Efficient Solution Criteria for a Coupled Fractional Laplacian System on Some Infinite Domains

Abstract

This article concerns a novel coupled implicit differential system under $\phi$–Riemann–Liouville ($\mathbb{RL}$) fractional derivatives with $\mathfrak{p}$-Laplacian operator and multi-point strip boundary conditions on unbounded domains. An applicable Banach space is introduced to define solutions on unbounded domains $[c,\infty )$. The explicit iterative solution’s existence and uniqueness ($\mathbb{E}a\mathbb{U}$) are established by employing the Banach fixed point strategy. The different types of Ulam–Hyers–Rassias ($\mathbb{UHR}$) stabilities are investigated. Ultimately, we provide a numerical application of a coupled $\phi$-$\mathbb{RL}$ fractional turbulent flow model to illustrate and test the effectiveness of our outcomes.

1. Introduction

The topic of fractional calculus has expanded from integer order to arbitrary order [1]. This has attracted the interest of many researchers [2,3]. Fractional differential equations ($FDEs$) have been used in numerous areas of science, such as physics, engineering, and finance, to represent complex models that exhibit memory effects and nonlocal behavior [4,5,6,7,8].

The $\mathfrak{p}$-Laplacian operator is an essential tool for modeling nonlinear phenomena in various fields, and its study has led to significant progress in both applied and theoretical mathematics [9]. It has numerous applications in engineering, physics, and mathematics due to its ability to describe real phenomena [10]. Furthermore, its study has presented several mathematical challenges due to its nonlinearity [11].

The nonlinearity acts challenges in the literature studies, such as authors in [12] explored bilinear Bäcklund transformations and various wave solutions, including N-soliton, breather, fission/fusion, and hybrid solutions, for a specific $(3+1)$-dimensional integrable wave equation relevant to fluid dynamics. Ref. [13] examined a complex $(3+1)$-dimensional shallow water wave equation that evolves, with applications that are relevant to oceans and rivers. Ref. [14] investigated the hetero-Bäcklund transformation, bilinear forms, and multisoliton solutions of a $(2+1)$ dimensional, generalized, modified, dispersive water wave system. Many research works have investigated $FDEs$, involving the $\mathfrak{p}$-Laplacian operator [15]. For example, Hasanov [16] studied the global existence of solutions for the initial Caputo fractional Laplacian differential equations with singular points. Ref. [17] applied the existence theorem to the $\Phi$-Caputo fractional Langevin differential system involving the $\mathfrak{p}$-Laplacian mapping. In addition, Derbazi et al. [18] established some qualitative results for the Caputo hybrid $FDEs$ with a $\mathfrak{p}$-Laplacian mapping. Ref. [19] used the Avery–Peterson fixed point theorem to investigate the existence of positive solutions for the Caputo $FDEs$ under some integral boundary conditions involving the $\mathfrak{p}$-Laplacian operator.

Unbounded solutions often arise in nonlinear ordinary, partial, and fractional differential equations ($FDEs$), as well as multi-order nonlinear problems with excitable behavior [20,21]. Their investigation is of great interest because it provides insights into the stability, limitations, and long-term behavior of dynamical systems [22]. Boutiara et al. [23] studied unbounded solutions of nonlinear fractional q-$\mathbb{RL}$ difference equations using classical fixed point theorems. Luca and Tudorache [21] investigated the existence of unbounded positive solutions for coupled nonlinear $FDEs$ involving Hadamard derivatives by employing the Leggett–Williams and Guo–Krasnoselskii fixed point theorems. In 2024, Nyamoradi and Ahmad [24] explored iterative positive solutions for Hadamard $FDEs$ with integral boundary conditions on the interval $(1,\infty )$. The authors of [25] discussed the $\mathbb{E}a\mathbb{U}$ of an unbounded solution for coupled $FDEs$. Furthermore, fixed point theorems have led to researchers studying integral equations and $FDEs$; we refer readers to the works [26,27,28,29].

In this study, motivated by the aforementioned works, we investigate the qualitative properties of an explicit iterative solution, such as the existence, uniqueness, and stability of various forms of the $\mathbb{UH}$ type, for the following novel coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system with multi-point strip boundary conditions on the unbounded domain $[c,\infty )$:

