf{B}}_{\mathbf{R}}$. For any $\mathbf{x},\mathbf{y}\in B_R$, using Lemma 7 and condition (44), we have for each j

$$\begin{array}{cc}\hfill \parallel T_j^{\left(1\right)}\left(\mathbf{x}\right)+T_j^{\left(2\right)}{\left(\mathbf{y}\right)\parallel }_{\infty }& \le \parallel T_j^{\left(1\right)}{\left(\mathbf{x}\right)\parallel }_{\infty }+{\parallel T_j^{\left(2\right)}\left(\mathbf{y}\right)\parallel }_{\infty }\hfill \\ \hfill & \le \parallel g_j{\parallel }_{\infty }{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}+{\parallel g_j\parallel }_{\infty }{\Omega }_j^{\left(2\right)}.\hfill \end{array}$$

Summing over j, we obtain $\parallel T^{\left(1\right)}\left(\mathbf{x}\right)+T^{\left(2\right)}{\left(\mathbf{y}\right)\parallel }_{\mathcal{X}}\le R$, so $T^{\left(1\right)}\left(\mathbf{x}\right)+T^{\left(2\right)}\left(\mathbf{y}\right)\in B_R$.

Step 2: ${\mathit{T}}^{\left(\mathbf{1}\right)}$ is continuous and compact. The continuity of $T^{\left(1\right)}$ follows from the continuity of $f_j$ and the fractional integral operators. For compactness, let $\left\{{\mathbf{x}}_n\right\}$ be a sequence in $B_R$. Using Lemma 7, we have

$$\parallel T^{\left(1\right)}\left({\mathbf{x}}_n\right){\parallel }_{\mathcal{X}}\le {\displaystyle \sum _{j=1}^m}{\parallel g_j\parallel }_{\infty }{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}},$$

so $T^{\left(1\right)}\left(B_R\right)$ is bounded. Moreover, using similar estimates to those in Lemma 8, $T^{\left(1\right)}\left(B_R\right)$ is equicontinuous. By the Arzelà–Ascoli theorem, $T^{\left(1\right)}\left(B_R\right)$ is relatively compact. Thus, $T^{\left(1\right)}$ is compact.

Step 3: ${\mathit{T}}^{\left(\mathbf{2}\right)}$ is a contraction. For any $\mathbf{x},\mathbf{y}\in B_R$, using the Lipschitz condition (32) and Lemma 7, we obtain for each j

$$\parallel T_j^{\left(2\right)}\left(\mathbf{x}\right)-T_j^{\left(2\right)}{\left(\mathbf{y}\right)\parallel }_{\infty }\le L_j{\Omega }_j^{\left(2\right)}\left(\parallel x_j-y_j{\parallel }_{\infty }+{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\right).$$

Summing over j and using the cyclic convention, we have

$$\begin{array}{cc}\hfill \parallel T^{\left(2\right)}\left(\mathbf{x}\right)-T^{\left(2\right)}{\left(\mathbf{y}\right)\parallel }_{\mathcal{X}}& \le {\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j^{\left(2\right)}\left(\parallel x_j-y_j{\parallel }_{\infty }+{\parallel x_{j+1}-y_{j+1}\parallel }_{\infty }\right)\hfill \\ \hfill & ={\displaystyle \sum _{j=1}^m}(L_{j-1}{\Omega }_{j-1}^{\left(2\right)}+L_j{\Omega }_j^{\left(2\right)}){\parallel x_j-y_j\parallel }_{\infty }\hfill \\ \hfill & \le 2\left({\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j^{\left(2\right)}\right){\parallel \mathbf{x}-\mathbf{y}\parallel }_{\mathcal{X}}.\hfill \end{array}$$

Condition (45) with $2{\sum }_{j=1}^m{L}_j{\Omega }_j^{\left(2\right)}<1$ implies that $T^{\left(2\right)}$ is a contraction mapping. All conditions of Krasnosel’skiĭ’s fixed-point theorem are satisfied, so there exists a fixed point $\mathbf{x}\in B_R$ such that $\mathbf{x}=T^{\left(1\right)}\left(\mathbf{x}\right)+T^{\left(2\right)}\left(\mathbf{x}\right)=T\left(\mathbf{x}\right)$. Hence, system (1) and (2) has at least one solution in $\mathcal{X}$. □

4.4. Existence via the Leray–Schauder Alternative

Finally, we present an existence result using the Leray–Schauder nonlinear alternative, which allows for more general growth conditions.

