On the Stability of Fractional Non-Autonomous Evolution Equations with Impulsive Effects

Abstract

This work addresses the Ulam–Hyers stability of mild solutions of fractional noninstantaneous evolution nonlinear equations subject to noninstantaneous impulses in the Banach space through an approach based on the measure of noncompactness. Mild solutions are formulated via operators generated by a closed linear operator and a probability density function. Existence results are derived by applying the fixed point theorem for k-set contractive operators. Furthermore, Ulam–Hyers stability is established under certain hypotheses, and an illustrative example is provided to validate the derived results.

1. Introduction

In recent years, fractional calculus has emerged as an effective tool for investigating complex phenomena in mathematics, physics, engineering, and various applied sciences. Unlike classical derivatives of integer order, fractional-order derivatives capture the hereditary and memory properties of different materials and processes. Because of this feature, $\mathcal{F}\mathcal{D}\mathcal{E}\mathcal{s}$ are often more suitable for modeling real-world phenomena than integer-order derivatives [1,2,3,4]. Within this discipline, a significant research direction is the study of fractional evolution equations, which provide abstract formulations for a wide range of problems arising in the applied sciences. These equations in their semilinear form have received increasing attention, and numerous results have been obtained regarding the existence and qualitative properties of their solutions [5,6,7,8,9,10]. While much of the existing research has been devoted to autonomous systems, in which governing operators are independent of time, many realistic problems are inherently time dependent. Nonautonomous fractional evolution equations thus provide a more suitable framework for analyzing parabolic-type problems where operators vary with time [11]. Recent advances have established results for nonautonomous systems with delays and nonlocal conditions, highlighting the growing importance of this research direction [12,13,14]. Pioneer contributions in this direction include the work of El-Borai et al. [15] on fundamental solutions for linear problems and Ouyang [16], who applied fixed-point techniques to solve time fractional reaction–diffusion equations with delay. Zhu and coauthors [17,18] extended these studies to abstract the Banach space and established results concerning its existence and uniqueness. In addition to time dependence, many processes in science and engineering are subject to impulsive effects. While instantaneous impulses occur at discrete moments and produce abrupt changes, noninstantaneous impulses act over finite time intervals, which makes them more realistic for modeling situations such as drug delivery, population dynamics, or control systems with gradual effects. As a result, fractional systems with noninstantaneous impulses have become a prominent topic of investigation. In addition to these developments, the concept of Ulam–Hyers stability has emerged as a useful tool for studying the robustness of solutions. Originating from a problem posed by Ulam [19] and later refined by Hyers [20]. This stability notion ensures that approximate solutions remain close to exact ones. The concept has been extended to various classes of $\mathcal{F}\mathcal{D}\mathcal{E}\mathcal{s}$ and has proven effective in understanding the stability properties of systems involving memory and nonlocal effects [21,22,23]. These contributions have enriched the theory and expanded its applications to real-world systems. Despite advances in $\mathcal{F}\mathcal{N}\mathcal{A}\mathcal{E}\mathcal{E}\mathcal{s}$, impulsive systems, and stability theory, the study of Ulam–Hyers stability for $\mathcal{F}\mathcal{N}\mathcal{A}\mathcal{E}\mathcal{E}\mathcal{s}$ subject to noninstantaneous impulses is still limited. The time-dependence of the operators, the fractional-order dynamics, and the noninstantaneous impulsive conditions raise new challenges that cannot be addressed by existing results for autonomous or instantaneous impulses cases. Motivated by these considerations, this paper is devoted to establishing Ulam–Hyers stability results for the following $\mathcal{F}\mathcal{N}\mathcal{A}\mathcal{E}\mathcal{E}\mathcal{s}$.

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill {}^{\mathfrak{c}}\mathcal{D}_{0+}^{\mathrm{♭}}\varpi \left(\mathcal{K}\right)+\mathrm{\aleph }\left(\mathcal{K}\right)\varpi \left(\mathcal{K}\right)& =\mathcal{Z}(\mathcal{K},\varpi \left({\mathcal{Q}}_1\left(\mathcal{K}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathcal{K}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathcal{K}\right)\right)),\hfill \\ & \mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}],\beta =0,1,2,\dots ,n;\hfill \\ \hfill \varpi \left(\mathcal{K}\right)& ={\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(\mathcal{K},\varpi \left({\mathcal{K}}_{\beta }^{-}\right)\right),\mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }],\beta =1,2,\dots ,n,\hfill \\ \hfill \varpi \left(0\right)& ={\mathrm{\aleph }}^{-1}\left(0\right){\varpi }_0.\hfill \end{array}\hfill \end{array}\right.$$

(1)

where ${}^{\mathfrak{c}}\mathcal{D}_{0+}^{\mathrm{♭}}$ represents the fractional-order derivative of Caputo’s type with respect to time $\mathcal{K}$, with fixed lower terminal 0, and $0<\mathrm{♭}\le 1$. $\Xi =[0,\mathrm{\top }]$, where $\mathrm{\top }>0$ is constant. Let $\mathrm{\aleph }\left(\mathcal{K}\right)$ be a family of closed linear operators with domain $D\left(\mathrm{\aleph }\right)$, dense in the Banach space ℷ and mapping ℷ into ℷ. $\mathcal{Z}:\Xi \times {\mathrm{\gimel }}^m⟶\mathrm{\gimel }$ is continuous in ℷ, and ${\mathcal{Q}}_{\mathsf{ȷ}}:\Xi ⟶\Xi$ are continuous functions such that $0\le {\mathcal{Q}}_{\mathsf{ȷ}}\left(\mathcal{K}\right)\le \mathcal{K}$, $\mathsf{ȷ}=1,2,\dots ,m$, where m is a positive integer. $0=s_0<{\mathcal{K}}_1<s_1<{\mathcal{K}}_2<\cdots <s_{\beta }<{\mathcal{K}}_{\beta +1}=\mathrm{\top }$. The mapping ${\mathrm{\wp }}_{\beta }:\Xi \times \mathrm{\gimel }⟶\mathrm{\gimel }$ denotes non-instantaneous impulsive function.

This paper is organized as follows: Section 2 outlines the basic definitions, symbols, and preliminaries associated with fractional calculus. In Section 3, we present the definition of mild solutions and demonstrate the existence results by applying the fixed point techniques. In Section 4, the main stability results are discussed along with essential concepts of stability. An illustrative example with non-instantaneous impulses is given in Section 5.

2. Preliminaries

This section introduces the key notations, definitions, and preliminaries related to fractional calculus, the Hausdorff measure of non-compactness, the fixed point theorem for k-set contractive operators and the operators $\kappa (\mathcal{K},s),\mathfrak{G}(\mathcal{K},\mathrm{♄})$ and $\mathrm{\vee }\left(\mathcal{K}\right)$ used in the subsequent analysis. The space $C(\Xi ,\mathrm{\gimel })$ denotes the collection of all continuous functions from the interval $\Xi$ into ℷ, which forms a Banach space under the supremum norm

$${\parallel \varpi \parallel }_C=\sup \{\parallel \varpi \left(\mathcal{K}\right)\parallel :\mathcal{K}\in \Xi \}\forall \varpi \in C(\Xi ,\mathrm{\gimel }).$$

The space $L\left(\mathrm{\gimel }\right)$ consists of all bounded linear operators defined on ℷ forming a Banach space under the operator norm topology. We denote by $L^1(\Xi ,\mathrm{\gimel })$ the complete normed linear space of all ℷ valued Bochner integrable functions defined on $\Xi$ equipped with the norm $\parallel {\varpi }_1\parallel ={\int }_0^{\mathcal{K}}\parallel \varpi \left(\mathcal{K}\right)\parallel d\mathcal{K}.$

Definition 1

([2]). Let $\mathcal{Z}\in L^1\left([0,\infty ),\mathbb{R})\right),$ the Riemann–Liouville fractional integral of order $\mathrm{♭}>0$ with lower limit zero is defined by

$$I_0^{\mathrm{♭}}={\displaystyle \frac{1}{\Gamma \left(\mathrm{♭}\right)}}{\int }_0^{\mathcal{K}}{(\mathcal{K}-s)}^{\mathrm{♭}-1}\mathcal{Z}\left(s\right)ds.$$

Definition 2

([2]). Let $\mathcal{Z}:[0,\infty )⟶\mathbb{R}$ be an n-times differentiable function. The fractional derivative of Caputo’s type of order ♭ with lower limit zero can be defined as:

$${}^{\mathfrak{c}}\mathcal{D}_0^{\mathrm{♭}}\mathcal{Z}\left(\mathcal{K}\right)={\displaystyle \frac{1}{\Gamma (n-\mathrm{♭})}}{\int }_0^{\mathcal{K}}{(\mathcal{K}-s)}^{n-\mathrm{♭}-1}{\mathcal{Z}}^{\left(n\right)}\left(s\right)ds,\mathit{where}n-1<\mathrm{♭}<n,n\in \mathbb{N}.$$

Remark 1

([13]). In Definitions 1 and 2 the integrals are taken as Bochner integrals. A measurable function $\mathcal{Z}:[0,\infty )⟶\mathrm{\gimel }$ is Bochner integrable provided that the scalar function $\parallel \mathcal{Z}\parallel$ is Lebesgue integrable on $[0,\infty ).$

The linear operator $-\mathrm{\aleph }\left(\mathcal{K}\right)$ is assumed to satisfy the following hypotheses in this study:

(${\mathcal{A}}_1$)

For any $\varrho$ with Re $\varrho \ge 0$, the inverse operator ${[\varrho I+\mathrm{\aleph }\left(\mathcal{K}\right)]}^{-1}$ exists and is bounded in $L\left(\mathrm{\gimel }\right)$.

$${\parallel [\varrho I+\mathrm{\aleph }\left(\mathcal{K}\right)]}^{-1}\parallel \le {\displaystyle \frac{𝐹}{\left|\varrho \right|+1}},$$

where $𝐹$ is positive constant, independent of the parameters $\mathcal{K}$ and $\varrho$.

(${\mathcal{A}}_2$)

For any $\mathcal{K},\mathcal{Q},s\in \Xi$, a constant $\rho \in (0,1]$ exists such that

$$\parallel [\mathrm{\aleph }\left(\mathcal{K}\right)-\mathrm{\aleph }\left(\mathcal{Q}\right)]{\mathrm{\aleph }}^{-1}{\left(s\right)\parallel \le 𝐹|\mathcal{K}-\mathcal{Q}|}^{\rho };$$

where $\rho$ and $𝐹$ are the positive constants independent of $\mathcal{K},\mathcal{Q}$ and s.

Remark 2

([13]). The operator $-\mathrm{\aleph }\left(s\right)$ generates an analytic semigroup $e^{-\mathcal{K}\mathrm{\aleph }\left(s\right)}$ for each $s\in \Xi$ and $\mathcal{K}>0$. There exists a constant $𝐹>0$ independent of $\mathcal{K}$ and s such that

$$\parallel {\mathrm{\aleph }}^n\left(s\right)e^{-\mathcal{K}\mathrm{\aleph }\left(s\right)}\parallel \le {\displaystyle \frac{𝐹}{{\mathcal{K}}^n}},\mathit{where}n=0,1;\mathcal{K}>0,s\in \Xi .$$

Remark 3

([13]). In assumption (${\mathcal{A}}_1$), if we take $\varrho =0$ and $\mathcal{K}=0$ then ∃ $𝐹>0$ independent of parameters $\mathcal{K}$ and $\varrho$ such that

$$\parallel {\mathrm{\aleph }}^{-1}\left(0\right)\parallel \le 𝐹.$$

If we take $\varrho =0$ and $\mathcal{K}=s_{\beta }$

$$\parallel {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right)\parallel \le 𝐹.$$

Definition 3.

