Existence and Stability Results for Fractional Hybrid Systems with Impulsive Effects

Abstract

This paper investigates the existence and stability of solutions for an impulsive hybrid fractional differential equation involving the Caputo derivative. By extending Dhage’s fixed-point theorem with two operators, we establish solution existence under explicitly derived conditions. Furthermore, we prove Ulam–Hyers stability, providing a quantitative bound that ensures robustness under small perturbations. Two illustrative examples with computed parameter bounds validate the theoretical results and highlight the applicability of the model in real-world systems with abrupt changes and memory effects.

1. Introduction

Impulsive hybrid fractional differential equations (IHFDEs) are a powerful mathematical tool for modeling systems that exhibit abrupt changes at specific moments while incorporating memory effects inherent in fractional calculus. Unlike ordinary fractional differential equations (FDEs), which describe continuous dynamical systems with hereditary properties, impulsive FDEs account for sudden state shifts, making them more suitable for real-world phenomena involving instantaneous perturbations. The primary distinction lies in their ability to model discontinuities at fixed or variable points, which ordinary FDEs cannot capture. These equations find extensive applications in applied sciences, including biological systems (e.g., modeling drug administration in pharmacokinetics), control theory (e.g., impulsive control in engineering), and epidemiology (e.g., disease transmission with sudden interventions). In particular, fractional-order models have been widely employed in computational and mathematical biology to describe complex epidemic and transmission dynamics [1,2,3,4]. Control theory (e.g., impulsive control in engineering), and epidemiology (e.g., disease transmission with sudden interventions). Their ability to integrate both memory effects and impulsive dynamics makes them highly relevant for complex, real-world systems (see [5,6,7,8,9,10,11,12]).

The Caputo fractional derivative is widely used in impulsive hybrid fractional differential equations due to its well-defined initial conditions and practical applicability in real-world problems. Unlike the Riemann–Liouville derivative, which requires fractional-order initial conditions that lack physical interpretation, the Caputo derivative allows for traditional integer-order initial conditions, making it more suitable for modeling applied systems. This characteristic is particularly important in impulsive hybrid systems, where abrupt changes occur at specific moments, and accurate initial values are crucial for solution stability. Additionally, the Caputo derivative effectively captures memory effects, which are essential for describing complex dynamical processes with hereditary properties. Its ability to balance mathematical rigor with real-world interpretability makes it a preferred choice for modeling physical, biological, and engineering systems subject to sudden perturbations. By using the Caputo derivative, researchers can ensure both theoretical accuracy and computational feasibility in studying impulsive hybrid fractional differential equations (see [13,14]).

The study of stability in impulsive hybrid fractional differential equations is crucial for ensuring the reliability and robustness of their solutions, particularly in applications involving abrupt changes and memory effects. Stability analysis determines whether small perturbations in initial conditions or external influences lead to significant deviations in system behavior over time. In this context, the Ulam–Hyers stability technique is highly appropriate due to its flexibility in handling perturbations and providing a generalized stability framework. Unlike Lyapunov stability, which often requires explicit Lyapunov functions that may be challenging to construct for complex hybrid fractional systems, Ulam–Hyers stability focuses on the approximate stability of solutions, making it more suitable for impulsive systems where discontinuities and sudden state changes occur. Moreover, Ulam–Hyers stability allows for a broader interpretation of stability in a functional sense, ensuring that even if exact solutions are difficult to obtain, small deviations remain bounded within a tolerable range. This makes it a powerful and effective tool for analyzing impulsive hybrid fractional differential equations in applied sciences (see [15,16,17,18]).

In [19], S. Sitho et al. established existence results for the following hybrid fractional integro-differential equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{c}{D}^{\gamma }\left({\displaystyle \frac{\varsigma \left(\widehat{x}\right)-{\sum }_{i=1}^m{I}^{{\nu }_i}{\chi }_i(\widehat{x},\varsigma \left(\widehat{x}\right))}{\omega (\widehat{x},\varsigma \left(\widehat{x}\right))}}\right)=\psi (\widehat{x},\varsigma \left(\widehat{x}\right)),\widehat{x}\in J=[0,T],0<\gamma \le 1,\hfill \\ \vartheta \left(0\right)=0,\hfill \end{array}\right.\end{array}$$

where $D^{\gamma }$ denotes the Riemann–Liouville fractional derivative of order $\gamma$.

In [20], K. Hilal et al. investigated boundary value problems related to hybrid fractional differential equations, utilizing Caputo differential operators of order $0<\iota <1,$

$$\left\{\begin{array}{c}{D}^{\iota }\left({\displaystyle \frac{\varsigma \left(\widehat{x}\right)}{\chi (\widehat{x},\varsigma \left(\widehat{x}\right))}}\right)=\psi (\widehat{x},\varsigma \left(\widehat{x}\right))a.e.\widehat{x}\in J=[0,T],\hfill \\ \hfill \\ a_1{\displaystyle \frac{\varsigma \left(0\right)}{\chi (0,\varsigma (0\left)\right)}}+b_1{\displaystyle \frac{\varsigma \left(T\right)}{\chi (T,\varsigma (T\left)\right)}}=c_1,\hfill \end{array}\right.$$

In [21], Dhage et al. discussed the existence of the solutions for the following first-order hybrid differential equation:

$$\left\{\begin{array}{c}{\displaystyle \frac{d}{dt}}\left[{\displaystyle \frac{\varsigma \left(\widehat{x}\right)}{\xi (\widehat{x},\varsigma \left(\widehat{x}\right))}}\right]=\phi (\widehat{x},\varsigma \left(\widehat{\varkappa }\right))a.e.\widehat{x}\in J=[0,T],\hfill \\ \hfill \\ \varsigma \left({\widehat{x}}_0\right)={\varsigma }_0\in \mathbb{R},\hfill \end{array}\right.$$

where $\xi \in C(J\times \mathbb{R},\mathbb{R}\setminus \{0\left\}\right)$ and $\phi \in \mathcal{C}(J\times \mathbb{R},\mathbb{R})$.

This paper builds upon the work presented in [20], focusing on the hybrid nonlinear Caputo fractional differential equation:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {}^{c}D^{{\gamma }_1}\left[{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right]={\Lambda }_1(\widehat{x},\vartheta \left(\widehat{x}\right)),\widehat{x}\in {\mathcal{J}}^{\prime }=\mathcal{J}\setminus \{{\widehat{x}}_1,\dots ,{\widehat{x}}_n\},1<{\gamma }_1\le 2,\hfill \\ \hfill & △\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_k\right)}{{\varpi }_1({\widehat{x}}_k,\vartheta \left({\widehat{x}}_k\right))}}\right]=I_k\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_k^{-}\right)}{{\varpi }_1({\widehat{x}}_k^{-},\vartheta \left({\widehat{x}}_k^{-}\right))}}\right],△\left[{\displaystyle \frac{{\vartheta }^{\prime }\left({\widehat{x}}_k\right)}{{\varpi }_1^{\prime }({\widehat{x}}_k,\vartheta \left({\widehat{x}}_k\right))}}\right]={\overline{I}}_k\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_k^{-}\right)}{{\varpi }_1({\widehat{x}}_k^{-},\vartheta \left({\widehat{x}}_k^{-}\right))}}\right],\hfill \\ & {\widehat{x}}_k\in (0,1),k=1,2,\dots ,n,\hfill \end{array}\right.\end{array}$$

(1)

supplemented with the boundary conditions:

$$\begin{array}{c}\hfill \begin{array}{cc}& \left[{\displaystyle \frac{{\vartheta }^{\prime }\left(0\right)}{{\varpi }_1^{\prime }(0,\vartheta \left(0\right))}}\right]+\mu \left[{\displaystyle \frac{\vartheta \left(\eta \right)}{{\varpi }_1(\eta ,\vartheta \left(\eta \right))}}\right]=0,\hfill \\ & \lambda \left[{\displaystyle \frac{{\vartheta }^{\prime }\left(1\right)}{{\varpi }_1^{\prime }(1,\vartheta \left(1\right))}}\right]+\left[{\displaystyle \frac{\vartheta \left(\xi \right)}{{\varpi }_1(\xi ,\vartheta \left(\xi \right))}}\right]=0,\eta ,\xi \in (0,1).\hfill \end{array}\end{array}$$

(2)

where ${}^{c}D^{{\gamma }_1}$ denotes the Caputo fractional derivative of order ${\gamma }_1$, and $J=[0,1]$. The functions ${\Lambda }_1:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}$, with $\lambda ,\mu \in \mathbb{R}$, and ${\varpi }_1:[0,1]\times \mathbb{R}\times \mathbb{R}\to \mathbb{R}\setminus \left\{0\right\}$ are continuous. Moreover, the impulse functions $I_k,{\overline{I}}_k:\mathbb{R}\to \mathbb{R}$ are continuous for all admissible k. $I_k,{\overline{I}}_k:\mathbb{R}⟶\mathbb{R}$ are continuous functions,

$$▵\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_k\right)}{{\varpi }_1({\widehat{x}}_k,\vartheta \left({\widehat{x}}_k\right))}}\right]={\displaystyle \frac{\vartheta \left({\widehat{x}}_k^{+}\right)}{{\varpi }_1({\widehat{x}}_k^{+},\vartheta \left({\widehat{x}}_k^{+}\right))}}-{\displaystyle \frac{\vartheta \left({\widehat{x}}_k^{-}\right)}{{\varpi }_1({\widehat{x}}_k^{-},\vartheta \left({\widehat{x}}_k^{-}\right))}},$$

with

$$\underset{h\to 0^{+}}{lim}\vartheta \left({\widehat{x}}_k^{+}\right)=\vartheta ({\widehat{x}}_k+h),\underset{h\to 0^{-}}{lim}\vartheta \left({\widehat{x}}_k^{-}\right)=\vartheta ({\widehat{x}}_k-h),k=1,2,\dots ,n,$$

for $0={\widehat{x}}_0<{\widehat{x}}_1<{\widehat{x}}_2<\dots <{\widehat{x}}_n<{\widehat{x}}_{n+1}=1.$

This study presents a novel approach to analyzing impulsive hybrid fractional differential equations by extending Dhage’s fixed-point theorem with two distinct operators, allowing for a more refined examination of solution properties. Unlike existing research, which often employs standard fixed-point techniques, this work enhances the theoretical framework by incorporating a broader perspective on solution existence. Additionally, the study utilizes the Ulam–Hyers stability method to rigorously assess the robustness of solutions under small perturbations, ensuring greater reliability for real-world applications. The inclusion of two illustrative examples not only validates the theoretical findings, but also highlights their practical applicability across various scientific and engineering domains. This research contributes significantly to the growing body of work on impulsive fractional systems by providing deeper mathematical insights and offering a more comprehensive stability analysis, thus paving the way for future advancements in fractional calculus and dynamic system modeling.

