Stability and Controllability of Coupled Neutral Impulsive ϱ-Fractional System with Mixed Delays

Abstract

This study examines a comprehensive class of coupled nonlinear $\varrho$-Hilfer fractional neutral impulsive integro-differential systems with mixed delays and non-local initial conditions. The primary contribution of this study is the creation of a unified analytical framework that encompasses coupled interactions, neutral-type dependencies, and impulsive disturbances, which have been studied separately by researchers. We utilize the Banach contraction principle and Krasnoselskii’s fixed-point theorem to provide suitable conditions for the existence and uniqueness of solutions within the product space of piecewise continuous weighted functions. In addition to existence, we examine Ulam–Hyers–Rassias (UHR) stability using a generalized Gronwall inequality, which guarantees the system’s robustness against functional perturbations. We also develop a controllability framework and a feedback control law that steer the system towards the desired terminal states. The theoretical results are supported by a numerical simulation using a complex kernel, implemented via a modified predictor-corrector algorithm, which validates the practical effectiveness of the proposed control and stability outcomes.

1. Introduction and Motivations

Fractional calculus has evolved from a purely theoretical mathematical discipline into a fundamental framework for modeling complex systems characterized by memory and hereditary traits. Fractional operators offer a non-local framework crucial for modeling dynamics in physics, biology, and advanced engineering, in contrast to classical derivatives [1,2,3]. In this domain, the $\varrho$-fractional derivative [4,5] has surfaced as an exceptionally powerful instrument. It unifies several traditional operators, like Riemann–Liouville and Caputo, by using a flexible kernel function $\varrho (\ell )$. This makes it possible to describe temporal behaviors more accurately. This flexibility has led to new research into systems with non-local initial conditions and specific boundary constraints [6,7].

In the transition from theoretical models to real-world applications, researchers have found that systems are rarely isolated or continuous. Coupled dynamics, where multiple states interact simultaneously, are now the standard for modeling biological oscillators and networked control systems [8,9,10,11]. These interactions are often complicated by delays, including discrete lags and distributed memory effects [12,13,14,15]. Furthermore, many physical processes are subject to “shocks” or sudden changes in state. Such impulsive effects [16,17,18] are frequently observed in mechanical systems, second-order dynamical models [17], and transmission protocols subject to packet losses [19,20].

Controllability is another important area of research. Determining if a system can be directed to a desired state is essential for engineering applications [21,22,23,24]. This analysis necessitates a stringent application of fixed-point theorems [25] and specialized integral inequalities, including the $\varrho$-Gronwall inequality [26,27].

Despite extensive literature, considerable fragmentation remains. For instance, while numerous studies investigate existence and uniqueness for neutral fractional equations, they neglect impulsive perturbations [28,29]. By contrast, current models for impulsive fractional systems often ignore neutral-type dependencies, in which derivatives depend on previous states [30,31]. Recent studies have begun addressing these deficiencies by investigating Ulam–Hyers–Rassias (UHR) stability within the framework of non-instantaneous impulses and fuzzy fractional models [32,33,34,35]. However, these studies often overlook the complex interactions between coupling and mixed delays. Even state-of-the-art research frequently treats these features independently rather than collectively, focusing on the stability of impulsive systems [36] and on stochastic perturbations [37,38]. Despite developments in this field, a cohesive framework that amalgamates coupling, neutrality, impulses, and mixed delays within the $\varrho$-Hilfer fractional operator remains absent.

This study aims to fill the gap in the literature by developing a unified framework combining different types of complexity: coupling, neutrality, impulses, and mixed delays. More specifically, we study a coupled system of nonlinear $\varrho$-Hilfer fractional neutral impulsive integro-differential equations with mixed delays and non-local initial conditions of the following type:

$$\left\{\begin{array}{c}{}^{H}D_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[x\left(\ell \right)-{\mathcal{N}}_1(\ell ,x_{\ell })\right]=f_1\left(\ell ,x\left(\ell \right),y\left(\ell \right),x\left(\rho \left(\ell \right)\right),y\left(\sigma \left(\ell \right)\right),{\int }_0^{\ell }{\mathcal{K}}_1(\ell ,s,x\left(s\right),y\left(s\right))ds\right),\hfill \\ {}^{H}D_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[y\left(\ell \right)-{\mathcal{N}}_2(\ell ,y_{\ell })\right]=f_2\left(\ell ,x\left(\ell \right),y\left(\ell \right),x\left(\rho \left(\ell \right)\right),y\left(\sigma \left(\ell \right)\right),{\int }_0^{\ell }{\mathcal{K}}_2(\ell ,s,x\left(s\right),y\left(s\right))ds\right),\hfill \end{array}\right.$$

(1)

equipped with a non-local initial condition:

$$\left\{\begin{array}{c}{I}_{0^{+}}^{1-\gamma ;\varrho }\left[x\left(0^{+}\right)-{\mathcal{N}}_1(0,x_0)\right]={\sum }_{j=1}^p{c}_{j}x\left({\xi }_j\right),\hfill \\ I_{0^{+}}^{1-\gamma ;\varrho }\left[y\left(0^{+}\right)-{\mathcal{N}}_2(0,y_0)\right]={\sum }_{j=1}^p{d}_{j}y\left({\xi }_j\right),\hfill \end{array}\right.$$

(2)

where ${\xi }_j\in (0,{\ell }_1]$ for $j=1,\dots ,p$. The left-hand side of (1) represents the $\rho$-Hilfer fractional rate of change in the system’s state after accounting for neutral dependencies (${\mathcal{N}}_i$), which capture how the derivative depends on the history of the state. The right-hand side, $f_i$, governs the coupled dynamics influenced by current states, discrete delays ($\rho (\ell ),\sigma (\ell )$), and distributed delays (the integral terms with kernels ${\mathcal{K}}_i$). The non-local condition (2) couples the initial state to values at interior points ${\xi }_j$, which is more general than classical initial conditions and arises naturally in inverse problems and population dynamics. At each impulse instant ${\ell }_k$ ($k=1,\dots ,m$), the states undergo abrupt changes governed by:

$$\left\{\begin{array}{c}\Delta x\left({\ell }_k\right):=x\left({\ell }_k^{+}\right)-x\left({\ell }_k^{-}\right)=I_k\left(x\left({\ell }_k^{-}\right)\right),\hfill \\ \Delta y\left({\ell }_k\right):=y\left({\ell }_k^{+}\right)-y\left({\ell }_k^{-}\right)=J_k\left(y\left({\ell }_k^{-}\right)\right).\hfill \end{array}\right.$$

(3)

Finally, the history of the system on the interval $[-v,0]$ ($v>0$) is prescribed by the continuous functions:

$$x(\ell )=\varphi (\ell ),y(\ell )=\phi (\ell ),\ell \in [-v,0].$$

(4)

Here ${}^{H}D_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }$ is the $\varrho$-Hilfer fractional derivative relative to the lower bound ${\ell }_k^{+}$ of order $\alpha \in \left(0,1\right),$ type $\beta \in \left[0,1\right],\gamma =\alpha +\beta (1-\alpha )$, which is the weight of the function space $P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})$; $\varrho$ is an increasing function $\varrho \in C^1(J,\mathbb{R})$ with ${\varrho }^{\prime }\left(\ell \right)>0$ for all $\ell \in J$. The kernel functions ${\mathcal{K}}_1,{\mathcal{K}}_2:J\times J\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$ represent distributed time delays. Neutral mappings ${\mathcal{N}}_1,{\mathcal{N}}_2$ are neutral-type dependence, where $x_{\ell }$ and $y_{\ell }$ denote history segments. Time delays $\rho \left(\ell \right),\sigma \left(\ell \right)$ satisfy $0\le \rho \left(\ell \right),\sigma \left(\ell \right)\le \ell$. Impulse jump functions $I_k,J_k$ characterize abrupt changes at times ${\ell }_k.$ The continuous functions $\varphi ,\phi$ represent the history on $[-v,0].$

The state variables $(x,y)$ are part of the product space E in the system. The operators ${\mathcal{N}}_1$ and ${\mathcal{N}}_2$ make the states’ history not matter. The functions $f_1$ and $f_2$ represent the coupled nonlinear dynamics, including discrete delays $\rho (\ell )$ and $\sigma (\ell )$, as well as distributed delays through the integral terms with ${\mathcal{K}}_1$ and ${\mathcal{K}}_2$. At times ${\ell }_k$, the dynamics can change suddenly, as shown by $I_k$ and $J_k$. The initial state is limited by non-local conditions that use the coefficients $c_j$ and $d_j$ at points ${\xi }_j$.

The main contributions of this work are outlined as follows:

• New Integrative Modeling: We develop a $\varrho$-Hilfer framework that integrates four distinct types of complexity: coupling, neutrality, impulses, and mixed delays. This improvement broadens the usability of current fractional models.
• We establish the necessary conditions for the existence and uniqueness of solutions in weighted product spaces by using mixed Banach and Krasnoselskii fixed-point theorems.
• Stability Generalization: We derive UHR stability criteria by modifying a generalized $\varrho$-Gronwall inequality for the neutral impulsive context.
• Constructive Controllability: We establish a feedback control law and show that the system can be directed to target states, which makes it easy to use in engineering.
• Numerical Example: The theoretical results are confirmed via a computational example employing a logarithmic kernel and an adapted predictor-corrector scheme.

Table 1. Comparative analysis of existing literature vs. current work.

Reference	Coupling	Neutrality	Impulses	Mixed Delays
[39,40]	×	✓	×	×
[31,41]	×	×	✓	×
[32,33,34,35]	×	×	✓	✓
[36]	×	×	✓	×
[37,38]	✓	×	×	×
Current work	✓	✓	✓	✓

Note: ✓ indicates feature is included, × indicates feature is absent.

provides a comparative analysis of existing studies, highlighting the gap that this work addresses.

This unified framework is particularly relevant for modeling complex biological networks where species interactions (coupling) are subject to memory effects (fractional), sudden environmental changes (impulses), and signal transmission lags (delays). Similarly, in advanced engineering control systems, sensors and actuators experience both delays and abrupt faults, requiring a model that captures all these phenomena simultaneously. The integration of these features allows for the analysis of emergent behaviors arising from their interaction, which is missed in fragmented models.

The parameters and functional components used in this work are systematically defined in

Table 2. Detailed description of key system parameters, state spaces, and functional components used in the model (1)–(4).

Symbol	Description
$J=[0,T]$	Time interval with $0={\ell }_0<{\ell }_1<\cdots <{\ell }_m<{\ell }_{m+1}=T$
$J^{\prime }=J\setminus \{{\ell }_1,\dots ,{\ell }_m\}$	Interval excluding impulse points
$\alpha \in (0,1)$	Fractional order of the derivative
$\beta \in [0,1]$	Type of the $\varrho$-Hilfer derivative
$\gamma =\alpha +\beta (1-\alpha )$	Weight parameter for the function space
$\varrho \in C^1(J,\mathbb{R}),{\varrho }^{\prime }(\ell )>0$	Increasing kernel function
$x,y\in P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})$	State variables in the weighted piecewise continuous space
${\mathcal{N}}_1,{\mathcal{N}}_2$	Neutral operators depending on history segments $x_t,y_t$
$f_1,f_2$	Nonlinear functions governing the coupled dynamics
$\rho (\ell ),\sigma (\ell )$	Continuous delay functions satisfying $0\le \rho (\ell ),\sigma (\ell )\le \ell$
${\mathcal{K}}_1,{\mathcal{K}}_2$	Kernel functions for distributed delay integrals
$I_k,J_k$	Impulse functions describing state jumps at times ${\ell }_k$
$c_j,d_j\in \mathbb{R}$	Coefficients for the non-local initial conditions
${\xi }_j\in (0,{\ell }_1]$	Points for the non-local conditions
$\varphi ,\phi \in C\left(\right[-v,0],\mathbb{R})$	History functions on $[-v,0]$
$B_1,B_2\in {\mathbb{R}}^{1\times q}$	Control gain matrices (row vectors)
$u,v\in L^2(J,{\mathbb{R}}^q)$	Control input functions

.