We define $\varpi \in C(\Xi ,\mathrm{\gimel })$ as a mild solution of $\mathcal{F}\mathcal{N}\mathcal{A}\mathcal{E}\mathcal{E}\mathcal{s}$ (1) if it satisfies the following condition:

$$\begin{array}{cc}\hfill \varpi \left(\mathcal{K}\right)& ={\mathrm{\aleph }}^{-1}\left(0\right){\varpi }_0+{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right){\varpi }_0,d\mathrm{♄}\hfill \\ \hfill & +{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\hfill \\ \hfill & +{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)ds,d\mathrm{♄},\hfill \end{array}$$

where the operators $\kappa (\mathcal{K},s)$, $\mathfrak{G}(\mathcal{K},\mathrm{♄})$ and $\vee \left(\mathcal{K}\right)$ are as follows:

$$\begin{array}{cc}\hfill \kappa (\mathcal{K},s)& =\mathrm{♭}{\int }_0^{\infty }\theta {\mathcal{K}}^{\mathrm{♭}-1}{\chi }_{\mathrm{♭}}\left(\theta \right)e^{-{\mathcal{K}}^{\mathrm{♭}}\theta \mathrm{\aleph }\left(s\right)}d\theta ,\hfill \\ \hfill \mathfrak{G}(\mathcal{K},\mathrm{♄})& ={\displaystyle \sum _{\beta =1}^{\infty }}\mathfrak{G}\ast \beta (\mathcal{K},\mathrm{♄}),\hfill \\ \hfill \vee \left(\mathcal{K}\right)& =-\mathrm{\aleph }\left(\mathcal{K}\right){\mathrm{\aleph }}^{-1}\left(0\right)-{\int }_0^{\mathcal{K}}\mathfrak{G}(\mathcal{K},s)\mathrm{\aleph }\left(s\right){\mathrm{\aleph }}^{-1}\left(0\right)ds.\hfill \end{array}$$

${\chi }_{\mathrm{♭}}$ is the probability density function defined on $[0,\infty )$ such that its Laplace transform is given by

$$\begin{array}{cc}\hfill {\int }_0^{\infty }{e}^{-\theta \mathfrak{x}}{\chi }_{\mathrm{♭}}\left(\theta \right)d\theta & ={\displaystyle \sum _{\mathsf{ȷ}=0}^{\infty }}{\displaystyle \frac{{(-\mathfrak{x})}^{\mathsf{ȷ}}}{\Gamma (1+\mathrm{♭}\mathsf{ȷ})}},0<\mathrm{♭}\le 1,\mathfrak{x}>0,\hfill \\ \hfill \mathfrak{G}\ast 1(\mathcal{K},\mathrm{♄})& =\left[\mathrm{\aleph }\left(\mathcal{K}\right)-\mathrm{\aleph }\left(\mathrm{♄}\right)\right]\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄}),\hfill \\ \hfill \mathfrak{G}\ast \beta +1(\mathcal{K},\mathrm{♄})& ={\int }_{\mathrm{♄}}^{\mathcal{K}}\mathfrak{G}\ast \beta (\mathcal{K},s)\mathfrak{G}\ast 1(s,\mathrm{♄})ds,\beta =1,2,\cdots ,n.\hfill \end{array}$$

For additional information on the definition and related properties of the function ${\chi }_{\mathrm{♭}}$, the reader is referred to the paper [24].

Lemma 1

([13]). For any $\mathcal{K}\in \Xi$ and $\mathfrak{g}\in L^1[0,\mathrm{\top }],$ we have

$${\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}\mathfrak{g}\left(s\right)dsd\mathrm{♄}=\mathbb{B}(\mathrm{♭},\rho ){\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}+\rho -1}\mathfrak{g}\left(\mathrm{♄}\right)d\mathrm{♄}.$$

Definition 4

([25]). For any bounded subset $\mathfrak{S}$ of the Banach space ℷ, the Hausdorff measure of non-compactness is defined by

$$\mathcal{H}\left(\mathfrak{S}\right)=\inf \left\{\xi >0:\mathfrak{S}={\cup }_{\beta =1}^n{\mathfrak{S}}_{\beta }\mathit{and}\mathit{radii}{\mathfrak{S}}_{\beta }\le \xi ,\beta =1,2,\cdots ,n\right\}.$$

Remark 4

([25]). The following properties are satisfied by the bounded subsets $\widehat{U}$ and $\widehat{V}$ of Banach space ℷ:

1. 

If $\widehat{U}\subset \widehat{V}$ then $\mathcal{H}\left(\widehat{U}\right)\le \mathcal{H}\left(\widehat{V}\right).$

2. 

$\mathcal{H}\left(\widehat{U}\right)=\mathcal{H}(\overline{\widehat{U}}),$ where $\overline{\widehat{U}}$ denotes the closure of $\widehat{U}$.

3. 

$\mathcal{H}\left(\widehat{U}\right)=0$ iff $\widehat{U}$ is relatively compact in ℷ.

4. 

$\mathcal{H}(\widehat{U}+\widehat{V})\le \mathcal{H}\left(\widehat{U}\right)+\mathcal{H}\left(\widehat{V}\right),$ where $\widehat{U}+\widehat{V}=\left\{x+y:x\in \widehat{U},y\in \widehat{V}\right\}.$

In this article, $\mathcal{H}(·)$ and ${\mathcal{H}}_C(·)$ denote the Hausdorff measure of non-compactness on the bounded set ℷ and $C(\Xi ,\mathrm{\gimel }),$ respectively. For any $\mathcal{z}\subset C(\Xi ,\mathrm{\gimel })$ and $\mathcal{K}\in \Xi ,\mathcal{z}\left(\mathcal{K}\right)=\left\{\varpi \left(\mathcal{K}\right):\varpi \in \mathcal{z}\right\}$ then $\mathcal{z}\left(\mathcal{K}\right)\subset \mathrm{\gimel }$. If $\mathcal{z}\subset \mathcal{C}(\Xi ,\mathrm{\gimel })$ is bounded with respect to the supremum norm ${\parallel ·\parallel }_{\infty }$, then $\mathcal{z}\left(\mathcal{K}\right)$ is bounded in ℷ.

Lemma 2

([25]). Assume that ℷ is a Banach space and that $\mathcal{z}\subset C(\Xi ,\mathrm{\gimel })$ be a bounded and equicontinuous set. Then $\mathcal{H}\left(\mathcal{z}\right(\mathcal{K}\left)\right)$ is continuous on Ξ and ${\mathcal{H}}_C\left(\mathcal{z}\right)={\max }_{\mathcal{K}\in \Xi }\left\{\mathcal{H}\left(\mathcal{z}\right(\mathcal{K}\left)\right)\right\}.$

Lemma 3

([25]). If $\mathcal{z}$ is a bounded subset of Banach space ℷ then ∃ a countable set ${\mathcal{z}}_0\subset \mathcal{z}$ for which ${\mathcal{H}}_C\left(\mathcal{z}\right)\le {\mathcal{H}}_C\left({\mathcal{z}}_0\right)$.

Lemma 4

([25,26]). If ${\mathcal{z}}_0={\left\{{\varpi }_n\right\}}_{n=1}^{\infty }\subset L^1(\Xi ,\mathrm{\gimel })$ is a bounded and countable subset of ℷ then $\mathcal{H}\left({\left\{{\varpi }_n\right\}}_{n=1}^{\infty }\right)$ is Lebesgue integrable on Ξ and

$$\mathcal{H}\left({\int }_{\Xi }{\varpi }_n\left(\mathcal{K}\right)d\mathcal{K}\right)\le {\int }_{\Xi }\mathcal{H}\left({\left\{{\varpi }_n\left(\mathcal{K}\right)\right\}}_{n=1}^{\infty }\right)d\mathcal{K}.$$

Definition 5.

Let $\tilde{\mathcal{F}}$ be a non-empty subset of Banach space ℷ. A continuous mapping $Q:\tilde{\mathcal{F}}⟶\mathrm{\gimel }$ is called contractive if ∃ a constant ℓ$\in [0,1)$ such that for every bounded set $\mathcal{z}\subset \tilde{\mathcal{F}}$

$$\mathcal{H}\left(Q\left(\mathcal{z}\right)\right)\le \mathcal{\ell }\mathcal{H}\left(\mathcal{z}\right).$$

Lemma 5

([27]). Let ℷ be a Banach space, and let $\mathcal{z}\subset \mathrm{\gimel }$ be a bounded, closed and convex set on ℷ. Suppose that the operator $⅁:\mathcal{z}⟶\mathcal{z}$ is k-set contractive. Then, ⅁ admits at least one fixed point in $\mathcal{z}$.

3. Existence Results

This section presents and demonstrates the existence results under the following set of assumptions.

• For an arbitrary $r>0,$∃ constants $0\le \mathbb{k}<\min \left\{\mathrm{♭},\rho \right\},{\zeta }_{\mathcal{Z}}>0$ such that for any $\mathcal{K}\in \Xi$ and ${\varpi }_{\mathsf{ȷ}}\in \mathrm{\gimel }$ satisfying $\parallel {\varpi }_{\mathsf{ȷ}}\parallel \le \mathsf{r};\mathsf{ȷ}=1,2,\cdots ,m$

  $$\parallel \mathcal{Z}(\mathcal{K},{\varpi }_1,{\varpi }_2,\cdots ,{\varpi }_m)\parallel \le {\varphi }_{\mathsf{r}}\left(\mathcal{K}\right)\mathrm{and}\underset{\mathsf{r}⟶\infty }{\lim }\inf {\displaystyle \frac{\parallel {\varphi }_{\mathsf{r}}{\parallel }_{L^{\frac{1}{\mathbb{k}}}[0,\mathrm{\top }]}}{\mathsf{r}}}={\zeta }_{\mathcal{Z}}.$$
• For an arbitrary $\mathsf{r}>0$ and all $\varpi \in {\mathrm{\complement }}_{\mathsf{r}}=\left\{\varpi \in C(\Xi ,\mathrm{\gimel }),\parallel \varpi \left(\mathcal{K}\right)\parallel \le \mathsf{r}\right\}$, ∃ a constant ${\zeta }_{\mathrm{\wp }}>0$ and a non-decreasing continuous function ${\mathrm{\Im }}_{{\mathrm{\wp }}_{\beta }}:{\mathbb{R}}^{+}⟶{\mathbb{R}}^{+}$ such that $\parallel {\mathrm{\wp }}_{\beta }(\mathcal{K},\varpi )\parallel \le {\mathrm{\Im }}_{{\mathrm{\wp }}_{\beta }}\left(\mathsf{r}\right)$. Let ${\mathrm{\Im }}_{\mathrm{\wp }}={\max }_{1\le \beta \le n}\left\{{\mathrm{\Im }}_{{\mathrm{\wp }}_{\beta }}\right\}$ then