Theorem 6

(Leray–Schauder Nonlinear Alternative). Let X be a Banach space, $C\subset X$ a convex closed set, and $U\subset C$ an open set with $0\in U$. If $T:\overline{U}\to C$ is a completely continuous operator, then either of the following applies:

• T has a fixed point in $\overline{U}$;
• There exist $u\in \partial U$ and $\varsigma \in (0,1)$ such that $u=\varsigma T\left(u\right)$.

Theorem 7

(Existence via the Leray–Schauder Alternative). Assume that $f_j:[0,b]\times {\mathbb{R}}^2\to \mathbb{R}$ for $j=1,2,\dots ,m$ are continuous functions satisfying the growth condition

$$|f_j(\varrho ,x_j,x_{j+1})|\le {\omega }_j\left(\varrho \right){\sigma }_j\left(\right|x_j|+|x_{j+1}\left|\right),j=1,2,\dots ,m,$$

(46)

for all $\varrho \in [0,b]$ and $x_j,x_{j+1}\in \mathbb{R}$, where ${\omega }_j\in C([0,b],{\mathbb{R}}^{+})$ and ${\sigma }_j:{\mathbb{R}}^{+}\to {\mathbb{R}}^{+}$ are continuous nondecreasing functions. If there exists a constant $K>0$ such that

$${\displaystyle \frac{K}{{\sum }_{j=1}^m{\parallel {\omega }_j\parallel }_{\infty }{\sigma }_j\left(K\right){\Omega }_j}}>1,$$

(47)

where $\parallel {\omega }_j{\parallel }_{\infty }={sup}_{\varrho \in [0,b]}{\omega }_j\left(\varrho \right)$ and ${\Omega }_j$ are defined by (31), then the m-cyclic coupled system (1) and (2) has at least one solution in $\mathcal{X}$ with ${\parallel \mathbf{x}\parallel }_{\mathcal{X}}<K$.

Proof. 

Consider the open set $U=\{\mathbf{x}\in \mathcal{X}:\parallel \mathbf{x}{\parallel }_{\mathcal{X}}<K\}$. Let $\mathbf{x}\in \overline{U}$ be such that $\mathbf{x}=\varsigma T\left(\mathbf{x}\right)$ for some $\varsigma \in (0,1)$. Then, for each $\varrho \in [0,b]$ and each j, using Lemma 7 and the growth condition (46), we have

$$\begin{array}{cc}\hfill \parallel x_j{\parallel }_{\infty }& \le \left({\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}|{\mathcal{G}}_j\left(b\right)|+|{\mathcal{G}}_j\left(\varrho \right)|\right)\left|{}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\right|\hfill \\ \hfill & +{\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}\left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\right|+\left|{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right)\right|\hfill \\ \hfill & \le \parallel {\omega }_j{\parallel }_{\infty }{\sigma }_j(\parallel x_j{\parallel }_{\infty }+\parallel x_{j+1}{\parallel }_{\infty }){\Omega }_j\hfill \\ \hfill & \le \parallel {\omega }_j{\parallel }_{\infty }{\sigma }_j{(\parallel \mathbf{x}\parallel }_{\mathcal{X}}){\Omega }_j.\hfill \end{array}$$

(48)

Summing over j, we obtain

$${\parallel \mathbf{x}\parallel }_{\mathcal{X}}\le {\displaystyle \sum _{j=1}^m}\parallel {\omega }_j{\parallel }_{\infty }{\sigma }_j{(\parallel \mathbf{x}\parallel }_{\mathcal{X}}){\Omega }_j.$$

(49)

If ${\parallel \mathbf{x}\parallel }_{\mathcal{X}}=K$, then, from (49), we obtain

$$K\le {\displaystyle \sum _{j=1}^m}{\parallel {\omega }_j\parallel }_{\infty }{\sigma }_j\left(K\right){\Omega }_j,$$

which contradicts condition (47). Hence, there is no $\mathbf{x}\in \partial U$ such that $\mathbf{x}=\varsigma T\left(\mathbf{x}\right)$ for any $\varsigma \in (0,1)$.