This is how the rest of the document is structured. Preliminary ideas and an analysis of the auxiliary lemma relating to the current problem are covered in Section 2. The main proof for the existence of a solution to problems (1) and (2) has been covered in Section 3. The Ulam–Hyers stability of the given fractional differential equations, (1) and (2), is examined in Section 4. Three examples are given in Section 5 to help further explain the objectives of the study. results. A conclusion and work for the future are introduced in Section 6.

2. Preliminaries

Recalling some preliminary facts, some basic definitions and properties of the fractional calculus.

Throughout this paper, ${\mathcal{J}}_0=[0,{\widehat{x}}_1]$, ${\mathcal{J}}_1=({\widehat{x}}_1,{\widehat{x}}_2]$, …, ${\mathcal{J}}_{n-1}=({\widehat{x}}_{n-1},{\widehat{x}}_n]$, ${\mathcal{J}}_n=({\widehat{x}}_n,1],n\in \mathbb{N},n>1.$

For ${\widehat{x}}_i\in (0,1)$, such that ${\widehat{x}}_1<{\widehat{x}}_2<\dots <{\widehat{x}}_n,$ we define the following spaces:

$${\mathcal{J}}^{\prime }=\mathcal{J}\setminus \{{\widehat{x}}_1,{\widehat{x}}_2,\dots ,{\widehat{x}}_n\},$$

$$\begin{array}{c}\hfill \widehat{\mathfrak{X}}=\{\vartheta \in \mathcal{C}(\mathcal{J},\mathbb{R}):\vartheta \in \mathcal{C}\left({\mathcal{J}}^{\prime }\right)\mathrm{and}\mathrm{left}\vartheta \left({\widehat{x}}_i^{+}\right)\mathrm{and}\mathrm{right}\mathrm{limit}\vartheta \left({\widehat{x}}_i^{-}\right))\mathrm{exist} \mathrm{and}\vartheta \left({\widehat{x}}_i^{-}\right)=\vartheta \left({\widehat{x}}_i\right),1\le i\le n\}.\end{array}$$

Clearly, $\widehat{\mathcal{X}}$ is a Banach algebra with the norm

$$\parallel \vartheta \parallel =sup\left\{\right|\vartheta \left(\widehat{x}\right)|,\widehat{x}\in \mathcal{J}\}.$$

We also define the space of piecewise continuous functions with the following impulses:

$$PC([0,1],\mathbb{R})=\left\{\vartheta :[0,1]\to \mathbb{R}\mid \vartheta \in \mathcal{C}(({\widehat{x}}_k,{\widehat{x}}_{k+1}],\mathbb{R}),\vartheta \left({\widehat{x}}_k^{+}\right),\vartheta \left({\widehat{x}}_k^{-}\right)\mathrm{exist},\vartheta \left({\widehat{x}}_k\right)=\vartheta \left({\widehat{x}}_k^{-}\right),k=1,\dots ,n\right\},$$

equipped with the same sup-norm $\parallel ·\parallel$. Note that $\widehat{\mathfrak{X}}\subset PC([0,1],\mathbb{R})$.

Definition 1  

([22]). The Riemann–Liouville (R-L) fractional integral of a function $\varphi \in L^1([a,b],{\mathbb{R}}^{+})$ of order $\widehat{\kappa }\in {\mathbb{R}}^{+}$ is given by the following:

$$I_a^{\widehat{\kappa }}\varphi \left(\widehat{x}\right)={\int }_a^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{\widehat{\kappa }-1}}{\Gamma \left(\widehat{\kappa }\right)}}\varphi \left(s\right)ds,$$

Definition 2  

([22]). For a function ϕ defined on the interval $[a,b]$, the Caputo fractional derivative of order $\widehat{\kappa }$ is expressed as follows:

$${(}^c{D}_{a^{+}}^{\widehat{\kappa }}\varphi )\left(\widehat{x}\right)={\displaystyle \frac{1}{\Gamma (n-\widehat{\kappa })}}{\int }_a^{\widehat{\kappa }}{\displaystyle \frac{{(\widehat{x}-s)}^{n-\widehat{\kappa }-1}}{\Gamma \left(\kappa \right)}}{\varphi }^{\left(n\right)}\left(s\right)ds,$$

where $n=\lfloor \widehat{\kappa }\rfloor +1$, with $\lfloor \widehat{\kappa }\rfloor$ denoting the integer part of $\widehat{x}$.

Lemma 1  

([22]). Let $\widehat{\kappa }>0$ and $\varsigma \in \mathcal{C}(0,T)\cap L(0,T)$. The corresponding fractional differential equation is expressed as follows:

$$D^{\widehat{\kappa }}\varsigma \left(\widehat{x}\right)=0,$$

has a unique solution

$$\varsigma \left(\widehat{x}\right)={\tau }_1{\widehat{x}}^{\widehat{\kappa }-1}+{\tau }_2{\widehat{x}}^{\widehat{\kappa }-2}+\dots +{\tau }_n{\widehat{x}}^{\widehat{\kappa }-n},$$

where ${\tau }_i\in \mathbb{R}$, $i=1,2,\dots ,n,$ and $n-1<\widehat{\kappa }<n$.

Lemma 2  

([22]). Let $\widehat{\iota }>0$. Then, for $\vartheta \in \mathcal{C}(0,T)\cap L(0,T)$ we have

$$I^{\widehat{\iota }}{D}^{\widehat{\iota }}\vartheta \left(\widehat{x}\right)=\vartheta \left(\widehat{x}\right)+{\upsilon }_0+{\upsilon }_{1}t+\dots +{\upsilon }_{n-1}{\widehat{x}}^{n-1},$$

fore some ${\upsilon }_j\in \mathbb{R}$, $j=1,2,\dots ,n-1$. Where $n=\left[\widehat{\iota }\right]+1$.

All operators in this work will be considered on the Banach space $\left(PC\left(\right[0,1],\mathbb{R}),\parallel ·\parallel \right)$ unless stated otherwise.

Theorem 1  

([23]). Let $\widehat{\mathfrak{S}}$ be a non-empty, closed, convex, and bounded subset of the Banach algebra $\widehat{\mathfrak{X}}$. Let $\widehat{\mathfrak{A}}:\widehat{\mathfrak{X}}\to \widehat{\mathfrak{X}}$ and $\widehat{\mathfrak{B}}:\widehat{\mathfrak{S}}\to \widehat{\mathfrak{X}}$ be two operators, such that:

(i) 

$\widehat{\mathfrak{A}}$ is Lipschitz continuous with a constant $\widehat{\alpha }$,

(ii) 

$\widehat{\mathfrak{B}}$ is completely continuous.

(iii) 

$\widehat{\tau }=\widehat{\mathfrak{A}}\widehat{\tau }\widehat{\mathfrak{B}}\widehat{\rho }⟹\widehat{\tau }\in \widehat{\mathfrak{S}}$ for all $\widehat{\rho }\in \widehat{\mathfrak{S}}$, and

(iv) 

$\widehat{M}\widehat{\alpha }<1$, where $\widehat{M}=\parallel \widehat{\mathfrak{B}}\left(\widehat{\mathfrak{S}}\right)\parallel =sup\{\parallel \widehat{\mathfrak{B}}\left(\widehat{\tau }\right)\parallel :\widehat{\tau }\in \widehat{\mathfrak{S}}\}$

Then, the operator equation $\widehat{\mathfrak{A}}\widehat{\tau }\widehat{\mathfrak{B}}\widehat{\tau }=\widehat{\tau }$ has a solution in $\widehat{\mathfrak{S}}$.

By a solution of the Equations (1) and (2), we refer to a function $\vartheta \in \mathcal{C}(\mathcal{J},\mathbb{R})$, such that:

(i) the function $\widehat{x}\mapsto {\displaystyle \frac{\vartheta }{{\varpi }_1(\widehat{x},\vartheta )}}$ is continuous for each $\vartheta \in \mathbb{R}$, and (ii) $\vartheta$ satisfies the Equations (1) and (2).

Lemma 3.  

Suppose ${\sigma }_1\in PC([0,1],\mathbb{R})$, the solution of the following hybrid system:

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {}^{c}D^{{\gamma }_1}\left[{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right]={\sigma }_1\left(\widehat{x}\right),\widehat{x}\in {\mathcal{J}}^{\prime }=\mathcal{J}\setminus \{{\widehat{x}}_1,\dots ,{\widehat{x}}_n\},1<{\gamma }_1\le 2,\hfill \\ \hfill & △\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_k\right)}{{\varpi }_1({\widehat{x}}_k,\vartheta \left({\widehat{x}}_k\right))}}\right]=I_k\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_k\right)}{{\varpi }_1({\widehat{x}}_k^{-},\vartheta \left({\widehat{x}}_k^{-}\right))}}\right],△\left[{\displaystyle \frac{{\vartheta }^{\prime }\left({\widehat{x}}_k\right)}{{\varpi }_1^{\prime }({\widehat{x}}_k,\vartheta \left({\widehat{x}}_k\right))}}\right]={\overline{I}}_k\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_k^{-}\right)}{{\varpi }_1({\widehat{x}}_k^{-},\vartheta \left({\widehat{x}}_k^{-}\right))}}\right],\hfill \\ & {\widehat{x}}_k\in (0,1),k=1,2,\dots ,n,\hfill \end{array}\right.\end{array}$$

(3)

supplemented with the boundary conditions;

$$\begin{array}{c}\hfill \begin{array}{cc}& \left[{\displaystyle \frac{{\vartheta }^{\prime }\left(0\right)}{{\varpi }_1^{\prime }(0,\vartheta \left(0\right))}}\right]+\mu \left[{\displaystyle \frac{\vartheta \left(\eta \right)}{{\varpi }_1(\eta ,\vartheta \left(\eta \right))}}\right]=0,\hfill \\ & \lambda \left[{\displaystyle \frac{{\vartheta }^{\prime }\left(1\right)}{{\varpi }_1^{\prime }(1,\vartheta \left(1\right))}}\right]+\left[{\displaystyle \frac{\vartheta \left(\xi \right)}{{\varpi }_1(\xi ,\vartheta \left(\xi \right))}}\right]=0,\hfill \end{array}\end{array}$$