  $$\underset{\mathsf{r}⟶\infty }{\lim }\inf {\displaystyle \frac{{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right)}{\mathsf{r}}}={\zeta }_{\mathrm{\wp }}.$$
• $\mathcal{Z}(\mathcal{K},{\varpi }_1,{\varpi }_2,\cdots ,{\varpi }_m)$ is Lipschitz continuous function with Lipschitz constant ${\mathsf{L}}_{\mathcal{Z}}>0$, i.e.,

  $$\parallel \mathcal{Z}(\mathcal{K},{\varpi }_1,{\varpi }_2,\cdots ,{\varpi }_m)-\mathcal{Z}(\mathcal{K},{\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_m)\parallel \le {\mathsf{L}}_{\mathcal{Z}}{\displaystyle \sum _{\mathsf{ȷ}=1}^m}\parallel {\varpi }_i-{\vartheta }_i\parallel ,$$

  $$\mathrm{for}({\varpi }_1,{\varpi }_2,\cdots ,{\varpi }_m);({\vartheta }_1,{\vartheta }_2,\cdots ,{\vartheta }_m)\in {\mathbb{R}}^m.$$
• ${\mathrm{\wp }}_{\beta }\left(\mathcal{K},\varpi \right),\beta =1,2,\cdots ,n$ is Lipschitz continuous function with Lipschitz constant ${\mathsf{L}}_{{\mathrm{\wp }}_{\beta }}>0$, i.e., $\parallel {\mathrm{\wp }}_{\beta }\left(\mathcal{K},\varpi \left(\mathcal{K}\right)\right)-{\mathrm{\wp }}_{\beta }\left(\mathcal{K},\vartheta \left(\mathcal{K}\right)\right)\parallel \le {\mathsf{L}}_{{\mathrm{\wp }}_{\beta }}\parallel \varpi \left(\mathcal{K}\right)-\vartheta \left(\mathcal{K}\right)\parallel ,\forall \varpi ,\vartheta \in \mathrm{\gimel }.$ $\mathrm{Let}{\mathsf{L}}_{\mathrm{\wp }}={\sup }_{1\le \beta \le n}\left\{{\mathsf{L}}_{{\mathrm{\wp }}_{\beta }}\right\}.$
• For any bounded, countable and equicontinuous sets ${\mathfrak{D}}_{\mathsf{ȷ}}\subset \mathrm{\gimel }$, ∃ positive constants $L_{\mathsf{ȷ}},\mathsf{ȷ}=1,2,\cdots ,m$ such that $\mathcal{H}\left(\mathcal{Z}(\mathcal{K},{\mathfrak{D}}_1,{\mathfrak{D}}_2,\cdots ,{\mathfrak{D}}_m)\right)\le {\sum }_{\mathsf{ȷ}=1}^m{L}_{\mathsf{ȷ}}\mathcal{H}\left({\mathfrak{D}}_{\mathsf{ȷ}}\right),\mathcal{K}\in \Xi .$

$$\begin{array}{cc}\hfill \mathrm{Let}& {\mathrm{♈}}_1=𝐹+{𝐹}^2{\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)\hfill \\ \hfill & {\mathrm{♈}}_2=𝐹{\mathrm{\top }}^{\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho -\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}+\rho -\mathbb{k}}}\right)}^{1-\mathbb{k}}\hfill \\ \hfill & {\mathrm{♈}}_3=𝐹{\mathrm{\top }}^{\mathrm{♭}}{\displaystyle \sum _{\mathsf{ȷ}=1}^m}{L}_{\mathsf{ȷ}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+𝐹{\mathrm{\top }}^{\rho }{\displaystyle \frac{\mathbb{B}(\mathrm{♭},\rho )}{\mathrm{♭}+\rho }}\right).\hfill \end{array}$$

Theorem 1.

Suppose the non-linear operators $\mathcal{Z}:\Xi \times {\mathrm{\gimel }}^m⟶\mathrm{\gimel }$ and ${\mathrm{\wp }}_{\beta }:\Xi \times \mathrm{\gimel }⟶\mathrm{\gimel }$ are continuous. Under the validity of the assumptions stated above the system of Equation (1) possesses at least one mild solution on the interval $\Xi =[0,\mathrm{\top }]$ provided that

$$\begin{array}{c}\hfill {\zeta }_{\mathrm{\wp }}{\mathsf{♈}}_1+{\zeta }_{\mathcal{Z}}{\mathsf{♈}}_2<1,{{\rm Y}}_{{\mathsf{♌}}_1}={\mathsf{L}}_{\mathrm{\wp }}{\mathsf{♈}}_1<1\mathit{and}{\mathsf{L}}_{\mathrm{\wp }}{\mathsf{♈}}_1+{\mathsf{♈}}_3<1.\end{array}$$

Proof. 

Consider the set ${\mathrm{\complement }}_{\mathsf{r}}=\left\{\varpi \in C(\Xi ,\mathrm{\gimel }):\parallel \varpi \left(\mathcal{K}\right)\parallel \le \mathsf{r}\right\}$. Define the operator $\mathrm{♌}:C(\Xi ,\mathrm{\gimel })⟶C(\Xi ,\mathrm{\gimel })$ as follows:

$$\mathrm{♌}={\mathrm{♌}}_1+{\mathrm{♌}}_2,\mathrm{where}$$

$${(}_{{\mathrm{♌}}_1}\varpi )\left(\mathcal{K}\right)=\left\{\begin{array}{c}{\mathrm{\aleph }}^{-1}\left(0\right){\varpi }_0+{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right){\varpi }_{0}d\mathrm{♄},\mathcal{K}\in (0,{\mathcal{K}}_1]\hfill \\ \\ {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(\mathcal{K},\varpi \left({\mathcal{K}}_{\beta }^{-}\right)\right),\mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }]\hfill \\ \\ {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(s_{\beta },\varpi \left({\mathcal{K}}_{\beta }^{-}\right)\right)+{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right){\mathrm{\wp }}_{\beta }(s_{\beta },\varpi \left({\mathcal{K}}_{\beta }^{-}\right)d\mathrm{♄},\mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}],\beta =1,2,\cdots ,n.\hfill \end{array}\right.$$

$${(}_{{\mathrm{♌}}_2}\varpi )\left(\mathcal{K}\right)=\left\{\begin{array}{c}{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\hfill \\ +{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄},\mathcal{K}\in (0,{\mathcal{K}}_1]\hfill \\ \\ 0,\mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }],\beta =1,2,\cdots ,n\hfill \\ \\ {\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\hfill \\ +{\int }_{s_{\beta }}^{\mathcal{K}}{\int }_{s_{\beta }}^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄},\mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}],\beta =1,2,\cdots ,n.\hfill \end{array}\right.$$

Step 1: We start by proving that ∃ a constant $\mathsf{r}>0$ for which the operator $\mathrm{♌}$ maps the set ${\mathrm{\complement }}_{\mathsf{r}}$ into itself.

$$\parallel \mathrm{♌}\parallel \le \parallel {\mathrm{♌}}_1\parallel +\parallel {\mathrm{♌}}_2\parallel .$$

For $\begin{array}{c}\hfill \mathcal{K}\in (0,{\mathcal{K}}_1]\end{array}$

$$\begin{array}{cc}\hfill \parallel \mathrm{♌}\parallel & \le \parallel {\mathrm{\aleph }}^{-1}\left(0\right){\varpi }_0\parallel +∥{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right){\varpi }_{0}d\mathrm{♄}∥\hfill \\ \hfill & +∥{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}∥\hfill \\ \hfill & +∥{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄}∥\hfill \\ & \le 𝐹\parallel {\varpi }_0\parallel +{𝐹}^2{\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}(1+{\mathrm{♄}}^{\rho })\parallel {\varpi }_0\parallel d\mathrm{♄}\hfill \\ \hfill & +𝐹{\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}+{𝐹}^2{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}{\varphi }_{\mathsf{r}}\left(s\right)dsd\mathrm{♄}\hfill \\ \hfill & \le 𝐹\parallel {\varpi }_0\parallel +{𝐹}^2\parallel {\varpi }_0\parallel {\mathcal{K}}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathcal{K}}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)+𝐹{\left({\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{{\displaystyle \frac{\mathrm{♭}-1}{1-\mathbb{k}}}}d\mathrm{♄}\right)}^{1-\mathbb{k}}{\left({\int }_0^{\mathcal{K}}{\varphi }_{\mathsf{r}}^{{\displaystyle \frac{1}{\mathbb{k}}}}\left(\mathrm{♄}\right)d\mathrm{♄}\right)}^{\mathbb{k}}\hfill \\ \hfill & +{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}+\rho -1}{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}\hfill \\ \hfill & \le 𝐹\parallel {\varpi }_0\parallel +{𝐹}^2\parallel {\varpi }_0\parallel {\mathcal{K}}_1^{\mathrm{♭}}\left[{\displaystyle \frac{1}{\mathrm{♭}}}+{\mathcal{K}}_1^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right]+𝐹{\mathcal{K}}_1^{\mathrm{♭}-\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}\hfill \\ \hfill & +{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\left({\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{{\displaystyle \frac{\mathrm{♭}+\rho -1}{1-\mathbb{k}}}}d\mathrm{♄}\right)}^{1-\mathbb{k}}{\left({\int }_0^{\mathcal{K}}{\varphi }_{\mathsf{r}}^{{\displaystyle \frac{1}{\mathbb{k}}}}\left(\mathrm{♄}\right)d\mathrm{♄}\right)}^{\mathbb{k}}\hfill \\ \hfill & \le 𝐹\parallel {\varpi }_0\parallel +{𝐹}^2\parallel {\varpi }_0\parallel {\mathcal{K}}_1^{\mathrm{♭}}\left[{\displaystyle \frac{1}{\mathrm{♭}}}+{\mathcal{K}}_1^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right]\hfill \\ \hfill & +𝐹{\mathcal{K}}_1^{\mathrm{♭}-\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}\parallel {\varphi }_{\mathsf{r}}{\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathcal{K}}_1^{\mathrm{♭}+\rho -\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}+\rho -\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}.\hfill \end{array}$$

For $\begin{array}{c}\hfill \mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }]\end{array},\beta =1,2,\cdots ,n$

$$\begin{array}{cc}\hfill \parallel \mathrm{♌}\parallel & \le \parallel {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(\mathcal{K},\varpi \left({\mathcal{K}}_{\beta }^{-}\right)\right)\parallel \le 𝐹{\mathfrak{G}}_{{\mathrm{\wp }}_{\beta }}\left(\mathsf{r}\right)\le 𝐹{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right).\hfill \end{array}$$