By Lemma 8, $T:\overline{U}\to \mathcal{X}$ is completely continuous. Applying the Leray–Schauder nonlinear alternative, T has a fixed point in $\overline{U}$, which is a solution of system (1) and (2) with ${\parallel \mathbf{x}\parallel }_{\mathcal{X}}<K$. □

Remark 4.

The four fixed-point theorems presented in this section provide a comprehensive toolkit for establishing the existence and uniqueness of solutions for the m-cyclic coupled system (1) and (2) under various conditions on the nonlinearities $f_j$.

5. Ulam–Hyers Stability Analysis

In this section, we investigate the Ulam–Hyers stability of the m-cyclic coupled system (1) and (2).

Definition 6

(Ulam–Hyers Stability). The m-cyclic coupled system (1) and (2) is said to be Ulam–Hyers stable if there exists a constant $\mathbf{C}=(C_1,C_2,\dots ,C_m)>0$ such that, for any $\epsilon =({\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_m)>0$ and for any functions $\mathbf{y}=(y_1,y_2,\dots ,y_m)\in \mathcal{X}$ satisfying the inequalities

$$\left\{\begin{array}{c}\left|{}_{k_j}{}^{H}D_{0^{+}}^{{\theta }_j,{\beta }_j;{\psi }_j}\left({}_{k_j}{}^{C}D_{0^{+}}^{{\zeta }_j;{\psi }_j}{y}_j\left(\varrho \right)\right)-f_j(\varrho ,y_j\left(\varrho \right),y_{j+1}\left(\varrho \right))\right|\le {\epsilon }_j,j=1,2,\dots ,m,\hfill \\ |{\varsigma }_{j1}{y}_j\left(0\right)+{\varsigma }_{j2}{y}_j\left(b\right)|\le {\epsilon }_j,j=1,2,\dots ,m,\hfill \\ |{\varsigma }_{j3}{}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{y}_j\left(0\right)+{\varsigma }_{j4}{}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{y}_j\left(b\right)|\le {\epsilon }_j,j=1,2,\dots ,m,\hfill \end{array}\right.$$

(50)

with $y_{m+1}=y_1$, there exists an exact solution $\mathbf{x}=(x_1,\dots ,x_m)\in \mathcal{X}$ of system (1) and (2) such that

$$\parallel y_j-x_j{\parallel }_{\infty }\le C_{j}max\{{\epsilon }_1,{\epsilon }_2,\dots ,{\epsilon }_m\},j=1,2,\dots ,m.$$

(51)

If the constants $C_j$ depend on ${\epsilon }_1,\dots ,{\epsilon }_m$ in such a way that $C_j\to 0$ as ${\epsilon }_j\to 0$, then the system is said to be generalized Ulam–Hyers stable.

Lemma 9

(Stability Estimates). Under the assumptions of Lemma 6, the following estimates hold for the operator T defined in (28):

$$\parallel T_j{\left(\mathbf{x}\right)\parallel }_{\infty }\le {\Omega }_j{\parallel f_j(·,x_j,x_{j+1})\parallel }_{\infty },j=1,2,\dots ,m,$$

(52)

where ${\Omega }_j$ are defined by (31).

Proof. 

The proof follows directly from the estimates in Lemma 7 and the definition of T. □

5.1. Ulam–Hyers Stability Under Lipschitz Conditions

We now prove the Ulam–Hyers stability of system (1) and (2) under the Lipschitz conditions assumed in Theorem 1.

Theorem 8

(Ulam–Hyers Stability). Assume that the conditions of Theorem 1 hold, i.e., $f_j$ satisfy the Lipschitz condition (32) with constants $L_j$, and

$$2{\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j<1,$$

(53)

where ${\Omega }_j$ are defined by (31). Then, the m-cyclic coupled system (1) and (2) is Ulam–Hyers stable in $\mathcal{X}$.

Proof. 

Let $\epsilon =({\epsilon }_1,\dots ,{\epsilon }_m)>0$ and let $\mathbf{y}=(y_1,\dots ,y_m)\in \mathcal{X}$ be an approximate solution satisfying inequalities (50). Define functions $g_j\in C([0,b],\mathbb{R})$ by

$$g_j\left(\varrho \right)={}_{k_j}{}^{H}D_{0^{+}}^{{\theta }_j,{\beta }_j;{\psi }_j}\left({}_{k_j}{}^{C}D_{0^{+}}^{{\zeta }_j;{\psi }_j}{y}_j\left(\varrho \right)\right)-f_j(\varrho ,y_j\left(\varrho \right),y_{j+1}\left(\varrho \right)),j=1,\dots ,m,$$

(54)

so that $\parallel g_j{\parallel }_{\infty }\le {\epsilon }_j$ for all $\varrho \in [0,b]$ and j.