(4)

is given by

$$\begin{array}{c}\hfill \vartheta \left(\widehat{x}\right)=\left\{\begin{array}{cc}\hfill & {\displaystyle {\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))[{\int }_0^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+{\displaystyle \frac{\mu (\lambda +\xi -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}{\int }_{{\widehat{x}}_k}^{\eta }{\displaystyle \frac{{(\eta -s)}^{{\gamma }_1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds}\hfill \\ \hfill & -{\displaystyle \frac{1+\mu (\eta -\widehat{x})}{[1-\mu (\lambda +\eta -\xi \left)\right]}}\left[{\displaystyle {\int }_{{\widehat{x}}_k}^{\xi }{\displaystyle \frac{{(\xi -s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+\lambda {\int }_{{\widehat{x}}_n}^1{\displaystyle \frac{{(1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds}\right]\hfill \\ \hfill & {\displaystyle -{\displaystyle \frac{1-\mu (\lambda +\xi +\widehat{x}-\eta )}{[1-\mu (\lambda +\eta -\xi \left)\right]}}\sum _{i=1}^n\lambda \left({\displaystyle {\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]}\right)}\hfill \\ \hfill & {\displaystyle -\sum _{i=1}^k\left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+I_i\left[{\displaystyle \frac{{\widehat{x}}_i^{-})}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right)}\hfill \\ \hfill & {\displaystyle -\sum _{i=1}^k\left({\displaystyle \frac{\mu \widehat{x}(\eta -\xi )+(\xi -\mu \lambda \eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}-{\widehat{x}}_i\right)\left({\displaystyle {\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i,\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]}\right)],}\hfill \\ \hfill & \widehat{x}\in [0,{\widehat{x}}_1],\hfill \\ \hfill & {\displaystyle {\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))[{\int }_{{\widehat{x}}_k}^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+{\displaystyle \frac{\mu (\lambda +\xi -\widehat{x})}{[1-\mu (\lambda +\zeta -\eta \left)\right]}}{\int }_{{\widehat{x}}_k}^{\eta }{\displaystyle \frac{{(\eta -s)}^{{\gamma }_1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds}\hfill \\ \hfill & -{\displaystyle \frac{1+\mu (\eta -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\left[{\displaystyle {\int }_{{\widehat{x}}_k}^{\xi }{\displaystyle \frac{{(\xi -s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+\lambda {\int }_{{\widehat{x}}_n}^1{\displaystyle \frac{{(1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds}\right]\hfill \\ \hfill & {\displaystyle -{\displaystyle \frac{1-\mu (\lambda +\xi +\widehat{x}-\eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\sum _{i=1}^n\lambda \left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right)}\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^k\left({\displaystyle \frac{\widehat{x}(1-\mu \lambda )+\mu \lambda \eta -\xi }{[1-\mu (\lambda +\eta -\xi \left)\right]}}\right)\left({\displaystyle {\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]}\right)],}\hfill \\ \hfill & \widehat{x}\in [{\widehat{x}}_k,{\widehat{x}}_{k+1}].k=1,2,\dots ,n\hfill \end{array}\right.\end{array}$$

(5)

Proof. 

Assume $\vartheta$ satisfies (3) and (4), then Lemma 2 implies:

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right]{=}^c{I}^{{\gamma }_1}{\sigma }_1\left(\widehat{x}\right)-{\alpha }_0-{\alpha }_1\widehat{x},\widehat{x}\in [0,{\widehat{x}}_1).\end{array}$$

(6)

For some constants ${\beta }_0,{\beta }_1\in \mathbb{R},$ we can write

$$\begin{array}{c}\hfill \left[{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right]{=}^c{I}^{{\gamma }_1}{\sigma }_1\left(\widehat{x}\right)-{\beta }_0-{\beta }_1(\widehat{x}-{\widehat{x}}_1)={\int }_{{\widehat{x}}_1}^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds-{\beta }_0-{\beta }_1(\widehat{x}-{\widehat{x}}_1),\widehat{x}\in ({\widehat{x}}_1,{\widehat{x}}_2].\end{array}$$

Using the impulse condition $△\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_1\right)}{{\varpi }_1({\widehat{x}}_1,\vartheta \left({\widehat{x}}_1\right))}}\right]={\displaystyle \frac{\vartheta \left({\widehat{x}}_1^{+}\right)}{{\varpi }_1({\widehat{x}}_1^{+},\vartheta \left({\widehat{x}}_1^{+}\right))}}-{\displaystyle \frac{\vartheta \left({\widehat{x}}_1^{-}\right)}{{\varpi }_1({\widehat{x}}_1^{-},\vartheta \left({\widehat{x}}_1^{-}\right))}}$ and $△\left[{\displaystyle \frac{{\vartheta }^{\prime }\left({\widehat{x}}_1\right)}{{\varpi }_1^{\prime }({\widehat{x}}_1,\vartheta \left({\widehat{x}}_1\right))}}\right]=\left[{\displaystyle \frac{{\vartheta }^{\prime }\left({\widehat{x}}_1^{+}\right)}{{\varpi }_1^{\prime }({\widehat{x}}_1^{+},\vartheta \left({\widehat{x}}_1^{+}\right))}}\right]-\left[{\displaystyle \frac{{\vartheta }^{\prime }\left({\widehat{x}}_1^{-}\right)}{{\varpi }_1^{\prime }({\widehat{x}}_1^{-},\vartheta \left({\widehat{x}}_1^{-}\right))}}\right],$ we find that

$$\begin{array}{c}\hfill {\displaystyle -{\beta }_0={\int }_0^{{\widehat{x}}_1}{\displaystyle \frac{{({\widehat{x}}_1-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds-{\alpha }_0-{\alpha }_1{\widehat{x}}_1+I_1\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_1^{-}\right)}{{\varpi }_1({\widehat{x}}_1^{-},\vartheta \left({\widehat{x}}_1^{-}\right))}}\right],}\end{array}$$

$$\begin{array}{c}\hfill {\displaystyle -{\beta }_1={\int }_0^{{\widehat{x}}_1}{\displaystyle \frac{{({\widehat{x}}_1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds-{\alpha }_1+{\overline{I}}_1\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_1^{-}\right)}{{\varpi }_1({\widehat{x}}_1^{-},\vartheta \left({\widehat{x}}_1^{-}\right))}}\right].}\end{array}$$

Thus

$$\begin{array}{cc}\hfill \left[{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right]& {\displaystyle ={\int }_{{\widehat{x}}_1}^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+{\int }_0^{{\widehat{x}}_1}{\displaystyle \frac{{({\widehat{x}}_1-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds-{\alpha }_0-{\alpha }_1\widehat{x}+I_1\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_1^{-}\right)}{{\varpi }_1({\widehat{x}}_1^{-},\vartheta \left({\widehat{x}}_1^{-}\right))}}\right]}\hfill \\ \hfill & +(\widehat{x}-{\widehat{x}}_1)\left({\displaystyle {\int }_0^{{\widehat{x}}_1}{\displaystyle \frac{{({\widehat{x}}_1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_1\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_1^{-}\right)}{{\varpi }_1({\widehat{x}}_1^{-},\vartheta \left({\widehat{x}}_1^{-}\right))}}\right]}\right),\widehat{x}\in ({\widehat{x}}_1,{\widehat{x}}_2].\hfill \end{array}$$

By repeating the process in this manner, the solution $\vartheta \left(\widehat{x}\right)$ for $\widehat{x}\in (t_k,t_{k+1}]$ can be expressed as:

$$\begin{array}{cc}\hfill \left[{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right]& {\displaystyle ={\int }_{{\widehat{x}}_k}^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds-{\alpha }_0-{\alpha }_1\widehat{x}+\sum _{i=1}^k\left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+I_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right)}\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^k(\widehat{x}-{\widehat{x}}_i)\left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right),\widehat{x}\in ({\widehat{x}}_k,{\widehat{x}}_{k+1}].}\hfill \end{array}$$

(7)

Applying the boundary conditions $\left[{\displaystyle \frac{{\vartheta }^{\prime }\left(0\right)}{{\varpi }_1(0,{\vartheta }^{\prime }\left(0\right))}}\right]+\mu \left[{\displaystyle \frac{\vartheta \left(\eta \right)}{{\varpi }_1(\eta ,\vartheta \left(\eta \right))}}\right]=0,\lambda \left[{\displaystyle \frac{{\vartheta }^{\prime }\left(1\right)}{{\varpi }_1^{\prime }(1,\vartheta \left(1\right))}}\right]+\left[{\displaystyle \frac{\vartheta \left(\xi \right)}{{\varpi }_1(\xi ,\vartheta \left(\xi \right))}}\right]=0,\eta ,\xi \in (0,1)$ the values of ${\alpha }_0,{\alpha }_1$ are given by

$$\begin{array}{cc}\hfill {\alpha }_0& {\displaystyle =-{\displaystyle \frac{\mu (\lambda +\xi )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}{\int }_{{\widehat{x}}_k}^{\eta }{\displaystyle \frac{{(\eta -s)}^{{\gamma }_1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+\sum _{i=1}^k\left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+I_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right)}\hfill \\ \hfill & +{\displaystyle \frac{(1+\mu \eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\left({\displaystyle {\int }_{{\widehat{x}}_k}^{\xi }{\displaystyle \frac{{(\xi -s)}^{{\gamma }_1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds+\lambda {\int }_{{\widehat{x}}_n}^1{\displaystyle \frac{{(1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds}\right)\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^n{\displaystyle \frac{\lambda (1+\mu \eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\left({\displaystyle {\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]}\right)}\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^n\left({\displaystyle \frac{(\xi -\mu \lambda \eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}-{\widehat{x}}_i\right)\left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right),}\hfill \end{array}$$

(8)

$$\begin{array}{cc}\hfill {\alpha }_1& {\displaystyle ={\displaystyle \frac{\mu }{[1-\mu (\lambda +\xi -\eta \left)\right]}}[{\int }_{{\widehat{x}}_k}^{\eta }{\displaystyle \frac{{(\eta -s)}^{{\gamma }_1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds-{\int }_{{\widehat{x}}_k}^{\xi }{\displaystyle \frac{{(\xi -s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\sigma }_1\left(s\right)ds}\hfill \\ \hfill & {\displaystyle -\lambda {\int }_{{\widehat{x}}_n}^1{\displaystyle \frac{{(1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds]-\sum _{i=1}^n\lambda \left({\displaystyle {\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]}\right)}\hfill \\ \hfill & {\displaystyle +\sum _{i=1}^n(\eta -\xi )\left({\displaystyle {\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\sigma }_1\left(s\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]}\right).}\hfill \end{array}$$

(9)

We derive (5) by substituting the values of ${\alpha }_0,{\alpha }_1$ in (6) and (7). This brings the proof is compltes. □

3. Key Findings

In this section, we prove the existence of at least one solution to problem (1) and (2) by applying Dhage’s fixed-point Theorem 1 on the Banach space

$$\mathcal{E}=\left(PC\left(\right[0,1],\mathbb{R}),\parallel ·\parallel \right),$$

where $\parallel \vartheta \parallel ={sup}_{\widehat{x}\in [0,1]}\left|\vartheta \left(\widehat{x}\right)\right|$ and $PC\left(\right[0,1],\mathbb{R})$ denotes the space of piecewise continuous functions with possible jumps at ${\widehat{x}}_1,\dots ,{\widehat{x}}_n$, as defined in Section 2.