For $\begin{array}{c}\hfill \mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}]\end{array},\beta =1,2,\cdots ,n$

$$\begin{array}{cc}\hfill \parallel \mathrm{♌}\parallel & {\le \parallel }_{{\mathrm{♌}}_1}{\parallel +\parallel }_{{\mathrm{♌}}_2}\parallel \hfill \\ \hfill & \le \parallel {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right)\parallel \parallel {\mathrm{\wp }}_{\beta }\left(s_{\beta },\varpi \left({\mathcal{K}}_{\beta }^{-}\right)\right)\parallel +∥{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄})\vee \left(\mathrm{♄}\right){\mathrm{\wp }}_{\beta }\left(s_{\beta },\varpi \left({\mathcal{K}}_{\beta }^{-}\right)\right)d\mathrm{♄}∥\hfill \\ \hfill & +∥{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}∥\hfill \\ \hfill & +∥{\int }_{s_{\beta }}^{\mathcal{K}}{\int }_{s_{\beta }}^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)dsd\mathrm{♄}∥\hfill \\ \hfill & \le 𝐹{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right)+{𝐹}^2{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right){\mathrm{\top }}^{\mathrm{♭}}\left[{\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right]+𝐹{\mathrm{\top }}^{\mathrm{♭}-\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}\hfill \\ \hfill & +{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho -\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}+\rho -\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}.\hfill \end{array}$$

Combining all the above terms

$$\begin{array}{cc}\hfill \parallel \mathrm{♌}\parallel & \le 𝐹\parallel {\varpi }_0\parallel +{𝐹}^2\parallel {\varpi }_0\parallel {\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)+𝐹{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right)\hfill \\ \hfill & +{𝐹}^2{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right){\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)+𝐹{\mathrm{\top }}^{\mathrm{♭}-\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}\hfill \\ \hfill & +{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho -\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}+\rho -\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}.\hfill \end{array}$$

If possible suppose $\parallel \mathrm{♌}\parallel >\mathsf{r}$. This implies

$$\begin{array}{cc}\hfill & 𝐹\parallel {\varpi }_0\parallel +{𝐹}^2\parallel {\varpi }_0\parallel {\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)+𝐹{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right)\hfill \\ \hfill & +{𝐹}^2{\mathrm{\Im }}_{\mathrm{\wp }}\left(\mathsf{r}\right){\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)+𝐹{\mathrm{\top }}^{\mathrm{♭}-\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}\hfill \\ \hfill & +{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho -\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}+\rho -\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}>\mathsf{r}.\hfill \end{array}$$

Divide both sides by $\mathsf{r}$ and take limit $\mathsf{r}⟶\infty$ on both sides, we have

$$\begin{array}{cc}\hfill & 𝐹{\zeta }_{\mathrm{\wp }}+{𝐹}^2{\zeta }_{\mathrm{\wp }}{\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)+𝐹{\mathrm{\top }}^{\mathrm{♭}-\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}{\zeta }_{\mathcal{Z}}\hfill \\ \hfill & +{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho -\mathbb{k}}{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}+\rho -\mathbb{k}}}\right)}^{1-\mathbb{k}}{\zeta }_{\mathcal{Z}}>1.\hfill \\ \hfill & \Rightarrow {\zeta }_{\mathrm{\wp }}{\mathrm{♈}}_1+{\zeta }_{\mathcal{Z}}{\mathrm{♈}}_2>1.\hfill \end{array}$$

which is a contradiction.

This proves that $\parallel \mathrm{♌}\parallel \le \mathsf{r}.$ This proves that $\mathcal{Z}\left({\mathrm{\complement }}_{\mathsf{r}}\right)\subset {\mathrm{\complement }}_{\mathsf{r}}.$

Step 2: To show that ${\mathrm{♌}}_1:{\mathrm{\complement }}_{\mathsf{r}}⟶{\mathrm{\complement }}_{\mathsf{r}}$ is Lipschitz continuous.

• For $\begin{array}{c}\hfill \mathcal{K}\in (0,{\mathcal{K}}_1]\end{array}$ and ${\varpi }_1,{\varpi }_2\in {\mathrm{\complement }}_{\mathsf{r}}$

  $$\parallel {\mathrm{♌}}_1{\varpi }_1-{\mathrm{♌}}_1{\varpi }_2\parallel =0=0·\parallel {\varpi }_1-{\varpi }_2\parallel .$$

  For $\begin{array}{c}\hfill \mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }]\end{array}\beta =1,2,\cdots ,n$

  $$\begin{array}{cc}\hfill \parallel {\mathrm{♌}}_1{\varpi }_1-{\mathrm{♌}}_1{\varpi }_2\parallel & \le \parallel {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right)\parallel \parallel {\mathrm{\wp }}_{\beta }\left(\mathcal{K},{\varpi }_1\left({\mathcal{K}}_{\beta }^{-}\right)\right)-{\mathrm{\wp }}_{\beta }\left(\mathcal{K},{\varpi }_2\left({\mathcal{K}}_{\beta }^{-}\right)\right)\parallel \hfill \\ \hfill & \le 𝐹{\mathsf{L}}_{{\mathrm{\wp }}_{\beta }}\parallel {\varpi }_1-{\varpi }_2\parallel \le 𝐹{\mathsf{L}}_{\mathrm{\wp }}\parallel {\varpi }_1-{\varpi }_2\parallel .\hfill \end{array}$$

  For $\begin{array}{c}\hfill \mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}]\end{array}\beta =1,2,\cdots ,n$

  $$\begin{array}{cc}\hfill \parallel {\mathrm{♌}}_1{\varpi }_1-{\mathrm{♌}}_1{\varpi }_2\parallel & \le \parallel {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right)\parallel \parallel {\mathrm{\wp }}_{\beta }\left(s_{\beta },{\varpi }_1\left({\mathcal{K}}_{\beta }^{-}\right)\right)-{\mathrm{\wp }}_{\beta }\left(s_{\beta },{\varpi }_2\left({\mathcal{K}}_{\beta }^{-}\right)\right)\parallel \hfill \\ \hfill & +\left|\right|{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right){\mathrm{\wp }}_{\beta }\left(s_{\beta },{\varpi }_1\left({\mathcal{K}}_{\beta }^{-}\right)\right)\hfill \\ \hfill & -{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right){\mathrm{\wp }}_{\beta }\left(s_{\beta },{\varpi }_2\left({\mathcal{K}}_{\beta }^{-}\right)\right)d\mathrm{♄}\left|\right|\hfill \\ \hfill & \le 𝐹{\mathsf{L}}_{{\mathrm{\wp }}_{\beta }}\parallel {\varpi }_1-{\varpi }_2\parallel +{\int }_{s_{\beta }}^{\mathcal{K}}\parallel \kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\parallel \parallel \vee \left(\mathrm{♄}\right)\parallel {\mathsf{L}}_{{\mathrm{\wp }}_{\beta }}\parallel {\varpi }_1-{\varpi }_2\parallel d\mathrm{♄}\hfill \\ \hfill & \le 𝐹{\mathsf{L}}_{\mathrm{\wp }}\parallel {\varpi }_1-{\varpi }_2\parallel +{𝐹}^2{\int }_{s_{\beta }}^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}(1+{\mathrm{♄}}^{\rho }){\mathsf{L}}_{{\mathrm{\wp }}_{\beta }}\parallel {\varpi }_1-{\varpi }_2\parallel d\mathrm{♄}\hfill \\ \hfill & \le 𝐹{\mathsf{L}}_{\mathrm{\wp }}\parallel {\varpi }_1-{\varpi }_2\parallel +{𝐹}^2{\mathsf{L}}_{\mathrm{\wp }}{\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)\parallel {\varpi }_1-{\varpi }_2\parallel \hfill \\ \hfill & \le \left[𝐹{\mathsf{L}}_{\mathrm{\wp }}+{𝐹}^2{\mathsf{L}}_{\mathrm{\wp }}{\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)\right]\parallel {\varpi }_1-{\varpi }_2\parallel .\hfill \end{array}$$

  By considering all the cases above, we have

  $$\begin{array}{cc}\hfill \parallel {\mathrm{♌}}_1{\varpi }_1-{\mathrm{♌}}_1{\varpi }_2\parallel \le & \left[𝐹{\mathsf{L}}_{\mathrm{\wp }}+{𝐹}^2{\mathsf{L}}_{\mathrm{\wp }}{\mathrm{\top }}^{\mathrm{♭}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right)\right]\parallel {\varpi }_1-{\varpi }_2\parallel ,\hfill \\ \hfill & \mathrm{for}\mathcal{K}\in {\cup }_{\beta =0}^n(s_{\beta },{\mathcal{K}}_{\beta +1}]\hfill \\ \hfill \le & {\mathsf{L}}_{\mathrm{\wp }}{\mathrm{♈}}_1\parallel {\varpi }_1-{\varpi }_2\parallel ={{\rm Y}}_{{\mathrm{♌}}_1}\parallel {\varpi }_1-{\varpi }_2\parallel .\hfill \\ \hfill \Rightarrow \parallel {\mathrm{♌}}_1{\varpi }_1-{\mathrm{♌}}_1{\varpi }_2\parallel \le & {{\rm Y}}_{{\mathrm{♌}}_1}\parallel {\varpi }_1-{\varpi }_2\parallel \mathrm{for}\mathrm{each}\mathcal{K}\in \Xi .\hfill \end{array}$$

  Since ${{\rm Y}}_{{\mathrm{♌}}_1}<1$. This proves the Lipschitz continuity of function ${\mathrm{♌}}_1$ with Lipschitz constant ${{\rm Y}}_{{\mathrm{♌}}_1}$.

Step 3: To prove that ${\mathrm{♌}}_2:{\mathrm{\complement }}_{\mathsf{r}}⟶{\mathrm{\complement }}_{\mathsf{r}}$ is continuous function

• For $\begin{array}{c}\hfill \mathcal{K}\in (0,{\mathcal{K}}_1]\end{array}$ Let $\left\{{\varpi }_n\right\}\subset {\mathrm{\complement }}_{\mathsf{r}}$ be a sequence such that ${\lim }_{n⟶\infty }{\varpi }_n=\varpi$ in ${\mathrm{\complement }}_{\mathsf{r}}$.

  $$\begin{array}{cc}\hfill & \parallel \left({\mathrm{♌}}_2{\varpi }_n\right)\left(\mathcal{K}\right)-\left({\mathrm{♌}}_2\varpi \right)\left(\mathcal{K}\right)\parallel \hfill \\ \hfill \le & \left|\right|{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})[\mathcal{Z}\left(\mathrm{♄},{\varpi }_n\left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\left]d\mathrm{♄}\right||\hfill \\ \hfill +& \left|\right|{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)[\mathcal{Z}\left(s,{\varpi }_n\left({\mathcal{Q}}_1\left(s\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(s\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)\left]dsd\mathrm{♄}\right||\hfill \\ \hfill \le & 𝐹{\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}\left|\right|\mathcal{Z}\left(\mathrm{♄},{\varpi }_n\left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\left|\right|d\mathrm{♄}\hfill \\ \hfill +& {𝐹}^2{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}\left|\right|\mathcal{Z}\left(s,{\varpi }_n\left({\mathcal{Q}}_1\left(s\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(s\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)\left|\right|dsd\mathrm{♄}.\hfill \end{array}$$

  In view of the above assumptions, we have

  $$\begin{array}{cc}\hfill & {\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}\left|\right|\mathcal{Z}\left(\mathrm{♄},{\varpi }_n\left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\left|\right|d\mathrm{♄}\le 2{\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}.\hfill \end{array}$$

  $$\begin{array}{cc}\hfill & {\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}\left|\right|\mathcal{Z}\left(s,{\varpi }_n\left({\mathcal{Q}}_1\left(s\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(s\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)\left|\right|dsd\mathrm{♄}\le 2\mathbb{B}(\mathrm{♭},\rho ){\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}+\rho -1}{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}.\hfill \end{array}$$

