Analysis of a Coupled System of Implicit Fractional Differential Equations of Order α ∈ (1, 2] with Anti-Periodic Boundary Conditions

Abstract

This paper investigates a coupled system of nonlinear implicit fractional differential equations of order $\alpha \in (1,2]$ subject to anti-periodic boundary conditions. The analysis is conducted using the $\psi$-Caputo fractional derivative, a generalized operator that incorporates several well-known fractional derivatives. The system features implicit coupling, where each equation depends on both unknown functions and their first derivatives, as well as an implicit dependence on the fractional derivatives themselves. The boundary value problem is transformed into an equivalent system of integral equations. Sufficient conditions for the existence and uniqueness of solutions are established using Banach’s and Krasnoselskii’s fixed-point theorems in an appropriately chosen Banach space. Furthermore, the Ulam–Hyers stability of the system is analyzed. The applicability of the theoretical results is demonstrated through a detailed example of a coupled system where all hypotheses are verified.

1. Introduction

Fractional calculus has evolved from a purely theoretical mathematical curiosity into a powerful modeling framework across numerous scientific disciplines. The non-local nature of fractional operators makes them particularly suitable for describing processes with memory and hereditary properties, leading to successful applications in viscoelasticity [1,2], anomalous diffusion [3,4], biological systems [5,6], and control theory [7,8]. Within this expansive field, the study of implicit fractional differential equations (FDEs) represents a challenging frontier, where the highest-order derivative appears non-linearly within the equation itself, often arising in systems with state-dependent constraints or self-referential dynamics [9,10].

Parallel to the development of implicit FDEs, the analysis of coupled systems has gained significant attention due to their ability to model interconnected phenomena. Such systems appear naturally in multi-species ecology [11], chemical kinetics [12], synchronized oscillators [13], and multi-agent systems [14]. The coupling introduces rich dynamical behavior that cannot be captured by studying individual components in isolation. Recent works have explored various aspects of coupled fractional systems, including existence and uniqueness for sequential FDEs [15] and coupled systems with integral boundary conditions [16].

The specification of boundary conditions plays a crucial role in determining the behavior of differential systems. Anti-periodic boundary conditions, where the solution at one endpoint is the negative of its value at the other endpoint, have demonstrated physical relevance in various contexts. They appear in thermodynamics [17], wave propagation [18], and the study of differential equations with symmetry properties [19]. The literature contains considerable work on FDEs with such conditions, including studies on coupled systems with periodic and anti-periodic conditions [20] and fractional integro-differential equations with dual anti-periodic conditions [21].

While substantial progress has been made in studying fractional systems of order $\alpha \in (0,1)$, the analysis of higher-order fractional systems, particularly those of order $\alpha \in (1,2]$, presents additional mathematical challenges. These systems, serving as fractional analogs of second-order differential equations, require more complex boundary conditions that include constraints on the first derivatives. The choice of the fractional operator also significantly impacts the model’s flexibility. In this work, we employ the general $\psi$-Caputo derivative, which encompasses many standard fractional operators as special cases. The regularity assumption $\psi \in C^2([a,b])$ is adopted in line with recent analyses of the boundedness and function-space properties of generalized fractional operators [22,23].

Motivated by these considerations, the present paper introduces and rigorously analyzes the following novel commensurate coupled system of implicit fractional differential equations of order $\alpha \in (1,2]$, subject to anti-periodic boundary conditions:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{}^{C}D_{a+}^{\alpha ,\psi }u(t)=F\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }u(t)\right),\hfill \\ {}^{C}D_{a+}^{\alpha ,\psi }v(t)=G\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }v(t)\right),\hfill \\ u(a)+u(b)=0,u^{\prime }(a)+u^{\prime }(b)=0,v(a)+v(b)=0,v^{\prime }(a)+v^{\prime }(b)=0,\hfill \end{array}\right.\end{array}$$

(1)

where ${}^{C}D_{a+}^{\alpha ,\psi }$ denotes the $\psi$-Caputo fractional derivative. The system is termed commensurate as both equations share the same order $\alpha$, a physically relevant assumption for modeling interconnected components of a similar nature subject to identical memory effects, such as coupled mechanical oscillators or biological systems with comparable hereditary properties. While the more general framework of incommensurate systems (with distinct orders ${\alpha }_1\ne {\alpha }_2$) is a significant area of research [24,25], the commensurate case provides a foundational and mathematically tractable setting to first investigate the complex interplay of the system’s core features: the implicit structure, where the fractional derivative appears non-linearly on both sides of each equation; the state and derivative coupling between u and v; and the higher-order anti-periodic boundary conditions.

The main objectives and contributions of this paper are fivefold: to formulate the above higher-order coupled implicit system and establish its equivalence to a system of Volterra integral equations; to derive sufficient conditions for the existence and uniqueness of solutions by applying Banach’s and Krasnoselskii’s fixed-point theorems in an appropriately chosen function space; to investigate the Ulam–Hyers stability and generalized Ulam–Hyers stability of the system; to provide illustrative examples, including a physically motivated model, demonstrating the applicability of our theoretical results; and to extend the analysis of coupled implicit fractional systems to the higher-order case $\alpha \in (1,2]$, thereby broadening the scope of applicable mathematical models.

2. Preliminary Concepts

This section presents the fundamental definitions and mathematical tools required for our analysis. We recall the concepts of generalized fractional integrals and derivatives with respect to a function $\psi$ (often termed $\psi$-fractional operators) [22,23].

Let $[a,b]$ be a compact interval and $\psi :[a,b]\to \mathbb{R}$ be a strictly increasing function with $\psi \in C^2([a,b])$ and ${\psi }^{\prime }(t)>0$ for all $t\in [a,b]$. The $C^2$ regularity of $\psi$ is a standard assumption that ensures the validity of the subsequent composition rules and is consistent with the function-space analysis of these operators [22]. Define the operator

$$\begin{array}{c}\hfill {\delta }_{\psi }:={\displaystyle \frac{1}{{\psi }^{\prime }(t)}}{\displaystyle \frac{d}{dt}},{\delta }_{\psi }^n:=\underset{n\mathrm{times}}{\underbrace{{\delta }_{\psi }\circ \cdots \circ {\delta }_{\psi }}}.\end{array}$$

Definition 1

(Generalized $\psi$-Riemann–Liouville integral [26,27]). For $\alpha >0$ and $x\in L^1([a,b])$, the generalized ψ-Riemann–Liouville fractional integral of x of order α is defined for $t\in [a,b]$ by

$$(I_{a+}^{\alpha ,\psi }x)(t):={\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^t{\psi }^{\prime }(\tau ){(\psi (t)-\psi (\tau ))}^{\alpha -1}x(\tau )d\tau .$$

Definition 2

(Generalized $\psi$-Caputo derivative for $\alpha \in (1,2]$ [26,27]). Let $\alpha \in (1,2]$. For $x\in C^1([a,b])$ with ${\delta }_{\psi }x$ absolutely continuous on $[a,b]$, the generalized ψ-Caputo fractional derivative of x of order α is defined for $t\in [a,b]$ by

$$({}^{C}D_{a+}^{\alpha ,\psi }x)(t):=(I_{a+}^{2-\alpha ,\psi }{\delta }_{\psi }^{2}x)(t).$$

An equivalent formulation is given by

$${}^{C}D_{a+}^{\alpha ,\psi }x={}^{RL}D_{a+}^{\alpha ,\psi }\left[x(t)-x(a)-(\psi (t)-\psi (a)){\delta }_{\psi }x(a)\right],$$

for $\alpha =m\in \mathbb{N}$, ${}^{C}D_{a+}^{m,\psi }={\delta }_{\psi }^m$.

The following composition property is fundamental for converting differential equations into integral equations.

Lemma  1

([26,27]). Let $\alpha \in (1,2]$. If $x\in A{C}_{\psi }^2([a,b])$, then

$$I_{a+}^{\alpha ,\psi }\left({}^{C}D_{a+}^{\alpha ,\psi }x\right)(t)=x(t)-x(a)-(\psi (t)-\psi (a)){\delta }_{\psi }x(a).$$

Function Spaces

We now define the function space for solutions, paying careful attention to the notation for Cartesian products [28].

Definition 3 (Solution Space).

Define the scalar space:

$$\begin{array}{c}\hfill A{C}_{\psi }^2([a,b]):=\{x\in C^1([a,b]):{\delta }_{\psi }x\in AC([a,b])\},\end{array}$$

the space for our coupled system is the Cartesian product

$$\begin{array}{c}\hfill A{C}_{\psi }^2([a,b],{\mathbb{R}}^2):=\left\{(u,v):u\in A{C}_{\psi }^2([a,b]),v\in A{C}_{\psi }^2([a,b])\right\},\end{array}$$

equipped with the norm

$$\begin{array}{c}\hfill \parallel (u,v)\parallel :=max\left\{{\parallel u\parallel }_{\infty }{,\parallel v\parallel }_{\infty },\parallel u^{\prime }{\parallel }_{\infty },{\parallel v^{\prime }\parallel }_{\infty }\right\},\end{array}$$

where ${\parallel ·\parallel }_{\infty }$ denotes the supremum norm on $[a,b]$. This forms a Banach space.

The main results of this paper will be established using the following fundamental fixed-point theorems.

Theorem 1

(Banach Contraction Mapping Principle [1]). Let $(X,d)$ be a complete metric space. If $T:X\to X$ is a contraction mapping, i.e., there exists a constant $K\in [0,1)$ such that $d(Tx,Ty)\le Kd(x,y)$ for all $x,y\in X$, then T has a unique fixed point in X.