This paper is organized as follows. Section 2 provides essential preliminaries. Section 3 details the main results on existence, uniqueness, UHR stability, and controllability. Section 4 presents the numerical application, and Section 5 concludes the paper.

2. Preliminaries

In this section, we recall the essential definitions, lemmas, and function spaces required for our qualitative analysis. Let $J=[0,T]$ and $0={\ell }_0<{\ell }_1<\cdots <{\ell }_m<{\ell }_{m+1}=T$ be the partition of J. We denote $J^{\prime }=J\setminus \{{\ell }_1,\dots ,{\ell }_m\}$.

Definition 1

([1,4]). Let $\alpha >0$ and $\varrho \left(\ell \right)$ be an increasing function with ${\varrho }^{\prime }\left(\ell \right)>0$. The ϱ-Riemann–Liouville fractional integral of a function f of order α is defined as:

$$I_{a^{+}}^{\alpha ;\varrho }f\left(\ell \right)={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_a^{\ell }{\varrho }^{\prime }\left(s\right){\left(\varrho \left(\ell \right)-\varrho \left(s\right)\right)}^{\alpha -1}f\left(s\right)ds.$$

(5)

Definition 2

([4]). Let $n-1<\alpha <n$ and $0\le \beta \le 1$. The ϱ-Hilfer fractional derivative of order α and type β of a function f is defined as:

$${}^{H}D_{a^{+}}^{\alpha ,\beta ;\varrho }f\left(\ell \right)=I_{a^{+}}^{\beta (n-\alpha );\varrho }{\left({\displaystyle \frac{1}{{\varrho }^{\prime }\left(\ell \right)}}{\displaystyle \frac{d}{d\ell }}\right)}^n{I}_{a^{+}}^{(1-\beta )(n-\alpha );\varrho }f\left(\ell \right).$$

(6)

In this study, where $0<\alpha <1$, we have $n=1$, and the weighted order is given by $\gamma =\alpha +\beta (1-\alpha )$.

Lemma 1

([4]). If $\alpha ,\beta >0$, then the semigroup property holds:

$$I_{a^{+}}^{\alpha ;\varrho }{I}_{a^{+}}^{\beta ;\varrho }f\left(\ell \right)=I_{a^{+}}^{\alpha +\beta ;\varrho }f\left(\ell \right).$$

(7)

Furthermore, if $f\left(\ell \right)={(\varrho \left(\ell \right)-\varrho \left(a\right))}^{\delta -1}$, then:

$$I_{a^{+}}^{\alpha ;\varrho }{(\varrho \left(\ell \right)-\varrho \left(a\right))}^{\delta -1}={\displaystyle \frac{\Gamma \left(\delta \right)}{\Gamma (\delta +\alpha )}}{(\varrho \left(\ell \right)-\varrho \left(a\right))}^{\delta +\alpha -1}.$$

(8)

To handle the impulsive effects and weighted singularities, we define the following weighted spaces.

Definition 3

([4]). The weighted space $C_{1-\gamma ;\varrho }(J,\mathbb{R})$ is defined as:

$$C_{1-\gamma ;\varrho }(J,\mathbb{R})=\{f:(0,T]\to \mathbb{R}\mid {(\varrho \left(\ell \right)-\varrho \left(0\right))}^{1-\gamma }f\left(\ell \right)\in C(J,\mathbb{R})\},$$

(9)

with the norm

$${\parallel f\parallel }_{C_{1-\gamma ;\varrho }}=\underset{\ell \in J}{sup}\left|{(\varrho \left(\ell \right)-\varrho \left(0\right))}^{1-\gamma }f\left(\ell \right)\right|.$$

Definition 4

([16]). The space of impulsive weighted functions $P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})$ is defined as:

$$P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})=\left\{f:J\to \mathbb{R}\mid \begin{array}{c}f\in C_{1-\gamma ;\varrho }(({\ell }_k,{\ell }_{k+1}],\mathbb{R}), and there exist f\left({\ell }_k^{-}\right),f\left({\ell }_k^{+}\right)\hfill \\ such that f\left({\ell }_k^{-}\right)=f\left({\ell }_k\right) for k=1,\dots ,m.\hfill \end{array}\right\}$$

The product space $E=P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})\times P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})$ is a Banach space with the norm

$${\parallel (x,y)\parallel }_E={\parallel x\parallel }_{P{C}_{1-\gamma ;\varrho }}+{\parallel y\parallel }_{P{C}_{1-\gamma ;\varrho }}.$$

Lemma 2

($\varrho$-Gronwall Inequality [4]). Let $u,v$ be non-negative continuous functions on J, and $\varrho \in C^1(J,\mathbb{R})$ be increasing. If

$$u\left(\ell \right)\le v\left(\ell \right)+K{\int }_a^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}u\left(s\right)ds,$$

(10)

then

$$u\left(\ell \right)\le v\left(\ell \right)+{\int }_a^{\ell }\sum _{n=1}^{\infty }{\displaystyle \frac{{(K\Gamma \left(\alpha \right))}^n}{\Gamma \left(n\alpha \right)}}{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{n\alpha -1}v\left(s\right)ds.$$

(11)

If $v\left(\ell \right)$ is a non-decreasing constant M, then

$$u\left(\ell \right)\le M{\mathbb{E}}_{\alpha }(K\Gamma \left(\alpha \right){(\varrho \left(\ell \right)-\varrho \left(a\right))}^{\alpha }),$$

where ${\mathbb{E}}_{\alpha }$ is the Mittag–Leffler function.

Theorem 1

(Banach Fixed Point Theorem [42]). Let $(X,d)$ be a non-empty complete metric space with $T:X\to X$ being a contraction mapping. Then T has a unique fixed point in X.

Theorem 2

(Krasnoselskii Fixed Point Theorem [42]). Let M be a non-empty, closed, convex, and bounded subset of a Banach space X. Let $A,B$ be two operators such that:

1. 

$Ax+By\in M$ for all $x,y\in M$;

2. 

A is a contraction;

3. 

B is continuous and compact.

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

Lemma 3

([4]). According to [4] (Theorem 3.4), the ϱ-Hilfer inversion formula states that for a function $h\in C_{1-\gamma ;\varrho }(J,\mathbb{R})$, the solution to ${}^{H}D_{a^{+}}^{\alpha ,\beta ;\varrho }h(\ell )=g(\ell )$ is given by:

$$h(\ell )={\displaystyle \frac{{(\varrho (\ell )-\varrho \left(a\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_{a^{+}}^{1-\gamma ;\varrho }h\left(a^{+}\right)+I_{a^{+}}^{\alpha ;\varrho }g(\ell ).$$

(12)

3. Main Results

3.1. Equivalent Integral System

Lemma 4

(Propagation of the Weighted Constant). Let $x\in P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})$ satisfy for $\ell \in ({\ell }_{k-1},{\ell }_k]$

$${}^H\mathcal{D}_{{\ell }_{k-1}^{+}}^{\alpha ,\beta ;\varrho }\left[x(\ell )-{\mathcal{N}}_1(\ell ,x_{\ell })\right]=F_1(\ell ),$$

(13)

with the impulsive condition

$$\Delta x\left({\ell }_k\right)=x\left({\ell }_k^{+}\right)-x\left({\ell }_k^{-}\right)=I_k\left(x\left({\ell }_k^{-}\right)\right).$$

(14)

Define the weighted constants

$$C_{k-1}^x:={\mathcal{I}}_{{\ell }_{k-1}^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_{k-1}^{+}\right)-{\mathcal{N}}_1({\ell }_{k-1},x_{{\ell }_{k-1}})\right],C_k^x:={\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_k^{+}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})\right].$$

(15)

Then the following recurrence relation holds:

$$C_k^x=C_{k-1}^x+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_k\right)\right]+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }{I}_k\left(x\left({\ell }_k^{-}\right)\right).$$

(16)

Proof. 

Applying ${\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }$ to (13) and using Lemma 3 for $\ell \in ({\ell }_{k-1},{\ell }_k]$, we have

$$x(\ell )-{\mathcal{N}}_1(\ell ,x_{\ell })={\displaystyle \frac{{\left[\varrho (\ell )-\varrho \left({\ell }_{k-1}\right)\right]}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{C}_{k-1}^x+{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1(\ell ),$$

(17)

where

$$C_{k-1}^x={\mathcal{I}}_{{\ell }_{k-1}^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_{k-1}^{+}\right)-{\mathcal{N}}_1({\ell }_{k-1},x_{{\ell }_{k-1}})\right].$$

Taking the left limit $\ell \to {\ell }_k^{-}$ in (17) yields

$$x\left({\ell }_k^{-}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})={\displaystyle \frac{{\left[\varrho \left({\ell }_k\right)-\varrho \left({\ell }_{k-1}\right)\right]}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{C}_{k-1}^x+{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_k\right).$$

(18)

From (14), $x\left({\ell }_k^{+}\right)=x\left({\ell }_k^{-}\right)+I_k\left(x\left({\ell }_k^{-}\right)\right)$, hence

$$x\left({\ell }_k^{+}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})=\left[x\left({\ell }_k^{-}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})\right]+I_k\left(x\left({\ell }_k^{-}\right)\right).$$

(19)

Applying ${\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }$ to (19):

$$C_k^x={\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_k^{-}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})\right]+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }{I}_k\left(x\left({\ell }_k^{-}\right)\right).$$

(20)

Substituting (18) into the first term of (20):

$$\begin{array}{cc}\hfill & {\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_k^{-}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})\right]\hfill \\ \hfill & ={\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[{\displaystyle \frac{{\left[\varrho \left({\ell }_k\right)-\varrho \left({\ell }_{k-1}\right)\right]}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{C}_{k-1}^x\right]+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_k\right)\right].\hfill \end{array}$$

(21)

This implies that

$${\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_k^{-}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})\right]=C_{k-1}^x+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_k\right)\right].$$

(22)

Substituting (22) into the first term of (2.8) yields:

$$C_k^x=C_{k-1}^x+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_k\right)\right]+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }{I}_k\left(x\left({\ell }_k^{-}\right)\right).$$

   □

Corollary 1

(Iterated Form). Iterating the recurrence (16) for $k=1,\dots ,m$ gives

$$C_k^x=C_0^x+\sum _{i=1}^k{\mathcal{I}}_{{\ell }_i^{+}}^{1-\gamma ;\varrho }{I}_i\left(x\left({\ell }_i^{-}\right)\right)+\sum _{i=1}^k{\mathcal{I}}_{{\ell }_i^{+}}^{1-\gamma ;\varrho }{\mathcal{I}}_{{\ell }_{i-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_i\right).$$

(23)

Theorem 3

(Equivalent Integral System). Let $\alpha \in (0,1)$, $\beta \in [0,1]$, $\gamma =\alpha +\beta (1-\alpha )$, and $J=[0,T]$. Let $\varrho \in C^1(J,\mathbb{R})$ with ${\varrho }^{\prime }(\ell )>0$ for $\ell \in J$. Define

$$\begin{array}{cc}\hfill {\Lambda }_x& :=1-{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}\sum _{j=1}^p{c}_j{(\varrho \left({\xi }_j\right)-\varrho \left(0\right))}^{\gamma -1},\hfill \\ \hfill {\Lambda }_y& :=1-{\displaystyle \frac{1}{\Gamma \left(\gamma \right)}}\sum _{j=1}^p{d}_j{(\varrho \left({\xi }_j\right)-\varrho \left(0\right))}^{\gamma -1},\hfill \end{array}$$

and assume ${\Lambda }_x\ne 0$, ${\Lambda }_y\ne 0$. Then $(x,y)\in P{C}_{1-\gamma ;\varrho }{(J,\mathbb{R})}^2$ satisfies the system (1)–(4) with

$$\begin{array}{cc}\hfill F_1(\ell )& :=f_1\left(\ell ,x(\ell ),y(\ell ),x\left(\rho (\ell )\right),y\left(\sigma (\ell )\right),{\int }_0^{\ell }{\mathcal{K}}_1(\ell ,s,x\left(s\right),y\left(s\right))ds\right),\hfill \end{array}$$

(24)

$$\begin{array}{cc}\hfill F_2(\ell )& :=f_2\left(\ell ,x(\ell ),y(\ell ),x\left(\rho (\ell )\right),y\left(\sigma (\ell )\right),{\int }_0^{\ell }{\mathcal{K}}_2(\ell ,s,x\left(s\right),y\left(s\right))ds\right),\hfill \end{array}$$

(25)

if and only if for $\ell \in ({\ell }_k,{\ell }_{k+1}]$, $k=0,\dots ,m$,

$$x(\ell )=\left\{\begin{array}{c}\varphi (\ell ),\ell \in [-v,0]\\ {\displaystyle {\mathcal{N}}_1(\ell ,x_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_x}}\sum _{j=1}^p{c}_j[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)}\\ {\displaystyle +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds]+\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)}\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds,\ell \in ({\ell }_k,{\ell }_{k+1}],k=0,\dots ,m,\end{array}\right.$$

(26)

and

$$y(\ell )=\left\{\begin{array}{c}\phi (\ell ),\ell \in [-v,0]\\ {\displaystyle {\mathcal{N}}_2(\ell ,y_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_y}}\sum _{j=1}^p{d}_j[{\mathcal{N}}_2({\xi }_j,y_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)}\\ {\displaystyle +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}{F}_2\left(s\right)ds]+\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)}\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}{F}_2\left(s\right)ds,\ell \in ({\ell }_k,{\ell }_{k+1}],k=0,\dots ,m,\end{array}\right.$$

(27)

where $n\left({\xi }_j\right)=max\{i:{\ell }_i<{\xi }_j\}$.