Now consider the linear FDEs

$$\left\{\begin{array}{c}{}_{k_j}{}^{H}D_{0^{+}}^{{\theta }_j,{\beta }_j;{\psi }_j}\left({}_{k_j}{}^{C}D_{0^{+}}^{{\zeta }_j;{\psi }_j}{y}_j\left(\varrho \right)\right)=f_j(\varrho ,y_j\left(\varrho \right),y_{j+1}\left(\varrho \right))+g_j\left(\varrho \right),j=1,\dots ,m,\hfill \end{array}\right.$$

(55)

with the boundary conditions

$$\begin{array}{cc}\hfill {\varsigma }_{j1}{y}_j\left(0\right)+{\varsigma }_{j2}{y}_j\left(b\right)& =r_{j1},\hfill \\ \hfill {\varsigma }_{j3}{}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{y}_j\left(0\right)+{\varsigma }_{j4}{}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{y}_j\left(b\right)& =r_{j2},j=1,\dots ,m,\hfill \end{array}$$

(56)

where $\parallel r_{j1}{\parallel }_{\infty },{\parallel r_{j2}\parallel }_{\infty }\le {\epsilon }_j$ by the approximate boundary conditions in (50).

Using Lemma 6 for the linear system with source terms $f_j+g_j$, we obtain the integral representation for $y_j$:

$$\begin{array}{cc}\hfill y_j\left(\varrho \right)& =\left({\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\mathcal{G}}_j\left(b\right)-{\mathcal{G}}_j\left(\varrho \right)\right){}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}(f_j+g_j)\left(b\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}(f_j+g_j)\left(b\right)+{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}(f_j+g_j)\left(\varrho \right)+{\Phi }_j\left(\varrho \right),\hfill \end{array}$$

(57)

where ${\Phi }_j\left(\varrho \right)$ are the contributions from the boundary perturbations $r_{j1},r_{j2}$. By linearity and the boundedness of the Green’s function, there exist constants $K_j>0$ such that

$$\parallel {\Phi }_j{\parallel }_{\infty }\le K_{j}max\{{\epsilon }_1,\dots ,{\epsilon }_m\},\forall \varrho \in [0,b],j=1,\dots ,m.$$

(58)

Now let $\mathbf{x}$ be the exact solution of system (1) and (2) corresponding to the same boundary conditions as $y_j$ but with zero perturbations. From Lemma 6, we have

$$\begin{array}{cc}\hfill x_j\left(\varrho \right)& =\left({\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{\mathcal{G}}_j\left(b\right)-{\mathcal{G}}_j\left(\varrho \right)\right){}_{k_j}I_{0^{+}}^{{\theta }_j-{\delta }_{k_j}+k_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)\hfill \\ \hfill & -{\displaystyle \frac{{\varsigma }_{j2}}{{\varsigma }_{j1}+{\varsigma }_{j2}}}{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(b\right)+{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}{f}_j(·,x_j,x_{j+1})\left(\varrho \right).\hfill \end{array}$$

(59)

Subtracting (59) from (57) and taking supremum norms, we obtain for each j

$$\begin{array}{cc}\hfill \parallel y_j-x_j{\parallel }_{\infty }& \le L_j{\Omega }_j(\parallel y_j-x_j{\parallel }_{\infty }+\parallel y_{j+1}-x_{j+1}{\parallel }_{\infty })+{\epsilon }_j{\Omega }_j^{\left(g\right)}+K_{j}max\{{\epsilon }_1,\dots ,{\epsilon }_m\},\hfill \end{array}$$

(60)

where

$$\begin{array}{cc}\hfill {\Omega }_j^{\left(g\right)}& =\left({\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}{\displaystyle \frac{|{\varsigma }_{j4}|}{|{\varsigma }_{j3}+{\varsigma }_{j4}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{|{\varsigma }_{j4}|}{|{\varsigma }_{j3}+{\varsigma }_{j4}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\delta }_{k_j}+{\zeta }_j}{k_j}-1}}{{\Gamma }_{k_j}({\delta }_{k_j}+{\zeta }_j)}}\right){\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j-{\delta }_{k_j}+k_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j-{\delta }_{k_j}+2k_j)}}\hfill \\ \hfill & +{\displaystyle \frac{|{\varsigma }_{j2}|}{|{\varsigma }_{j1}+{\varsigma }_{j2}|}}{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}+{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}.\hfill \end{array}$$