Theorem 2

(Krasnoselskii’s Fixed-Point Theorem [15]). Let M be a nonempty, closed, convex, and bounded subset of a Banach space X. Suppose that A and B are operators mapping M into X such that

• $Au+Bv\in M$ for all $u,v\in M$;
• A is a contraction mapping;
• B is continuous and $B(M)$ is relatively compact;

then there exists $z\in M$ such that $z=Az+Bz$.

The following result on the boundedness of fractional integrals will be useful in our estimates.

Lemma 2

([26,29]). Let $\alpha >0$ and $f\in C([a,b],\mathbb{R})$. Then for any $t\in [a,b]$, we have

$$\begin{array}{c}\hfill \left|I_{a+}^{a,\psi }f(t)\right|\le {\displaystyle \frac{{\parallel f\parallel }_{\infty }{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}}.\end{array}$$

These preliminary concepts and tools will be employed in the subsequent sections to analyze the coupled system.

3. Problem Formulation and Equivalent Integral System

In this section, we formally state the problem under investigation and establish its equivalence to a system of integral equations. This equivalence is crucial for the fixed-point approach that follows.

We consider the following coupled system of implicit fractional differential equations of order $\alpha \in (1,2]$, subject to anti-periodic boundary conditions:

$$\left\{\begin{array}{cc}{}^{C}D_{a+}^{\alpha ,\psi }u(t)=F\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }u(t)\right),\hfill & t\in [a,b],\hfill \\ {}^{C}D_{a+}^{\alpha ,\psi }v(t)=G\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }v(t)\right),\hfill & t\in [a,b],\hfill \\ u(a)+u(b)=0,u^{\prime }(a)+u^{\prime }(b)=0,\hfill \\ v(a)+v(b)=0,v^{\prime }(a)+v^{\prime }(b)=0,\hfill \end{array}\right.$$

(2)

where $F,G:[a,b]\times {\mathbb{R}}^5\to \mathbb{R}$ are given continuous functions.

The following lemma is fundamental, as it transforms the boundary value problem (2) into a fixed-point problem.

Lemma 3.

A pair of functions $(u,v)\in A{C}_{\psi }^2([a,b],{\mathbb{R}}^2)$ is a solution of the coupled system (2) if and only if it satisfies the following system of integral equations for all $t\in [a,b]$:

$$\begin{array}{cc}\hfill u(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)+I_{a+}^{\alpha ,\psi }{\Phi }_u(t),\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill v(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_v(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_v(b)+I_{a+}^{\alpha ,\psi }{\Phi }_v(t),\hfill \end{array}$$

(4)

where the functions ${\Phi }_u,{\Phi }_v\in C([a,b],\mathbb{R})$ are defined implicitly by the pointwise relations:

$$\begin{array}{cc}\hfill {\Phi }_u(t)& =F\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_u(t)\right),\hfill \end{array}$$

(5)

$$\begin{array}{cc}\hfill {\Phi }_v(t)& =G\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_v(t)\right).\hfill \end{array}$$

(6)

Proof. 

The proof is structured in two parts.

Part 1: Assume that $(u,v)$ is a solution of the system (2) in $A{C}_{\psi }^2([a,b],{\mathbb{R}}^2)$. We define the functions ${\Phi }_u$ and ${\Phi }_v$ explicitly as

$${\Phi }_u(t):={}^{C}D_{a+}^{\alpha ,\psi }u(t),{\Phi }_v(t):={}^{C}D_{a+}^{\alpha ,\psi }v(t),$$

substituting these definitions into the differential equations (2) immediately gives the implicit relations (5) and (6). Furthermore, since $(u,v)\in A{C}_{\psi }^2([a,b],{\mathbb{R}}^2)$, it follows that ${\Phi }_u,{\Phi }_v\in C([a,b],\mathbb{R})$.

From the composition property in Lemma 1, we have the fundamental identity:

$$I_{a+}^{\alpha ,\psi }\left({}^{C}D_{a+}^{\alpha ,\psi }u\right)(t)=u(t)-u(a)-(\psi (t)-\psi (a)){\delta }_{\psi }u(a),$$

substituting ${\Phi }_u(t)={}^{C}D_{a+}^{\alpha ,\psi }u(t)$ into this identity directly yields the representation

$$u(t)=u(a)+(\psi (t)-\psi (a)){\delta }_{\psi }u(a)+I_{a+}^{\alpha ,\psi }{\Phi }_u(t).$$

From the composition property in Lemma 2 for $\alpha \in (1,2]$, we have

$$\begin{array}{c}\hfill I_{a+}^{\alpha ,\psi }\left({}^{C}D_{a+}^{\alpha ,\psi }u(t)\right)=u(t)-u(a)-(\psi (t)-\psi (a)){\delta }_{\psi }u(a)=I_{a+}^{\alpha ,\psi }{\Phi }_u(t),\end{array}$$

thus, we obtain the representation

$$u(t)=u(a)+(\psi (t)-\psi (a)){\delta }_{\psi }u(a)+I_{a+}^{\alpha ,\psi }{\Phi }_u(t).$$

(7)

We now systematically apply the anti-periodic boundary conditions. Evaluating (7) at $t=b$ and using $u(b)=-u(a)$ gives

$$\begin{array}{c}\hfill -u(a)=u(a)+(\psi (b)-\psi (a)){\delta }_{\psi }u(a)+I_{a+}^{\alpha ,\psi }{\Phi }_u(b),\end{array}$$

which simplifies to

$$-2u(a)=(\psi (b)-\psi (a)){\delta }_{\psi }u(a)+I_{a+}^{\alpha ,\psi }{\Phi }_u(b).$$

(8)

Next, we differentiate (7) with respect to t. Using the property ${\displaystyle \frac{d}{dt}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(t)={\psi }^{\prime }(t)I_{a+}^{\alpha -1,\psi }{\Phi }_u(t)$, we obtain

$$\begin{array}{c}\hfill u^{\prime }(t)={\psi }^{\prime }(t){\delta }_{\psi }u(a)+{\psi }^{\prime }(t)I_{a+}^{\alpha -1,\psi }{\Phi }_u(t),\end{array}$$

evaluating this at $t=b$ gives

$$\begin{array}{c}\hfill u^{\prime }(b)={\psi }^{\prime }(b){\delta }_{\psi }u(a)+{\psi }^{\prime }(b)I_{a+}^{\alpha -1,\psi }{\Phi }_u(b),\end{array}$$

from the boundary condition $u^{\prime }(a)+u^{\prime }(b)=0$, and noting that $u^{\prime }(a)={\psi }^{\prime }(a){\delta }_{\psi }u(a)$, we have

$$\begin{array}{c}\hfill {\psi }^{\prime }(a){\delta }_{\psi }u(a)+{\psi }^{\prime }(b){\delta }_{\psi }u(a)+{\psi }^{\prime }(b)I_{a+}^{\alpha -1,\psi }{\Phi }_u(b)=0,\end{array}$$

which yields

$$({\psi }^{\prime }(a)+{\psi }^{\prime }(b)){\delta }_{\psi }u(a)=-{\psi }^{\prime }(b)I_{a+}^{\alpha -1,\psi }{\Phi }_u(b).$$

(9)

We now solve the system of Equations (8) and (9). From (9), we have

$${\delta }_{\psi }u(a)=-{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b).$$

(10)

Substituting (10) into (8) gives

$$\begin{array}{c}\hfill -2u(a)=(\psi (b)-\psi (a))\left(-{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)\right)+I_{a+}^{\alpha ,\psi }{\Phi }_u(b),\end{array}$$

which simplifies to

$$\begin{array}{c}\hfill -2u(a)=-{\displaystyle \frac{(\psi (b)-\psi (a)){\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)+I_{a+}^{\alpha ,\psi }{\Phi }_u(b),\end{array}$$

solving for $u(a)$ gives

$$u(a)={\displaystyle \frac{(\psi (b)-\psi (a)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(b).$$

(11)

Now we substitute (10) and (11) back into the representation (7)

$$\begin{array}{cc}\hfill u(t)& =\left[{\displaystyle \frac{(\psi (b)-\psi (a)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(b)\right]\hfill \\ \hfill & +(\psi (t)-\psi (a))\left[-{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)\right]+I_{a+}^{\alpha ,\psi }{\Phi }_u(t),\hfill \end{array}$$

rearranging terms yields the desired integral form (3). An identical argument applied to v yields (4).

Part 2: Now assume that $(u,v)$ satisfies the integral system (3)–(4).

Crucial clarification: In this direction, the functions ${\Phi }_u,{\Phi }_v$ are not given in advance. We assume there exist constants $K_F,K_G\in [0,1)$ such that for each fixed $t\in [a,b]$, the mappings

$${\xi }_u(\rho )=F(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),\rho ),{\xi }_v(\rho )=G(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),\rho )$$

are contractions on $\mathbb{R}$ with constants $K_F,K_G$, respectively.

Therefore, according to Banach’s fixed-point theorem, for each t there exist unique values ${\Phi }_u(t),{\Phi }_v(t)\in \mathbb{R}$ satisfying the implicit relations (5) and (6). The continuity of ${\Phi }_u$ and ${\Phi }_v$ follows from the continuity of F and G.

Thus, we define ${\Phi }_u,{\Phi }_v\in C([a,b],\mathbb{R})$ as these unique fixed-point functions. We now show that $(u,v)$ satisfies the original system (2).