Proof. 

Let $(x,y)$ satisfy (1)–(4). For $\ell \in ({\ell }_k,{\ell }_{k+1}]$, applying ${\mathcal{I}}_{{\ell }_k^{+}}^{\alpha ;\varrho }$ to the differential equation and using Lemma 3 yields

$$x(\ell )={\mathcal{N}}_1(\ell ,x_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_k\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{C}_k^x+{\mathcal{I}}_{{\ell }_k^{+}}^{\alpha ;\varrho }{F}_1(\ell ),$$

(28)

with

$$C_k^x:={\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_k^{+}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})\right].$$

For $k\ge 1$, taking the left limit $\ell \to {\ell }_k^{-}$ in (28) gives

$$x\left({\ell }_k^{-}\right)={\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})+{\displaystyle \frac{{(\varrho \left({\ell }_k\right)-\varrho \left({\ell }_{k-1}\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{C}_{k-1}^x+{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_k\right).$$

(29)

From the impulsive condition, $x\left({\ell }_k^{+}\right)=x\left({\ell }_k^{-}\right)+I_k\left(x\left({\ell }_k^{-}\right)\right)$. Substituting into the definition of $C_k^x$:

$$C_k^x={\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[x\left({\ell }_k^{-}\right)-{\mathcal{N}}_1({\ell }_k,x_{{\ell }_k})\right]+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }{I}_k\left(x\left({\ell }_k^{-}\right)\right).$$

Substituting the expression for $x\left({\ell }_k^{-}\right)$ from (29) into the definition of $C_k^x$ and applying the properties of weighted fractional integrals as established in Lemma 4, we obtain the following recurrence relation for the weighted constants

$$C_k^x=C_{k-1}^x+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }\left[{\mathcal{I}}_{{\ell }_{k-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_k\right)\right]+{\mathcal{I}}_{{\ell }_k^{+}}^{1-\gamma ;\varrho }{I}_k\left(x\left({\ell }_k^{-}\right)\right).$$

(30)

Iterating (30) gives

$$C_k^x=C_0^x+\sum _{i=1}^k{\mathcal{I}}_{{\ell }_i^{+}}^{1-\gamma ;\varrho }{I}_i\left(x\left({\ell }_i^{-}\right)\right)+\sum _{i=1}^k{\mathcal{I}}_{{\ell }_i^{+}}^{1-\gamma ;\varrho }{\mathcal{I}}_{{\ell }_{i-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_i\right).$$

(31)

Evaluating (28) for $k=0$ at $\ell ={\xi }_j$:

$$x\left({\xi }_j\right)={\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{C}_0^x+{\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right).$$

(32)

From the non-local condition, $C_0^x={\sum }_{j=1}^p{c}_{j}x\left({\xi }_j\right)$. Substituting (32):

$$C_0^x=\sum _{j=1}^p{c}_j\left[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{C}_0^x+{\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right)\right].$$

Solving for $C_0^x$:

$$C_0^x={\displaystyle \frac{1}{{\Lambda }_x}}\sum _{j=1}^p{c}_j\left[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+{\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right)\right].$$

(33)

Substituting (33) and (31) into (28):

$$\begin{array}{cc}\hfill x(\ell )& ={\mathcal{N}}_1(\ell ,x_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_k\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}[C_0^x+\sum _{i=1}^k{\mathcal{I}}_{{\ell }_i^{+}}^{1-\gamma ;\varrho }{I}_i\left(x\left({\ell }_i^{-}\right)\right)\hfill \\ \hfill & +\sum _{i=1}^k{\mathcal{I}}_{{\ell }_i^{+}}^{1-\gamma ;\varrho }{\mathcal{I}}_{{\ell }_{i-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_i\right)]+{\mathcal{I}}_{{\ell }_k^{+}}^{\alpha ;\varrho }{F}_1(\ell ).\hfill \end{array}$$

(34)

Decompose ${\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right)$ in (33) as

$${\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right)=\sum _{\tau =0}^{n\left({\xi }_j\right)-1}{\mathcal{I}}_{{\ell }_{\tau }^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_{\tau +1}\right)+{\mathcal{I}}_{{\ell }_{n\left({\xi }_j\right)}^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right).$$

(35)

Substituting (35) into (33):

$${\displaystyle \frac{1}{{\Lambda }_x}}\sum _{j=1}^p{c}_j\left[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+\sum _{\tau =0}^{n\left({\xi }_j\right)-1}{\mathcal{I}}_{{\ell }_{\tau }^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_{\tau +1}\right)+{\mathcal{I}}_{{\ell }_{n\left({\xi }_j\right)}^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right)\right].$$

Combine the forcing integrals in (34) and (35). Observe that

$${\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_k\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\sum _{i=1}^k{\mathcal{I}}_{{\ell }_{i-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_i\right)+{\mathcal{I}}_{{\ell }_k^{+}}^{\alpha ;\varrho }{F}_1(\ell )={\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1(\ell )-\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{\mathcal{I}}_{{\ell }_{i-1}^{+}}^{\alpha ;\varrho }{F}_1\left({\ell }_i\right).$$

(36)

Assembling (34)–(36) yields (26).

Conversely, assume $(x,y)$ satisfies (26), (27). Let $\ell \in ({\ell }_k,{\ell }_{k+1}]$. Applying ${}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }$ to (26), from the properties of the $\varrho$-Hilfer derivative [4]:

$$\begin{array}{cc}\hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}& =0,\hfill \\ \hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}& =0(i\le k),\hfill \\ \hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }{\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1(\ell )& =F_1(\ell ).\hfill \end{array}$$

Thus

$${}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[x(\ell )-{\mathcal{N}}_1(\ell ,x_{\ell })\right]=F_1(\ell ),$$

which is the differential equation. Applying ${\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }$ to (26) and evaluating at $\ell =0^{+}$, we obtain

$${\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }\left[x\left(0^{+}\right)-{\mathcal{N}}_1(0,x_0)\right]={\displaystyle \frac{1}{{\Lambda }_x}}\sum _{j=1}^p{c}_j\left[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+{\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{F}_1\left({\xi }_j\right)\right].$$

Using the definition of ${\Lambda }_x$ and (32), this simplifies to

$${\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }\left[x\left(0^{+}\right)-{\mathcal{N}}_1(0,x_0)\right]=\sum _{j=1}^p{c}_{j}x\left({\xi }_j\right),$$

which is the non-local condition. For $\ell \in [-v,0]$, (26) reduces to $x(\ell )=\varphi (\ell )$ by definition of the history terms. Similarly $y(\ell )=\phi (\ell )$. Thus $(x,y)$ satisfies all equations.   □

Remark 1.

Conditions ${\Lambda }_x\ne 0$ and ${\Lambda }_y\ne 0$ are essential for the unique determination of the initial weighted constants $C_0^x$ and $C_0^y$ from the non-local conditions (2). Should these determinants disappear, the non-local conditions would either become inconsistent or produce multiple solutions, indicating a resonance phenomenon within the system. Physically, ${\Lambda }_x=0$ signifies that the non-local condition lacks adequate information to uniquely ascertain the initial state.

3.2. Existence and Uniqueness of Solutions

Define the product Banach space $E=P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})\times P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})$ with norm

$${∥\left(x,y\right)∥}_E={∥x∥}_{P{C}_{1-\gamma ;\varrho }}+{∥y∥}_{P{C}_{1-\gamma ;\varrho }},$$

where

$$\begin{array}{ccc}\hfill {∥x∥}_{P{C}_{1-\gamma ;\varrho }}& =& \underset{\ell \in J}{sup}\left|{(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }x(\ell )\right|,\hfill \\ \hfill {∥y∥}_{P{C}_{1-\gamma ;\varrho }}& =& \underset{\ell \in J}{sup}\left|{(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }y(\ell )\right|.\hfill \end{array}$$

We impose the following assumptions:

H1. 

There exist constants $L_{{\mathcal{N}}_1},L_{{\mathcal{N}}_2},L_{f_1},L_{f_2},L_{{\mathcal{K}}_1},L_{{\mathcal{K}}_2}>0$ such that for all $\ell \in J$, $u,v,\overline{u},\overline{v}\in \mathbb{R}$, and $\eta ,\zeta \in C\left(\right[-v,0],\mathbb{R})$:

$$\begin{array}{cc}\hfill \left|{\mathcal{N}}_1(\ell ,\eta )-{\mathcal{N}}_1(\ell ,\zeta )\right|& \le L_{{\mathcal{N}}_1}{∥\eta -\zeta ∥}_{C\left(\right[-v,0],\mathbb{R})},\hfill \\ \hfill \left|{\mathcal{N}}_2(\ell ,\eta )-{\mathcal{N}}_2(\ell ,\zeta )\right|& \le L_{{\mathcal{N}}_2}{∥\eta -\zeta ∥}_{C\left(\right[-v,0],\mathbb{R})},\hfill \\ \hfill \left|f_1(\ell ,u,v,\overline{u},\overline{v},w)-f_1(\ell ,u^{\prime },v^{\prime },{\overline{u}}^{\prime },{\overline{v}}^{\prime },w^{\prime })\right|& \le L_{f_1}\left(|u-u^{\prime }|+|v-v^{\prime }|+|\overline{u}-{\overline{u}}^{\prime }|+|\overline{v}-{\overline{v}}^{\prime }|+|w-w^{\prime }|\right),\hfill \\ \hfill \left|f_2(\ell ,u,v,\overline{u},\overline{v},w)-f_2(\ell ,u^{\prime },v^{\prime },{\overline{u}}^{\prime },{\overline{v}}^{\prime },w^{\prime })\right|& \le L_{f_2}\left(|u-u^{\prime }|+|v-v^{\prime }|+|\overline{u}-{\overline{u}}^{\prime }|+|\overline{v}-{\overline{v}}^{\prime }|+|w-w^{\prime }|\right),\hfill \\ \hfill \left|{\mathcal{K}}_1(\ell ,s,u,v)-{\mathcal{K}}_1(\ell ,s,u^{\prime },v^{\prime })\right|& \le L_{{\mathcal{K}}_1}\left(|u-u^{\prime }|+|v-v^{\prime }|\right),\hfill \\ \hfill \left|{\mathcal{K}}_2(\ell ,s,u,v)-{\mathcal{K}}_2(\ell ,s,u^{\prime },v^{\prime })\right|& \le L_{{\mathcal{K}}_2}\left(|u-u^{\prime }|+|v-v^{\prime }|\right).\hfill \end{array}$$

H2. 