To simplify the subsequent estimates, we first introduce the following constants derived from the boundary parameters:

$$\begin{array}{cccc}\hfill {\omega }_1& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{\mu (\lambda +\xi -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,\hfill & \hfill {\omega }_2& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1+\mu (\eta -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,\hfill \\ \hfill {\omega }_3& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1-\mu (\lambda +\xi +\widehat{x}-\eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,\hfill & \hfill {\omega }_4& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{(1-\mu \lambda )\widehat{x}+\mu \lambda \eta -\xi }{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,\hfill \\ \hfill {\omega }_4^{\prime }& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{\mu \widehat{x}(\eta -\xi )+(\xi -\mu \lambda \eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,\hfill & \hfill {\omega }_5& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|.\hfill \end{array}$$

(10)

We also record the following fractional integral bounds, which will be used repeatedly in the compactness and equicontinuity arguments:

$$\begin{array}{cc}\hfill {\int }_{{\widehat{x}}_k}^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}ds& ={\displaystyle \frac{{(\widehat{x}-{\widehat{x}}_k)}^{{\gamma }_1}}{\Gamma ({\gamma }_1+1)}}\le {\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}},\widehat{x}\in [{\widehat{x}}_k,{\widehat{x}}_{k+1}],\hfill \\ \hfill {\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}ds& ={\displaystyle \frac{{({\widehat{x}}_i-{\widehat{x}}_{i-1})}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}\le {\displaystyle \frac{1}{\Gamma \left({\gamma }_1\right)}},i=1,\dots ,n.\hfill \end{array}$$

(11)

The assumptions required for our main result are:

$\left(K1\right)$ There exist constants $L_1,L_2>0$, such that

$$|{\varpi }_1(\widehat{x},\vartheta )|\le L_1,\left|{\Lambda }_1(\widehat{x},\vartheta )\right|\le L_2,$$

for all $(\widehat{x},\vartheta )\in [0,1]\times \mathbb{R}$.

$\left(K2\right)$ The functions ${\varpi }_1,{\Lambda }_1:[0,1]\times \mathbb{R}\to \mathbb{R}$ are continuous and satisfy the Lipschitz conditions

$$|{\varpi }_1(\widehat{x},u)-{\varpi }_1(\widehat{x},v)|\le {\pi }_1|u-v|,|{\Lambda }_1(\widehat{x},u)-{\Lambda }_1(\widehat{x},v)|\le {\pi }_2|u-v|,$$

for all $\widehat{x}\in [0,1]$ and $u,v\in \mathbb{R}$, where ${\pi }_1,{\pi }_2>0$ are Lipschitz constants.

For the impulse terms, we denote

$${\nu }_1=\underset{1\le i\le n}{max}\underset{\vartheta \in \mathbb{R}}{sup}\left|{\overline{I}}_i\left[{\displaystyle \frac{\vartheta }{{\varpi }_1({\widehat{x}}_i^{-},\vartheta )}}\right]\right|,{\nu }_2=\underset{1\le i\le n}{max}Lip\left({\overline{I}}_i\left[{\displaystyle \frac{·}{{\varpi }_1({\widehat{x}}_i^{-},·)}}\right]\right),$$

where “Lip” denotes the Lipschitz constant of the indicated map.

Theorem 2.  

Assume that $\left(K1\right)$–$\left(K2\right)$ hold. If

$$L_1\left({\displaystyle \frac{({\omega }_1+{\omega }_2+1)L_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{({\omega }_2+{\omega }_{3}n)\left|\lambda \right|L_2+{\omega }_{4}n{L}_2}{\Gamma \left({\gamma }_1\right)}}+({\omega }_3\left|\lambda \right|+{\omega }_4)n{\nu }_1\right)<1,$$

(12)

then the impulsive hybrid system (1) and (2) possesses at least one solution on $[0,1]$.

Proof. 

Let

$$r={\displaystyle \frac{({\omega }_1+{\omega }_2+1)L_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{({\omega }_2+{\omega }_{3}n)\left|\lambda \right|L_2+{\omega }_{4}n{L}_2}{\Gamma \left({\gamma }_1\right)}}+({\omega }_3\left|\lambda \right|+{\omega }_4)n{\nu }_1,$$

and define the closed ball

$$\widehat{\mathfrak{S}}=\{\vartheta \in PC([0,1],\mathbb{R}):\parallel \vartheta \parallel \le r\}.$$

Clearly, $\widehat{\mathfrak{S}}$ is a non-empty, closed, convex, and bounded subset of $\mathcal{E}$.

We now introduce two operators $\widehat{\mathfrak{A}},\widehat{\mathfrak{B}}$ on $\mathcal{E}$. For $\vartheta \in PC\left(\right[0,1],\mathbb{R})$ and $\widehat{x}\in [0,1]$, set

$$\left(\widehat{\mathfrak{A}}\vartheta \right)\left(\widehat{x}\right)={\varpi }_1\left(\widehat{x},\vartheta \left(\widehat{x}\right)\right),$$

(13)

and for $\vartheta \in \widehat{\mathfrak{S}}$

$$\begin{array}{cc}\hfill \left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)=& {\int }_{{\widehat{x}}_k}^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\Lambda }_1\left(s,\vartheta \left(s\right)\right)ds\hfill \\ & +{\displaystyle \frac{\mu (\lambda +\xi -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}{\int }_{{\widehat{x}}_k}^{\eta }{\displaystyle \frac{{(\eta -s)}^{{\gamma }_1}}{\Gamma \left({\gamma }_1\right)}}{\Lambda }_1\left(s,\vartheta \left(s\right)\right)ds\hfill \\ & -{\displaystyle \frac{1+\mu (\eta -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\left[{\int }_{{\widehat{x}}_k}^{\xi }{\displaystyle \frac{{(\xi -s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}{\Lambda }_1\left(s,\vartheta \left(s\right)\right)ds+\lambda {\int }_{{\widehat{x}}_n}^1{\displaystyle \frac{{(1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\Lambda }_1\left(s,\vartheta \left(s\right)\right)ds\right]\hfill \\ & -{\displaystyle \frac{1-\mu (\lambda +\xi +\widehat{x}-\eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\sum _{i=1}^n\lambda \left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\Lambda }_1\left(s,\vartheta \left(s\right)\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right)\hfill \\ & -\sum _{i=1}^k{\displaystyle \frac{\widehat{x}(1-\mu \lambda )+\mu \lambda \eta -\xi }{[1-\mu (\lambda +\xi -\eta \left)\right]}}\left({\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}{\Lambda }_1\left(s,\vartheta \left(s\right)\right)ds+{\overline{I}}_i\left[{\displaystyle \frac{\vartheta \left({\widehat{x}}_i^{-}\right)}{{\varpi }_1({\widehat{x}}_i^{-},\vartheta \left({\widehat{x}}_i^{-}\right))}}\right]\right),\hfill \end{array}$$

(14)

where the index k is chosen so that $\widehat{x}\in ({\widehat{x}}_k,{\widehat{x}}_{k+1}]$ (for $\widehat{x}\in [0,{\widehat{x}}_1]$, we take $k=0$ and the empty sum is interpreted as zero). It follows from Lemma 3 that the integral Equation (5) can be written as the fixed-point equation

$$\vartheta \left(\widehat{x}\right)=\left(\widehat{\mathfrak{A}}\vartheta \right)\left(\widehat{x}\right)\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right),\widehat{x}\in [0,1].$$

(15)

We shall verify that the operators $\widehat{\mathfrak{A}}$ and $\widehat{\mathfrak{B}}$ satisfy all four conditions of Theorem 1 (Dhage’s theorem) on the set $\widehat{\mathfrak{S}}$.

Claim 1—Lipschitz continuity of $\widehat{\mathfrak{A}}$. For any $\vartheta ,\phi \in \mathcal{E}$ and $\widehat{x}\in [0,1]$, assumption $\left(K2\right)$ gives

$$|\left(\widehat{\mathfrak{A}}\vartheta \right)\left(\widehat{x}\right)-\left(\widehat{\mathfrak{A}}\phi \right)\left(\widehat{x}\right)|=|{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))-{\varpi }_1(\widehat{x},\phi \left(\widehat{x}\right))|\le {\pi }_1|\vartheta \left(\widehat{x}\right)-\phi \left(\widehat{x}\right)|\le {\pi }_1\parallel \vartheta -\phi \parallel .$$

Taking the supremum over $\widehat{x}$ yields

$$\parallel \widehat{\mathfrak{A}}\vartheta -\widehat{\mathfrak{A}}\phi \parallel \le {\pi }_1\parallel \vartheta -\phi \parallel ,$$

hence $\widehat{\mathfrak{A}}:\mathcal{E}\to \mathcal{E}$ is a Lipschitz continuous with constant $\widehat{\alpha }={\pi }_1$.

Claim 2—Continuity of $\widehat{\mathfrak{B}}$ on $\widehat{\mathfrak{S}}$. Let $\left\{{\vartheta }_m\right\}\subset \widehat{\mathfrak{S}}$ converge uniformly to $\vartheta \in \widehat{\mathfrak{S}}$. Because ${\Lambda }_1$ is continuous and bounded (by $\left(K1\right)$), the Lebesgue-dominated convergence theorem applies to every integral appearing in (14). Moreover, the impulse terms are continuous by assumption. Consequently, for each fixed $\widehat{x}$,

$$\underset{m\to \infty }{lim}\left(\widehat{\mathfrak{B}}{\vartheta }_m\right)\left(\widehat{x}\right)=\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right).$$

Uniform convergence on $[0,1]$ follows from the uniform boundedness of the kernels and the compactness of the parameter intervals. Thus, $\widehat{\mathfrak{B}}:\widehat{\mathfrak{S}}\to \mathcal{E}$ is continuous.