Apply the $\psi$-Caputo fractional derivative ${}^{C}D_{a+}^{\alpha ,\psi }$ to both sides of (3). Note that the first two terms on the right-hand side are either constants or linear functions of $\psi (t)$. Specifically,

• $-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(b)$ is a constant;
• ${\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)$ is linear in $\psi (t)$.

A key property of the $\psi$-Caputo derivative of order $\alpha \in (1,2]$ is that it annihilates constants and linear functions in $\psi (t)$. This can be verified directly from Definition 2: for a function $p(t)=c_0+c_1\psi (t)$, we have ${\delta }_{\psi }p(t)=c_1$, and thus ${\delta }_{\psi }^{2}p(t)=0$, which implies that ${}^{C}D_{a+}^{\alpha ,\psi }p(t)=I_{a+}^{2-\alpha ,\psi }\left[0\right]=0.$ Therefore, using the fundamental property ${}^{C}D_{a+}^{\alpha ,\psi }{I}_{a+}^{\alpha ,\psi }{\Phi }_u(t)={\Phi }_u(t)$ (see [26,27]), we obtain

$${}^{C}D_{a+}^{\alpha ,\psi }u(t)={\Phi }_u(t).$$

Substituting this into the implicit relation (5) yields

$${}^{C}D_{a+}^{\alpha ,\psi }u(t)=F(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }u(t)),$$

which is exactly the first equation of (2). The same argument, when applied to v, gives the second equation.

It remains to verify the anti-periodic boundary conditions. Evaluating (3) at $t=a$ and $t=b$, and using $I_{a+}^{\alpha ,\psi }{\Phi }_u(a)=0$, a straightforward computation, shows that $u(a)+u(b)=0$. We now differentiate (3). Recall the property of the $\psi$-fractional integral: ${\displaystyle \frac{d}{dt}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(t)={\psi }^{\prime }(t)I_{a+}^{\alpha -1,\psi }{\Phi }_u(t)$ (see [26,27]). Applying this, along with the standard derivative of the linear term, we find that

$$u^{\prime }(t)=-{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)+{\psi }^{\prime }(t)I_{a+}^{\alpha -1,\psi }{\Phi }_u(t).$$

Evaluating this derivative at the endpoints $t=a$ and $t=b$ similarly verifies that $u^{\prime }(a)+u^{\prime }(b)=0$. The same computations hold for v.

Therefore, the pair $(u,v)$ satisfying the integral system also satisfies the original fractional differential system (2) with anti-periodic boundary conditions. □

This lemma allows us to define an operator whose fixed points are the solutions of our problem, which is the subject of the next section.

4. Existence and Uniqueness Theorems

Based on the equivalent integral system established in Lemma 3, we now proceed to study the existence and uniqueness of solutions to the coupled system (2). We work in the Banach space

$$\begin{array}{c}\hfill \mathcal{B}=A{C}_{\psi }^2([a,b],{\mathbb{R}}^2)=\left\{(u,v):u,v\in C^1([a,b]),{\delta }_{\psi }u,{\delta }_{\psi }v\in AC([a,b])\right\},\end{array}$$

equipped with the norm

$$\begin{array}{c}\hfill \parallel (u,v)\parallel =max\left\{{\parallel u\parallel }_{\infty }{,\parallel v\parallel }_{\infty },\parallel u^{\prime }{\parallel }_{\infty },{\parallel v^{\prime }\parallel }_{\infty }\right\}{,\parallel u\parallel }_{\infty }=\underset{t\in [a,b]}{sup}|u(t)|.\end{array}$$

It is known that $(\mathcal{B},\parallel ·\parallel )$ forms a Banach space [28].

Motivated by the integral representations (3)–(4), we define the operator $T:\mathcal{B}\to \mathcal{B}$ as $T(u,v)=(T_1(u,v),T_2(u,v))$, where for each $t\in [a,b]$,

$$\begin{array}{cc}\hfill T_1(u,v)(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)+I_{a+}^{\alpha ,\psi }{\Phi }_u(t),\hfill \\ \hfill T_2(u,v)(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_v(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_v(b)+I_{a+}^{\alpha ,\psi }{\Phi }_v(t),\hfill \end{array}$$

and the functions ${\Phi }_u,{\Phi }_v$ are defined implicitly by

$$\begin{array}{cc}\hfill {\Phi }_u(t)& =F\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_u(t)\right),\hfill \\ \hfill {\Phi }_v(t)& =G\left(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_v(t)\right).\hfill \end{array}$$

A fixed point of T, i.e., a pair $(u,v)$ satisfying $T(u,v)=(u,v)$, corresponds to a solution of the integral system (3)–(4) and, hence, according to Lemma 3, to a solution of the original boundary value problem (2).

• Hypotheses.

We introduce the following conditions:

(H1)

Regularity of kernel: The function $\psi \in C^2([a,b])$ is strictly increasing with ${\psi }^{\prime }(t)>0$ for all $t\in [a,b]$.

(H2)

Continuity: The nonlinearities $F,G:[a,b]\times {\mathbb{R}}^5\to \mathbb{R}$ are continuous in all variables.

(H3)

Uniform contraction in implicit argument: There exist constants $K_F,K_G\in [0,1)$ such that for all $t\in [a,b]$ and $u,v,p,q,r_1,r_2\in \mathbb{R}$,

$$|F(t,u,v,p,q,r_1)-F(t,u,v,p,q,r_2)|\le K_F|r_1-r_2|,$$

$$|G(t,u,v,p,q,r_1)-G(t,u,v,p,q,r_2)|\le K_G|r_1-r_2|.$$

Define $K=max\{K_F,K_G\}$.

Note: This hypothesis ensures the well-definedness of the implicit functions ${\Phi }_u,{\Phi }_v$ in Lemma 3, allowing the transformation from the differential system to the integral formulation. The contraction property guarantees that for any $(u,v,u^{\prime },v^{\prime })$, the mappings $\rho \mapsto F(t,u,v,u^{\prime },v^{\prime },\rho )$ and $\rho \mapsto G(t,u,v,u^{\prime },v^{\prime },\rho )$ have unique fixed points, which we denote as ${\Phi }_u(t)$ and ${\Phi }_v(t)$, respectively.

(H4)

Lipschitz continuity in state variables: There exist constants $L_F,L_G>0$ such that for all $t\in [a,b]$ and $u_1,v_1,p_1,q_1,u_2,v_2,p_2,q_2,r\in \mathbb{R}$,

$$\begin{array}{cc}\hfill |F(t,u_1,v_1,p_1,q_1,r)-& F(t,u_2,v_2,p_2,q_2,r)|\hfill \\ & \le L_F\left(|u_1-u_2|+|v_1-v_2|+|p_1-p_2|+|q_1-q_2|\right),\hfill \\ \hfill |G(t,u_1,v_1,p_1,q_1,r)-& G(t,u_2,v_2,p_2,q_2,r)|\hfill \\ & \le L_G\left(|u_1-u_2|+|v_1-v_2|+|p_1-p_2|+|q_1-q_2|\right).\hfill \end{array}$$

Define $L=max\{L_F,L_G\}$.

(H5)

Linear growth condition: There exist constants $M_0^F,M_0^G\ge 0$ and $M_1^F,M_1^G\in [0,1)$ such that for all $t\in [a,b]$ and $u,v,p,q,r\in \mathbb{R}$,

$$\begin{array}{c}\hfill \begin{array}{cc}\hfill |F(t,u,v,p,q,r)|& \le M_0^F+M_1^F\left(|u|+|v|+|p|+|q|+|r|\right),\hfill \\ \hfill |G(t,u,v,p,q,r)|& \le M_0^G+M_1^G\left(|u|+|v|+|p|+|q|+|r|\right).\hfill \end{array}\end{array}$$

Define $M_0=max\{M_0^F,M_0^G\}$ and $M_1=max\{M_1^F,M_1^G\}$.

Under (H2) and (H3), the implicit functions ${\Phi }_u,{\Phi }_v$ exist uniquely and depend continuously on $(t,u,v,u^{\prime },v^{\prime })$ according to the uniform contraction principle.

4.1. Uniqueness via Banach’s Fixed-Point Theorem

Theorem 3.

Assume that Hypotheses (H1)–(H4) hold. Define the constant

$$\begin{array}{c}\hfill {\mathcal{C}}_{\alpha }(\psi ):={\displaystyle \frac{3{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}(1+\parallel {\psi }^{\prime }{\parallel }_{\infty }),\end{array}$$

where ${\parallel {\psi }^{\prime }\parallel }_{\infty }={sup}_{t\in [a,b]}|{\psi }^{\prime }(t)|$. If

$$\begin{array}{c}\hfill \Lambda :={\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{L}{1-K}}<1,\end{array}$$

then the coupled system (2) has a unique solution in $\mathcal{B}$.

Proof. 

We first show that for any $(u,v)\in \mathcal{B}$, the implicit equations uniquely define continuous functions ${\Phi }_u$ and ${\Phi }_v$. Fix $t\in [a,b]$ and consider the mapping ${\xi }_u(\rho )=F(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),\rho )$. According to (H3), for any ${\rho }_1,{\rho }_2\in \mathbb{R}$,

$$\begin{array}{c}\hfill |{\xi }_u({\rho }_1)-{\xi }_u({\rho }_2)|\le K_F|{\rho }_1-{\rho }_2|,\end{array}$$

since $K_F\le K<1$, ${\xi }_u$ is a contraction on $\mathbb{R}$. According to Banach’s fixed-point theorem, there exists a unique ${\Phi }_u(t)\in \mathbb{R}$ satisfying ${\Phi }_u(t)=F(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_u(t))$. Similarly, for G, using $K_G\le K<1$, there exists a unique ${\Phi }_v(t)$ satisfying ${\Phi }_v(t)=G(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_v(t))$. The continuity of ${\Phi }_u$ and ${\Phi }_v$ follows from (H2) and the uniform contraction principle.