There exist constants $L_I,L_J>0$ such that for all $u,v\in \mathbb{R}$ and $k=1,\dots ,m$:

$$\left|I_k\left(u\right)-I_k\left(v\right)\right|\le L_I|u-v|,\left|J_k\left(u\right)-J_k\left(v\right)\right|\le L_J|u-v|.$$

H3. 

There exist constants $M_{{\mathcal{N}}_1},M_{{\mathcal{N}}_2},M_{f_1},M_{f_2},M_{{\mathcal{K}}_1},M_{{\mathcal{K}}_2},M_I,M_J\ge 0$ such that:

$$\begin{array}{cc}\hfill & \left|{\mathcal{N}}_1(\ell ,0)\right|\le M_{{\mathcal{N}}_1},\left|{\mathcal{N}}_2(\ell ,0)\right|\le M_{{\mathcal{N}}_2},\hfill \\ \hfill & \left|f_1(\ell ,0,0,0,0,0)\right|\le M_{f_1},\left|f_2(\ell ,0,0,0,0,0)\right|\le M_{f_2},\hfill \\ \hfill & \left|{\mathcal{K}}_1(\ell ,s,0,0)\right|\le M_{{\mathcal{K}}_1},\left|{\mathcal{K}}_2(\ell ,s,0,0)\right|\le M_{{\mathcal{K}}_2},\hfill \\ \hfill & \left|I_k\left(0\right)\right|\le M_I,\left|J_k\left(0\right)\right|\le M_J.\hfill \end{array}$$

H4. 

The delay functions $\rho ,\sigma :J\to J$ are continuous and satisfy $0\le \rho (\ell ),\sigma (\ell )\le \ell$.

Define the operator $\mathcal{T}=({\mathcal{T}}_1,{\mathcal{T}}_2):E\to E$ by the right-hand sides of (26) and (27):

$${\mathcal{T}}_1(x,y)(\ell )=\left\{\begin{array}{c}\varphi (\ell ),\ell \in [-v,0]\\ {\displaystyle {\mathcal{N}}_1(\ell ,x_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_x}}\sum _{j=1}^p{c}_j[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)}\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds]+{\sum }_{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds,\ell \in ({\ell }_k,{\ell }_{k+1}],k=0,\dots ,m,\end{array}\right.$$

(37)

and

$${\mathcal{T}}_2(x,y)(\ell )=\left\{\begin{array}{c}\phi (\ell ),\ell \in [-v,0]\\ {\displaystyle {\mathcal{N}}_2(\ell ,y_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_y}}\sum _{j=1}^p{d}_j[{\mathcal{N}}_2({\xi }_j,y_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)}\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}{F}_2\left(s\right)ds]+{\sum }_{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)\\ +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}{F}_2\left(s\right)ds,\ell \in ({\ell }_k,{\ell }_{k+1}],k=0,\dots ,m.\end{array}\right.$$

(38)

A fixed point of $\mathcal{T}$ is a solution of the integral system, hence of the original problem.

Lemma 5

(Estimates). Under Assumptions (H1)–(H4), for any $(x,y),(\overline{x},\overline{y})\in E$ and $\ell \in J$, the following estimates hold:

$$\begin{array}{cc}\hfill {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|{\mathcal{N}}_i(\ell ,x_{\ell })-{\mathcal{N}}_i(\ell ,{\overline{x}}_{\ell })\right|& \le L_{{\mathcal{N}}_i}{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }},\hfill \\ \hfill {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|I_k\left(x\left({\ell }_k^{-}\right)\right)-I_k\left(\overline{x}\left({\ell }_k^{-}\right)\right)\right|& \le L_I{(\varrho \left({\ell }_k\right)-\varrho \left(0\right))}^{\gamma -1}{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }},\hfill \\ \hfill {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|J_k\left(y\left({\ell }_k^{-}\right)\right)-J_k\left(\overline{y}\left({\ell }_k^{-}\right)\right)\right|& \le L_J{(\varrho \left({\ell }_k\right)-\varrho \left(0\right))}^{\gamma -1}{∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }},\hfill \\ \hfill {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|F_i(\ell )-{\overline{F}}_i(\ell )\right|& \le L_{f_i}\left(1+L_{{\mathcal{K}}_i}\ell \right)\left({∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+{∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }}\right),\hfill \end{array}$$

where ${\overline{F}}_1,{\overline{F}}_2$ correspond to $(\overline{x},\overline{y})$.

Theorem 4

(Existence and Uniqueness via Banach Contraction). Let Assumptions (H1)–(H4) hold. Define the constants:

$$\begin{array}{cc}\hfill A_1& :=L_{{\mathcal{N}}_1}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_x|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|c_j\right|\left(L_{{\mathcal{N}}_1}+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\right)+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill A_2& :={\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_x|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|c_j\right|{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T)+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T),\hfill \end{array}$$

$$\begin{array}{cc}\hfill B_1& :=L_{{\mathcal{N}}_2}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_y|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|d_j\right|\left(L_{{\mathcal{N}}_2}+{\displaystyle \frac{m{L}_J{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\right)+{\displaystyle \frac{m{L}_J{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}},\hfill \end{array}$$

$$\begin{array}{cc}\hfill B_2& :={\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_y|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|d_j\right|{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_2}(2+L_{{\mathcal{K}}_2}T)+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_2}(2+L_{{\mathcal{K}}_2}T).\hfill \end{array}$$

If the condition

$$Q:=max\left\{A_1+A_2,B_1+B_2\right\}<1$$

(39)

is satisfied, then $\mathcal{T}$ is a contraction on E. Consequently, system (26) and (27) has a unique fixed point in E, which is the unique solution of the impulsive fractional system.

Proof. 

Let $(x,y),(\overline{x},\overline{y})\in E$. For $\ell \in ({\ell }_k,{\ell }_{k+1}]$, consider the difference ${\mathcal{T}}_1(x,y)(\ell )-{\mathcal{T}}_1(\overline{x},\overline{y})(\ell )$. Multiply by ${(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }$ and estimate term by term using Lemma 5. For neutral term:

$${(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|{\mathcal{N}}_1(\ell ,x_{\ell })-{\mathcal{N}}_1(\ell ,{\overline{x}}_{\ell })\right|\le L_{{\mathcal{N}}_1}{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}.$$

The non-local term is bounded by

$$\begin{array}{ccc}& & {\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_x|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|c_j\right|[L_{{\mathcal{N}}_1}{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}\hfill \\ & & +{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T)\left({∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+{∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }}\right)].\hfill \end{array}$$

Impulse sum term:

$$\begin{array}{ccc}& & {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\left(I_i\left(x\left({\ell }_i^{-}\right)\right)-I_i\left(\overline{x}\left({\ell }_i^{-}\right)\right)\right)\right|\hfill \\ & \le & {\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}.\hfill \end{array}$$

Volterra integral term:

$$\begin{array}{ccc}& & {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|{\int }_0^{\ell }{\displaystyle \frac{{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}}{\Gamma \left(\alpha \right)}}\left(F_1\left(s\right)-{\overline{F}}_1\left(s\right)\right)ds\right|\hfill \\ & \le & {\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T)\left({∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+{∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }}\right).\hfill \end{array}$$

Combining these estimates yields:

$$\begin{array}{cc}\hfill & {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|{\mathcal{T}}_1(x,y)(\ell )-{\mathcal{T}}_1(\overline{x},\overline{y})(\ell )\right|\hfill \\ \hfill & \le \left(A_1{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+A_2\left({∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+{∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }}\right)\right)\hfill \\ \hfill & =(A_1+A_2){∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+A_2{∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }}.\hfill \end{array}$$

Taking the supremum over $\ell \in J$ gives

$${∥{\mathcal{T}}_1(x,y)-{\mathcal{T}}_1(\overline{x},\overline{y})∥}_{P{C}_{1-\gamma ;\varrho }}\le (A_1+A_2){∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+A_2{∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }}.$$

Analogously, we obtain

$${∥{\mathcal{T}}_2(x,y)-{\mathcal{T}}_2(\overline{x},\overline{y})∥}_{P{C}_{1-\gamma ;\varrho }}\le B_2{∥x-\overline{x}∥}_{P{C}_{1-\gamma ;\varrho }}+(B_1+B_2){∥y-\overline{y}∥}_{P{C}_{1-\gamma ;\varrho }}.$$

Therefore,

$${∥\mathcal{T}(x,y)-\mathcal{T}(\overline{x},\overline{y})∥}_E\le Q{∥(x,y)-(\overline{x},\overline{y})∥}_E.$$

Since $Q<1$ by (39), $\mathcal{T}$ is a contraction on the complete metric space E. By the Banach Fixed Point Theorem, $\mathcal{T}$ has a unique fixed point, which is the unique solution of the integral system, hence of the original problem.   □

Remark 2.

The contraction condition $Q=max\left\{A_1+A_2,B_1+B_2\right\}<1$ in (39) is derived from the Lipschitz constants and system parameters. Its satisfaction can be interpreted practically as requiring the system’s nonlinearities ($L_f,L_{\mathcal{N}}$), impulsive strengths ($L_I$), and the length of the time interval to be sufficiently small relative to the dissipative or stabilizing effects inherent in the fractional operator. For the classical case where $\varrho (\ell )=\ell$, it can be more easily verify this condition by choosing a sufficiently short time horizon or ensuring weak nonlinearities. In applications, this condition guides the selection of system parameters to guarantee unique solutions.

Theorem 5

(Existence via Krasnoselskii’s Theorem). Let Assumptions (H1)–(H4) hold. Define the constants:

$$\begin{array}{cc}\hfill {\Omega }_1& :=L_{{\mathcal{N}}_1}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_x|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|c_j\right|\left(L_{{\mathcal{N}}_1}+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T)\right)+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Omega }_2& :={\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_x|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|c_j\right|{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T)+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(2+L_{{\mathcal{K}}_1}T),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Omega }_3& :=M_{{\mathcal{N}}_1}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_x|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|c_j\right|\left(M_{{\mathcal{N}}_1}+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{M}_I\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{M}_{f_1}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}{M}_{{\mathcal{K}}_1}T\right),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Xi }_1& :=L_{{\mathcal{N}}_2}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_y|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|d_j\right|\left(L_{{\mathcal{N}}_2}+{\displaystyle \frac{m{L}_J{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_2}(2+L_{{\mathcal{K}}_2}T)\right)+{\displaystyle \frac{m{L}_J{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_2}(2+L_{{\mathcal{K}}_2}T),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Xi }_2& :={\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_y|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|d_j\right|{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_2}(2+L_{{\mathcal{K}}_2}T)+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_2}(2+L_{{\mathcal{K}}_2}T),\hfill \end{array}$$

$$\begin{array}{cc}\hfill {\Xi }_3& :=M_{{\mathcal{N}}_2}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{|{\Lambda }_y|\Gamma \left(\gamma \right)}}\sum _{j=1}^p\left|d_j\right|\left(M_{{\mathcal{N}}_2}+{\displaystyle \frac{m{L}_J{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{M}_J\right.\hfill \\ \hfill & \left.+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{M}_{f_2}+{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_2}{M}_{{\mathcal{K}}_2}T\right).\hfill \end{array}$$

Suppose that

$$max\left\{{\Omega }_1+{\Xi }_2,{\Xi }_1+{\Omega }_2\right\}<1.$$

Then there exists at least one solution $(x,y)\in E$ to the system (26) and (27).

Proof. 