Claim 3—Compactness of $\widehat{\mathfrak{B}}$. We prove that $\widehat{\mathfrak{B}}\left(\widehat{\mathfrak{S}}\right)$ is uniformly bounded and equicontinuous; the Arzelà–Ascoli theorem then guarantees that $\widehat{\mathfrak{B}}$ is compact.

Uniform boundedness. For $\vartheta \in \widehat{\mathfrak{S}}$ and $\widehat{x}\in [0,1]$, we estimate using $\left(K1\right)$ the definitions of ${\omega }_1,\dots ,{\omega }_4$, and the bounds (11):

$$\begin{array}{cc}\hfill |\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)|& \le L_2{\int }_{{\widehat{x}}_k}^{\widehat{x}}{\displaystyle \frac{{(\widehat{x}-s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}ds+L_2{\omega }_1{\int }_{{\widehat{x}}_k}^{\eta }{\displaystyle \frac{{(\eta -s)}^{{\gamma }_1}}{\Gamma \left({\gamma }_1\right)}}ds\hfill \\ & +L_2{\omega }_2\left({\int }_{{\widehat{x}}_k}^{\xi }{\displaystyle \frac{{(\xi -s)}^{{\gamma }_1-1}}{\Gamma \left({\gamma }_1\right)}}ds+\left|\lambda \right|{\int }_{{\widehat{x}}_n}^1{\displaystyle \frac{{(1-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}ds\right)\hfill \\ & +{\omega }_3\left|\lambda \right|\sum _{i=1}^n\left(L_2{\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}ds+{\nu }_1\right)\hfill \\ & +{\omega }_4\sum _{i=1}^n\left(L_2{\int }_{{\widehat{x}}_{i-1}}^{{\widehat{x}}_i}{\displaystyle \frac{{({\widehat{x}}_i-s)}^{{\gamma }_1-2}}{\Gamma ({\gamma }_1-1)}}ds+{\nu }_1\right)\hfill \\ & \le L_2\left({\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left({\gamma }_1\right)}}\right)+{\displaystyle \frac{{\omega }_3\left|\lambda \right|n}{\Gamma \left({\gamma }_1\right)}}+{\displaystyle \frac{{\omega }_{4}n}{\Gamma \left({\gamma }_1\right)}}\right)\hfill \\ & +({\omega }_3\left|\lambda \right|+{\omega }_4)n{\nu }_1\hfill \\ & ={\displaystyle \frac{({\omega }_1+{\omega }_2+1)L_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{({\omega }_2+{\omega }_{3}n)\left|\lambda \right|L_2+{\omega }_{4}n{L}_2}{\Gamma \left({\gamma }_1\right)}}+({\omega }_3\left|\lambda \right|+{\omega }_4)n{\nu }_1=r.\hfill \end{array}$$

(16)

Hence, $\parallel \widehat{\mathfrak{B}}\vartheta \parallel \le r$ for every $\vartheta \in \widehat{\mathfrak{S}}$; i.e., $\widehat{\mathfrak{B}}\left(\widehat{\mathfrak{S}}\right)$ is uniformly bounded.

Equicontinuity. Let $\widehat{x},\widehat{y}\in [{\widehat{x}}_k,{\widehat{x}}_{k+1}]$ with $\widehat{x}<\widehat{y}$. Using the mean-value theorem and the uniform bound on the derivative

$$\begin{array}{cc}\hfill |{\left(\widehat{\mathfrak{B}}\vartheta \right)}^{\prime }\left(\widehat{z}\right)|& \le L_2\left({\displaystyle \frac{1}{\Gamma \left({\gamma }_1\right)}}+{\displaystyle \frac{{\omega }_5}{\Gamma ({\gamma }_1+1)}}+{\omega }_5\left|\mu \right|\left({\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left({\gamma }_1\right)}}\right)+{\displaystyle \frac{{\omega }_5\left|\mu \lambda \right|n}{\Gamma \left({\gamma }_1\right)}}+{\displaystyle \frac{{\omega }_5(1+|\mu \lambda \left|\right)n}{\Gamma \left({\gamma }_1\right)}}\right)\hfill \\ & +{\omega }_5\left(\left|\mu \lambda \right|+1+\left|\mu \lambda \right|\right)n{\nu }_1=:M,\hfill \end{array}$$

which holds for all $\vartheta \in \widehat{\mathfrak{S}}$ and $\widehat{z}\in ({\widehat{x}}_k,{\widehat{x}}_{k+1})$, we obtain

$$|\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{y}\right)-\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)|\le {\int }_{\widehat{x}}^{\widehat{y}}\left|{\left(\widehat{\mathfrak{B}}\vartheta \right)}^{\prime }\left(s\right)\right|ds\le M(\widehat{y}-\widehat{x}).$$

Because M does not depend on $\vartheta$ or on the particular subinterval $[{\widehat{x}}_k,{\widehat{x}}_{k+1}]$, the family $\widehat{\mathfrak{B}}\left(\widehat{\mathfrak{S}}\right)$ is equicontinuous on the whole interval $[0,1]$.

By the Arzelà–Ascoli theorem, $\widehat{\mathfrak{B}}:\widehat{\mathfrak{S}}\to \mathcal{E}$ is a compact operator.

Claim 4—Invariance condition. Let $\vartheta \in \mathcal{E}$ and $\upsilon \in \widehat{\mathfrak{S}}$ satisfy $\vartheta =\widehat{\mathfrak{A}}\vartheta \widehat{\mathfrak{B}}\upsilon$. Then, using $\left(K1\right)$ and the bound (16),

$$|\vartheta \left(\widehat{x}\right)|=|\left(\widehat{\mathfrak{A}}\vartheta \right)\left(\widehat{x}\right)\left|\right|\left(\widehat{\mathfrak{B}}\upsilon \right)\left(\widehat{x}\right)|\le L_{1}r,$$

so that $\parallel \vartheta \parallel \le L_{1}r$. By definition of r and hypothesis (12) we have $L_{1}r<1$; consequently, $\vartheta \in \widehat{\mathfrak{S}}$. Thus, condition (iii) of Theorem 1 is satisfied.

Claim 5—Norm inequality. From (16) we have $\parallel \widehat{\mathfrak{B}}\vartheta \parallel \le r$ for all $\vartheta \in \widehat{\mathfrak{S}}$, hence

$$\widehat{M}:=sup\{\parallel \widehat{\mathfrak{B}}\vartheta \parallel :\vartheta \in \widehat{\mathfrak{S}}\}\le r.$$

Since $\widehat{\alpha }={\pi }_1$, condition (iv) of Theorem 1 reads ${\pi }_1\widehat{M}<1$. But ${\pi }_1\widehat{M}\le {\pi }_{1}r$, and by (12) we indeed have ${\pi }_{1}r<1$. All hypotheses of Dhage’s fixed-point theorem are fulfilled; therefore, the operator equation $\vartheta =\widehat{\mathfrak{A}}\vartheta \widehat{\mathfrak{B}}\vartheta$ possesses at least one solution $\vartheta \in \widehat{\mathfrak{S}}$. By construction, such a solution satisfies the impulsive hybrid system (1) and (2). □

4. Stability Results for the Problem

In this section, we establish the Ulam–Hyers (U–H) stability of the impulsive hybrid system (1) and (2). Let us first recall the constants that will appear in the stability estimates, which were introduced in Section 3:

$$\begin{array}{cc}\hfill {\omega }_1& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{\mu (\lambda +\xi -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,{\omega }_2=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1+\mu (\eta -\widehat{x})}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,\hfill \\ \hfill {\omega }_3& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1-\mu (\lambda +\xi +\widehat{x}-\eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,{\omega }_4=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{(1-\mu \lambda )\widehat{x}+\mu \lambda \eta -\xi }{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,\hfill \\ \hfill {\omega }_4^{\prime }& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{\mu \widehat{x}(\eta -\xi )+(\xi -\mu \lambda \eta )}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|,{\omega }_5=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1}{[1-\mu (\lambda +\xi -\eta \left)\right]}}\right|.\hfill \end{array}$$

We also recall the fractional kernel bounds:

$${\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}},{\displaystyle \frac{1}{\Gamma \left({\gamma }_1\right)}},{\displaystyle \frac{1}{\Gamma ({\gamma }_1-1)}},$$

as well as the Lipschitz constants ${\pi }_1,{\pi }_2$ from assumption $\left(K2\right)$ and the impulse Lipschitz constant

$${\nu }_2=\underset{1\le i\le n}{max}Lip\left({\overline{I}}_i\left[{\displaystyle \frac{·}{{\varpi }_1({\widehat{x}}_i^{-},·)}}\right]\right).$$

Define the nonlinear operator ${\mathcal{Z}}_1:PC([0,1],\mathbb{R})\to PC([0,1],\mathbb{R})$ by

$${\mathcal{Z}}_1\left(\vartheta \right)\left(\widehat{x}\right)={}^{c}D^{{\gamma }_1}\left[{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\overline{\omega }}_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right]-{\Lambda }_1(\widehat{x},\vartheta \left(\widehat{x}\right)),\widehat{x}\in [0,1],1<{\gamma }_1\le 2.$$

(17)

For a given ${\varsigma }_1>0$, we consider the inequality

$$\parallel {\mathcal{Z}}_1\left(\vartheta \right)\parallel \le {\varsigma }_1.$$

(18)

Definition 3  

(Ulam–Hyers stability). The impulsive hybrid system (1) and (2) is said to be Ulam–Hyers stable if there exists a constant $\mathcal{C}>0$, such that for every ${\varsigma }_1>0$ and for every function $\vartheta \in PC\left(\right[0,1],\mathbb{R})$ satisfying (18), there exists a solution $\widehat{\vartheta }\in PC([0,1],\mathbb{R})$ of (1) and (2) with

$$\parallel \vartheta -\widehat{\vartheta }\parallel \le \mathcal{C}{\varsigma }_1.$$

(19)

Theorem 3.  