(61)

Summing these inequalities over $j=1,\dots ,m$ and using the cyclic convention $y_{m+1}=y_1$, $x_{m+1}=x_1$, we obtain

$$\begin{array}{cc}\hfill {\displaystyle \sum _{j=1}^m}{\parallel y_j-x_j\parallel }_{\infty }& \le {\displaystyle \sum _{j=1}^m}{L}_j{\Omega }_j(\parallel y_j-x_j{\parallel }_{\infty }+\parallel y_{j+1}-x_{j+1}{\parallel }_{\infty })\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^m}{\epsilon }_j{\Omega }_j^{\left(g\right)}+{\displaystyle \sum _{j=1}^m}{K}_{j}max\{{\epsilon }_1,\dots ,{\epsilon }_m\}\hfill \\ \hfill & ={\displaystyle \sum _{j=1}^m}(L_{j-1}{\Omega }_{j-1}+L_j{\Omega }_j){\parallel y_j-x_j\parallel }_{\infty }\hfill \\ \hfill & +{\displaystyle \sum _{j=1}^m}{\epsilon }_j{\Omega }_j^{\left(g\right)}+\left({\displaystyle \sum _{j=1}^m}{K}_j\right)max\{{\epsilon }_1,\dots ,{\epsilon }_m\}.\hfill \end{array}$$

(62)

Let $\Delta =2{\sum }_{j=1}^m{L}_j{\Omega }_j$. By condition (53), we have $\Delta <1$. Rearranging (62) gives

$$(1-\Delta ){\displaystyle \sum _{j=1}^m}{\parallel y_j-x_j\parallel }_{\infty }\le {\displaystyle \sum _{j=1}^m}{\epsilon }_j{\Omega }_j^{\left(g\right)}+\left({\displaystyle \sum _{j=1}^m}{K}_j\right)max\{{\epsilon }_1,\dots ,{\epsilon }_m\}.$$

(63)

Therefore,

$${\displaystyle \sum _{j=1}^m}{\parallel y_j-x_j\parallel }_{\infty }\le {\displaystyle \frac{{\sum }_{j=1}^m{\Omega }_j^{\left(g\right)}+{\sum }_{j=1}^m{K}_j}{1-\Delta }}max\{{\epsilon }_1,\dots ,{\epsilon }_m\}.$$

(64)

From this, we can deduce the individual estimates

$$\parallel y_j-x_j{\parallel }_{\infty }\le C_{j}max\{{\epsilon }_1,\dots ,{\epsilon }_m\},j=1,\dots ,m,$$

(65)

with

$$C_j={\displaystyle \frac{{\sum }_{j=1}^m{\Omega }_j^{\left(g\right)}+{\sum }_{j=1}^m{K}_j}{1-\Delta }},j=1,\dots ,m.$$

(66)

Thus, the m-cyclic coupled system (1) and (2) is Ulam–Hyers stable in $\mathcal{X}$. □

Corollary 1

(Generalized Ulam–Hyers Stability). Under the same assumptions as in Theorem 8, the m-cyclic coupled system (1) and (2) is also generalized Ulam–Hyers stable.

5.2. Ulam–Hyers–Rassias Stability

We now consider a more general form of stability, known as Ulam–Hyers–Rassias stability, where the constant $\epsilon$ is replaced by a function.