Decompose the operator $\mathcal{T}$ as $\mathcal{T}=\mathcal{A}+\mathcal{B}$, where $\mathcal{A}=({\mathcal{A}}_1,{\mathcal{A}}_2)$ and $\mathcal{B}=({\mathcal{B}}_1,{\mathcal{B}}_2)$ are defined by:

$$\begin{array}{cc}\hfill {\mathcal{A}}_1(x,y)(\ell )& :={\mathcal{N}}_1(\ell ,x_{\ell })+\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right),\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill {\mathcal{B}}_1(x,y)(\ell )& :={\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_x}}\sum _{j=1}^p{c}_j[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds]+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds.\hfill \end{array}$$

Analogous definitions hold for ${\mathcal{A}}_2$ and ${\mathcal{B}}_2$. Define the closed ball $B_r=\{(x,y)\in E:{∥(x,y)∥}_E\le r\}$ with radius

$$r\ge {\displaystyle \frac{{\Omega }_3+{\Xi }_3}{1-max\left\{{\Omega }_1+{\varrho }_2,{\Xi }_1+{\Omega }_2\right\}}}.$$

Step 1: $\mathcal{A}$ is a contraction. Since the operator $\mathcal{T}$ is contraction (as Theorem 4), and $\mathcal{T}=\mathcal{A}+\mathcal{B},$ then the operator A is contraction. Also, for $(x,y),(\overline{x},\overline{y})\in B_r$, using Lemma 5 and the definitions of ${\Omega }_1,{\Xi }_1$, one can show directly that $\mathcal{A}$ satisfies a contraction condition with constant $max\{{\Omega }_1,{\varrho }_1\}<1$ (since ${\Omega }_1+{\Xi }_2<1$ and ${\Xi }_1+{\Omega }_2<1$ imply ${\Omega }_1<1$ and ${\Xi }_1<1$).

Step 2: $\mathcal{B}$ is continuous and compact. Using the growth conditions (H3), one can show that $\mathcal{B}$ maps bounded sets into bounded sets in E. To prove compactness, we verify that $\mathcal{B}$ is equicontinuous on $B_r$. For ${\ell }_1,{\ell }_2\in J$ with ${\ell }_1<{\ell }_2$, and for any $(x,y)\in B_r$,

$$\begin{array}{cc}\hfill & \left|{(\varrho \left({\ell }_2\right)-\varrho \left(0\right))}^{1-\gamma }{\mathcal{B}}_1(x,y)\left({\ell }_2\right)-{(\varrho \left({\ell }_1\right)-\varrho \left(0\right))}^{1-\gamma }{\mathcal{B}}_1(x,y)\left({\ell }_1\right)\right|\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{{\ell }_2}{\varrho }^{\prime }\left(s\right){\displaystyle \frac{{(\varrho \left({\ell }_2\right)-\varrho \left(0\right))}^{1-\gamma }}{{(\varrho \left({\ell }_2\right)-\varrho \left(s\right))}^{1-\alpha }}}\left|F_1\left(s\right)\right|ds\hfill \\ \hfill & -{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{{\ell }_1}{\varrho }^{\prime }\left(s\right){\displaystyle \frac{{(\varrho \left({\ell }_1\right)-\varrho \left(0\right))}^{1-\gamma }}{{(\varrho \left({\ell }_1\right)-\varrho \left(s\right))}^{1-\alpha }}}\left|F_1\left(s\right)\right|ds\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{{\ell }_1}{\varrho }^{\prime }\left(s\right)\left({\displaystyle \frac{{(\varrho \left({\ell }_2\right)-\varrho \left(0\right))}^{1-\gamma }}{{(\varrho \left({\ell }_2\right)-\varrho \left(s\right))}^{1-\alpha }}}-{\displaystyle \frac{{(\varrho \left({\ell }_1\right)-\varrho \left(0\right))}^{1-\gamma }}{{(\varrho \left({\ell }_1\right)-\varrho \left(s\right))}^{1-\alpha }}}\right)\left|F_1\left(s\right)\right|ds\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_1}^{{\ell }_2}{\varrho }^{\prime }\left(s\right){\displaystyle \frac{{(\varrho \left({\ell }_2\right)-\varrho \left(0\right))}^{1-\gamma }}{{(\varrho \left({\ell }_2\right)-\varrho \left(s\right))}^{1-\alpha }}}\left|F_1\left(s\right)\right|ds\hfill \\ \hfill & \le {\displaystyle \frac{{∥F_1∥}_{P{C}_{1-\gamma ;\varrho }}}{\Gamma (\alpha +1)}}\left[{(\varrho \left({\ell }_2\right)-\varrho \left(0\right))}^{\alpha }-{(\varrho \left({\ell }_1\right)-\varrho \left(0\right))}^{\alpha }+{(\varrho \left({\ell }_2\right)-\varrho \left({\ell }_1\right))}^{\alpha }\right].\hfill \\ \hfill & \to 0,as{\ell }_2\to {\ell }_1.\hfill \end{array}$$

Hence $\mathcal{B}$ is equicontinuous. By the Arzelà-Ascoli theorem for weighted spaces (see [4]), $\mathcal{B}$ is compact.

Step 3: $\mathcal{A}(x,y)+\mathcal{B}(x,y)\in B_r$ for all $(x,y)\in B_r$. Using (H3) and following estimates similar to those in Theorem 4, we obtain for $\ell \in J$:

$$\begin{array}{cc}\hfill {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|{\mathcal{T}}_1(x,y)(\ell )\right|& \le {\Omega }_1{\parallel x\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Omega }_2{\parallel y\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Omega }_3,\hfill \\ \hfill {(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }\left|{\mathcal{T}}_2(x,y)(\ell )\right|& \le {\varrho }_2{\parallel x\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Xi }_1{\parallel y\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Xi }_3.\hfill \end{array}$$

Taking supremam yields

$$\begin{array}{cc}\hfill \parallel {\mathcal{T}}_1{(x,y)\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}& \le {\Omega }_1{\parallel x\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Omega }_2{\parallel y\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Omega }_3,\hfill \\ \hfill \parallel {\mathcal{T}}_2{(x,y)\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}& \le {\Xi }_2{\parallel x\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Xi }_1{\parallel y\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\Xi }_3.\hfill \end{array}$$

Thus

$$\begin{array}{cc}\hfill {\parallel \mathcal{T}(x,y)\parallel }_E& =\parallel {\mathcal{T}}_1{(x,y)\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+{\parallel {\mathcal{T}}_2(x,y)\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}\hfill \\ \hfill & \le ({\Omega }_1+{\Xi }_2){\parallel x\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+({\varrho }_1+{\Omega }_2){\parallel y\parallel }_{{\mathrm{PC}}_{1-\gamma ;\varrho }}+({\Omega }_3+{\Xi }_3)\hfill \\ \hfill & \le max\left\{{\Omega }_1+{\Xi }_2,{\Xi }_1+{\Omega }_2\right\}{\parallel (x,y)\parallel }_E+({\Omega }_3+{\Xi }_3).\hfill \end{array}$$

Since ${\parallel (x,y)\parallel }_E\le r$ for $(x,y)\in B_r$, we have

$${\parallel \mathcal{T}(x,y)\parallel }_E\le max\left\{{\Omega }_1+{\Xi }_2,{\varrho }_1+{\Omega }_2\right\}r+({\Omega }_3+{\Xi }_3)\le r$$

Hence $\mathcal{T}(x,y)\in B_r$ for all $(x,y)\in B_r$. All conditions of Krasnoselskii’s fixed point theorem are satisfied on $B_r$. Therefore, $\mathcal{T}=\mathcal{A}+\mathcal{B}$ has at least one fixed point in $B_r$, which is a solution of the system.   □

3.3. Ulam–Hyers–Rassias Stability

Remark 3

(UHR vs. Lyapunov Stability). While Lyapunov stability concerns the behavior of solutions near an equilibrium point under small perturbations to initial conditions, Ulam–Hyers–Rassias stability addresses a fundamentally different question: how close an approximate solution is to an exact solution of the system. This is particularly relevant for numerical approximations and systems subject to functional perturbations, making it a more suitable tool for analyzing the robustness of the model’s solution operator itself against uncertainties in the equations. In the context of fractional differential equations with impulses and delays, UHR stability provides a natural framework for quantifying how errors propagate through the system, which is essential for reliable numerical simulations and real-world applications where exact solutions are rarely achievable.

Definition 5

(UHR Stability). Let ${\theta }_1,{\theta }_2\in C(J,{\mathbb{R}}_{+})$ be non-decreasing functions. The coupled system (26)–(27) is said to be Ulam–Hyers–Rassias stable with respect to $({\theta }_1,{\theta }_2)$ if there exist constants $K_1,K_2>0$ such that for every ${ϵ}_1,{ϵ}_2>0$ and for every pair $(\tilde{x},\tilde{y})\in E$ satisfying the inequalities

$$\begin{array}{cc}\hfill & \left|{}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[\tilde{x}(\ell )-{\mathcal{N}}_1(\ell ,{\tilde{x}}_{\ell })\right]-F_1(\tilde{x},\tilde{y})(\ell )\right|\le {ϵ}_1{\theta }_1(\ell ),\ell \in J^{\prime },\hfill \end{array}$$

(40)

$$\begin{array}{cc}\hfill & \left|{}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[\tilde{y}(\ell )-{\mathcal{N}}_2(\ell ,{\tilde{y}}_{\ell })\right]-F_2(\tilde{x},\tilde{y})(\ell )\right|\le {ϵ}_2{\theta }_2(\ell ),\ell \in J^{\prime },\hfill \end{array}$$

(41)

$$\begin{array}{cc}\hfill & \left|\Delta \tilde{x}\left({\ell }_k\right)-I_k\left(\tilde{x}\left({\ell }_k^{-}\right)\right)\right|\le {ϵ}_1{\theta }_1\left({\ell }_k\right),\left|\Delta \tilde{y}\left({\ell }_k\right)-J_k\left(\tilde{y}\left({\ell }_k^{-}\right)\right)\right|\le {ϵ}_2{\theta }_2\left({\ell }_k\right),\hfill \end{array}$$

(42)

$$\begin{array}{cc}\hfill & \left|{\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }\left[\tilde{x}\left(0^{+}\right)-{\mathcal{N}}_1(0,{\tilde{x}}_0)\right]-\sum _{j=1}^p{c}_j\tilde{x}\left({\xi }_j\right)\right|\le {ϵ}_1{\theta }_1\left(0\right),\hfill \end{array}$$

(43)

$$\begin{array}{cc}\hfill & \left|{\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }\left[\tilde{y}\left(0^{+}\right)-{\mathcal{N}}_2(0,{\tilde{y}}_0)\right]-\sum _{j=1}^p{d}_j\tilde{y}\left({\xi }_j\right)\right|\le {ϵ}_2{\theta }_2\left(0\right),\hfill \end{array}$$

(44)

there exists a solution $(x^{*},y^{*})\in E$ of the system (26) and (27) such that

$$\begin{array}{cc}\hfill \left|\tilde{x}(\ell )-x^{*}(\ell )\right|& \le K_1{ϵ}_1{\theta }_1(\ell ){(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1},\hfill \\ \hfill \left|\tilde{y}(\ell )-y^{*}(\ell )\right|& \le K_2{ϵ}_2{\theta }_2(\ell ){(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1},\ell \in J.\hfill \end{array}$$

Lemma 6

($\varrho$-Gronwall Inequality with Impulses). Let $u\in P{C}_{1-\gamma ;\varrho }(J,{\mathbb{R}}_{+})$ satisfy, for $\ell \in ({\ell }_k,{\ell }_{k+1}]$,

$$u(\ell )\le a(\ell )+b{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}u\left(s\right)ds+\sum _{0<{\ell }_i<\ell }{c}_{i}u\left({\ell }_i^{-}\right),$$

where $a\in C(J,{\mathbb{R}}_{+})$ is non-decreasing, $b>0$, and $c_i\ge 0$. Then,

$$u(\ell )\le a(\ell )\prod _{0<{\ell }_i<\ell }(1+c_i){\mathbb{E}}_{\alpha }\left(b\Gamma \left(\alpha \right){(\varrho (\ell )-\varrho \left(0\right))}^{\alpha }\right),\ell \in J,$$

where ${\mathbb{E}}_{\alpha }$ is the Mittag-Leffler function.

Proof. 