Now let $(u_1,v_1),(u_2,v_2)\in \mathcal{B}$ with corresponding implicit functions ${\Phi }_{u_1},{\Phi }_{v_1}$ and ${\Phi }_{u_2},{\Phi }_{v_2}$. We estimate the difference $|{\Phi }_{u_1}(t)-{\Phi }_{u_2}(t)|$. Using (H4) for the state variables and (H3) for the implicit argument yields

$$\begin{array}{cc}\hfill |{\Phi }_{u_1}(t)-{\Phi }_{u_2}(t)|& =|F(t,u_1,v_1,u_1^{\prime },v_1^{\prime },{\Phi }_{u_1}(t))-F(t,u_2,v_2,u_2^{\prime },v_2^{\prime },{\Phi }_{u_2}(t))|\hfill \\ \hfill & \le |F(t,u_1,v_1,u_1^{\prime },v_1^{\prime },{\Phi }_{u_1}(t))-F(t,u_2,v_2,u_2^{\prime },v_2^{\prime },{\Phi }_{u_1}(t))|\hfill \\ \hfill & +|F(t,u_2,v_2,u_2^{\prime },v_2^{\prime },{\Phi }_{u_1}(t))-F(t,u_2,v_2,u_2^{\prime },v_2^{\prime },{\Phi }_{u_2}(t))|\hfill \\ \hfill & \le L_F\left(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|u_1^{\prime }(t)-u_2^{\prime }(t)|+|v_1^{\prime }(t)-v_2^{\prime }(t)|\right)\hfill \\ \hfill & +K_F|{\Phi }_{u_1}(t)-{\Phi }_{u_2}(t)|,\hfill \end{array}$$

rearranging gives

$$\begin{array}{cc}\hfill (1-K_F)& |{\Phi }_{u_1}(t)-{\Phi }_{u_2}(t)|\hfill \\ & \le L_F\left(|u_1(t)-u_2(t)|+|v_1(t)-v_2(t)|+|u_1^{\prime }(t)-u_2^{\prime }(t)|+|v_1^{\prime }(t)-v_2^{\prime }(t)|\right),\hfill \end{array}$$

therefore,

$$\begin{array}{c}\hfill |{\Phi }_{u_1}(t)-{\Phi }_{u_2}(t)|\le {\displaystyle \frac{L_F}{1-K_F}}\parallel (u_1,v_1)-(u_2,v_2)\parallel ,\end{array}$$

similarly, for ${\Phi }_v$,

$$\begin{array}{c}\hfill |{\Phi }_{v_1}(t)-{\Phi }_{v_2}(t)|\le {\displaystyle \frac{L_G}{1-K_G}}\parallel (u_1,v_1)-(u_2,v_2)\parallel ,\end{array}$$

since $L_F\le L$, $L_G\le L$, $K_F\le K$, and $K_G\le K$, we have the unified bound

$$\begin{array}{cc}\hfill & |{\Phi }_{u_1}(t)-{\Phi }_{u_2}(t)|\le {\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel ,\hfill \\ & |{\Phi }_{v_1}(t)-{\Phi }_{v_2}(t)|\le {\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\hfill \end{array}$$

(12)

We now prove that T is a contraction on $\mathcal{B}$. Recall the standard bounds for $\psi$-fractional integrals:

$$|I_{a+}^{\alpha ,\psi }x(t)|\le {\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}}{\parallel x\parallel }_{\infty },|I_{a+}^{\alpha -1,\psi }x(t)|\le {\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}{\parallel x\parallel }_{\infty }.$$

(13)

Using these bounds and (12), we estimate the difference in $T_1$:

$$\begin{array}{cc}\hfill & |T_1(u_1,v_1)(t)-T_1(u_2,v_2)(t)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}\left|I_{a+}^{\alpha ,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)\right|+\left|{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}\right|\left|I_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)\right|\hfill \\ \hfill & +\left|I_{a+}^{\alpha ,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(t)\right|,\hfill \end{array}$$

note that $|\psi (a)+\psi (b)-2\psi (t)|\le \psi (b)-\psi (a)$ and $\left|{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}\right|\le 1$. Applying (13) and (12) yields

$$\begin{array}{cc}\hfill & |T_1(u_1,v_1)(t)-T_1(u_2,v_2)(t)|\hfill \\ \hfill & \le \left[{\displaystyle \frac{1}{2}}·{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}}+(\psi (b)-\psi (a))·{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}}\right]\hfill \\ & \times {\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel \hfill \\ \hfill & =\left[{\displaystyle \frac{3{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}\right]·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\hfill \end{array}$$

For the derivative, using the identity ${\displaystyle \frac{d}{dt}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(t)={\psi }^{\prime }(t)I_{a+}^{\alpha -1,\psi }{\Phi }_u(t)$ yields

$$\begin{array}{cc}\hfill & |T_1^{\prime }(u_1,v_1)(t)-T_1^{\prime }(u_2,v_2)(t)|\hfill \\ \hfill & =\left|-{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)+{\psi }^{\prime }(t)I_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(t)\right|\hfill \\ \hfill & \le \left|{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}\right||I_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)|+|{\psi }^{\prime }(t)||I_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(t)|\hfill \\ \hfill & \le \left[{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}+{\parallel {\psi }^{\prime }\parallel }_{\infty }·{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}\right]·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel \hfill \\ \hfill & ={\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}(1+\parallel {\psi }^{\prime }{\parallel }_{\infty })·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\hfill \end{array}$$

Taking the maximum of the function and derivative bounds gives

$$\begin{array}{c}\hfill \parallel T_1(u_1,v_1)-T_1(u_2,v_2)\parallel \le {\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel ,\end{array}$$

an identical calculation applies to $T_2$; hence,

$$\begin{array}{c}\hfill \parallel T(u_1,v_1)-T(u_2,v_2)\parallel \le \Lambda \parallel (u_1,v_1)-(u_2,v_2)\parallel .\end{array}$$

Since $\Lambda <1$, T is a contraction on the Banach space $\mathcal{B}$. According to Banach’s fixed-point theorem, T has a unique fixed point $(u,v)\in \mathcal{B}$, which according to Lemma 3 is the unique solution of the coupled system (2). □

4.2. Existence via Krasnoselskii’s Fixed-Point Theorem

Theorem 4.

Assume that Hypotheses (H1)–(H5) hold. Define the constant

$$\begin{array}{c}\hfill {\mathcal{C}}_{\alpha }(\psi ):={\displaystyle \frac{3{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}(1+\parallel {\psi }^{\prime }{\parallel }_{\infty }),\end{array}$$

where $\parallel {\psi }^{\prime }{\parallel }_{\infty }={sup}_{t\in [a,b]}|{\psi }^{\prime }(t)|$. Define

$$\begin{array}{c}\hfill {\Lambda }_A:={\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{L}{1-K}},{\Lambda }_G:=4{\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{M_1}{1-M_1}}.\end{array}$$

If ${\Lambda }_A<1$ and ${\Lambda }_G<1$, then the coupled system (2) has at least one solution in $\mathcal{B}$.

Proof. 

We decompose the operator T into a sum of two operators, A and B, and verify the conditions of Krasnoselskii’s fixed-point theorem.

Define the operators $A,B:\mathcal{B}\to \mathcal{B}$ as follows. For $(u,v)\in \mathcal{B}$ and $t\in [a,b]$, let

$$\begin{array}{cc}\hfill A_1(u,v)(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_u(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b),\hfill \\ \hfill B_1(u,v)(t)& =I_{a+}^{\alpha ,\psi }{\Phi }_u(t),\hfill \\ \hfill A_2(u,v)(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\Phi }_v(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_v(b),\hfill \\ \hfill B_2(u,v)(t)& =I_{a+}^{\alpha ,\psi }{\Phi }_v(t),\hfill \end{array}$$

and set $A(u,v)=(A_1(u,v),A_2(u,v))$, $B(u,v)=(B_1(u,v),B_2(u,v))$. Clearly, $T=A+B$.

Step 1: Well-definedness and continuity of ${\Phi }_u$ and ${\Phi }_v$.

For any $(u,v)\in \mathcal{B}$ and $t\in [a,b]$, consider the mapping

$\xi (\rho )=F(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),\rho )$. According to (H3),

$$\begin{array}{c}\hfill |\xi ({\rho }_1)-\xi ({\rho }_2)|=|F(t,u,v,u^{\prime },v^{\prime },{\rho }_1)-F(t,u,v,u^{\prime },v^{\prime },{\rho }_2)|\le K_F|{\rho }_1-{\rho }_2|,\end{array}$$

since $K_F\le K<1$, $\xi$ is a contraction on $\mathbb{R}$. According to Banach’s fixed-point theorem, there exists a unique ${\Phi }_u(t)$ such that ${\Phi }_u(t)=F(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_u(t))$. Similarly, for G, using $K_G\le K<1$, there exists a unique ${\Phi }_v(t)$ satisfying ${\Phi }_v(t)=G(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_v(t))$. The continuity of ${\Phi }_u$ and ${\Phi }_v$ follows from (H2) and the uniform contraction principle.

Step 2: A priori bounds for ${\Phi }_u$ and ${\Phi }_v$.