Definition 7

(Ulam–Hyers–Rassias Stability). The m-cyclic coupled system (1) and (2) is said to be Ulam–Hyers–Rassias stable with respect to $\phi \in C([0,b],{\mathbb{R}}^{+})$ if there exists a constant $C_{\phi }>0$ such that, for any $\epsilon >0$ and for any functions $\mathbf{y}\in \mathcal{X}$ satisfying

$$\left\{\begin{array}{c}\left|{}_{k_j}{}^{H}D_{0^{+}}^{{\theta }_j,{\beta }_j;{\psi }_j}\left({}_{k_j}{}^{C}D_{0^{+}}^{{\zeta }_j;{\psi }_j}{y}_j\left(\varrho \right)\right)-f_j(\varrho ,y_j\left(\varrho \right),y_{j+1}\left(\varrho \right))\right|\le \epsilon \phi \left(\varrho \right),j=1,\dots ,m,\hfill \\ |{\varsigma }_{j1}{y}_j\left(0\right)+{\varsigma }_{j2}{y}_j\left(b\right)|\le \epsilon \phi \left(0\right),j=1,\dots ,m,\hfill \\ |{\varsigma }_{j3}{}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{y}_j\left(0\right)+{\varsigma }_{j4}{}_{k_j}{}^{C}D_{0^{+}}^{{\delta }_{k_j}+{\zeta }_j-k_j;{\psi }_j}{y}_j\left(b\right)|\le \epsilon \phi \left(0\right),j=1,\dots ,m,\hfill \end{array}\right.$$

(67)

there exists an exact solution $\mathbf{x}\in \mathcal{X}$ of system (1) and (2) such that

$$\parallel y_j-x_j{\parallel }_{\infty }\le C_{\phi }\epsilon {\parallel \phi \parallel }_{\infty },j=1,\dots ,m.$$

(68)

Theorem 9

(Ulam–Hyers–Rassias Stability). Assume that the conditions of Theorem 8 hold. Let $\phi \in C([0,b],{\mathbb{R}}^{+})$ be a nondecreasing function. Then, the m-cyclic coupled system (1) and (2) is Ulam–Hyers–Rassias stable with respect to φ.

Proof. 

The proof follows a structure similar to that of Theorem 8, but with the constants ${\epsilon }_j$ replaced by $\epsilon \phi \left(\varrho \right)$ and $\epsilon \phi \left(0\right)$. Using Lemma 7 and the properties of the fractional integral, we obtain estimates of the form

$${∥{}_{k_j}I_{0^{+}}^{{\theta }_j+{\zeta }_j;{\psi }_j}\left(\epsilon \phi \right)\left(\varrho \right)∥}_{\infty }\le \epsilon {\parallel \phi \parallel }_{\infty }{\displaystyle \frac{{({\psi }_j\left(b\right)-{\psi }_j\left(0\right))}^{\frac{{\theta }_j+{\zeta }_j}{k_j}}}{{\Gamma }_{k_j}({\theta }_j+{\zeta }_j+k_j)}}.$$

(69)

Proceeding as in the proof of Theorem 8, we arrive at

$${\displaystyle \sum _{j=1}^m}\parallel y_j-x_j{\parallel }_{\infty }\le C_{\phi }\epsilon {\parallel \phi \parallel }_{\infty },$$

(70)

with

$$C_{\phi }={\displaystyle \frac{{\sum }_{j=1}^m{\tilde{\Omega }}_j^{\left(g\right)}+{\sum }_{j=1}^m{\tilde{K}}_j}{1-\Delta }},$$

(71)

where ${\tilde{\Omega }}_j^{\left(g\right)}={\Omega }_j^{\left(g\right)}{\parallel \phi \parallel }_{\infty }$ and ${\tilde{K}}_j=K_j{\parallel \phi \parallel }_{\infty }$ are the modified constants accounting for the function $\phi$. This establishes Ulam–Hyers–Rassias stability. □

6. Numerical Examples

Example 1

(Uniqueness via the Banach Contraction Principle). Consider a 2-cyclic coupled system ($m=2$) on $[0,1]$ with the same parameters as in the original Example 1.

Define the nonlinearities:

$$\begin{array}{cc}\hfill f_1(\varrho ,x_1,x_2)& ={\displaystyle \frac{0.05}{1+{\varrho }^2}}sin\left(x_1\right)+{\displaystyle \frac{0.03}{2+\varrho }}cos\left(x_2\right),\hfill \\ \hfill f_2(\varrho ,x_2,x_1)& =0.04x_2.\hfill \end{array}$$

Here, $f_1$ is bounded, but $f_2$ is unbounded (linear).

-

$f_1$ is Lipschitz with constant $0.05+0.03=0.08$.

-

$f_2$ is Lipschitz with constant $0.04$.

Thus, $L_1=0.08$, $L_2=0.04$. With ${\Omega }_1={\Omega }_2\approx 0.3892$, we have

$$2{\displaystyle \sum _{j=1}^2}{L}_j{\Omega }_j=2(0.08+0.04)\times 0.3892=0.0934<1.$$

Hence, the Banach theorem applies, guaranteeing a unique solution.