Thus for every $\mathrm{♄}\in [0,\mathcal{K}]$ and $\mathcal{K}\in \Xi$, the functions $2{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)$ and $2\mathbb{B}(\mathrm{♭},\rho ){(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}+\rho -1}{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)$ are Lebesgue integrable. Since ${\varpi }_n\to \varpi$ in $\mathcal{C}(\Xi ,\mathrm{\gimel })$ with respect to the supremum norm, and $Q_j:\Xi \to \Xi$ is continuous, we have, for every $\mathrm{♄}\in \Xi$ and each j,

$${∥{\varpi }_n\left(Q_j\left(\mathrm{♄}\right)\right)-\varpi \left(Q_j\left(\mathrm{♄}\right)\right)∥}_{\mathrm{\gimel }}\le {\parallel {\varpi }_n-\varpi \parallel }_{\infty }⟶0\mathrm{as}n\to \infty .$$

Thus, ${\varpi }_n\left(Q_j\left(\mathrm{♄}\right)\right)\to \varpi \left(Q_j\left(\mathrm{♄}\right)\right)$ uniformly with respect to $\mathrm{♄}$. Therefore, by Lebesgue dominated convergence theorem,

$$\parallel \left({\mathrm{♌}}_2{\varpi }_n\right)\left(\mathcal{K}\right)-\left({\mathrm{♌}}_2\varpi \right)\left(\mathcal{K}\right)\parallel ⟶0\mathrm{as}n⟶\infty ,$$

which proves that ${\mathrm{♌}}_2$ is continuous for $\mathcal{K}\in (0,{\mathcal{K}}_1]$.

Step 4: To prove that ${\mathrm{♌}}_2$ is equicontinuous function on ${\mathrm{\complement }}_{\mathsf{r}}.$ For $\begin{array}{c}\hfill \mathcal{K}\in (0,{\mathcal{K}}_1]\end{array}$

For any $\varpi \in {\mathrm{\complement }}_{\mathsf{r}}$ and $0\le {\mathcal{K}}^{\prime }\le {\mathcal{K}}^{″}\le {\mathcal{K}}_1.$

$$\begin{array}{cc}\hfill & \left|\right|\left({\mathrm{♌}}_2\varpi \right)\left({\mathcal{K}}^{″}\right)-\left({\mathrm{♌}}_2\varpi \right)\left({\mathcal{K}}^{\prime }\right)\left|\right|\hfill \\ & \\ \hfill \le & \left|\right|{\int }_0^{{\mathcal{K}}^{″}}\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\hfill \\ \hfill +& {\int }_0^{{\mathcal{K}}^{″}}{\int }_0^{\mathrm{♄}}\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄}\hfill \\ \hfill -& {\int }_0^{{\mathcal{K}}^{\prime }}\kappa ({\mathcal{K}}^{\prime }-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\hfill \\ \hfill -& {\int }_0^{{\mathcal{K}}^{\prime }}{\int }_0^{\mathrm{♄}}\kappa ({\mathcal{K}}^{\prime }-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄}\left|\right|\hfill \\ \hfill \le & \left|\right|{\int }_0^{{\mathcal{K}}^{\prime }}\left[\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})-\kappa ({\mathcal{K}}^{\prime }-\mathrm{♄},\mathrm{♄})\right]\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\left|\right|\hfill \\ \hfill +& \left|\right|{\int }_0^{{\mathcal{K}}^{\prime }}{\int }_0^{\mathrm{♄}}\left[\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})-\kappa ({\mathcal{K}}^{\prime }-\mathrm{♄},\mathrm{♄})\right]\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄}\left|\right|\hfill \\ \hfill +& \left|\right|{\int }_{{\mathcal{K}}^{\prime }}^{{\mathcal{K}}^{″}}\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\left|\right|\hfill \\ \hfill +& \left|\right|{\int }_{{\mathcal{K}}^{\prime }}^{{\mathcal{K}}^{″}}{\int }_0^{\mathrm{♄}}\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄}\left|\right|\hfill \\ \hfill =& I_1+I_2+I_3+I_4.\hfill \end{array}$$

We only need to establish that $I_k⟶0,1\le k\le 4,$ independently of $\varpi \in {\mathrm{\complement }}_{\mathsf{r}}$ as ${\mathcal{K}}^{″}-{\mathcal{K}}^{\prime }⟶0$.

By observation that the function $\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})$ is continuous in the uniform operator topology about the variables $\mathcal{K}$ and $\mathrm{♄},$ hold, then

$$\begin{array}{cc}\hfill I_1& \le {({\mathcal{K}}^{\prime }-\upsilon )}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}\underset{\mathrm{♄}\in [0,{\mathcal{K}}^{\prime }-\upsilon ]}{\sup }∥\kappa ({\mathcal{K}}^{″}-\mathrm{♄})-\kappa ({\mathcal{K}}^{\prime }-\mathrm{♄})∥\hfill \\ \hfill & +𝐹{\int }_{{\mathcal{K}}^{\prime }-\upsilon }^{{\mathcal{K}}^{\prime }}\left[{({\mathcal{K}}^{″}-\mathrm{♄})}^{\mathrm{♭}-1}+{({\mathcal{K}}^{\prime }-\mathrm{♄})}^{\mathrm{♭}-1}\right]{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}\hfill \\ \hfill & ⟶0\mathrm{as}{\mathcal{K}}^{″}-{\mathcal{K}}^{\prime }⟶0\mathrm{and}\upsilon ⟶0.\hfill \\ \hfill I_2& \le \underset{\mathrm{♄}\in [0,{\mathcal{K}}^{\prime }-\upsilon ]}{\sup }∥\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})-\kappa ({\mathcal{K}}^{\prime }-\mathrm{♄},\mathrm{♄})∥·𝐹{\int }_0^{{\mathcal{K}}^{\prime }-\upsilon }{\int }_0^{\mathrm{♄}}{(\mathrm{♄}-s)}^{\rho -1}{\varphi }_{\mathsf{r}}\left(s\right)dsd\mathrm{♄}\hfill \\ \hfill & +{𝐹}^2{\int }_{{\mathcal{K}}^{\prime }-\upsilon }^{{\mathcal{K}}^{\prime }}{\int }_0^{\mathrm{♄}}\left[{({\mathcal{K}}^{″}-\mathrm{♄})}^{\mathrm{♭}-1}+{({\mathcal{K}}^{\prime }-\mathrm{♄})}^{\mathrm{♭}-1}\right]{(\mathrm{♄}-s)}^{\rho -1}{\varphi }_{\mathsf{r}}\left(s\right)dsd\mathrm{♄}\hfill \\ \hfill & \le {\left({\displaystyle \frac{1-\mathbb{k}}{\rho -\mathbb{k}}}\right)}^{1-\mathbb{k}}{\displaystyle \frac{𝐹{({\mathcal{K}}^{\prime }-\upsilon )}^{\rho }}{\rho }}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}\underset{\mathrm{♄}\in [0,{\mathcal{K}}^{\prime }-\upsilon ]}{\sup }∥\kappa ({\mathcal{K}}^{″}-\mathrm{♄},\mathrm{♄})-\kappa ({\mathcal{K}}^{\prime }-\mathrm{♄},\mathrm{♄})∥\hfill \\ \hfill & +{𝐹}^2\Gamma \left(\rho \right){\int }_{{\mathcal{K}}^{\prime }-\upsilon }^{{\mathcal{K}}^{\prime }}\left[{({\mathcal{K}}^{″}-\mathrm{♄})}^{\mathrm{♭}-1}+({\mathcal{K}}^{\prime }-\mathrm{♄})\right]I_{\mathrm{♄}}^{\rho }{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}\hfill \\ \hfill & ⟶0\mathrm{as}{\mathcal{K}}^{″}-{\mathcal{K}}^{\prime }⟶0\mathrm{and}\upsilon ⟶0.\hfill \\ \hfill I_3& \le {\int }_{{\mathcal{K}}^{\prime }}^{{\mathcal{K}}^{″}}{({\mathcal{K}}^{″}-\mathrm{♄})}^{\mathrm{♭}-1}{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}\le 𝐹{\left({\int }_{{\mathcal{K}}^{\prime }}^{{\mathcal{K}}^{″}}{({\mathcal{K}}^{″}-\mathrm{♄})}^{{\displaystyle \frac{\mathrm{♭}-1}{1-\mathbb{k}}}}\right)}^{1-\mathbb{k}}{\left({\int }_{{\mathcal{K}}^{\prime }}^{{\mathcal{K}}^{″}}{\varphi }_{\mathsf{r}}^{{\displaystyle \frac{1}{\mathbb{k}}}}\left(\mathrm{♄}\right)d\mathrm{♄}\right)}^{\mathbb{k}}\hfill \\ \hfill & \le 𝐹{\left({\displaystyle \frac{1-\mathbb{k}}{\mathrm{♭}-\mathbb{k}}}\right)}^{1-\mathbb{k}}{\parallel {\varphi }_{\mathsf{r}}\parallel }_{L^{{\displaystyle \frac{1}{\mathbb{k}}}}[0,\mathrm{\top }]}{({\mathcal{K}}^{″}-{\mathcal{K}}^{\prime })}^{\mathrm{♭}-\mathbb{k}}⟶0\mathrm{as}{t}^{″}-{\mathcal{K}}^{\prime }⟶0.\hfill \\ \hfill I_4& \le {𝐹}^2{\int }_{{\mathcal{K}}^{\prime }}^{{\mathcal{K}}^{″}}{\int }_0^{\mathrm{♄}}{({\mathcal{K}}^{″}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}{\varphi }_{\mathsf{r}}\left(s\right)dsd\mathrm{♄}\hfill \\ \hfill & \le {𝐹}^2{\int }_{{\mathcal{K}}^{\prime }}^{{\mathcal{K}}^{″}}{({\mathcal{K}}^{″}-\mathrm{♄})}^{\mathrm{♭}-1}{I}_{\mathrm{♄}}^{\rho }{\varphi }_{\mathsf{r}}\left(\mathrm{♄}\right)d\mathrm{♄}⟶0\mathrm{as}{\mathcal{K}}^{\prime }-{\mathcal{K}}^{″}⟶0.\hfill \end{array}$$

In the same way, it can be established that

$$∥{\mathrm{♌}}_2\varpi \left({\mathcal{K}}^{″}\right)-{\mathrm{♌}}_2\varpi \left({\mathcal{K}}^{\prime }\right)∥⟶0\mathrm{as}{\mathcal{K}}^{″}-{\mathcal{K}}^{\prime }⟶0;\mathrm{for}\mathcal{K}\in {\cup }_{\beta =1}^n(s_{\beta },{\mathcal{K}}_{\beta +1}].$$

⇒${\mathrm{♌}}_2$ is equicontinuous for $\mathcal{K}\in \Xi .$ Step 5: To prove that $\mathrm{♌}={\mathrm{♌}}_1+{\mathrm{♌}}_2$ is a contraction mapping.