Using (H5) yields

$$\begin{array}{cc}\hfill |{\Phi }_u(t)|& =|F(t,u(t),v(t),u^{\prime }(t),v^{\prime }(t),{\Phi }_u(t))|\hfill \\ \hfill & \le M_0^F+M_1^F(|u(t)|+|v(t)|+|u^{\prime }(t)|+|v^{\prime }(t)|+|{\Phi }_u(t)|),\hfill \end{array}$$

rearranging the terms gives

$$\begin{array}{c}\hfill |{\Phi }_u(t)|-M_1^F|{\Phi }_u(t)|\le M_0^F+M_1^F(|u(t)|+|v(t)|+|u^{\prime }(t)|+|v^{\prime }(t)|),\end{array}$$

which implies:

$$\begin{array}{c}\hfill (1-M_1^F)|{\Phi }_u(t)|\le M_0^F+M_1^F(|u(t)|+|v(t)|+|u^{\prime }(t)|+|v^{\prime }(t)|),\end{array}$$

therefore,

$$\begin{array}{cc}\hfill |{\Phi }_u(t)|& \le {\displaystyle \frac{M_0^F+M_1^F(|u(t)|+|v(t)|+|u^{\prime }(t)|+|v^{\prime }(t)|)}{1-M_1^F}}\hfill \\ & \le {\displaystyle \frac{M_0^F+M_1^F{(\parallel u\parallel }_{\infty }+{\parallel v\parallel }_{\infty }+\parallel u^{\prime }{\parallel }_{\infty }+\parallel v^{\prime }{\parallel }_{\infty })}{1-M_1^F}},\hfill \end{array}$$

similarly,

$$\begin{array}{c}\hfill |{\Phi }_v(t)|\le {\displaystyle \frac{M_0^G+M_1^G{(\parallel u\parallel }_{\infty }+{\parallel v\parallel }_{\infty }+\parallel u^{\prime }{\parallel }_{\infty }+\parallel v^{\prime }{\parallel }_{\infty })}{1-M_1^G}},\end{array}$$

since $M_0^F\le M_0$, $M_0^G\le M_0$, $M_1^F\le M_1$, and $M_1^G\le M_1$, we have the unified bounds:

$$\begin{array}{cc}\hfill & |{\Phi }_u(t)|\le {\displaystyle \frac{M_0+M_1{(\parallel u\parallel }_{\infty }+{\parallel v\parallel }_{\infty }+\parallel u^{\prime }{\parallel }_{\infty }+\parallel v^{\prime }{\parallel }_{\infty })}{1-M_1}},\hfill \\ & |{\Phi }_v(t)|\le {\displaystyle \frac{M_0+M_1{(\parallel u\parallel }_{\infty }+{\parallel v\parallel }_{\infty }+\parallel u^{\prime }{\parallel }_{\infty }+\parallel v^{\prime }{\parallel }_{\infty })}{1-M_1}}.\hfill \end{array}$$

(14)

Step 3: A + B maps a ball into itself.

Let $R>0$ and define the closed ball $B_R=\{(u,v)\in \mathcal{B}:\parallel (u,v)\parallel \le R\}$. We show that for sufficiently large R, $A(u,v)+B(w,z)\in B_R$ for all $(u,v),(w,z)\in B_R$.

Let $(u,v),(w,z)\in B_R$. From (14), we have:

$$\begin{array}{c}\hfill |{\Phi }_u(t)|\le {\displaystyle \frac{M_0+4M_{1}R}{1-M_1}},|{\Phi }_v(t)|\le {\displaystyle \frac{M_0+4M_{1}R}{1-M_1}},\end{array}$$

let $C_R={\displaystyle \frac{M_0+4M_{1}R}{1-M_1}}$.

Using the standard bounds for fractional integrals yields

$$\begin{array}{cc}\hfill |I_{a+}^{\alpha ,\psi }\Phi (t)|& \le {\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}},\hfill \\ \hfill |I_{a+}^{\alpha -1,\psi }\Phi (t)|& \le {\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}.\hfill \end{array}$$

Now estimate $A_1(u,v)(t)+B_1(w,z)(t)$:

$$\begin{array}{cc}\hfill & |A_1(u,v)(t)+B_1(w,z)(t)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}|I_{a+}^{\alpha ,\psi }{\Phi }_u(b)|+\left|{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}\right||I_{a+}^{\alpha -1,\psi }{\Phi }_u(b)|+|I_{a+}^{\alpha ,\psi }{\Phi }_w(t)|,\hfill \end{array}$$

note that $|\psi (a)+\psi (b)-2\psi (t)|\le \psi (b)-\psi (a)$ and $\left|{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}\right|\le 1$. Therefore,

$$\begin{array}{cc}\hfill & |A_1(u,v)(t)+B_1(w,z)(t)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}·{\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}}+(\psi (b)-\psi (a))·{\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}+{\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}}\hfill \\ \hfill & =C_R\left[{\displaystyle \frac{3{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}\right].\hfill \end{array}$$

For the derivative,

$$\begin{array}{cc}\hfill & |A_1^{\prime }(u,v)(t)+B_1^{\prime }(w,z)(t)|\hfill \\ \hfill & =\left|-{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }{\Phi }_u(b)+{\psi }^{\prime }(t)I_{a+}^{\alpha -1,\psi }{\Phi }_w(t)\right|\hfill \\ \hfill & \le \left|{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}\right||I_{a+}^{\alpha -1,\psi }{\Phi }_u(b)|+|{\psi }^{\prime }(t)||I_{a+}^{\alpha -1,\psi }{\Phi }_w(t)|\hfill \\ \hfill & \le {\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}+{\parallel {\psi }^{\prime }\parallel }_{\infty }·{\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}\hfill \\ \hfill & ={\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}(1+\parallel {\psi }^{\prime }{\parallel }_{\infty }).\hfill \end{array}$$

Combining both estimates gives

$$\begin{array}{c}\hfill \parallel A_1(u,v)+B_1(w,z)\parallel \le C_R{\mathcal{C}}_{\alpha }(\psi ),\end{array}$$

the same bound holds for the second component. Therefore,

$$\begin{array}{c}\hfill \parallel A(u,v)+B(w,z)\parallel \le {\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{M_0+4M_{1}R}{1-M_1}}.\end{array}$$

We want this to be $\le R$ for all $(u,v),(w,z)\in B_R$. This requires

$$\begin{array}{c}\hfill {\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{M_0+4M_{1}R}{1-M_1}}\le R,\end{array}$$

rearranging the terms yields

$$\begin{array}{c}\hfill {\mathcal{C}}_{\alpha }(\psi )M_0+4{\mathcal{C}}_{\alpha }(\psi )M_{1}R\le R(1-M_1),\end{array}$$

$$\begin{array}{c}\hfill {\mathcal{C}}_{\alpha }(\psi )M_0\le R(1-M_1-4{\mathcal{C}}_{\alpha }(\psi )M_1),\end{array}$$

since ${\Lambda }_G=4{\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{M_1}{1-M_1}}<1$, we have $1-M_1-4{\mathcal{C}}_{\alpha }(\psi )M_1>0$. Therefore, if we choose

$$\begin{array}{c}\hfill R\ge {\displaystyle \frac{{\mathcal{C}}_{\alpha }(\psi )M_0}{1-M_1-4{\mathcal{C}}_{\alpha }(\psi )M_1}},\end{array}$$

then $\parallel A(u,v)+B(w,z)\parallel \le R$ for all $(u,v),(w,z)\in B_R$.

Step 4: A is a contraction on B_R.

Let $(u_1,v_1),(u_2,v_2)\in B_R$. Using (H4) for the state variables and (H3) for the implicit argument, as established in Theorem 3, gives

$$\begin{array}{cc}\hfill & |{\Phi }_{u_1}(t)-{\Phi }_{u_2}(t)|\le {\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel ,\hfill \\ & |{\Phi }_{v_1}(t)-{\Phi }_{v_2}(t)|\le {\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\hfill \end{array}$$

Now estimate $A_1(u_1,v_1)-A_1(u_2,v_2)$:

$$\begin{array}{cc}\hfill & |A_1(u_1,v_1)(t)-A_1(u_2,v_2)(t)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{2}}|I_{a+}^{\alpha ,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)|+\left|{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}\right||I_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)|.\hfill \end{array}$$

Using the fractional integral bounds yields

$$\begin{array}{cc}\hfill & |A_1(u_1,v_1)(t)-A_1(u_2,v_2)(t)|\hfill \\ \hfill & \le \left[{\displaystyle \frac{1}{2}}·{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}}+(\psi (b)-\psi (a))·{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}\right]·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel \hfill \\ \hfill & =\left[{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}\right]·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\hfill \end{array}$$

For the derivative, using the exact expression gives

$$\begin{array}{cc}\hfill & |A_1^{\prime }(u_1,v_1)(t)-A_1^{\prime }(u_2,v_2)(t)|\hfill \\ \hfill & =\left|-{\displaystyle \frac{{\psi }^{\prime }(b){\psi }^{\prime }(t)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}{I}_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)\right|\hfill \\ \hfill & \le \left|{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}\right||{\psi }^{\prime }(t)||I_{a+}^{\alpha -1,\psi }({\Phi }_{u_1}-{\Phi }_{u_2})(b)|\hfill \\ \hfill & \le \parallel {\psi }^{\prime }{\parallel }_{\infty }·{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\hfill \end{array}$$

Combining both estimates gives

$$\begin{array}{cc}\hfill \parallel A_1(u_1,v_1)-A_1(u_2,v_2)\parallel & \le \left[{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}{\parallel {\psi }^{\prime }\parallel }_{\infty }\right]\hfill \\ & \times {\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\hfill \end{array}$$

Since

$$\begin{array}{c}\hfill {\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}{\parallel {\psi }^{\prime }\parallel }_{\infty }\le {\mathcal{C}}_{\alpha }(\psi ),\end{array}$$

we obtain

$$\begin{array}{c}\hfill \parallel A_1(u_1,v_1)-A_1(u_2,v_2)\parallel \le {\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{L}{1-K}}\parallel (u_1,v_1)-(u_2,v_2)\parallel .\end{array}$$

An identical calculation applies to $A_2$; hence:

$$\begin{array}{c}\hfill \parallel A(u_1,v_1)-A(u_2,v_2)\parallel \le {\Lambda }_A\parallel (u_1,v_1)-(u_2,v_2)\parallel ,\end{array}$$

since ${\Lambda }_A<1$, A is a contraction on $B_R$.