• Why other theorems are not needed (or fail):

Theorem	Applicable?	Reason
Banach	✓ Yes	Contraction condition holds; gives uniqueness.
Schaefer	No	$f_2\left(x_2\right)=0.04x_2$ is unbounded, so condition (40) fails.
Krasnosel’skiĭ	✓ Yes	Could be applied, but Banach is stronger (uniqueness).
Leray–Schauder	✓ Yes	Could be applied, but Banach is stronger (uniqueness).

The Banach theorem is the only one among the four that guarantees uniqueness, and it is essential here because Schaefer fails due to the unboundedness of $f_2$.

Example 2

(Existence via Schaefer’s Fixed-Point Theorem). Consider a 3-cyclic coupled system ($m=3$) on $[0,2]$ with the same parameters as in the original Example 2.

Define the nonlinearities:

$$\begin{array}{cc}\hfill f_1(\varrho ,x_1,x_2)& ={\displaystyle \frac{2}{1+{\varrho }^2}}·{\displaystyle \frac{x_1}{1+|x_1|}},\hfill \\ \hfill f_2(\varrho ,x_2,x_3)& =e^{-\varrho }·{\displaystyle \frac{x_2}{1+|x_2|}},\hfill \\ \hfill f_3(\varrho ,x_3,x_1)& ={\displaystyle \frac{\varrho }{1+\varrho }}·arctan(x_3^2+x_1^2).\hfill \end{array}$$

Each $|f_j|$ is uniformly bounded: $|f_1|\le 2$, $|f_2|\le 1$, $|f_3|\le \pi /2$. Thus, Schaefer’s theorem (Theorem 3) applies, guaranteeing at least one solution.

• Why the Banach theorem fails: The function ${\displaystyle \frac{x_1}{1+|x_1|}}$ has Lipschitz constant 1 (maximum slope at $x_1=0$). To make the Banach theorem fail, we replace ${\displaystyle \frac{x_1}{1+|x_1|}}$ with $2.5·{\displaystyle \frac{x_1}{1+|x_1|}}$, which has Lipschitz constant $2.5$ and remains bounded. Then, with $L_1=2.5$, $L_2=1$, $L_3\approx 0.5$, and ${\Omega }_j\approx 0.1892$, we obtain

  $$2{\displaystyle \sum _{j=1}^3}{L}_j{\Omega }_j\approx 2(2.5+1+0.5)\times 0.1892=2\times 4\times 0.1892=1.5136>1,$$

  so the Banach contraction condition fails.

• Comparison of theorems:

Theorem	Applicable?	Reason
Banach	No	Contraction condition fails ($2\sum L_j{\Omega }_j>1$).
Schaefer	✓ Yes	Uniformly bounded; directly applicable.
Krasnosel’skiĭ	Possible	Splitting not natural; Schaefer is simpler.
Leray–Schauder	✓ Yes	Would also work, but Schaefer is more direct.

Schaefer is chosen because the nonlinearities are bounded but not contractive.

Example 3

(Existence via Krasnosel’skiĭ’s Fixed-Point Theorem). Consider a 2-cyclic coupled system on $[0,1]$ with $k_j=1$ (classical derivatives) and the same parameters as in the original Example 3.

Split the nonlinearities into contractive and compact parts:

$$\begin{array}{cc}\hfill f_1(\varrho ,x_1,x_2)& =\underset{\mathrm{compact}}{\underbrace{{\displaystyle \frac{\varrho }{1+\varrho }}cos\left(x_2\right)}}+\underset{\mathrm{contractive}}{\underbrace{2x_1}},\hfill \\ \hfill f_2(\varrho ,x_2,x_1)& =\underset{\mathrm{compact}}{\underbrace{e^{-\varrho }{\displaystyle \frac{x_2^2}{1+x_2^2}}}}+\underset{\mathrm{contractive}}{\underbrace{{\displaystyle \frac{1}{25}}arctan\left(x_1\right)}}.\hfill \end{array}$$

Contractive Lipschitz constants $L_1^{\left(2\right)}=2$, $L_2^{\left(2\right)}=0.04$. Computing ${\Omega }_1^{\left(2\right)}={\Omega }_2^{\left(2\right)}\approx 0.1845$ yields

$$2{\displaystyle \sum _{j=1}^2}{L}_j{\Omega }_j^{\left(2\right)}=2(2\times 0.1845+0.04\times 0.1845)=0.753<1.$$

By Theorem 5, at least one solution exists.