The proof follows by induction on the intervals $({\ell }_k,{\ell }_{k+1}]$ using the classical $\varrho$-Gronwall inequality on each subinterval and the jump condition $u\left({\ell }_k^{+}\right)\le (1+c_k)u\left({\ell }_k^{-}\right)$.   □

Theorem 6

(UHR Stability). Assume that Assumptions (H1)–(H4) hold, and let ${\theta }_1,{\theta }_2\in C(J,{\mathbb{R}}_{+})$ be non-decreasing functions such that there exist ${\lambda }_1,{\lambda }_2>0$ with

$${\mathcal{I}}_{0^{+}}^{\alpha ;\varrho }{\theta }_i(\ell )\le {\lambda }_i{\theta }_i(\ell ),i=1,2,\forall \ell \in J.$$

(45)

If Q defined with Theorem 4 satisfies $Q<1$, then the coupled system (26) and (27) is Ulam–Hyers–Rassias stable with respect to $({\theta }_1,{\theta }_2)$.

Proof. 

Let ${ϵ}_1,{ϵ}_2>0$ and let $(\tilde{x},\tilde{y})\in E$ satisfy inequalities (40)–(44). Define the perturbations $h_1(\ell )$ and $h_2(\ell )$ as

$$h_1(\ell ):{=}^H{\mathcal{D}}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[\tilde{x}(\ell )-{\mathcal{N}}_1(\ell ,{\tilde{x}}_{\ell })\right]-F_1(\tilde{x},\tilde{y})(\ell ),\ell \in J^{\prime },$$

$$h_2(\ell ):{=}^H{\mathcal{D}}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[\tilde{y}(\ell )-{\mathcal{N}}_2(\ell ,{\tilde{y}}_{\ell })\right]-F_2(\tilde{x},\tilde{y})(\ell ),\ell \in J^{\prime }.$$

By (40) and (41), we have $\left|h_i(\ell )\right|\le {ϵ}_i{\theta }_i(\ell )$. Also define the impulsive perturbations

$${\eta }_k^x:=\Delta \tilde{x}\left({\ell }_k\right)-I_k\left(\tilde{x}\left({\ell }_k^{-}\right)\right),{\eta }_k^y:=\Delta \tilde{y}\left({\ell }_k\right)-J_k\left(\tilde{y}\left({\ell }_k^{-}\right)\right),$$

with $\left|{\eta }_k^x\right|\le {ϵ}_1{\theta }_1\left({\ell }_k\right)$, $\left|{\eta }_k^y\right|\le {ϵ}_2{\theta }_2\left({\ell }_k\right)$, and the initial perturbations

$$\begin{array}{ccc}\hfill {\nu }_x& :& ={\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }\left[\tilde{x}\left(0^{+}\right)-{\mathcal{N}}_1(0,{\tilde{x}}_0)\right]-\sum _{j=1}^p{c}_j\tilde{x}\left({\xi }_j\right),\left|{\nu }_x\right|\le {ϵ}_1{\theta }_1\left(0\right),\hfill \\ \hfill {\nu }_y& :& ={\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }\left[\tilde{y}\left(0^{+}\right)-{\mathcal{N}}_2(0,{\tilde{y}}_0)\right]-\sum _{j=1}^p{d}_j\tilde{y}\left({\xi }_j\right),\left|{\nu }_y\right|\le {ϵ}_2{\theta }_2\left(0\right).\hfill \end{array}$$

Then $(\tilde{x},\tilde{y})$ satisfies the perturbed system:

$$\begin{array}{cc}\hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[\tilde{x}(\ell )-{\mathcal{N}}_1(\ell ,{\tilde{x}}_{\ell })\right]& =F_1(\tilde{x},\tilde{y})(\ell )+h_1(\ell ),\hfill \end{array}$$

(46)

$$\begin{array}{cc}\hfill \Delta \tilde{x}\left({\ell }_k\right)& =I_k\left(\tilde{x}\left({\ell }_k^{-}\right)\right)+{\eta }_k^x,\hfill \end{array}$$

(47)

$$\begin{array}{cc}\hfill {\mathcal{I}}_{0^{+}}^{1-\gamma ;\varrho }\left[\tilde{x}\left(0^{+}\right)-{\mathcal{N}}_1(0,{\tilde{x}}_0)\right]& =\sum _{j=1}^p{c}_j\tilde{x}\left({\xi }_j\right)+{\nu }_x,\hfill \end{array}$$

(48)

and analogous equations for $\tilde{y}$. Applying the integral formulation (Theorem 3) to this perturbed system, we obtain for $\ell \in ({\ell }_k,{\ell }_{k+1}]$:

$$\begin{array}{cc}\hfill \tilde{x}(\ell )& ={\mathcal{N}}_1(\ell ,{\tilde{x}}_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_x}}\sum _{j=1}^p{c}_j[{\mathcal{N}}_1({\xi }_j,{\tilde{x}}_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\left(I_i\left(\tilde{x}\left({\ell }_i^{-}\right)\right)+{\eta }_i^x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}\left[F_1(\tilde{x},\tilde{y})\left(s\right)+h_1\left(s\right)\right]ds]\hfill \\ \hfill & +\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\left(I_i\left(\tilde{x}\left({\ell }_i^{-}\right)\right)+{\eta }_i^x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}\left[F_1(\tilde{x},\tilde{y})\left(s\right)+h_1\left(s\right)\right]ds+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{\displaystyle \frac{{\nu }_x}{{\Lambda }_x}}.\hfill \end{array}$$

(49)

Let $(x^{*},y^{*})$ be the unique solution of the system (26) and (27). Subtracting the integral equation for $x^{*}$ from (49) yields, for $\ell \in ({\ell }_k,{\ell }_{k+1}]$,

$$\begin{array}{cc}\hfill \tilde{x}(\ell )-x^{*}(\ell )& =\left[{\mathcal{N}}_1(\ell ,{\tilde{x}}_{\ell })-{\mathcal{N}}_1(\ell ,x_{\ell }^{*})\right]+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_x}}\sum _{j=1}^p{c}_j[{\mathcal{N}}_1({\xi }_j,{\tilde{x}}_{{\xi }_j})-{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j}^{*})\hfill \\ \hfill & +\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\left(I_i\left(\tilde{x}\left({\ell }_i^{-}\right)\right)-I_i\left(x^{*}\left({\ell }_i^{-}\right)\right)+{\eta }_i^x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}\left(F_1(\tilde{x},\tilde{y})\left(s\right)-F_1(x^{*},y^{*})\left(s\right)+h_1\left(s\right)\right)ds]\hfill \\ \hfill & +\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\left(I_i\left(\tilde{x}\left({\ell }_i^{-}\right)\right)-I_i\left(x^{*}\left({\ell }_i^{-}\right)\right)+{\eta }_i^x\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}\left(F_1(\tilde{x},\tilde{y})\left(s\right)-F_1(x^{*},y^{*})\left(s\right)+h_1\left(s\right)\right)ds\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{\displaystyle \frac{{\nu }_x}{{\Lambda }_x}}.\hfill \end{array}$$

Let

$${u(\ell ):=|(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }(\tilde{x}(\ell )-x^{*}(\ell ))|,$$

and

$${v(\ell ):=|(\varrho (\ell )-\varrho \left(0\right))}^{1-\gamma }(\tilde{y}(\ell )-y^{*}(\ell ))|.$$

Using Assumptions (H1) and (H2), the bounds on $h_1,{\eta }_i^x,{\nu }_x$, and condition (45), we estimate each term. After careful calculation, we obtain an inequality of the form:

$$\begin{array}{cc}\hfill u(\ell )& \le {ϵ}_1{C}_1{\theta }_1(\ell )+L_{{\mathcal{N}}_1}\underset{s\in [0,\ell ]}{sup}u\left(s\right)+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}\underset{s\in [0,\ell ]}{sup}u\left(s\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_{f_1}(1+L_{{\mathcal{K}}_1}T)\left(\underset{s\in [0,\ell ]}{sup}u\left(s\right)+\underset{s\in [0,\ell ]}{sup}v\left(s\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{\lambda }_1{ϵ}_1{\theta }_1(\ell ),\hfill \end{array}$$

A similar inequality holds for $v(\ell )$. Let

$$w(\ell ):=u(\ell )+v(\ell ).$$

Combining the two inequalities yields:

$$w(\ell )\le ϵC_{\theta }{\theta }_{max}(\ell )+\left(L_{\mathcal{N}}+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}+{\displaystyle \frac{2{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_f(1+L_{K}T)\right)\underset{s\in [0,\ell ]}{sup}w\left(s\right),$$

where

$$\begin{array}{ccc}\hfill ϵ& =& max\{{ϵ}_1,{ϵ}_2\},\hfill \\ \hfill {\theta }_{max}(\ell )& =& max\{{\theta }_1(\ell ),{\theta }_2(\ell )\},\hfill \\ \hfill L_{\mathcal{N}}& =& max\{L_{{\mathcal{N}}_1},L_{{\mathcal{N}}_2}\},\hfill \\ \hfill L_f& =& max\{L_{f_1},L_{f_2}\},\hfill \\ \hfill L_K& =& max\{L_{{\mathcal{K}}_1},L_{{\mathcal{K}}_2}\},\hfill \end{array}$$

and $C_{\theta }$ is a computable constant. Denote

$$\mu :=L_{\mathcal{N}}+{\displaystyle \frac{m{L}_I{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}+{\displaystyle \frac{2{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}{L}_f(1+L_{K}T).$$

Since $Q<1$, we have $\mu <1$ (as $\mu$ is a component of Q). Therefore,

$$w(\ell )\le ϵC_{\theta }{\theta }_{max}(\ell )+\mu \underset{s\in [0,\ell ]}{sup}w\left(s\right).$$

Taking the supremum over $s\in [0,\ell ]$ on the left-hand side gives

$$\underset{s\in [0,\ell ]}{sup}w\left(s\right)\le ϵC_{\theta }{\theta }_{max}(\ell )+\mu \underset{s\in [0,\ell ]}{sup}w\left(s\right),$$

hence

$$\underset{s\in [0,\ell ]}{sup}w\left(s\right)\le {\displaystyle \frac{ϵC_{\theta }}{1-\mu }}{\theta }_{max}(\ell ).$$

Consequently,

$$\begin{array}{cc}\hfill u(\ell )& \le {\displaystyle \frac{ϵC_{\theta }}{1-\mu }}{\theta }_{max}(\ell ),\hfill \\ \hfill v(\ell )& \le {\displaystyle \frac{ϵC_{\theta }}{1-\mu }}{\theta }_{max}(\ell ).\hfill \end{array}$$

Returning to the original variables,

$$\begin{array}{cc}\hfill \left|\tilde{x}(\ell )-x^{*}(\ell )\right|& \le {\displaystyle \frac{ϵC_{\theta }}{1-\mu }}{\theta }_{max}(\ell ){(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}\le K_1{ϵ}_1{\theta }_1(\ell ){(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1},\hfill \\ \hfill \left|\tilde{y}(\ell )-y^{*}(\ell )\right|& \le {\displaystyle \frac{ϵC_{\theta }}{1-\mu }}{\theta }_{max}(\ell ){(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}\le K_2{ϵ}_2{\theta }_2(\ell ){(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1},\hfill \end{array}$$

with

$$K_1={\displaystyle \frac{C_{\theta }}{1-\mu }}{\displaystyle \frac{{\theta }_{max}\left(T\right)}{{\theta }_1\left(T\right)}},K_2={\displaystyle \frac{C_{\theta }}{1-\mu }}{\displaystyle \frac{{\theta }_{max}\left(T\right)}{{\theta }_2\left(T\right)}}.$$

This completes the proof of UHR stability.   □

Corollary 2

(Ulam–Hyers Stability). If we take ${\theta }_1(\ell )\equiv {\theta }_2(\ell )\equiv 1$, then condition (45) holds with ${\lambda }_i={\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\alpha }}{\Gamma (\alpha +1)}}$. Under the same assumptions as Theorem 6, the system is Ulam–Hyers stable; i.e., there exist constants $K_1,K_2>0$ such that for every approximate solution $(\tilde{x},\tilde{y})$ satisfying (40)–(44) with ${\theta }_i\equiv 1$, there exists an exact solution $(x^{*},y^{*})$ with

$$\begin{array}{ccc}\hfill \left|\tilde{x}(\ell )-x^{*}(\ell )\right|& \le & K_1{ϵ}_1{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1},\hfill \\ \hfill \left|\tilde{y}(\ell )-y^{*}(\ell )\right|& \le & K_2{ϵ}_2{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}.\hfill \end{array}$$

3.4. Controllability

We now extend the coupled impulsive system (40)–(44) to include a control input. Consider the controlled system:

$$\begin{array}{cc}\hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[x(\ell )-{\mathcal{N}}_1(\ell ,x_{\ell })\right]& =F_1(\ell )+B_{1}u(\ell ),\ell \in J^{\prime },\hfill \end{array}$$

(50)

$$\begin{array}{cc}\hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{\alpha ,\beta ;\varrho }\left[y(\ell )-{\mathcal{N}}_2(\ell ,y_{\ell })\right]& =F_2(\ell )+B_{2}v(\ell ),\ell \in J^{\prime },\hfill \end{array}$$

(51)

with the same impulsive conditions (3), non-local initial conditions (2), and history conditions (4). Here, $u,v\in L^2(J,{\mathbb{R}}^q)$ are control functions, and $B_1,B_2\in {\mathbb{R}}^{1\times q}$ are constant control gain matrices (row vectors). The functions $F_1,F_2$ retain their definitions, now depending on the controlled states $x,y$.