Assume that conditions $\left(K1\right)$–$\left(K2\right)$ hold, and let $\widehat{\mathfrak{S}}=\{\vartheta \in PC([0,1],\mathbb{R}):\parallel \vartheta \parallel \le r\}$ be the invariant ball from Theorem 2. Define

$$\begin{array}{cc}\hfill {\mathcal{L}}_1& ={\pi }_{1}r+{\pi }_1{L}_1\left[{\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|{\pi }_2}{\Gamma \left({\gamma }_1\right)}}\right)+{\omega }_3\left|\lambda \right|n{\displaystyle \frac{{\pi }_2}{\Gamma \left({\gamma }_1\right)}}+k{\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+k({\omega }_4^{\prime }+1){\displaystyle \frac{{\pi }_2}{\Gamma \left({\gamma }_1\right)}}\right.\hfill \\ & \left.+\left({\omega }_3\left|\lambda \right|n+k+k({\omega }_4^{\prime }+1)\right){\nu }_2\right],\hfill \\ \hfill {\mathcal{L}}_2& ={\pi }_{1}r+{\pi }_1{L}_1\left[{\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|{\pi }_2}{\Gamma \left({\gamma }_1\right)}}\right)+{\omega }_3\left|\lambda \right|n{\displaystyle \frac{{\pi }_2}{\Gamma \left({\gamma }_1\right)}}+n({\omega }_4+1){\displaystyle \frac{{\pi }_2}{\Gamma \left({\gamma }_1\right)}}\right.\hfill \\ & \left.+\left({\omega }_3\left|\lambda \right|n+n({\omega }_4+1)\right){\nu }_2\right],\hfill \end{array}$$

where r is the radius defined in Theorem 2, and

$$\begin{array}{cc}\hfill {\Phi }_1& =L_1\left[{\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left({\gamma }_1\right)}}\right)+{\displaystyle \frac{{\omega }_3\left|\lambda \right|n}{\Gamma \left({\gamma }_1\right)}}+{\displaystyle \frac{k}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{k({\omega }_4^{\prime }+1)}{\Gamma \left({\gamma }_1\right)}}\right],\hfill \\ \hfill {\Phi }_2& =L_1\left[{\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left({\gamma }_1\right)}}\right)+{\displaystyle \frac{{\omega }_3\left|\lambda \right|n}{\Gamma \left({\gamma }_1\right)}}+{\displaystyle \frac{n({\omega }_4+1)}{\Gamma \left({\gamma }_1\right)}}\right].\hfill \end{array}$$

If

$${\mathcal{L}}_1<1and{\mathcal{L}}_2<1,$$

(20)

then the impulsive hybrid system (1) and (2) is Ulam–Hyers stable on $[0,1]$, and the stability constant in (19) can be taken as

$$\mathcal{C}=max\left\{{\displaystyle \frac{{\Phi }_1}{1-{\mathcal{L}}_1}},{\displaystyle \frac{{\Phi }_2}{1-{\mathcal{L}}_2}}\right\}.$$

(21)

Proof. 

Let $\vartheta \in \widehat{\mathfrak{S}}$ satisfy inequality (18). Using Lemma 3, inequality (18) implies that $\vartheta$ satisfies the equivalent integral equation with an additive perturbation:

$$\vartheta \left(\widehat{x}\right)={\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))\left[\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)+\delta \left(\widehat{x}\right)\right],$$

(22)

where $\widehat{\mathfrak{B}}$ is defined in (14) and the perturbation term satisfies

$$|\delta \left(\widehat{x}\right)|\le {\varsigma }_1\left[{\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left({\gamma }_1\right)}}\right)+{\displaystyle \frac{{\omega }_3\left|\lambda \right|n}{\Gamma \left({\gamma }_1\right)}}+{\displaystyle \frac{k}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{k({\omega }_4^{\prime }+1)}{\Gamma \left({\gamma }_1\right)}}\right]\mathrm{for}\widehat{x}\in [0,{\widehat{x}}_1],$$

and a similar bound (with k replaced by n and ${\omega }_4^{\prime }$ by ${\omega }_4$) for $\widehat{x}\in [{\widehat{x}}_k,{\widehat{x}}_{k+1}]$.

Let $\widehat{\vartheta }\in PC([0,1],\mathbb{R})$ be an exact solution of (1) and (2), which satisfies

$$\widehat{\vartheta }\left(\widehat{x}\right)={\varpi }_1(\widehat{x},\widehat{\vartheta }\left(\widehat{x}\right))\left(\widehat{\mathfrak{B}}\widehat{\vartheta }\right)\left(\widehat{x}\right),\widehat{x}\in [0,1].$$

(23)

Define the error function $e\left(\widehat{x}\right)=\vartheta \left(\widehat{x}\right)-\widehat{\vartheta }\left(\widehat{x}\right)$. Subtracting (23) from (22) gives

$$e\left(\widehat{x}\right)=\left[{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))-{\varpi }_1(\widehat{x},\widehat{\vartheta }\left(\widehat{x}\right))\right]\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)+{\varpi }_1(\widehat{x},\widehat{\vartheta }\left(\widehat{x}\right))\left[\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)-\left(\widehat{\mathfrak{B}}\widehat{\vartheta }\right)\left(\widehat{x}\right)\right]+{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))\delta \left(\widehat{x}\right).$$

(24)

Case 1: $\widehat{x}\in [0,{\widehat{x}}_1]$. From (24), using the Lipschitz condition $\left(K2\right)$, the boundedness $|{\varpi }_1|\le L_1$, and the uniform bound $\parallel \widehat{\mathfrak{B}}\vartheta \parallel \le r$ (which holds for $\vartheta \in \widehat{\mathfrak{S}}$ by Theorem 2), we obtain

$$\begin{array}{cc}\hfill \left|e\right(\widehat{x}\left)\right|& \le {\pi }_1\parallel e\parallel ·|\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)|+L_1|\left(\widehat{\mathfrak{B}}\vartheta \right)\left(\widehat{x}\right)-\left(\widehat{\mathfrak{B}}\widehat{\vartheta }\right)\left(\widehat{x}\right)|+L_1\left|\delta \left(\widehat{x}\right)\right|\hfill \\ & \le {\pi }_{1}r\parallel e\parallel +L_1\parallel \left(\widehat{\mathfrak{B}}\vartheta \right)-\left(\widehat{\mathfrak{B}}\widehat{\vartheta }\right)\parallel +L_1\left|\delta \left(\widehat{x}\right)\right|.\hfill \end{array}$$

The difference $\parallel \left(\widehat{\mathfrak{B}}\vartheta \right)-\left(\widehat{\mathfrak{B}}\widehat{\vartheta }\right)\parallel$ can be estimated exactly as in the proof of Claim 3 in Section 3, yielding

$$\begin{array}{cc}\hfill \parallel \left(\widehat{\mathfrak{B}}\vartheta \right)-\left(\widehat{\mathfrak{B}}\widehat{\vartheta }\right)\parallel & \le [{\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|{\pi }_2}{\Gamma \left({\gamma }_1\right)}}\right)+{\omega }_3\left|\lambda \right|n{\displaystyle \frac{{\pi }_2}{\Gamma \left({\gamma }_1\right)}}+k{\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}\hfill \\ \hfill & +k({\omega }_4^{\prime }+1){\displaystyle \frac{{\pi }_2}{\Gamma \left({\gamma }_1\right)}}.+\left({\omega }_3\left|\lambda \right|n+k+k({\omega }_4^{\prime }+1)\right){\nu }_2]\parallel e\parallel .\hfill \end{array}$$

Therefore,

$$|e\left(\widehat{x}\right)|\le {\mathcal{L}}_1\parallel e\parallel +{\Phi }_1{\varsigma }_1,$$

with ${\mathcal{L}}_1$ and ${\Phi }_1$ as defined in the theorem statement.

Taking the supremum over $\widehat{x}\in [0,{\widehat{x}}_1]$ gives

$${\parallel e\parallel }_{[0,{\widehat{x}}_1]}\le {\mathcal{L}}_1\parallel e\parallel +{\Phi }_1{\varsigma }_1.$$

(25)

Case 2: $\widehat{x}\in [{\widehat{x}}_k,{\widehat{x}}_{k+1}],k=1,\dots ,n$. An analogous computation yields

$${\parallel e\parallel }_{[{\widehat{x}}_k,{\widehat{x}}_{k+1}]}\le {\mathcal{L}}_2\parallel e\parallel +{\Phi }_2{\varsigma }_1.$$

(26)

Since $\parallel e\parallel =max\{\parallel e{\parallel }_{[0,{\widehat{x}}_1]},\parallel e{\parallel }_{[{\widehat{x}}_1,{\widehat{x}}_2]},\dots ,\parallel e{\parallel }_{[{\widehat{x}}_n,1]}\}$, inequalities (25) and (26) together imply

$$\parallel e\parallel \le max\{{\mathcal{L}}_1,{\mathcal{L}}_2\}\parallel e\parallel +max\{{\Phi }_1,{\Phi }_2\}{\varsigma }_1.$$

By hypothesis (20), $max\{{\mathcal{L}}_1,{\mathcal{L}}_2\}<1$, hence

$$\parallel e\parallel \le {\displaystyle \frac{max\{{\Phi }_1,{\Phi }_2\}}{1-max\{{\mathcal{L}}_1,{\mathcal{L}}_2\}}}{\varsigma }_1.$$

Taking $\mathcal{C}=max\left\{{\displaystyle \frac{{\Phi }_1}{1-{\mathcal{L}}_1}},{\displaystyle \frac{{\Phi }_2}{1-{\mathcal{L}}_2}}\right\}$ gives exactly (19), which completes the proof. □

Quantitative Interpretation of Ulam–Hyers Stability

The inequality $\parallel \vartheta -\widehat{\vartheta }\parallel \le \mathcal{C}{\varsigma }_1$ provides a quantitative reliability measure for the impulsive hybrid fractional system. The stability constant $\mathcal{C}$ depends explicitly on the Lipschitz constants ${\pi }_1,{\pi }_2$ of the nonlinearities ${\varpi }_1$ and ${\Lambda }_1$; the fractional kernel bounds through $\Gamma ({\gamma }_1+1),\Gamma \left({\gamma }_1\right),\Gamma ({\gamma }_1-1)$; the impulse sensitivity coefficient ${\nu }_2$; the boundary parameters $\lambda ,\mu ,\eta ,\xi$ via the weights ${\omega }_1,\dots ,{\omega }_5$; and the number and location of impulse points $n,{\widehat{x}}_1,\dots ,{\widehat{x}}_n$. A smaller value of $\mathcal{C}$ indicates stronger robustness of the system against perturbations: if the residual ${\mathcal{Z}}_1\left(\vartheta \right)$ is bounded by ${\varsigma }_1$, then every approximate solution remains within a distance $\mathcal{C}{\varsigma }_1$ from an exact solution. This explicit bound enables the assessment of model sensitivity directly from the system parameters, without requiring numerical simulation.

5. Examples

In this section, we present three concrete examples that validate our theoretical results. All examples respect the standing assumptions $1<{\gamma }_1\le 2$, and all constants are computed explicitly to verify the conditions of Theorems 2 and 3.