• Comparison of theorems:

Theorem	Applicable?	Reason
Banach	No	Full $f_1$ has Lipschitz constant $\approx 2.5$ , so $2\sum L_j{\Omega }_j>1$.
Schaefer	No	$f_1$ contains $2x_1$ , which is unbounded.
Krasnosel’skiĭ	✓ Yes	Contractive part ($2x_1$) handled by $T^{\left(2\right)}$; compact part bounded.
Leray–Schauder	Possible	Requires a priori bound; Krasnosel’skiĭ is more natural here.

Krasnosel’skiĭ is chosen because the operator naturally splits into a contraction ($2x_1$) and a compact part.

Example 4

(Existence via the Leray–Schauder Alternative). Consider a 2-cyclic coupled system on $[0,1]$ with the same parameters as in the original Example 4.

Define nonlinearities with sublinear growth:

$$\begin{array}{cc}\hfill f_1(\varrho ,x_1,x_2)& ={\displaystyle \frac{\sqrt{1+{\varrho }^2}}{1+\varrho }}·{\displaystyle \frac{|x_1{|}^{0.5}+{\left|x_2\right|}^{0.5}}{1+|x_1{|}^{0.5}+{\left|x_2\right|}^{0.5}}},\hfill \\ \hfill f_2(\varrho ,x_2,x_1)& ={\displaystyle \frac{e^{-\varrho }}{1+\varrho }}·ln(1+|x_2|+|x_1\left|\right).\hfill \end{array}$$

Here, $\parallel {\omega }_1{\parallel }_{\infty }={\parallel {\omega }_2\parallel }_{\infty }\approx 1$, ${\sigma }_1\left(t\right)=\sqrt{t}/(1+\sqrt{t})$, ${\sigma }_2\left(t\right)=ln(1+t)$. For $K=10$, ${\sigma }_1\left(10\right)=0.76$, ${\sigma }_2\left(10\right)\approx 2.398$, and

$${\displaystyle \frac{K}{{\sum }_{j=1}^2{\parallel {\omega }_j\parallel }_{\infty }{\sigma }_j\left(K\right){\Omega }_j}}={\displaystyle \frac{10}{0.3102\times 3.158}}\approx 10.21>1.$$

By Theorem 7, at least one solution exists with ${\parallel \mathbf{x}\parallel }_{\mathcal{X}}<10$.

• Comparison of theorems:

Theorem	Applicable?	Reason
Banach	No	$|x_1{|}^{0.5}$ is not Lipschitz near 0 (derivative $\to \infty$).
Schaefer	No	$f_2$ contains $ln(1+|x_2|+|x_1\left|\right)$ , which is unbounded.
Krasnosel’skiĭ	No	No natural splitting into small contraction + compact part.
Leray–Schauder	✓ Yes	Sublinear growth condition holds; a priori bound exists.

Leray–Schauder is the only theorem among the four that applies here.

7. Conclusions

In this paper, we have investigated an m-cyclic coupled system of sequential $(k,\psi )$-Hilfer and $(k,\psi )$-Caputo fractional differential equations with boundary conditions. The cyclic coupling structure $f_j(\varrho ,x_j,x_{j+1})$ with $x_{m+1}=x_1$ provides a natural framework for modeling circular interactions and feedback loops, while the boundary conditions for each component are non-separated but decoupled between components. We used several fixed-point theorems, including the Banach contraction principle, Schaefer’s fixed-point theorem, Krasnosel’skiĭ’s fixed-point theorem, and the Leray–Schauder alternative, to establish sufficient conditions for existence and uniqueness. A comprehensive Ulam–Hyers stability analysis was conducted, and explicit stability constants were derived. The relevance of the theoretical results was illustrated through numerical examples.

Future Research Directions:

• Fractional differential inclusions: Extending the analysis to multi-valued maps $F_j(\varrho ,x_j,x_{j+1})$ with convex and nonconvex values.
• Applications: Implementing the theoretical framework for real-world problems such as feedback control systems with fractional controllers and cyclic population dynamics with memory effects.