Since ${\mathrm{♌}}_1$ is Lipschitz continuous function with Lipschitz constant ${{\rm Y}}_{{\mathrm{♌}}_1},$ it follows that

$$\mathcal{H}\left({\mathrm{♌}}_1\left(\mathcal{z}\right)\right)\le {{\rm Y}}_{{\mathrm{♌}}_1}\mathcal{H}\left(\mathcal{z}\right).$$

Consequently, ${\mathrm{♌}}_1$ is a contraction mapping. Let $B=\overline{co}\mathcal{Z}\left({\mathrm{\complement }}_{\mathsf{r}}\right)$, where $\overline{co}$ is the closure of convex hull. For any $\mathcal{z}\subset B,$ there exists a countable set ${\mathcal{z}}_0=\left\{{\varpi }_n\right\}\subset \mathcal{z}$ such that

$${\mathcal{H}}_C\left({\mathrm{♌}}_2\left(\mathcal{z}\right)\right)\le {\mathcal{H}}_C\left({\mathrm{♌}}_2\left({\mathcal{z}}_0\right)\right)$$

(2)

For $\begin{array}{c}\hfill \mathcal{K}\in {\cup }_{\beta =1}^n(s_{\beta },{\mathcal{K}}_{\beta +1}]\end{array}$

$$\begin{array}{cc}\hfill \mathcal{H}\left({\mathrm{♌}}_2\left({\mathcal{z}}_0\right)\right)\le & \mathcal{H}\left[{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathcal{Z}\left(\mathrm{♄},{\varpi }_n\left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)d\mathrm{♄}\right]\hfill \\ \hfill +& \mathcal{H}\left[{\int }_{s_{\beta }}^{\mathcal{K}}{\int }_{s_{\beta }}^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\mathcal{Z}\left(s,{\varpi }_n\left({\mathcal{Q}}_1\left(s\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(s\right)\right)\right)dsd\mathrm{♄}\right]\hfill \\ \hfill \le & 𝐹{\int }_{s_{\beta }}^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}\mathcal{H}\left(\mathcal{Z}\left(\mathrm{♄},{\varpi }_n\left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),{\varpi }_n\left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,{\varpi }_n\left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\right)d\mathrm{♄}\hfill \\ \hfill +& {𝐹}^2{\int }_{s_{\beta }}^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}\times \hfill \\ \hfill & \left[{\mathcal{L}}_1\mathcal{H}\left({\mathcal{z}}_0\left({\mathcal{Q}}_1\left(s\right)\right)\right)+{\mathcal{L}}_2\mathcal{H}\left({\mathcal{z}}_0\left({\mathcal{Q}}_2\left(s\right)\right)\right)+\cdots +{\mathcal{L}}_m\mathcal{H}\left({\mathcal{z}}_0\left({\mathcal{Q}}_m\left(s\right)\right)\right)\right]dsd\mathrm{♄}\hfill \\ \hfill \le & 𝐹{\displaystyle \sum _{\mathsf{ȷ}=1}^m}{\mathcal{L}}_{\mathsf{ȷ}}{\int }_{s_{\beta }}^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}d\mathrm{♄}{\mathcal{H}}_C\left(\mathcal{z}\right)+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \sum _{\mathsf{ȷ}=1}^m}{\mathcal{L}}_{\mathsf{ȷ}}{\int }_{s_{\beta }}^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}+\rho -1}d\mathrm{♄}{\mathcal{H}}_C\left(\mathcal{z}\right)\hfill \\ \hfill \le & 𝐹{\displaystyle \sum _{\mathsf{ȷ}=1}^m}{\displaystyle \frac{{\mathcal{L}}_{\mathsf{ȷ}}{\mathrm{\top }}^{\mathrm{♭}}}{\mathrm{♭}}}{\mathcal{H}}_C\left(\mathcal{z}\right)+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \sum _{\mathsf{ȷ}=1}^m}{\displaystyle \frac{{\mathcal{L}}_{\mathsf{ȷ}}{\mathrm{\top }}^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}{\mathcal{H}}_C\left(\mathcal{z}\right)\hfill \\ \hfill =& 𝐹{\mathrm{\top }}^{\mathrm{♭}}{\displaystyle \sum _{\mathsf{ȷ}=1}^m}{\mathcal{L}}_{\mathsf{ȷ}}\left({\displaystyle \frac{1}{\mathrm{♭}}}+𝐹{\mathrm{\top }}^{\rho }{\displaystyle \frac{\mathbb{B}(\mathrm{♭},\rho )}{\mathrm{♭}+\rho }}\right){\mathcal{H}}_C\left(\mathcal{z}\right)={\mathrm{♈}}_3{\mathcal{H}}_C\left(\mathcal{z}\right)\hfill \end{array}$$

(3)

$${\mathcal{H}}_C\left({\mathrm{♌}}_2\left({\mathcal{z}}_0\right)\right)=\underset{\mathcal{K}\in [0,\mathrm{\top }]}{\max }\mathcal{H}\left({\mathrm{♌}}_2\left({\mathcal{z}}_0\right)\left(\mathcal{K}\right)\right)$$

(4)

Therefore, from Equations (2)–(4) one gets that

$$\begin{array}{cc}\hfill {\mathcal{H}}_C\left({\mathrm{♌}}_2\left(\mathcal{z}\right)\right)& \le {\mathrm{♈}}_3{\mathcal{H}}_C\left(\mathcal{z}\right)\hfill \end{array}$$

(5)

We have

$$\begin{array}{cc}\hfill {\mathcal{H}}_C\left(\left(\mathrm{♌}\right(\mathcal{z})\right)& \le {\mathcal{H}}_C\left({\mathrm{♌}}_1\left(\mathcal{z}\right)\right)+{\mathcal{H}}_C\left({\mathrm{♌}}_2\left(\mathcal{z}\right)\right)\le \left({{\rm Y}}_{{\mathrm{♌}}_1}+{\mathrm{♈}}_3\right){\mathcal{H}}_C\left(\mathcal{z}\right)=\left({\mathsf{L}}_{\mathrm{\wp }}{\mathrm{♈}}_1+{\mathrm{♈}}_3\right){\mathcal{H}}_C\left(\mathcal{z}\right)\hfill \end{array}$$

Since $\left({\mathsf{L}}_{\mathrm{\wp }}{\mathrm{♈}}_1+{\mathrm{♈}}_3\right)<1$. Therefore, by the corresponding k-set contractive fixed-point theorem, the operator $\mathrm{♌}$ possesses at least one fixed point in ${\mathrm{\complement }}_{\mathsf{r}}$, which is a mild solution of the system of Equation (1). It is worth showing that the three smallness conditions imposed in Theorem 1 are independent in their roles and are required at different stages of the fixed-point argument. The condition ${\zeta }_{\mathrm{\wp }}{\mathrm{♈}}_1+{\zeta }_{\mathcal{Z}}{\mathrm{♈}}_2<1$ guarantees that the operator preserves the closed ball ${\mathrm{\complement }}_{\mathsf{r}}$, which is essential for establishing $\mathcal{Z}\left({\mathrm{\complement }}_{\mathsf{r}}\right)\subseteq {\mathrm{\complement }}_{\mathsf{r}}$. The condition ${{\rm Y}}_{{\mathrm{♌}}_1}={\mathsf{L}}_{\mathrm{\wp }}{\mathrm{♈}}_1<1$ is subsequently used to ensure that ${\mathrm{♌}}_1$ is Lipschitz continuous with a contraction constant strictly less than one. Finally, the condition ${\mathsf{L}}_{\mathrm{\wp }}{\mathrm{♈}}_1+{\mathrm{♈}}_3<1$ guarantees the required k-set contractive property of the combined operator $\mathrm{♌}={\mathrm{♌}}_1+{\mathrm{♌}}_2$, which allows the fixed-point theorem to be applied. Hence, these assumptions are not redundant: the first controls the invariance of the working ball, the second establishes the contraction behavior of the momentum component, and the third provides the contractive/condensing property of the full operator required to conclude the existence of a mild solution. □

4. Stability Analysis

Consider the inequalities

$$\left\{\begin{array}{c}\left|{}^{\mathfrak{c}}\mathcal{D}_{\mathcal{K}}^{\mathrm{♭}}\sigma \left(\mathcal{K}\right)+\mathrm{\aleph }\left(\mathcal{K}\right)\sigma \left(\mathcal{K}\right)-\mathcal{Z}\left(\mathcal{K},\sigma \left({\mathcal{Q}}_1\left(\mathcal{K}\right)\right),\sigma \left({\mathcal{Q}}_2\left(\mathcal{K}\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(\mathcal{K}\right)\right)\right)\right|\le ϵ\hfill \\ \mathrm{for}\mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}],\beta =0,1,2,\dots ,n;\hfill \\ \\ \left|\sigma \left(\mathcal{K}\right)-{\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(\mathcal{K},\sigma \left({\mathcal{K}}_{\beta }^{-}\right)\right)\right|\le ϵ\mathrm{for}\mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }],\beta =1,2,\dots ,n.\hfill \end{array}\right.$$

(6)

A function $\sigma \left(\mathcal{K}\right)$ is a solution of above equation iff there is a real number $\mathrm{♇}\left(\mathcal{K}\right)$ such that $∥\mathrm{♇}\left(\mathcal{K}\right)∥\le ϵ$ and

$$\left\{\begin{array}{c}{}^{\mathfrak{c}}\mathcal{D}_{\mathcal{K}}^{\mathrm{♭}}\sigma \left(\mathcal{K}\right)+\mathrm{\aleph }\left(\mathcal{K}\right)\sigma \left(\mathcal{K}\right)=\mathcal{Z}\left(\mathcal{K},\sigma \left({\mathcal{Q}}_1\left(\mathcal{K}\right)\right),\sigma \left({\mathcal{Q}}_2\left(\mathcal{K}\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(\mathcal{K}\right)\right)\right)+\mathrm{♇}\left(\mathcal{K}\right)\hfill \\ \mathrm{for}\mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}],\beta =0,1,2,\dots ,n;\hfill \\ \\ \sigma \left(\mathcal{K}\right)={\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(\mathcal{K},\sigma \left({\mathcal{K}}_{\beta }^{-}\right)\right)+\mathrm{♇}\left(\mathcal{K}\right),\mathrm{for}\mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }],\beta =1,2,\dots ,n.\hfill \end{array}\right.$$

Definition 6.

The system of equations is said to exhibit Ulam–Hyers stability if ∃ a real number $\mathrm{\Re }>0$ such that for given $ϵ>0$ and for each solution $\sigma \left(\mathcal{K}\right)$ of Equation (6) there exists a solution $\varpi \left(\mathcal{K}\right)$ of Equation (1) with

$$\left|\sigma \right(\mathcal{K})-\varpi (\mathcal{K}\left)\right|\le ϵ\mathrm{\Re }.$$

$$\begin{array}{cc}\hfill \mathit{Let}{\Theta }_1& =\left[{\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}}{\mathrm{♭}}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right]{\mathsf{L}}_{\mathcal{Z}}.\hfill \\ \hfill {\Theta }_2& =\left[𝐹{\mathsf{L}}_{\mathrm{\wp }}+{𝐹}^2{\mathsf{L}}_{\mathrm{\wp }}{\mathrm{\top }}^{\mathrm{♭}}\left\{{\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right\}+{\displaystyle \frac{{𝐹}^2{\mathrm{\top }}^{\mathrm{♭}}}{\mathrm{♭}}}{\mathsf{L}}_{\mathcal{Z}}+{\displaystyle \frac{{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}{\mathsf{L}}_{\mathcal{Z}}\right].\hfill \end{array}$$

Theorem 2.