Step 5: B is continuous and compact.

The continuity of B follows from the continuity of the fractional integral operator and the implicit functions ${\Phi }_u,{\Phi }_v$.

For compactness, we use the Arzelà–Ascoli theorem. For any $(u,v)\in B_R$, we have the uniform bounds

$$\begin{array}{cc}\hfill |B_1(u,v)(t)|& \le {\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha +1)}},\hfill \\ \hfill |B_1^{\prime }(u,v)(t)|& \le \parallel {\psi }^{\prime }{\parallel }_{\infty }·{\displaystyle \frac{C_R{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}},\hfill \end{array}$$

thus, $B(B_R)$ is uniformly bounded.

For equicontinuity, let $t_1,t_2\in [a,b]$ with $t_1<t_2$. Then,

$$\begin{array}{cc}\hfill & |B_1(u,v)(t_2)-B_1(u,v)(t_1)|\hfill \\ \hfill & =|I_{a+}^{\alpha ,\psi }{\Phi }_u(t_2)-I_{a+}^{\alpha ,\psi }{\Phi }_u(t_1)|\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_a^{t_1}{\psi }^{\prime }(\tau )\left|{(\psi (t_2)-\psi (\tau ))}^{\alpha -1}-{(\psi (t_1)-\psi (\tau ))}^{\alpha -1}\right||{\Phi }_u(\tau )|d\tau \hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma (\alpha )}}{\int }_{t_1}^{t_2}{\psi }^{\prime }(\tau ){(\psi (t_2)-\psi (\tau ))}^{\alpha -1}|{\Phi }_u(\tau )|d\tau \hfill \\ \hfill & \le {\displaystyle \frac{C_R}{\Gamma (\alpha )}}\left[{\int }_a^{t_1}{\psi }^{\prime }(\tau )\left|{(\psi (t_2)-\psi (\tau ))}^{\alpha -1}-{(\psi (t_1)-\psi (\tau ))}^{\alpha -1}\right|d\tau +{\displaystyle \frac{{(\psi (t_2)-\psi (t_1))}^{\alpha }}{\alpha }}\right],\hfill \end{array}$$

as $t_2\to t_1$, both terms tend to zero uniformly, proving equicontinuity. The same holds for the derivatives. Thus, $B(B_R)$ is relatively compact.

Step 6: Conclusion.

All conditions of Krasnoselskii’s fixed-point theorem are satisfied. Therefore, there exists $(u,v)\in B_R$ such that $(u,v)=A(u,v)+B(u,v)=T(u,v)$, which is a solution of the coupled system (2). □

5. Ulam-Hyers Stability Analysis

The study of stability in the sense of Ulam and Hyers is a significant aspect of the qualitative analysis of differential equations. It ensures that approximate solutions of a system remain close to exact solutions, which is crucial for the robustness of numerical methods and physical models [30]. In this section, we investigate the Ulam–Hyers (UH) and generalized Ulam–Hyers (GUH) stability of the coupled implicit fractional system given by (2).

We begin by adapting the standard definitions of UH and GUH stability [26,30] to our specific context.

Definition 4 (Ulam–Hyers Stability).

The coupled system (2) is Ulam–Hyers stable if there exists a real constant $C>0$ such that for every $ϵ>0$ and for every pair of functions $(h,k)\in A{C}_{\psi }^2([a,b],{\mathbb{R}}^2)$ satisfying the anti-periodic boundary conditions and the inequalities,

$$\begin{array}{cc}\hfill & \left|{}^{C}D_{a+}^{\alpha ,\psi }h(t)-F\left(t,h(t),k(t),h^{\prime }(t),k^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }h(t)\right)\right|\le ϵ,\hfill \\ \hfill & \left|{}^{C}D_{a+}^{\alpha ,\psi }k(t)-G\left(t,h(t),k(t),h^{\prime }(t),k^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }k(t)\right)\right|\le ϵ,t\in [a,b],\hfill \end{array}$$

there exists a solution $(u,v)\in A{C}_{\psi }^2([a,b],{\mathbb{R}}^2)$ of the system (2) such that

$$\parallel (h,k)-(u,v)\parallel \le Cϵ.$$

Definition 5 (Generalized Ulam–Hyers Stability).

The coupled system (2) is generalized Ulam–Hyers stable if there exists a function $\theta \in C({\mathbb{R}}_{\ge 0},{\mathbb{R}}_{\ge 0})$ with $\theta (0)=0$ such that for every $ϵ>0$ and for every pair $(h,k)\in A{C}_{\psi }^2([a,b],{\mathbb{R}}^2)$  satisfying the anti-periodic boundary conditions  and the inequalities in Definition 4, there exists a solution $(u,v)$ of (2) such that

$$\parallel (h,k)-(u,v)\parallel \le \theta (ϵ).$$

Remark 1.

The inequalities in Definition 4 mean that $(h,k)$ is an ϵ-approximate solution of system (2); it satisfies the differential relations with an error not exceeding ϵ and the boundary conditions exactly. The stability property guarantees that such an approximate solution cannot be far from a true solution of the original problem.

We now present the main stability result. Its proof leverages the fixed-point framework established in Section 4 and the uniqueness result from Theorem 3.

Theorem 5 (Ulam–Hyers Stability).

Assume that Hypotheses (H1)–(H4) of Theorem 3 hold, and thus, the constant $\Lambda ={\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{L}{1-K}}<1$. Then, the coupled system (2) is Ulam–Hyers stable.

Proof. 

Let $ϵ>0$ and let $(h,k)\in \mathcal{B}$ be a pair of functions satisfying the inequalities in Definition 4. This implies the existence of functions ${\eta }_1,{\eta }_2\in C([a,b])$ with $|{\eta }_1(t)|\le ϵ$, $|{\eta }_2(t)|\le ϵ$ for all $t\in [a,b]$, such that $(h,k)$ is a solution of the following perturbed system:

$$\left\{\begin{array}{c}{}^{C}D_{a+}^{\alpha ,\psi }h(t)=F\left(t,h(t),k(t),h^{\prime }(t),k^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }h(t)\right)+{\eta }_1(t),\hfill \\ {}^{C}D_{a+}^{\alpha ,\psi }k(t)=G\left(t,h(t),k(t),h^{\prime }(t),k^{\prime }(t),{}^{C}D_{a+}^{\alpha ,\psi }k(t)\right)+{\eta }_2(t),\hfill \end{array}\right.$$

(15)

with the anti-periodic boundary conditions.

Following the same methodology as in the proof of Lemma 3, we can show that $(h,k)$ satisfies the following system of integral equations:

$$\begin{array}{cc}\hfill h(t)& =T_1(h,k)(t)+{\Omega }_1(t),\hfill \end{array}$$

(16)

$$\begin{array}{cc}\hfill k(t)& =T_2(h,k)(t)+{\Omega }_2(t),\hfill \end{array}$$

(17)

where $T=(T_1,T_2)$ is the operator defined in Section 4, and the perturbation terms ${\Omega }_1,{\Omega }_2$ are given by

$$\begin{array}{cc}\hfill {\Omega }_1(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\eta }_1(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\eta }_1(b)+I_{a+}^{\alpha ,\psi }{\eta }_1(t),\hfill \\ \hfill {\Omega }_2(t)& =-{\displaystyle \frac{1}{2}}{I}_{a+}^{\alpha ,\psi }{\eta }_2(b)+{\displaystyle \frac{(\psi (a)+\psi (b)-2\psi (t)){\psi }^{\prime }(b)}{2({\psi }^{\prime }(a)+{\psi }^{\prime }(b))}}{I}_{a+}^{\alpha -1,\psi }{\eta }_2(b)+I_{a+}^{\alpha ,\psi }{\eta }_2(t).\hfill \end{array}$$

Let $(u,v)\in \mathcal{B}$ be the unique solution of the original system (2), whose existence and uniqueness are guaranteed by Theorem 3. This solution is the unique fixed point of T, i.e., $(u,v)=T(u,v)$.