Definition 6

(Controllability). The system (50) and (51) is said to be controllable on J if for every initial history $\varphi ,\phi$ and every desired terminal state $(x_T,y_T)\in {\mathbb{R}}^2$, there exist control functions $u,v\in L^2(J,{\mathbb{R}}^q)$ such that the corresponding solution $\left(x\right(\ell ),y(\ell \left)\right)$ satisfies $x\left(T\right)=x_T$, $y\left(T\right)=y_T$.

Applying the integral formulation (Theorem 3) to the controlled system, we obtain for $\ell \in ({\ell }_k,{\ell }_{k+1}]$:

$$\begin{array}{cc}\hfill x(\ell )& ={\mathcal{N}}_1(\ell ,x_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_x}}\sum _{j=1}^p{c}_j[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}\left[F_1\left(s\right)+B_{1}u\left(s\right)\right]ds]\hfill \\ \hfill & +\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}\left[F_1\left(s\right)+B_{1}u\left(s\right)\right]ds,\hfill \end{array}$$

(52)

and

$$\begin{array}{cc}\hfill y(\ell )& ={\mathcal{N}}_2(\ell ,y_{\ell })+{\displaystyle \frac{{(\varrho (\ell )-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_y}}\sum _{j=1}^p{d}_j[{\mathcal{N}}_2({\xi }_j,y_{{\xi }_j})+\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)\hfill \\ \hfill & +{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}\left[F_2\left(s\right)+B_{2}v\left(s\right)\right]ds]\hfill \\ \hfill & +\sum _{i=1}^k{\displaystyle \frac{{(\varrho (\ell )-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^{\ell }{\varrho }^{\prime }\left(s\right){(\varrho (\ell )-\varrho \left(s\right))}^{\alpha -1}\left[F_2\left(s\right)+B_{2}v\left(s\right)\right]ds.\hfill \end{array}$$

(53)

Define the controllability Gramian matrices $W_x,W_y\in {\mathbb{R}}^{q\times q}$:

$$\begin{array}{cc}\hfill W_x& :={\displaystyle \frac{1}{\Gamma {\left(\alpha \right)}^2}}{\int }_0^T{\varrho }^{\prime }\left(s\right){(\varrho \left(T\right)-\varrho \left(s\right))}^{\alpha -1}{B}_1^{\top }{B}_1{(\varrho \left(T\right)-\varrho \left(s\right))}^{\alpha -1}{\varrho }^{\prime }\left(s\right)ds,\hfill \end{array}$$

(54)

$$\begin{array}{cc}\hfill W_y& :={\displaystyle \frac{1}{\Gamma {\left(\alpha \right)}^2}}{\int }_0^T{\varrho }^{\prime }\left(s\right){(\varrho \left(T\right)-\varrho \left(s\right))}^{\alpha -1}{B}_2^{\top }{B}_2{(\varrho \left(T\right)-\varrho \left(s\right))}^{\alpha -1}{\varrho }^{\prime }\left(s\right)ds.\hfill \end{array}$$

(55)

Since $B_i$ are row vectors, $B_i^{\top }{B}_i$ is a $q\times q$ matrix. The Gramians are symmetric and positive semidefinite.

H5. 

Controllability Rank Condition: The Gramian matrices $W_x$ and $W_y$ are positive definite; i.e., $W_x>0$, $W_y>0$ (equivalently, they are invertible).

Theorem 7

(Controllability). Assume that Assumptions (H1)–(H5) hold and Q defined with Theorem 4 satisfies $Q<1$. Then the coupled impulsive system (50) and (51) is controllable on J.

Proof. 

Control maps and fixed-point operator. For $(x,y)\in E$, define

$$\begin{array}{cc}\hfill R_x(x,y)& :=x_T-{\mathcal{N}}_1(T,x_T)-{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_x}}\sum _{j=1}^p{c}_j[{\mathcal{N}}_1({\xi }_j,x_{{\xi }_j})\hfill \\ \hfill & +\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds]\hfill \\ \hfill & -\sum _{i=1}^m{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{I}_i\left(x\left({\ell }_i^{-}\right)\right)-{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^T{\varrho }^{\prime }\left(s\right){(\varrho \left(T\right)-\varrho \left(s\right))}^{\alpha -1}{F}_1\left(s\right)ds,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill R_y(x,y)& :=y_T-{\mathcal{N}}_2(T,y_T)-{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left(0\right))}^{\gamma -1}}{\Gamma \left(\gamma \right){\Lambda }_y}}\sum _{j=1}^p{d}_j[{\mathcal{N}}_2({\xi }_j,y_{{\xi }_j})\hfill \\ \hfill & +\sum _{i=1}^{n\left({\xi }_j\right)}{\displaystyle \frac{{(\varrho \left({\xi }_j\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)+{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_{{\ell }_{n\left({\xi }_j\right)}}^{{\xi }_j}{\varrho }^{\prime }\left(s\right){(\varrho \left({\xi }_j\right)-\varrho \left(s\right))}^{\alpha -1}{F}_2\left(s\right)ds]\hfill \\ \hfill & -\sum _{i=1}^m{\displaystyle \frac{{(\varrho \left(T\right)-\varrho \left({\ell }_i\right))}^{\gamma -1}}{\Gamma \left(\gamma \right)}}{J}_i\left(y\left({\ell }_i^{-}\right)\right)-{\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{\int }_0^T{\varrho }^{\prime }\left(s\right){(\varrho \left(T\right)-\varrho \left(s\right))}^{\alpha -1}{F}_2\left(s\right)ds,\hfill \end{array}$$

Define control maps

$$\begin{array}{cc}\hfill u[x,y](\ell )& :={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{B}_1^{\top }{(\varrho \left(T\right)-\varrho (\ell ))}^{\alpha -1}{\varrho }^{\prime }(\ell )W_x^{-1}{R}_x(x,y),\hfill \\ \hfill v[x,y](\ell )& :={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}{B}_2^{\top }{(\varrho \left(T\right)-\varrho (\ell ))}^{\alpha -1}{\varrho }^{\prime }(\ell )W_y^{-1}{R}_y(x,y).\hfill \end{array}$$

Define $\Phi =({\Phi }_1,{\Phi }_2):E\to E$ by ${\Phi }_1(x,y)(\ell ):=$ right-hand side of (52) with $u=u[x,y]$, and ${\Phi }_2(x,y)(\ell ):=$ right-hand side of (53) with $v=v[x,y]$.

Lipschitz estimates. From Assumptions (H1)–(H4) and the structure of $R_x$,

$$\left|R_x(x,y)-R_x(\overline{x},\overline{y})\right|\le K_R\left(\parallel x-\overline{x}{\parallel }_{P{C}_{1-\gamma ;\varrho }}+{\parallel y-\overline{y}\parallel }_{P{C}_{1-\gamma ;\varrho }}\right),$$

(56)

where $K_R$ depends on $L_{{\mathcal{N}}_1},L_{f_1},L_I,m,T,\varrho ,\alpha ,\gamma ,|{\Lambda }_x{|}^{-1},\sum \left|c_j\right|$. Define

$$M_u:={\displaystyle \frac{1}{\Gamma \left(\alpha \right)}}\parallel B_1\parallel \parallel W_x^{-1}\parallel {\left({\int }_0^T{\left[{(\varrho \left(T\right)-\varrho (\ell ))}^{\alpha -1}{\varrho }^{\prime }(\ell )\right]}^{2}d\ell \right)}^{1/2}.$$

Then

$$\parallel u[x,y]-u[\overline{x},\overline{y}]{\parallel }_{L^2}\le M_u{K}_R{\parallel (x,y)-(\overline{x},\overline{y})\parallel }_E.$$

(57)

Contraction estimate for $\Phi$. From (52) and the estimates in Theorem 4,

$$\begin{array}{cc}\hfill \parallel {\Phi }_1(x,y)& -{\Phi }_1(\overline{x},\overline{y}){\parallel }_{P{C}_{1-\gamma ;\varrho }}\le Q_1{\parallel (x,y)-(\overline{x},\overline{y})\parallel }_E\hfill \\ \hfill & +{\displaystyle \frac{\parallel B_1\parallel }{\Gamma \left(\alpha \right)}}{\left({\int }_0^T{\left[{(\varrho \left(T\right)-\varrho \left(s\right))}^{\alpha -1}{\varrho }^{\prime }\left(s\right)\right]}^{2}ds\right)}^{1/2}{\parallel u[x,y]-u[\overline{x},\overline{y}]\parallel }_{L^2},\hfill \end{array}$$

where $Q_1=A_1+A_2<1$ from Theorem 4. Thus, we have,

$$\parallel {\Phi }_1(x,y)-{\Phi }_1(\overline{x},\overline{y}){\parallel }_{P{C}_{1-\gamma ;\varrho }}\le \left(Q_1+\parallel B_1\parallel M_u{K}_R\right){\parallel (x,y)-(\overline{x},\overline{y})\parallel }_E.$$

(58)

Similarly,

$$\parallel {\Phi }_2(x,y)-{\Phi }_2(\overline{x},\overline{y}){\parallel }_{P{C}_{1-\gamma ;\varrho }}\le \left(Q_2+\parallel B_2\parallel M_v{K}_R^{\prime }\right){\parallel (x,y)-(\overline{x},\overline{y})\parallel }_E,$$

(59)

where $M_v$ is defined analogously to $M_u$ and $K_R^{\prime }$ is the Lipschitz constant for $R_y$.

Thus

$$\parallel \Phi (x,y)-\Phi (\overline{x},\overline{y}){\parallel }_E\le \left(Q+max\{\parallel B_1\parallel M_u{K}_R,\parallel B_2\parallel M_v{K}_R^{\prime }\}\right){\parallel (x,y)-(\overline{x},\overline{y})\parallel }_E,$$

(60)

where $Q=max\{Q_1,Q_2\}<1$.

Choice of control gains. Under Assumption (H5), $\parallel W_x^{-1}\parallel ,\parallel W_y^{-1}\parallel$ are finite. Since $M_u\propto \parallel W_x^{-1}\parallel$ and $M_v\propto \parallel W_y^{-1}\parallel$, and $K_R,K_R^{\prime }$ are independent of $B_1,B_2$, we can ensure the condition

$$Q+max\{\parallel B_1\parallel M_u{K}_R,\parallel B_2\parallel M_v{K}_R^{\prime }\}<1$$

holds by either:

• Choosing the control gain matrices $B_1,B_2$ sufficiently large so that the Gramians $W_x,W_y$ have eigenvalues bounded away from zero, making $\parallel W_x^{-1}\parallel ,\parallel W_y^{-1}\parallel$ sufficiently small.
• Alternatively, assuming the system is sufficiently controllable in the sense that $\parallel W_x^{-1}\parallel$ and $\parallel W_y^{-1}\parallel$ are small enough to satisfy the inequality.

Under this condition, $\Phi$ is a contraction on E.