The given system of Equation (1) is Ulam–Hyers-stable if all assumptions of Theorem 1 are satisfied and, additionally, ${\Theta }_1<1$ and ${\Theta }_2<1$.

Proof. 

From the preceding theorem, above system of equations is equivalent to

$$\sigma \left(\mathcal{K}\right)=\left\{\begin{array}{c}{\mathrm{\aleph }}^{-1}\left(0\right){\varpi }_0+{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right){\varpi }_{0}d\mathrm{♄}\hfill \\ +{\int }_0^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\left\{\mathcal{Z}\left(\mathrm{♄},\sigma \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\sigma \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)+\mathrm{♇}\left(\mathrm{♄}\right)\right\}d\mathrm{♄}\hfill \\ +{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\left\{\mathcal{Z}\left(s,\sigma \left({\mathcal{Q}}_1\left(s\right)\right),\sigma \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(s\right)\right)\right)+\mathrm{♇}\left(s\right)\right\}dsd\mathrm{♄},\mathcal{K}\in (0,{\mathcal{K}}_1]\hfill \\ \\ {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(\mathcal{K},\sigma \left({\mathcal{K}}_{\beta }^{-}\right)\right)+\mathrm{♇}\left(\mathcal{K}\right),\mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }],\beta =1,2,\cdots ,n\hfill \\ \\ {\mathrm{\aleph }}^{-1}\left(s_{\beta }\right){\mathrm{\wp }}_{\beta }\left(s_{\beta },\sigma \left({\mathcal{K}}_{\beta }^{-}\right)\right)+\mathrm{♇}\left(s_{\beta }\right)+{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\vee \left(\mathrm{♄}\right)\left\{{\mathrm{\wp }}_{\beta }\left(s_{\beta },\sigma \left({\mathcal{K}}_{\beta }^{-}\right)\right)+\mathrm{♇}\left(\mathrm{♄}\right)\right\}d\mathrm{♄}\hfill \\ +{\int }_{s_{\beta }}^{\mathcal{K}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\left\{\mathcal{Z}\left(\mathrm{♄},\sigma \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\sigma \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)+\mathrm{♇}\left(\mathrm{♄}\right)\right\}d\mathrm{♄}\hfill \\ +{\int }_{s_{\beta }}^{\mathcal{K}}{\int }_{s_{\beta }}^{\mathrm{♄}}\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})\mathfrak{G}(\mathrm{♄},s)\left\{\mathcal{Z}\left(s,\sigma \left({\mathcal{Q}}_1\left(s\right)\right),\sigma \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(s\right)\right)\right)+\left(s\right)\right\}dsd\mathrm{♄},\mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}],\beta =1,2,\cdots ,n.\hfill \end{array}\right.$$

(7)

For $\begin{array}{c}\hfill \mathcal{K}\in [0,{\mathcal{K}}_1]\end{array}$

$$\begin{array}{cc}\hfill ∥\sigma \left(\mathcal{K}\right)-\varpi \left(\mathcal{K}\right)∥& \le {\int }_0^{\mathcal{K}}∥\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})∥\left|\right|\mathcal{Z}\left(\mathrm{♄},\sigma \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\sigma \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(\mathrm{♄},\varpi \left({\mathcal{Q}}_1\left(\mathrm{♄}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathrm{♄}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathrm{♄}\right)\right)\right)\left|\right|d\mathrm{♄}\hfill \\ \hfill & +{\int }_0^{\mathcal{K}}∥\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})∥∥\mathrm{♇}∥d\mathrm{♄}\hfill \\ \hfill & +{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}∥\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})∥∥\mathfrak{G}(\mathrm{♄},s)∥\left|\right|\mathcal{Z}\left(s,\sigma \left({\mathcal{Q}}_1\left(s\right)\right),\sigma \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\sigma \left({\mathcal{Q}}_m\left(s\right)\right)\right)\hfill \\ \hfill & -\mathcal{Z}\left(s,\varpi \left({\mathcal{Q}}_1\left(s\right)\right),\varpi \left({\mathcal{Q}}_2\left(s\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(s\right)\right)\right)\left|\right|dsd\mathrm{♄}\hfill \\ \hfill & +{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}∥\kappa (\mathcal{K}-\mathrm{♄},\mathrm{♄})∥∥\mathrm{♇}∥dsd\mathrm{♄}\hfill \\ \hfill & \le 𝐹{\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{\mathsf{L}}_{\mathcal{Z}}∥\sigma -\varpi ∥d\mathrm{♄}+𝐹{\int }_0^{\mathcal{K}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}ϵd\mathrm{♄}\hfill \\ \hfill & +{𝐹}^2{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}{\mathsf{L}}_{\mathcal{Z}}∥\sigma -\varpi ∥dsd\mathrm{♄}\hfill \\ \hfill & +{𝐹}^2{\int }_0^{\mathcal{K}}{\int }_0^{\mathrm{♄}}{(\mathcal{K}-\mathrm{♄})}^{\mathrm{♭}-1}{(\mathrm{♄}-s)}^{\rho -1}ϵdsd\mathrm{♄}\hfill \\ \hfill & \le \left[{\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}}{\mathrm{♭}}}{\mathsf{L}}_{\mathcal{Z}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathsf{L}}_{\mathcal{Z}}{\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right]∥\sigma -\varpi ∥+\left[{\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}}{\mathrm{♭}}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right]ϵ\hfill \\ & \\ \hfill \Rightarrow ∥\sigma -\varpi ∥& \le {\displaystyle \frac{ϵ\left({\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}}{\mathrm{♭}}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right)}{\left(1-{\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}{\mathsf{L}}_{\mathcal{Z}}}{\mathrm{♭}}}-{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }{\mathsf{L}}_{\mathcal{Z}}}{\mathrm{♭}+\rho }}\right)}}\hfill \\ & \le {\displaystyle \frac{ϵ}{1-{\Theta }_1}}\left({\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}}{\mathrm{♭}}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right),\mathrm{where}{\Theta }_1<1.\hfill \\ & \Rightarrow ∥\sigma -\varpi ∥\le ϵ{\mathrm{\Re }}_1,\mathrm{where},{\mathrm{\Re }}_1=\left({\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}}{\mathrm{♭}}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right)>0.\hfill \end{array}$$

For $\begin{array}{c}\hfill \mathcal{K}\in ({\mathcal{K}}_{\beta },s_{\beta }]\end{array},\beta =1,2,\cdots ,n.$

$$\begin{array}{cc}\hfill ∥\sigma -\varpi ∥\le & 𝐹{\mathsf{L}}_{\mathrm{\wp }}∥\sigma -\varpi ∥+𝐹ϵ\hfill \\ \hfill ∥\sigma -\varpi ∥\le & {\displaystyle \frac{ϵ𝐹}{1-𝐹{\mathsf{L}}_{\mathrm{\wp }}}}={𝐹}_1ϵ,\mathrm{where}{𝐹}_1={\displaystyle \frac{𝐹}{1-𝐹{\mathsf{L}}_{\mathrm{\wp }}}}\mathrm{is}\mathrm{some}\mathrm{constant}.\hfill \end{array}$$

For $\begin{array}{c}\hfill \mathcal{K}\in (s_{\beta },{\mathcal{K}}_{\beta +1}]\end{array},\beta =1,2,\cdots ,n.$ Following similar procedure and above mentioned hypotheses, it is easy to check that

$$\begin{array}{cc}\hfill ∥\sigma -\varpi ∥\le & [𝐹{\mathsf{L}}_{\mathrm{\wp }}+{𝐹}^2{\mathsf{L}}_{\mathrm{\wp }}{\mathrm{\top }}^{\mathrm{♭}}\left\{{\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right\}+{\displaystyle \frac{{𝐹}^2{\mathrm{\top }}^{\mathrm{♭}}}{\mathrm{♭}}}{\mathsf{L}}_{\mathcal{Z}}\hfill \\ \hfill +& {\displaystyle \frac{{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}{\mathsf{L}}_{\mathcal{Z}}]∥\sigma -\varpi ∥+ϵ\left[1+{\displaystyle \frac{𝐹{\mathrm{\top }}^{\mathrm{♭}}}{\mathrm{♭}}}+{\displaystyle \frac{{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right]\hfill \\ & \\ \hfill \Rightarrow ∥\sigma -\varpi ∥\le & {\displaystyle \frac{ϵ}{1-{\Theta }_2}}\left(1+{\displaystyle \frac{𝐹{\mathrm{\top }}^{\mathrm{♭}}}{\mathrm{♭}}}+{\displaystyle \frac{{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right),\mathrm{where}{\Theta }_2<1.\hfill \\ & \\ \hfill \Rightarrow ∥\sigma -\varpi ∥\le & ϵ{\mathrm{\Re }}_2,\mathrm{where}{\mathrm{\Re }}_2=\left(1+{\displaystyle \frac{𝐹{\mathrm{\top }}^{\mathrm{♭}}}{\mathrm{♭}}}+{\displaystyle \frac{{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right).\hfill \end{array}$$

Thus $∥\sigma -\varpi ∥\le \mathrm{\Re }ϵ$, where $\mathrm{\Re }=\max \left\{{\mathrm{\Re }}_1,{\mathrm{\Re }}_2\right\}$ for all $\mathcal{K}\in {\cup }_{\beta =0}^n(s_{\beta },{\mathcal{K}}_{\beta +1}]$. This proves that system of Equation (1) is Ulam–Hyers-stable. □

5. Applications

Consider the following non-autonomous partial $\mathcal{F}\mathcal{D}\mathcal{E}\mathcal{s}$ with non-instantaneous impulses:

$$\left\{\begin{array}{c}\begin{array}{cc}\hfill {\mathcal{D}}_0^{{\displaystyle \frac{3}{5}}}\varpi (\mathcal{x},\mathcal{K})-(1+\mathcal{K}){\displaystyle \frac{{\partial }^2\varpi (\mathcal{x},\mathcal{K})}{\partial {\mathcal{x}}^2}}& ={\displaystyle \frac{\sin \mathcal{K}}{25\left(1+|\varpi \left(\mathcal{x},{\mathcal{Q}}_1\left(\mathcal{K}\right)\right)|+\cdots +|\varpi \left(\mathcal{x},{\mathcal{Q}}_m\left(\mathcal{K}\right)\right)|\right)}},\mathcal{K}\in \left(0,{\displaystyle \frac{1}{3}}\right]\hfill \\ & \\ \hfill \varpi (\mathcal{x},\mathcal{K})& ={\displaystyle \frac{\cos \mathcal{K}}{10}}\varpi \left(\mathcal{x},{{\displaystyle \frac{1}{3}}}^{-}\right),\mathcal{K}\in \left({\displaystyle \frac{1}{3}},{\displaystyle \frac{2}{3}}\right]\hfill \\ & \\ \hfill {\mathcal{D}}_0^{{\displaystyle \frac{3}{5}}}\varpi (\mathcal{x},\mathcal{K})-(1+\mathcal{K}){\displaystyle \frac{{\partial }^2\varpi (\mathcal{x},\mathcal{K})}{\partial {\mathcal{x}}^2}}& ={\displaystyle \frac{\sin \mathcal{K}}{25\left(1+|\varpi \left(\mathcal{x},{\mathcal{Q}}_1\left(\mathcal{K}\right)\right)|+\cdots +|\varpi \left(\mathcal{x},{\mathcal{Q}}_m\left(\mathcal{K}\right)\right)|\right)}},\mathcal{K}\in \left({\displaystyle \frac{2}{3}},1\right]\hfill \\ & \\ \hfill \varpi (\mathcal{x},0)& =0.\hfill \end{array}\hfill \end{array}\right.$$

where ${\mathcal{D}}_0^{{\displaystyle \frac{3}{5}}}$ signifies the fractional partial derivative w.r.t time of order ${\displaystyle \frac{3}{5}}$ with lower limit 0 in the Caputo’s sense. Let $\mathrm{\gimel }=L^2\left([0,1],\mathbb{R}\right)$ be a complete normed space with $L^2-$ norm $\parallel ·\parallel$. Let us define an operator $\mathrm{\aleph }\left(\mathcal{K}\right)$ in Banach space ℷ by

$$\mathrm{\aleph }\left(\mathcal{K}\right)=-(1+\mathcal{K}){\displaystyle \frac{{\partial }^{2}u}{\partial x^2}}.$$