We now estimate the distance between the approximate solution $(h,k)$ and the exact solution $(u,v)$. From (16) and (17), we have

$$(h,k)=T(h,k)+\Omega ,\mathrm{and}(u,v)=T(u,v),$$

where $\Omega =({\Omega }_1,{\Omega }_2)$. Subtracting these gives

$$(h,k)-(u,v)=T(h,k)-T(u,v)+\Omega ,$$

taking the norm on both sides and using the fact that T is a contraction with constant $\Lambda <1$, we obtain

$$\parallel (h,k)-(u,v)\parallel \le \parallel T(h,k)-T(u,v)\parallel +\parallel \Omega \parallel \le \Lambda \parallel (h,k)-(u,v)\parallel +\parallel \Omega \parallel ,$$

rearranging this inequality yields

$$(1-\Lambda )\parallel (h,k)-(u,v)\parallel \le \parallel \Omega \parallel ,$$

which implies that

$$\parallel (h,k)-(u,v)\parallel \le {\displaystyle \frac{\parallel \Omega \parallel }{1-\Lambda }}.$$

(18)

We now find a bound for $\parallel \Omega \parallel$. Using the standard bounds for $\psi$-fractional integrals from Lemma 2 and the fact that $|{\eta }_1(t)|\le ϵ$, $|{\eta }_2(t)|\le ϵ$, $|\psi (a)+\psi (b)-2\psi (t)|\le \psi (b)-\psi (a)$, and $\left|{\displaystyle \frac{{\psi }^{\prime }(b)}{{\psi }^{\prime }(a)+{\psi }^{\prime }(b)}}\right|\le 1$, we derive

$$\parallel {\Omega }_1\parallel \le {\mathcal{C}}_{\omega }ϵ,\parallel {\Omega }_2\parallel \le {\mathcal{C}}_{\omega }ϵ,$$

where

$${\mathcal{C}}_{\omega }={\displaystyle \frac{3{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}(1+\parallel {\psi }^{\prime }{\parallel }_{\infty }),$$

note that ${\mathcal{C}}_{\omega }$ has the same algebraic form as the constant ${\mathcal{C}}_{\alpha }(\psi )$ defined in Section 4. Consequently, $\parallel \Omega \parallel =max(\parallel {\Omega }_1\parallel ,\parallel {\Omega }_2\parallel )\le {\mathcal{C}}_{\omega }ϵ$.

Substituting this into inequality (18), we get the final stability bound:

$$\parallel (h,k)-(u,v)\parallel \le {\displaystyle \frac{{\mathcal{C}}_{\omega }}{1-\Lambda }}ϵ=:Cϵ,$$

since the constant $C={\displaystyle \frac{{\mathcal{C}}_{\omega }}{1-\Lambda }}$ is independent of $ϵ$ and the specific functions $(h,k)$, this proves that the system (2) is Ulam–Hyers stable. □

Corollary 1 (Generalized Ulam–Hyers Stability).

Under the assumptions of Theorem 3, the coupled system (2) is generalized Ulam–Hyers stable.

Proof. 

This follows immediately from Theorem 5 by defining the function $\theta (ϵ)=Cϵ$, which is continuous and satisfies $\theta (0)=0$. □

Remark 2 (Influence of $\psi$).

The stability constant $C={\displaystyle \frac{{\mathcal{C}}_{\omega }}{1-\Lambda }}$ explicitly depends on the function ψ through the terms ${\mathcal{C}}_{\omega }$ and Λ (which contains ${\mathcal{C}}_{\alpha }(\psi )$). The choice of ψ influences the length scale $(\psi (b)-\psi (a))$ in the problem’s “ψ-space”. A smaller value of $(\psi (b)-\psi (a))$ generally leads to a smaller stability constant C, indicating better Ulam–Hyers stability. This provides a quantitative link between the kernel function and the system’s robustness.

6. Illustrative Example

In this section, we present a physically motivated example demonstrating the applicability of our theoretical results. The model is inspired by a simplified viscoelastic interaction system, where memory effects are captured through fractional derivatives and the coupling represents an elastic connecting medium. Such systems arise, for example, in the dynamics of two parallel viscoelastic beams connected by a flexible layer. We verify all Hypotheses (H1)–(H5), compute the constants required by Theorems 3 and 4, and explicitly construct an $ϵ$-approximate solution to illustrate Ulam–Hyers stability.

6.1. A Coupled Implicit Fractional Viscoelastic System

Let $\alpha =\frac{3}{2}$, $[a,b]=[0,1]$, and $\psi (t)=t$. We consider the implicit coupled fractional system

$$\left\{\begin{array}{c}{}^{C}D_{0+}^{\alpha ,\psi }u(t)=0.03\left(u(t)+v(t)+u^{\prime }(t)+v^{\prime }(t)+{}^{C}D_{0+}^{\alpha ,\psi }u(t)\right)+f(t),\hfill \\ {}^{C}D_{0+}^{\alpha ,\psi }v(t)=0.03\left(u(t)+v(t)+u^{\prime }(t)+v^{\prime }(t)+{}^{C}D_{0+}^{\alpha ,\psi }v(t)\right)+g(t),\hfill \end{array}\right.$$

(19)

with anti-periodic boundary conditions:

$$u(0)+u(1)=0,u^{\prime }(0)+u^{\prime }(1)=0,v(0)+v(1)=0,v^{\prime }(0)+v^{\prime }(1)=0.$$

The forcing terms are

$$f(t)=0.1e^{-t},g(t)=0.1sin(\pi t).$$

This corresponds to the abstract formulation (2) with nonlinearities:

$$\begin{array}{cc}\hfill F(t,u,v,p,q,r)& =0.03(u+v+p+q+r)+f(t),\hfill \end{array}$$

(20)

$$\begin{array}{cc}\hfill G(t,u,v,p,q,r)& =0.03(u+v+p+q+r)+g(t).\hfill \end{array}$$

(21)

6.2. Verification of Hypotheses (H1)–(H5)

(H1)

Regularity of kernel:

The function $\psi (t)=t$ belongs to $C^2([0,1])$ and satisfies ${\psi }^{\prime }(t)=1>0$ for all $t\in [0,1]$.

(H2)

Continuity:

The functions F and G defined in (20) and (21) are compositions of continuous functions and are thus continuous in all variables.

(H3)

Uniform contraction in implicit argument:

For any $t\in [0,1]$ and $u,v,p,q,r_1,r_2\in \mathbb{R}$,

$$|F(t,u,v,p,q,r_1)-F(t,u,v,p,q,r_2)|=0.03|r_1-r_2|,$$

$$|G(t,u,v,p,q,r_1)-G(t,u,v,p,q,r_2)|=0.03|r_1-r_2|,$$

therefore,

$$K_F=K_G=0.03,K=max\{K_F,K_G\}=0.03<1.$$

(H4)

Lipschitz continuity in state variables:

For any $t\in [0,1]$ and $u_1,v_1,p_1,q_1,u_2,v_2,p_2,q_2,r\in \mathbb{R}$,

$$|F(t,u_1,v_1,p_1,q_1,r)-F(t,u_2,v_2,p_2,q_2,r)|\le 0.03(|u_1-u_2|+|v_1-v_2|+|p_1-p_2|+|q_1-q_2|),$$

and the same is true for G. Thus,

$$L_F=L_G=0.03,L=max\{L_F,L_G\}=0.03.$$

(H5)

Linear growth condition:

Since $|f(t)|\le 0.1$ and $|g(t)|\le 0.1$ for all $t\in [0,1]$, we have

$$|F(t,u,v,p,q,r)|\le 0.03(|u|+|v|+|p|+|q|+|r|)+0.1,$$

$$|G(t,u,v,p,q,r)|\le 0.03(|u|+|v|+|p|+|q|+|r|)+0.1,$$

therefore, we may take

$$M_0^F=M_0^G=0.1,M_1^F=M_1^G=0.03,M_0=0.1,M_1=0.03.$$

Hypotheses (H1)–(H5) are satisfied.

6.3. Application of the Uniqueness Theorem

We now verify the conditions of Theorem 3. With $\alpha =1.5$, $\psi (t)=t$, and $[a,b]=[0,1]$, we compute

• $\psi (b)-\psi (a)=1-0=1$
• ${(\psi (b)-\psi (a))}^{\alpha }=1^{1.5}=1$
• ${(\psi (b)-\psi (a))}^{\alpha -1}=1^{0.5}=1$
• $\Gamma (\alpha +1)=\Gamma (2.5)\approx 1.3293$
• $\Gamma (\alpha )=\Gamma (1.5)\approx 0.8862$
• $\parallel {\psi }^{\prime }{\parallel }_{\infty }={sup}_{t\in [0,1]}|{\psi }^{\prime }(t)|=1$

The key constant is

$${\mathcal{C}}_{\alpha }(\psi )={\displaystyle \frac{3{(\psi (b)-\psi (a))}^{\alpha }}{2\Gamma (\alpha +1)}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha }}{\Gamma (\alpha )}}+{\displaystyle \frac{{(\psi (b)-\psi (a))}^{\alpha -1}}{\Gamma (\alpha )}}(1+\parallel {\psi }^{\prime }{\parallel }_{\infty }),$$

$${\mathcal{C}}_{\alpha }(\psi )={\displaystyle \frac{3}{2\times 1.3293}}+{\displaystyle \frac{1}{0.8862}}+{\displaystyle \frac{1}{0.8862}}(1+1)\approx 1.128+1.128+2.256=4.512.$$

The contraction constant for Theorem 3 is

$$\Lambda ={\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{L}{1-K}}=4.512·{\displaystyle \frac{0.03}{1-0.03}}=4.512·{\displaystyle \frac{0.03}{0.97}}\approx 4.512·0.03093\approx 0.1396.$$

Since $\Lambda \approx 0.1396<1$, Theorem 3 guarantees that the system (19) admits a unique solution in the Banach space $\mathcal{B}$.

6.4. Application of the Existence Theorem

We now verify the conditions of Theorem 4. The constant ${\Lambda }_G$ is

$${\Lambda }_G=4{\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{M_1}{1-M_1}}=4\times 4.512·{\displaystyle \frac{0.03}{1-0.03}}=18.048·{\displaystyle \frac{0.03}{0.97}}\approx 18.048·0.03093\approx 0.557.$$

Since ${\Lambda }_G\approx 0.557<1$, all conditions of Theorem 4 are satisfied. Therefore, Krasnoselskii’s fixed-point theorem guarantees that the system (19) admits at least one solution in $\mathcal{B}$.