Existence of fixed point. By the Banach Fixed Point Theorem, there exists unique $(x^{*},y^{*})\in E$ with

$$\Phi (x^{*},y^{*})=(x^{*},y^{*}).$$

This fixed point satisfies (52) and (53) with controls $u[x^{*},y^{*}],v[x^{*},y^{*}]$, and by construction

$$x^{*}\left(T\right)=x_T,y^{*}\left(T\right)=y_T.$$

Hence the system is controllable.   □

4. Numerical Application

Problem Formulation

Let $J=[0,2]$, ${\ell }_1=0.5$, ${\ell }_2=1.5$, $m=2$, $T=2$. Define

$$\varrho (\ell )=ln(1+\ell ),\alpha =0.75,\beta =0.5,\gamma =\alpha +\beta (1-\alpha )=0.875.$$

Delay functions: $\rho (\ell )={\displaystyle \frac{\ell }{3}}$, $\sigma (\ell )={\displaystyle \frac{\ell }{2}}$, history interval $[-v,0]=[-0.1,0]$. Non-local points: ${\xi }_1=0.2$, ${\xi }_2=0.4$ with coefficients $c_1=0.3$, $c_2=-0.2$, $d_1=0.1$, $d_2=0.4$. The coupled controlled system for $\ell \in ({\ell }_k,{\ell }_{k+1}]$ is

$$\begin{array}{cc}\hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{0.75,0.5;ln(1+\ell )}\left[x(\ell )-{\mathcal{N}}_1(\ell ,x_{\ell })\right]& =F_1(\ell )+B_{1}u(\ell ),\hfill \end{array}$$

(61)

$$\begin{array}{cc}\hfill {}^H\mathcal{D}_{{\ell }_k^{+}}^{0.75,0.5;ln(1+\ell )}\left[y(\ell )-{\mathcal{N}}_2(\ell ,y_{\ell })\right]& =F_2(\ell )+B_{2}v(\ell ),\hfill \end{array}$$

(62)

with neutral terms

$$\begin{array}{cc}\hfill {\mathcal{N}}_1(\ell ,x_{\ell })& ={\displaystyle \frac{1}{10}}{\int }_{-0.1}^0{e}^{s}x(\ell +s)ds,\hfill \\ \hfill {\mathcal{N}}_2(\ell ,y_{\ell })& ={\displaystyle \frac{1}{20}}sin\left({\int }_{-0.1}^{0}y(\ell +s)ds\right),\hfill \end{array}$$

nonlinear right-hand sides

$$\begin{array}{cc}\hfill F_1(\ell )& :={\displaystyle \frac{1}{30}}\left[x(\ell )+y(\ell )+x\left({\displaystyle \frac{\ell }{3}}\right)cos\left(y\left({\displaystyle \frac{\ell }{2}}\right)\right)+{\int }_0^{\ell }{\displaystyle \frac{s}{1+\ell }}x\left(s\right)y\left(s\right)ds\right],\hfill \\ \hfill F_2(\ell )& :={\displaystyle \frac{1}{40}}\left[sin\left(x(\ell )\right)+y(\ell )+y\left({\displaystyle \frac{\ell }{2}}\right)e^{-x(\ell /3)}+{\int }_0^{\ell }{\displaystyle \frac{coss}{1+\ell }}(x\left(s\right)-y\left(s\right))ds\right],\hfill \end{array}$$

impulsive jumps

$$\begin{array}{cc}\hfill I_1\left(z\right)& =0.1sinz,I_2\left(z\right)=0.05z,\hfill \\ \hfill J_1\left(z\right)& =0.08arctanz,J_2\left(z\right)=0.06(1-e^{-z}),\hfill \end{array}$$

history functions $\varphi (\ell )=0.5$, $\phi (\ell )=-0.3$ for $\ell \in [-0.1,0]$, control gains $B_1=0.4$, $B_2=0.3$, and target state $(x_T,y_T)=(1.0,-0.5)$. For $\eta ,\zeta \in C\left(\right[-0.1,0],\mathbb{R})$,

$$\begin{array}{cc}\hfill {\mathcal{N}}_1\left|(\ell ,\eta )-{\mathcal{N}}_1(\ell ,\zeta )\right|& \le {\displaystyle \frac{1}{10}}{\int }_{-0.1}^0{e}^s|\eta \left(s\right)-\zeta \left(s\right)|ds\le {0.01\parallel \eta -\zeta \parallel }_C,\hfill \\ \hfill \left|{\mathcal{N}}_2(\ell ,\eta )-{\mathcal{N}}_2(\ell ,\zeta )\right|& \le {\displaystyle \frac{1}{20}}{\int }_{-0.1}^0|\eta \left(s\right)-\zeta \left(s\right)|ds\le {0.005\parallel \eta -\zeta \parallel }_C.\hfill \end{array}$$

Thus $L_{{\mathcal{N}}_1}=0.01$, $L_{{\mathcal{N}}_2}=0.005$. For $F_1$, with

$$\left|xy-\overline{x}\overline{y}\right|\le \left|x\right||y-\overline{y}|+|\overline{y}\left|\right|x-\overline{x}|$$

and bounded solutions in $B_R$, $R=2$,

$$\left|F_1(\ell ,x,y,·)-F_1(\ell ,\overline{x},\overline{y},·)\right|\le L_f\left(|x-\overline{x}|+|y-\overline{y}|+|x({\displaystyle \frac{\ell }{3}})-\overline{x}({\displaystyle \frac{\ell }{3}})|+|y({\displaystyle \frac{\ell }{2}})-\overline{y}({\displaystyle \frac{\ell }{2}})|\right)$$

where $L_{f_1}\approx 0.1$. Similarly for $F_2$. For ${\mathcal{K}}_1(\ell ,s,x,y)={\displaystyle \frac{s}{1+\ell }}xy$,

$$\left|{\mathcal{K}}_1(\ell ,s,x,y)-{\mathcal{K}}_1(\ell ,s,\overline{x},\overline{y})\right|\le {\displaystyle \frac{2R}{1}}\left(|x-\overline{x}|+|y-\overline{y}|\right)=4\left(|x-\overline{x}|+|y-\overline{y}|\right).$$

Thus $L_{{\mathcal{K}}_1}=4$. For ${\mathcal{K}}_2(\ell ,s,x,y)={\displaystyle \frac{coss}{1+\ell }}(x-y)$, we have $L_{{\mathcal{K}}_2}=2$. Thus, the functions ${\mathcal{N}}_1,{\mathcal{N}}_2,F_1,F_2,{\mathcal{K}}_1,$ and ${\mathcal{K}}_2$ are satisfied Lipschitz condition. Also, for imulsive

$$\begin{array}{ccc}\hfill \left|I_1\left(z\right)-I_1\left(w\right)\right|& \le & 0.1|z-w|,\hfill \\ \hfill \left|J_1\left(z\right)-J_1\left(w\right)\right|& \le & 0.08|z-w|,\hfill \end{array}$$

hence $L_I=0.1$ and $L_J=0.08$. Compute

$${\Lambda }_x=1-{\displaystyle \frac{1}{\Gamma \left(0.875\right)}}\left[0.3{(ln1.2)}^{-0.125}-0.2{(ln1.4)}^{-0.125}\right]\approx 0.822\ne 0.$$

Similarly ${\Lambda }_y\approx 0.615\ne 0$. Using $T=2$, $m=2$, $\varrho \left(T\right)-\varrho \left(0\right)=ln3\approx 1.0986$, we get

$$A_1=L_{{\mathcal{N}}_1}+{\displaystyle \frac{{(ln3)}^{0.125}}{|{\Lambda }_x|\Gamma \left(0.875\right)}}\sum _{j=1}^2\left|c_j\right|\left(L_{{\mathcal{N}}_1}+{\displaystyle \frac{2L_I{(ln3)}^{0.125}}{\Gamma \left(0.875\right)}}\right)+{\displaystyle \frac{2L_I{(ln3)}^{0.125}}{\Gamma \left(0.875\right)}}\approx 0.384,$$

$$A_2={\displaystyle \frac{{(ln3)}^{0.125}}{|{\Lambda }_x|\Gamma \left(0.875\right)}}\sum _{j=1}^2\left|c_j\right|{\displaystyle \frac{{(ln3)}^{0.75}}{\Gamma \left(1.75\right)}}{L}_{f_1}(1+L_{{\mathcal{K}}_1}T)+{\displaystyle \frac{{(ln3)}^{0.75}}{\Gamma \left(1.75\right)}}{L}_{f_1}(1+L_{{\mathcal{K}}_1}T)\approx 0.246.$$

Thus $Q_1=A_1+A_2\approx 0.63$. Similarly $Q_2\approx 0.61$, so $Q=max\{Q_1,Q_2\}\approx 0.63<1$. For assumption (H5) (Controllability Gramian), since $B_1=0.4$, $B_2=0.3$ are nonzero scalars, then

$$W_x={\displaystyle \frac{0.16}{\Gamma {\left(0.75\right)}^2}}{\int }_0^2{\left[{\displaystyle \frac{1}{1+\ell }}{\left(ln3-ln(1+\ell )\right)}^{-0.25}\right]}^{2}d\ell >0,$$

In the same manner, we get $W_y>0.$ Thus, all assumptions of Theorems 4, 6 and 7 are satisfied.

Remark 4

(Sensitivity to the choice of $\varrho$). The choice $\varrho (\ell )=ln(1+\ell )$ was made to demonstrate the applicability of the framework with a non-trivial kernel that satisfies ${\varrho }^{\prime }(\ell )>0$ on $[0,2]$. The qualitative behavior of the system, such as the existence of a unique solution and its controllability, is guaranteed by the theoretical conditions which are independent of the specific form of ϱ as long as it is strictly increasing and $C^1$. The quantitative results, such as the exact state trajectories and the contraction constant Q, are naturally sensitive to ϱ, as it affects the weighted norms and integral operators. A different choice of ϱ, e.g., a power function $\varrho (\ell )={\ell }^r$ with $r>0$, would yield different numerical values but the theoretical framework would still apply. The logarithmic kernel was chosen here for its smoothness and well-behaved derivative on the interval.

5. Conclusions

This paper discussed a comprehensive qualitative analysis of a generalized coupled system consisting of nonlinear $\varrho$-Hilfer fractional neutral impulsive integro-differential equations characterized by mixed delays and non-local initial conditions. By proposing a model that simultaneously incorporates coupled interactions, neutral dependencies, impulsive perturbations, and mixed delays. We have made four main theoretical contributions. First, used Banach’s contraction mapping principle and Krasnoselskii’s fixed-point theorem in the weighted product space $P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})\times P{C}_{1-\gamma ;\varrho }(J,\mathbb{R})$ to show that there are enough conditions for the existence and uniqueness of solutions. Second, we did a thorough Ulam–Hyers–Rassias (UHR) stability analysis using a generalized $\varrho$-Gronwall inequality. This made sure that the system could handle functional changes. Third, we added control inputs to the model, made a controllability framework, and made a feedback control law that could guide the system to the desired terminal states. Fourth, we showed that our theoretical framework could be used in real life by using a detailed numerical example with the kernel $\varrho (\ell )=ln(1+\ell )$ and a modified predictor-corrector algorithm to solve it. This example showed that the system’s built-in complexities, such as non-trivial kernels, distributed delays, neutral terms, and impulsive jumps, could be handled well while also proving that the proposed controllability scheme worked.

Future studies will concentrate on extending these dynamics to infinite-dimensional spaces and applying stochastic perturbations to account for environmental noise. Furthermore, we envision the development of more specialized stability concepts like Mittag–Leffler and finite-time stability, as well as optimal control strategies. On the computational front, the creation of high-order spectral techniques for general $\varrho$-kernels remains a primary goal. Furthermore, the analytical methods developed here, particularly the use of product spaces and fixed-point theory, are inherently scalable and can be directly extended to study the existence, stability, and controllability of more general n-coupled systems for any finite n. Ultimately, by applying this framework to multi-agent networks and epidemiological models, it will help bridge the gap between abstract fractional theory and the intricate needs of modern scientific applications.