5.1. Existence for a Well-Posed Hybrid System

Consider the impulsive hybrid fractional differential equation:

$$\begin{array}{cc}\hfill {}^{c}D^{1.5}\left({\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_1(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right)& ={\displaystyle \frac{1}{40}}{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{1+{\vartheta }^2\left(\widehat{x}\right)}},\widehat{x}\in [0,1],\widehat{x}\ne {\displaystyle \frac{1}{4}},\hfill \\ \hfill \Delta \left[{\displaystyle \frac{\vartheta \left(\frac{1}{4}\right)}{{\varpi }_1\left(\frac{1}{4},\vartheta \left(\frac{1}{4}\right)\right)}}\right]& ={\displaystyle \frac{1}{20}}{\displaystyle \frac{\vartheta \left(\frac{1}{4}\right)}{2+{\vartheta }^2\left(\frac{1}{4}\right)}},\hfill \end{array}$$

(27)

$$\begin{array}{cc}\hfill \Delta \left[{\displaystyle \frac{{\vartheta }^{\prime }\left(\frac{1}{4}\right)}{{\varpi }_1^{\prime }\left(\frac{1}{4},\vartheta \left(\frac{1}{4}\right)\right)}}\right]& ={\displaystyle \frac{1}{30}}{\displaystyle \frac{\vartheta \left(\frac{1}{4}\right)}{3+|\vartheta \left(\frac{1}{4}\right)|}},\hfill \end{array}$$

(28)

with nonlocal boundary conditions

$${\displaystyle \frac{{\vartheta }^{\prime }\left(0\right)}{{\varpi }_1^{\prime }(0,\vartheta \left(0\right))}}+{\displaystyle \frac{\vartheta \left(0.6\right)}{{\varpi }_1(0.6,\vartheta \left(0.6\right))}}=0.$$

(29)

Parameter specification:

$${\gamma }_1=1.5,\lambda =1,\mu =1,\eta =0.6,\xi =0,n=1,{\widehat{x}}_1={\displaystyle \frac{1}{4}}.$$

Function definitions:

$${\varpi }_1(\widehat{x},\vartheta )=1.2+{\displaystyle \frac{1}{8}}{\displaystyle \frac{\vartheta }{1+{\vartheta }^2}},{\Lambda }_1(\widehat{x},\vartheta )={\displaystyle \frac{1}{40}}{\displaystyle \frac{\vartheta }{1+{\vartheta }^2}}.$$

Step 1: Verification of $\left(K1\right)$–$\left(K2\right)$. For all $\vartheta \in \mathbb{R}$:

$$\begin{array}{cc}\hfill |{\varpi }_1(\widehat{x},\vartheta )|& \le 1.2+{\displaystyle \frac{1}{8}}·{\displaystyle \frac{1}{2}}=1.2625\Rightarrow L_1=1.2625,\hfill \\ \hfill |{\Lambda }_1(\widehat{x},\vartheta )|& \le {\displaystyle \frac{1}{40}}·{\displaystyle \frac{1}{2}}=0.0125\Rightarrow L_2=0.0125.\hfill \end{array}$$

The Lipschitz constants are bounded by the maximum derivatives:

$$\left|{\displaystyle \frac{\partial {\varpi }_1}{\partial \vartheta }}\right|\le {\displaystyle \frac{1}{8}}·{\displaystyle \frac{1}{{(1+{\vartheta }^2)}^{3/2}}}\le {\displaystyle \frac{1}{8}}\Rightarrow {\pi }_1=0.125,$$

$$\left|{\displaystyle \frac{\partial {\Lambda }_1}{\partial \vartheta }}\right|\le {\displaystyle \frac{1}{40}}·{\displaystyle \frac{1}{{(1+{\vartheta }^2)}^{3/2}}}\le {\displaystyle \frac{1}{40}}\Rightarrow {\pi }_2=0.025.$$

Step 2: Computation of ${\omega }_i$. First, compute the denominator:

$$[1-\mu (\lambda +\xi -\eta \left)\right]=[1-1(1+0-0.6\left)\right]=1-0.4=0.6.$$

Then:

$$\begin{array}{cc}\hfill {\omega }_1& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{\mu (\lambda +\xi -\widehat{x})}{0.6}}\right|=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1(1+0-\widehat{x})}{0.6}}\right|={\displaystyle \frac{1}{0.6}}=1.6667,\hfill \\ \hfill {\omega }_2& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1+\mu (\eta -\widehat{x})}{0.6}}\right|=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1+1(0.6-\widehat{x})}{0.6}}\right|={\displaystyle \frac{1.6}{0.6}}=2.6667,\hfill \\ \hfill {\omega }_3& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1-\mu (\lambda +\xi +\widehat{x}-\eta )}{0.6}}\right|=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1-1(1+0+\widehat{x}-0.6)}{0.6}}\right|\hfill \\ \hfill & =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{0.4-\widehat{x}}{0.6}}\right|={\displaystyle \frac{0.4}{0.6}}=0.6667(\mathrm{at}\widehat{x}=0),\hfill \\ \hfill {\omega }_4& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{(1-\mu \lambda )\widehat{x}+\mu \lambda \eta -\xi }{0.6}}\right|=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{(1-1)\widehat{x}+1\times 0.6-0}{0.6}}\right|={\displaystyle \frac{0.6}{0.6}}=1.\hfill \end{array}$$

Step 3: Impulse bounds.

$${\nu }_1=max\left\{\underset{\vartheta \in \mathbb{R}}{sup}\left|{\displaystyle \frac{1}{20}}{\displaystyle \frac{\vartheta }{2+{\vartheta }^2}}\right|,\underset{\vartheta \in \mathbb{R}}{sup}\left|{\displaystyle \frac{1}{30}}{\displaystyle \frac{\vartheta }{3+\left|\vartheta \right|}}\right|\right\}=max\left\{{\displaystyle \frac{1}{20\sqrt{2}}},{\displaystyle \frac{1}{30}}\right\}=max\{0.03536,0.03333\}=0.03536.$$

Step 4: Verification of Theorem 2 conditions. We need the Gamma function values: $\Gamma ({\gamma }_1+1)=\Gamma \left(2.5\right)=1.3293$, $\Gamma \left({\gamma }_1\right)=\Gamma \left(1.5\right)=0.8862$.

Compute the radius r:

$$\begin{array}{cc}\hfill r& ={\displaystyle \frac{({\omega }_1+{\omega }_2+1)L_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{({\omega }_2+{\omega }_{3}n)\left|\lambda \right|L_2+{\omega }_{4}n{L}_2}{\Gamma \left({\gamma }_1\right)}}+({\omega }_3\left|\lambda \right|+{\omega }_4)n{\nu }_1\hfill \\ & ={\displaystyle \frac{(1.6667+2.6667+1)\times 0.0125}{1.3293}}+{\displaystyle \frac{(2.6667+0.6667\times 1)\times 1\times 0.0125+1\times 1\times 0.0125}{0.8862}}+(0.6667\times 1+1)\times 1\times 0.03536\hfill \\ & ={\displaystyle \frac{5.3334\times 0.0125}{1.3293}}+{\displaystyle \frac{3.3334\times 0.0125+0.0125}{0.8862}}+1.6667\times 0.03536\hfill \\ & ={\displaystyle \frac{0.06667}{1.3293}}+{\displaystyle \frac{0.04167+0.0125}{0.8862}}+0.05893\hfill \\ & =0.05015+{\displaystyle \frac{0.05417}{0.8862}}+0.05893\hfill \\ & =0.05015+0.06112+0.05893=0.1702.\hfill \end{array}$$

Now check condition (12) of Theorem 2:

$$L_{1}r=1.2625\times 0.1702=0.2148<1.$$

Additionally, ${\pi }_{1}r=0.125\times 0.1702=0.02128<1$.

Since all conditions of Theorem 2 are satisfied, the impulsive hybrid system (27)–(29) possesses at least one solution on $[0,1]$.

5.2. Alternative System Satisfying Existence Conditions

Consider a different configuration:

$$\begin{array}{cc}\hfill {}^{c}D^{1.3}\left({\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_2(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right)& ={\displaystyle \frac{1}{50}}{\displaystyle \frac{arctan\left(\vartheta \left(\widehat{x}\right)\right)}{2}},\widehat{x}\in [0,1],\widehat{x}\ne {\displaystyle \frac{1}{3}},\hfill \end{array}$$

(30)

$$\begin{array}{cc}\hfill \Delta \left[{\displaystyle \frac{\vartheta \left(\frac{1}{3}\right)}{{\varpi }_2\left(\frac{1}{3},\vartheta \left(\frac{1}{3}\right)\right)}}\right]& ={\displaystyle \frac{1}{25}}{\displaystyle \frac{\vartheta \left(\frac{1}{3}\right)}{4+{\vartheta }^2\left(\frac{1}{3}\right)}},\hfill \end{array}$$

(31)

with boundary conditions

$$2{\displaystyle \frac{{\vartheta }^{\prime }\left(0\right)}{{\varpi }_2^{\prime }(0,\vartheta \left(0\right))}}+{\displaystyle \frac{\vartheta \left(0.7\right)}{{\varpi }_2(0.7,\vartheta \left(0.7\right))}}=0.$$

(32)

Parameters: ${\gamma }_1=1.3$, $\lambda =2$, $\mu =1$, $\eta =0.7$, $\xi =0$, $n=1$, ${\widehat{x}}_1=1/3$.

Functions: ${\varpi }_2(\widehat{x},\vartheta )=1.1+0.05sin\left(\vartheta \right)$, ${\Lambda }_2(\widehat{x},\vartheta )={\displaystyle \frac{1}{50}}{\displaystyle \frac{arctan\left(\vartheta \right)}{2}}$.

Bounds: $L_1=1.15$, $L_2={\displaystyle \frac{\pi }{200}}\approx 0.01571$, ${\pi }_1=0.05$, ${\pi }_2={\displaystyle \frac{1}{100}}=0.01$.

${\omega }_i$ constants: $[1-\mu (\lambda +\xi -\eta \left)\right]=1-1(2+0-0.7)=0.3$,

$$\begin{array}{cc}\hfill {\omega }_1& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1(2+0-\widehat{x})}{0.3}}\right|={\displaystyle \frac{2}{0.3}}=6.6667,\hfill \\ \hfill {\omega }_2& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1+1(0.7-\widehat{x})}{0.3}}\right|={\displaystyle \frac{1.7}{0.3}}=5.6667,\hfill \\ \hfill {\omega }_3& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1-1(2+0+\widehat{x}-0.7)}{0.3}}\right|=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{-0.3-\widehat{x}}{0.3}}\right|={\displaystyle \frac{1.3}{0.3}}=4.3333,\hfill \\ \hfill {\omega }_4& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{(1-2)\widehat{x}+2\times 0.7-0}{0.3}}\right|=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{-\widehat{x}+1.4}{0.3}}\right|={\displaystyle \frac{1.4}{0.3}}=4.6667.\hfill \end{array}$$

Impulse bound: ${\nu }_1={\displaystyle \frac{1}{25}}·{\displaystyle \frac{1}{4}}=0.01$.