It is well established in [28] that the operator $-\mathrm{\aleph }\left(s\right)$ is the infinitesimal generator of an analytic semigroup $e^{-\mathcal{K}\mathrm{\aleph }\left(s\right)}$ on the Banach space ℷ. Furthermore, one can easily verify that the linear operator $-\mathrm{\aleph }\left(\mathcal{K}\right)$ fulfills the conditions $\left({\mathcal{A}}_1\right)$ and $\left({\mathcal{A}}_2\right)$.

For any $\mathcal{K}\in [0,1]$

$$0=s_o<{\displaystyle \frac{1}{3}}={\mathcal{K}}_1<{\displaystyle \frac{2}{3}}=s_1<1={\mathcal{K}}_2.$$

Let $\varpi \left(\mathcal{K}\right)x=\varpi (x,\mathcal{K})$. The functions $\mathcal{Z}$ and ℘ are defined as

$$\mathcal{Z}\left(\mathcal{K},\varpi \left({\mathcal{Q}}_1\left(\mathcal{K}\right)\right),\varpi \left({\mathcal{Q}}_2\left(\mathcal{K}\right)\right),\cdots ,\varpi \left({\mathcal{Q}}_m\left(\mathcal{K}\right)\right)\right)={\displaystyle \frac{\sin \mathcal{K}}{25\left(1+|\varpi \left(x,{\mathcal{Q}}_1\left(\mathcal{K}\right)\right)|+\cdots +|\varpi \left(x,{\mathcal{Q}}_m\left(\mathcal{K}\right)\right)|\right)}}$$

$${\mathrm{\wp }}_1\left(\mathcal{K},\varpi \left({{\displaystyle \frac{1}{3}}}^{-}\right)\right)={\displaystyle \frac{\cos \mathcal{K}}{3}}\varpi \left(x,{{\displaystyle \frac{1}{3}}}^{-}\right)$$

$${\varpi }_0=0.$$

By the definitions of non-linear functions $\mathcal{Z}$ and ${\mathrm{\wp }}_1$, one can easily verify that hypotheses $\left(1\right)$ and $\left(2\right)$ are satisfied with $\mathbb{k}={\zeta }_{\mathcal{Z}}={\zeta }_{\mathrm{\wp }}=0.$

In addition, it is easy to check that $\mathcal{Z}$ and ${\mathrm{\wp }}_1$ are functions satisfying the Lipschitz condition with Lipschitz constants ${\mathsf{L}}_{\mathcal{Z}}={\displaystyle \frac{1}{25}}$ and ${\mathsf{L}}_{{\mathrm{\wp }}_1}={\displaystyle \frac{1}{10}}$. Furthermore, we have

$$\mathrm{\top }=1,\mathrm{♭}={\displaystyle \frac{3}{5}},\rho =1,𝐹=1,\mathbb{B}\left({\displaystyle \frac{3}{5}},2\right)=1.04.$$

Consequently, all the hypotheses of Theorem 1 are satisfied. Therefore, the considered non-autonomous fractional partial differential system admits at least one mild solution

$$\varpi \in C(\Xi ,\mathrm{\gimel }).$$

Now, using

$${\Theta }_1=\left[{\displaystyle \frac{𝐹{\mathcal{K}}_1^{\mathrm{♭}}}{\mathrm{♭}}}+{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\displaystyle \frac{{\mathcal{K}}_1^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}\right]{\mathsf{L}}_{\mathcal{Z}},$$

we obtain

$$\begin{array}{cc}\hfill {\Theta }_1& =\left[{\displaystyle \frac{\left(1\right){\left(\frac{1}{3}\right)}^{3/5}}{3/5}}+{\left(1\right)}^2\mathbb{B}\left({\displaystyle \frac{3}{5}},1\right){\displaystyle \frac{{\left(\frac{1}{3}\right)}^{8/5}}{8/5}}\right]{\displaystyle \frac{1}{25}}\hfill \\ \hfill & =\left[{\displaystyle \frac{5}{3}}{\left({\displaystyle \frac{1}{3}}\right)}^{3/5}+{\displaystyle \frac{5}{3}}{\displaystyle \frac{5}{8}}{\left({\displaystyle \frac{1}{3}}\right)}^{8/5}\right]{\displaystyle \frac{1}{25}}\hfill \\ \hfill & =\left[{\displaystyle \frac{5}{3}}\left(0.517281857\right)+{\displaystyle \frac{25}{24}}\left(0.172427286\right)\right]{\displaystyle \frac{1}{25}}\hfill \\ \hfill & =\left[0.862136429+0.179611756\right]{\displaystyle \frac{1}{25}}\hfill \\ \hfill & ={\displaystyle \frac{1.041748185}{25}}\approx 0.04166993.\hfill \end{array}$$

Therefore,

$$\begin{array}{|c|}\hline {\Theta }_1\approx 0.04166993\\ \hline\end{array}.$$

Next, using

$$\begin{array}{cc}\hfill {\Theta }_2=& [𝐹{\mathsf{L}}_{\mathrm{\wp }}+{𝐹}^2{\mathsf{L}}_{\mathrm{\wp }}{\mathrm{\top }}^{\mathrm{♭}}\left\{{\displaystyle \frac{1}{\mathrm{♭}}}+{\mathrm{\top }}^{\rho }\mathbb{B}(\mathrm{♭},\rho +1)\right\}\hfill \\ \hfill & +{\displaystyle \frac{{𝐹}^2{\mathrm{\top }}^{\mathrm{♭}}}{\mathrm{♭}}}{\mathsf{L}}_{\mathcal{Z}}+{\displaystyle \frac{{𝐹}^2\mathbb{B}(\mathrm{♭},\rho ){\mathrm{\top }}^{\mathrm{♭}+\rho }}{\mathrm{♭}+\rho }}{\mathsf{L}}_{\mathcal{Z}}],\hfill \end{array}$$

we have

$$\begin{array}{cc}\hfill {\Theta }_2=& [{\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{10}}\left\{{\displaystyle \frac{1}{3/5}}+\mathbb{B}\left({\displaystyle \frac{3}{5}},2\right)\right\}\hfill \\ \hfill & +{\displaystyle \frac{1}{3/5}}{\displaystyle \frac{1}{25}}+{\displaystyle \frac{\mathbb{B}\left(\frac{3}{5},1\right)}{8/5}}{\displaystyle \frac{1}{25}}]\hfill \\ \hfill =& \left[{\displaystyle \frac{1}{10}}+{\displaystyle \frac{1}{10}}\left\{{\displaystyle \frac{5}{3}}+{\displaystyle \frac{25}{24}}\right\}+{\displaystyle \frac{5}{3}}{\displaystyle \frac{1}{25}}+{\displaystyle \frac{\frac{5}{3}}{\frac{8}{5}}}{\displaystyle \frac{1}{25}}\right]\hfill \\ \hfill =& \left[0.1+0.270833333+0.066666667+0.041666667\right]\hfill \\ \hfill & =0.479166667.\hfill \end{array}$$

Hence,

$$\begin{array}{|c|}\hline {\Theta }_2\approx 0.47916667\\ \hline\end{array}.$$

Finally,

$$\begin{array}{cc}\hfill {\Theta }_1+{\Theta }_2& =0.04166993+0.47916667\hfill \\ \hfill & =0.52083660<1.\hfill \end{array}$$

Therefore,

$$\begin{array}{|c|}\hline {\Theta }_1+{\Theta }_2=0.52083660<1.\\ \hline\end{array}$$

Thus, the required condition of Theorem 2 is satisfied, and consequently the considered system is Ulam–Hyers-stable. In particular, because the Caputo derivative is taken globally with the fixed lower terminal 0, the mild solution on $\left(s_1,{\mathcal{K}}_2\right]$ retains the complete memory generated on $[0,s_1]$; hence, the corresponding integral terms in the mild formulation have lower limit 0 rather than $s_1$.

Finally, the Lipschitz constants

$${\mathsf{L}}_{\mathcal{Z}}={\displaystyle \frac{1}{25}},{\mathsf{L}}_{{\mathrm{\wp }}_1}={\displaystyle \frac{1}{10}}$$

are sufficiently small for the stability condition in Theorem 2. Hence, there exists a constant $\mathrm{\Re }>0$ such that, for every $ϵ>0$, every $ϵ$-approximate solution $\tilde{\varpi }$ of the above system satisfies

$$\parallel \tilde{\varpi }{-\varpi \parallel }_{C(\Xi ,\mathrm{\gimel })}\le \mathrm{\Re }ϵ.$$

Therefore, the considered fractional partial differential system is Ulam–Hyers-stable.

6. Conclusions

This study addresses the Ulam–Hyers stability of fractional nonautonomous evolution equations subject to delayed arguments, fractional memory, time-dependent nonlinear operators, noninstantaneous impulses. By employing suitable fixed-point theorems, we derive sufficient conditions ${\zeta }_{\mathrm{\wp }}{\mathrm{♈}}_1+{\zeta }_{\mathcal{Z}}{\mathrm{♈}}_2<1$, which ensures the invariance of the prescribed closed ball, and ${{\rm Y}}_{{\mathrm{♌}}_1}={\mathsf{L}}_{\mathrm{\wp }}{\mathrm{♈}}_1<1$ provides the required contraction property and Lipschitz control of the momentum component, while ${\mathsf{L}}_{\mathrm{\wp }}{\mathrm{♈}}_1+{\mathrm{♈}}_3<1$ guarantees the required k-set contractive property of the combined operator. These conditions collectively establish the well-posedness framework needed for the mild solution analysis. Furthermore, the obtained estimates yield the Ulam–Hyers stability of the mild solution to ensure the existence and stability of mild solutions. The results extend earlier studies on autonomous systems and provide a broader framework for analyzing time-dependent operators. An illustrative example is provided to validate the applicability of the obtained results.