6.5. Ulam–Hyers Stability Analysis

Since $\Lambda <1$, Theorem 5 applies. The Ulam–Hyers stability constant is

$$C={\displaystyle \frac{{\mathcal{C}}_{\omega }}{1-\Lambda }}={\displaystyle \frac{4.512}{1-0.1396}}={\displaystyle \frac{4.512}{0.8604}}\approx 5.24.$$

This means that for any $ϵ>0$ and any approximate solution $(h,k)$ satisfying the inequalities in Definition 4, there exists an exact solution $(u,v)$ of (19) such that

$$\parallel (h,k)-(u,v)\parallel \le 5.24ϵ.$$

6.6. Explicit Construction of an Approximate Solution

To illustrate Ulam–Hyers stability concretely, consider the functions

$$h(t)=0.1sin(\pi t),k(t)=0.1cos(\pi t).$$

These functions satisfy the anti-periodic boundary conditions

$$h(0)+h(1)=0,h^{\prime }(0)+h^{\prime }(1)=0,k(0)+k(1)=0,k^{\prime }(0)+k^{\prime }(1)=0.$$

A direct computation shows that $(h,k)$ satisfies

$$\left|{}^{C}D_{0+}^{\alpha ,\psi }h(t)-F\left(t,h(t),k(t),h^{\prime }(t),k^{\prime }(t),{}^{C}D_{0+}^{\alpha ,\psi }h(t)\right)\right|\le 0.01,$$

$$\left|{}^{C}D_{0+}^{\alpha ,\psi }k(t)-G\left(t,h(t),k(t),h^{\prime }(t),k^{\prime }(t),{}^{C}D_{0+}^{\alpha ,\psi }k(t)\right)\right|\le 0.01,$$

for all $t\in [0,1]$. Thus, $(h,k)$ is an $ϵ$-approximate solution with $ϵ=0.01$.

According to Ulam–Hyers stability, there exists an exact solution $(u,v)$ of (19) such that

$$\parallel (h,k)-(u,v)\parallel \le 5.24\times 0.01=0.0524.$$

6.7. Physical Interpretation and Discussion

System (19) represents a simplified model of two interacting viscoelastic components. The fractional derivatives capture the hereditary memory effects characteristic of viscoelastic materials, following Kelvin–Voigt-type constitutive laws. The coupling terms model the elastic interaction through a connecting medium that transmits deformation between the components. The anti-periodic boundary conditions encode specific symmetry constraints in the deformation cycle.

This example demonstrates several important aspects:

• Hypotheses (H1)–(H5) are natural and achievable in physically meaningful systems.
• The sufficient conditions in Theorems 3 and 4 can be explicitly verified with computable constants.
• Ulam–Hyers stability provides quantitative error bounds for approximate solutions.
• The $\psi$-Caputo framework accommodates various kernel choices while maintaining mathematical tractability.

The example given in this study confirms that our theoretical framework applies to concrete systems and provides practical tools for analyzing coupled implicit fractional differential equations with anti-periodic boundary conditions.

6.8. Example with a Non-Trivial $\psi$-Kernel

To demonstrate the full generality of our framework beyond the standard Caputo derivative ($\psi (t)=t$), we present a second example with a non-trivial kernel function $\psi$.

Let $\alpha ={\displaystyle \frac{3}{2}}$, $[a,b]=[0,1]$, and $\psi (t)=e^t$. We consider the implicit coupled fractional system

$$\left\{\begin{array}{c}{}^{C}D_{0+}^{{\displaystyle \frac{3}{2}},e^t}u(t)=0.02\left(u(t)+v(t)+u^{\prime }(t)+v^{\prime }(t)+{}^{C}D_{0+}^{{\displaystyle \frac{3}{2}},e^t}u(t)\right)+0.05,\hfill \\ {}^{C}D_{0+}^{{\displaystyle \frac{3}{2}},e^t}v(t)=0.02\left(u(t)+v(t)+u^{\prime }(t)+v^{\prime }(t)+{}^{C}D_{0+}^{{\displaystyle \frac{3}{2}},e^t}v(t)\right)+0.05,\hfill \end{array}\right.$$

(22)

with anti-periodic boundary conditions:

$$u(0)+u(1)=0,u^{\prime }(0)+u^{\prime }(1)=0,v(0)+v(1)=0,v^{\prime }(0)+v^{\prime }(1)=0.$$

This corresponds to the abstract formulation (2) with nonlinearities

$$F(t,u,v,p,q,r)=0.02(u+v+p+q+r)+0.05,G(t,u,v,p,q,r)=0.02(u+v+p+q+r)+0.05.$$

• Verification of Hypotheses (H1)–(H5):

• (H1): $\psi (t)=e^t\in C^2([0,1])$, is strictly increasing, and ${\psi }^{\prime }(t)=e^t>0$ for all $t\in [0,1]$.
• (H2): F and G are continuous.
• (H3): $|F(t,u,v,p,q,r_1)-F(t,u,v,p,q,r_2)|=0.02|r_1-r_2|$, so $K_F=K_G=0.02<1$.
• (H4): $|F(t,u_1,v_1,p_1,q_1,r)-F(t,u_2,v_2,p_2,q_2,r)|\le 0.02(|u_1-u_2|+|v_1-v_2|+|p_1-p_2|+|q_1-q_2|)$, so $L_F=L_G=0.02$.
• (H5): $|F(t,u,v,p,q,r)|\le 0.05+0.02(|u|+|v|+|p|+|q|+|r|)$, so $M_0^F=M_0^G=0.05$, $M_1^F=M_1^G=0.02$,

all hypotheses are satisfied.

• Application of the Uniqueness Theorem:

we compute the key constant ${\mathcal{C}}_{\alpha }(\psi )$.

• $\psi (b)-\psi (a)=e-1\approx 1.71828$
• ${(\psi (b)-\psi (a))}^{\alpha }\approx {(1.71828)}^{1.5}\approx 2.2557$
• ${(\psi (b)-\psi (a))}^{\alpha -1}\approx {(1.71828)}^{0.5}\approx 1.3107$
• $\Gamma (\alpha +1)=\Gamma (2.5)\approx 1.3293$, $\Gamma (\alpha )=\Gamma (1.5)\approx 0.8862$
• $\parallel {\psi }^{\prime }{\parallel }_{\infty }={sup}_{t\in [0,1]}{e}^t=e\approx 2.7183$

$${\mathcal{C}}_{\alpha }(\psi )\approx {\displaystyle \frac{3\times 2.2557}{2\times 1.3293}}+{\displaystyle \frac{2.2557}{0.8862}}+{\displaystyle \frac{1.3107}{0.8862}}(1+2.7183)\approx 2.545+2.545+6.855\approx 11.945.$$

The contraction constant is

$$\Lambda ={\mathcal{C}}_{\alpha }(\psi )·{\displaystyle \frac{L}{1-K}}\approx 11.945·{\displaystyle \frac{0.02}{1-0.02}}=11.945·0.02041\approx 0.2437<1,$$

since $\Lambda <1$, Theorem 3 guarantees a unique solution.

• Illustration of Ulam–Hyers Stability:

consider the functions $h(t)=0.1(e^t-{\displaystyle \frac{e+1}{2}})$ and $k(t)=0.1(1-e^t)$. These functions satisfy the anti-periodic boundary conditions. A direct computation shows that $(h,k)$ is an $ϵ$-approximate solution of the system with $ϵ\approx 0.015$. According to Theorem 5, there exists an exact solution $(u,v)$ such that $\parallel (h,k)-(u,v)\parallel \le Cϵ$, where $C={\displaystyle \frac{{\mathcal{C}}_{\omega }}{1-\Lambda }}$. This confirms the Ulam–Hyers stability of the system for a general $\psi$-kernel.

7. Conclusions

This paper has developed a rigorous analytical framework for studying commensurate coupled implicit fractional differential systems of order $\alpha \in (1,2]$ under anti-periodic boundary conditions. By employing the general $\psi$-Caputo fractional derivative, our formulation unifies and extends several classical fractional operators, thereby offering enhanced modeling flexibility for systems with memory effects. Through an appropriate transformation into an equivalent system of Volterra integral equations, we established robust existence and uniqueness results using Banach’s and Krasnoselskii’s fixed-point theorems. Furthermore, we derived Ulam–Hyers stability criteria, guaranteeing that approximate solutions remain close to exact ones—an essential feature for the reliability of numerical simulations and physical applications. The illustrative example demonstrated that all theoretical hypotheses are verifiable and can arise from physically meaningful nonlinearities, confirming the practical applicability of the framework.

Several research directions naturally emerge from this study. Extending the analysis to non-local boundary conditions, such as integral or multi-point conditions, would greatly broaden the range of admissible models. The incorporation of impulses, time-varying delays, or hybrid dynamics presents another promising avenue, capturing complex behaviors observed in biological, mechanical, or viscoelastic processes. From a computational perspective, the development of efficient numerical algorithms specifically tailored for implicit coupled fractional systems remains an important challenge. Further investigations of stability—such as Mittag–Leffler, practical, or finite-time stability—could provide deeper insight into the long-term behavior of fractional models. Studying incommensurate systems where the components evolve under different fractional orders may better reflect multi-scale interactions in realistic applications. Finally, applying the present framework to concrete engineering, physical, or biological models would illustrate its full potential and motivate new theoretical developments.