Verification: $\Gamma ({\gamma }_1+1)=\Gamma \left(2.3\right)=1.1667$, $\Gamma \left({\gamma }_1\right)=\Gamma \left(1.3\right)=0.8975$.

$$\begin{array}{cc}\hfill r& ={\displaystyle \frac{(6.6667+5.6667+1)\times 0.01571}{1.1667}}+{\displaystyle \frac{(5.6667+4.3333\times 1)\times 2\times 0.01571+4.6667\times 1\times 0.01571}{0.8975}}+(4.3333\times 2+4.6667)\times 1\times 0.01\hfill \\ & ={\displaystyle \frac{13.3334\times 0.01571}{1.1667}}+{\displaystyle \frac{10\times 0.03142+0.07328}{0.8975}}+13.3333\times 0.01\hfill \\ & =0.1796+0.4357+0.1333=0.7486.\hfill \end{array}$$

Check: $L_{1}r=1.15\times 0.7486=0.8609<1$. All conditions satisfied.

5.3. Explicit Verification of Ulam–Hyers Stability

We now construct an example specifically designed to verify Theorem 3. Consider:

$$\begin{array}{cc}\hfill {}^{c}D^{1.7}\left({\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{{\varpi }_3(\widehat{x},\vartheta \left(\widehat{x}\right))}}\right)& ={\displaystyle \frac{1}{60}}{\displaystyle \frac{\vartheta \left(\widehat{x}\right)}{3+\left|\vartheta \right(\widehat{x}\left)\right|}},\widehat{x}\in [0,1],\widehat{x}\ne {\displaystyle \frac{1}{2}},\hfill \end{array}$$

(33)

$$\begin{array}{cc}\hfill \Delta \left[{\displaystyle \frac{\vartheta \left(\frac{1}{2}\right)}{{\varpi }_3\left(\frac{1}{2},\vartheta \left(\frac{1}{2}\right)\right)}}\right]& ={\displaystyle \frac{1}{40}}{\displaystyle \frac{\vartheta \left(\frac{1}{2}\right)}{5+{\vartheta }^2\left(\frac{1}{2}\right)}},\hfill \end{array}$$

(34)

with

$${\displaystyle \frac{{\vartheta }^{\prime }\left(0\right)}{{\varpi }_3^{\prime }(0,\vartheta \left(0\right))}}+{\displaystyle \frac{\vartheta \left(0.5\right)}{{\varpi }_3(0.5,\vartheta \left(0.5\right))}}=0.$$

(35)

Parameters: ${\gamma }_1=1.7$, $\lambda =1$, $\mu =1$, $\eta =0.5$, $\xi =0$, $n=1$, ${\widehat{x}}_1=1/2$.

Functions: ${\varpi }_3(\widehat{x},\vartheta )=1.3+0.06{\displaystyle \frac{\vartheta }{1+{\vartheta }^2}}$, ${\Lambda }_3(\widehat{x},\vartheta )={\displaystyle \frac{1}{60}}{\displaystyle \frac{\vartheta }{3+\left|\vartheta \right|}}$.

Bounds: $L_1=1.36$, $L_2={\displaystyle \frac{1}{60}}·{\displaystyle \frac{1}{3}}=0.005556$, ${\pi }_1=0.06$, ${\pi }_2={\displaystyle \frac{1}{60}}·{\displaystyle \frac{1}{9}}=0.001852$, ${\nu }_2={\displaystyle \frac{1}{40}}·{\displaystyle \frac{1}{5}}=0.005$.

${\omega }_i$ constants: $[1-\mu (\lambda +\xi -\eta \left)\right]=1-1(1+0-0.5)=0.5$,

$$\begin{array}{cc}\hfill {\omega }_1& ={\displaystyle \frac{1(1+0-0)}{0.5}}=2,\hfill \\ \hfill {\omega }_2& ={\displaystyle \frac{1+1(0.5-0)}{0.5}}=3,\hfill \\ \hfill {\omega }_3& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{1-1(1+0+\widehat{x}-0.5)}{0.5}}\right|=\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{0.5-\widehat{x}}{0.5}}\right|=1,\hfill \\ \hfill {\omega }_4& =\underset{\widehat{x}\in [0,1]}{sup}\left|{\displaystyle \frac{(1-1)\widehat{x}+1\times 0.5-0}{0.5}}\right|=1.\hfill \end{array}$$

Step 1: Compute the invariant ball radius r. $\Gamma ({\gamma }_1+1)=\Gamma \left(2.7\right)=1.6765$, $\Gamma \left({\gamma }_1\right)=\Gamma \left(1.7\right)=0.9086$, ${\nu }_1={\displaystyle \frac{1}{40}}·{\displaystyle \frac{1}{5}}=0.005$.

$$\begin{array}{cc}\hfill r& ={\displaystyle \frac{(2+3+1)\times 0.005556}{1.6765}}+{\displaystyle \frac{(3+1\times 1)\times 1\times 0.005556+1\times 1\times 0.005556}{0.9086}}+(1\times 1+1)\times 1\times 0.005\hfill \\ & ={\displaystyle \frac{6\times 0.005556}{1.6765}}+{\displaystyle \frac{4\times 0.005556+0.005556}{0.9086}}+2\times 0.005\hfill \\ & =0.01989+0.03056+0.01=0.06045.\hfill \end{array}$$

Step 2: Compute stability constants for $\widehat{x}\in [0,{\displaystyle \frac{1}{2}}]$ ($k=0$). First compute ${\Psi }_1$:

$$\begin{array}{cc}\hfill {\Psi }_1& ={\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{{\pi }_2}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|{\pi }_2}{\Gamma \left({\gamma }_1\right)}}\right)+{\omega }_3\left|\lambda \right|n{\displaystyle \frac{{\pi }_2}{\Gamma \left({\gamma }_1\right)}}\hfill \\ & +\left[{\omega }_3\left|\lambda \right|n\right]{\nu }_2\hfill \\ & ={\displaystyle \frac{0.001852}{1.6765}}+{\displaystyle \frac{2\times 0.001852}{1.6765}}+3\left({\displaystyle \frac{0.001852}{1.6765}}+{\displaystyle \frac{1\times 0.001852}{0.9086}}\right)+1\times 1\times 1\times {\displaystyle \frac{0.001852}{0.9086}}\hfill \\ & +[1\times 1\times 1]\times 0.005\hfill \\ & =0.001105+0.002210+3(0.001105+0.002039)+0.002039+0.005\hfill \\ & =0.001105+0.002210+0.009432+0.002039+0.005=0.019786.\hfill \end{array}$$

Now compute ${\mathcal{L}}_1$:

$${\mathcal{L}}_1={\pi }_{1}r+{\pi }_1{L}_1{\Psi }_1=0.06\times 0.06045+0.06\times 1.36\times 0.019786=0.003627+0.001614=0.005241.$$

Compute ${\Phi }_1$:

$$\begin{array}{cc}\hfill {\Phi }_1& =L_1\left[{\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{{\omega }_1}{\Gamma ({\gamma }_1+1)}}+{\omega }_2\left({\displaystyle \frac{1}{\Gamma ({\gamma }_1+1)}}+{\displaystyle \frac{\left|\lambda \right|}{\Gamma \left({\gamma }_1\right)}}\right)+{\displaystyle \frac{{\omega }_3\left|\lambda \right|n}{\Gamma \left({\gamma }_1\right)}}\right]\hfill \\ & =1.36\left[{\displaystyle \frac{1}{1.6765}}+{\displaystyle \frac{2}{1.6765}}+3\left({\displaystyle \frac{1}{1.6765}}+{\displaystyle \frac{1}{0.9086}}\right)+{\displaystyle \frac{1\times 1\times 1}{0.9086}}\right]\hfill \\ & =1.36\left[0.5965+1.1930+3(0.5965+1.1005)+1.1005\right]\hfill \\ & =1.36\left[0.5965+1.1930+5.0910+1.1005\right]=1.36\times 7.9810=10.8542.\hfill \end{array}$$

Step 3: Stability constant and interpretation. Since ${\mathcal{L}}_1=0.005241<1$, the stability constant is as follows:

$$\mathcal{C}={\displaystyle \frac{{\Phi }_1}{1-{\mathcal{L}}_1}}={\displaystyle \frac{10.8542}{0.994759}}=10.912.$$

Interpretation: For any approximate solution $\vartheta$ satisfying $\parallel {\mathcal{Z}}_1\left(\vartheta \right)\parallel \le {\varsigma }_1$, Theorem 3 guarantees the existence of an exact solution $\widehat{\vartheta }$ with $\parallel \vartheta -\widehat{\vartheta }\parallel \le 10.912{\varsigma }_1$. For instance, if the residual is bounded by ${\varsigma }_1=0.01$, then the deviation from an exact solution is at most $0.10912$, demonstrating the quantitative stability guaranteed by our theory.

6. Conclusions

This study has successfully established the existence and Ulam–Hyers stability of solutions for a class of impulsive hybrid fractional differential equations involving the Caputo derivative. By extending Dhage’s fixed-point theorem within a two-operator framework, we derived sufficient conditions ensuring the existence of at least one solution. Furthermore, we provided a quantitative stability analysis through the computation of explicit stability constants, guaranteeing that small perturbations lead to proportionally bounded deviations from exact solutions. The theoretical framework was rigorously validated through three carefully constructed examples, each demonstrating the applicability of our results under different parameter configurations and nonlinear structures.

Future research may focus on extending the present framework to more general fractional operators, such as the $\psi$-Caputo or Hadamard derivatives, and to systems with non-instantaneous impulses or variable-time impulses. From an applied perspective, integrating the Ulam–Hyers stability analysis with data-driven methods—such as statistical validation of stability bounds using experimental or simulated data, or image-based analysis of solution sensitivity—could enhance the method’s practical applicability and provide richer interpretation for real-world systems. Developing numerical algorithms for efficient computation of stability constants and implementing them in software tools would further bridge the gap between theoretical guarantees and practical implementation in fields such as control theory, biological modeling, and engineering systems.