$$\left\{\begin{array}{c}{\displaystyle {\mathcal{D}}_{c^{+}}^{{\theta }_1,\phi }{\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{D}}_{c^{+}}^{{\vartheta }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)={\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right),\mathsf{t}\in \mathcal{T}=[c,\infty ),}\hfill \\ {\displaystyle {\mathcal{D}}_{c^{+}}^{{\theta }_2,\phi }{\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{D}}_{c^{+}}^{{\vartheta }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)={\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right),\mathsf{t}\in \mathcal{T}=[c,\infty ),}\hfill \\ {\displaystyle \mathcal{U}\left(c\right)=0,{\mathcal{D}}_{c^{+}}^{{\vartheta }_1,\phi }\mathcal{U}\left(c\right)=0,\underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_1-1,\phi }\mathcal{U}\left(\mathsf{t}\right)=\sum _{j=1}^n{\eta }_j{\mathcal{D}}_{c^{+}}^{1,\phi }\mathcal{U}\left({ϵ}_j\right),n\in \mathbb{N},{ϵ}_j\in [c,\infty ),}\hfill \\ {\displaystyle \mathcal{V}\left(c\right)=0,{\mathcal{D}}_{c^{+}}^{{\vartheta }_2,\phi }\mathcal{V}\left(c\right)=0,\underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_2-1,\phi }\mathcal{V}\left(\mathsf{t}\right)=\sum _{i=1}^m{\zeta }_i{\mathcal{D}}_{c^{+}}^{1,\phi }\mathcal{V}\left({\epsilon }_i\right),m\in \mathbb{N},{\epsilon }_i\in [c,\infty ),}\hfill \end{array}\right.$$

(1)

where ${\mathcal{D}}_{c^{+}}^{x,\phi }$ is the $\phi$-fractional $\mathbb{RL}$ derivatives of order $x\in \{{\theta }_1,{\theta }_2,{\vartheta }_1,{\vartheta }_2,{\gamma }_1,{\gamma }_2\}$, such that ${\theta }_1,{\theta }_2\in (0,1],$ ${\vartheta }_1,{\vartheta }_2\in (1,2)$, $1<{\gamma }_1,{\gamma }_2<{\vartheta }_1,{\vartheta }_2<2$; and $\phi \left(\mathsf{t}\right)\in {\mathfrak{C}}^2(\mathcal{T},{\mathbb{R}}^{+})$ be an increasing continuous positive function, with ${\phi }^{\prime }\left(\mathsf{t}\right)\ne 0,\forall \mathsf{t}\in \mathcal{T}.$ Furthermore, $\mathcal{U},\mathcal{V}\in \mathfrak{C}(\mathcal{T},\mathbb{K}),$ and ${\mathcal{W}}_1,{\mathcal{W}}_2\in \mathfrak{C}(\mathcal{T}\times \mathbb{K}\times \mathbb{K},\mathbb{K})$ are continuous functions, and $\mathbb{K}$ is a real space; ${\eta }_j,{\zeta }_i\in \mathbb{K},c<{ϵ}_1<{ϵ}_2<\cdots <{ϵ}_n<\infty ,c<{\epsilon }_1<{\epsilon }_2<\cdots <{\epsilon }_m<\infty ,$ such that $({\vartheta }_1-1){\sum }_{j=1}^n{\eta }_j{\phi }^{{\vartheta }_1-2}({ϵ}_j;c)\ne 0$, and $({\vartheta }_2-1){\sum }_{i=1}^m{\zeta }_i{\phi }^{{\vartheta }_2-2}({\epsilon }_i;c)\ne 0$, where we adopted the notation ${\phi }^{\delta }(\mathsf{t};c)={\left(\phi \left(\mathsf{t}\right)-\phi \left(c\right)\right)}^{\delta }$. Moreover, the $\mathfrak{p}$-Laplacian operator ${\mathsf{\psi }}_{\mathfrak{p}}\left(\mathcal{U}\right)$ is given as ${\mathsf{\psi }}_{\mathfrak{p}}\left(\mathcal{U}\right)={\left|\mathcal{U}\right|}^{\mathfrak{p}-2}\mathcal{U},\mathcal{U}\ne 0,{\mathsf{\psi }}_{\mathfrak{p}}\left(0\right)=0$, with the inverse operator defined by ${\mathsf{\psi }}_{\mathfrak{p}}^{-1}\left(\mathcal{U}\right)={\mathsf{\psi }}_{\mathfrak{q}}\left(\mathcal{U}\right)={\left|\mathcal{U}\right|}^{\mathfrak{q}-2}\mathcal{U}$, such that ${\displaystyle \frac{1}{\mathfrak{p}}}+{\displaystyle \frac{1}{\mathfrak{q}}}=1.$

The main contributions of this work are as follows:

• A new coupled $\phi$-$\mathbb{RL}$ fractional implicit differential system is considered, with $\mathfrak{p}$-Laplacian operator and multi-point strip boundary conditions on unbounded domains.
• An applicable Banach space is introduced to allow solutions to be defined on unbounded domains $[c,\infty )$.
• The $\mathbb{E}a\mathbb{U}$ of explicit iterative solution for the proposed system (1) is investigated by employing the Banach fixed point strategy.
• Some types of the $\mathbb{UHR}$ stabilities are established for the suggested coupled system (1).
• The main results are applied to a coupled $\phi$-$\mathbb{RL}$ fractional turbulent flow model.
• The proposed problem is more generalized than those existing in the literature.

The remainder of this article is organized as follows: Section 2 introduces the fundamental concepts of fractional calculus. Section 3 explores the $\mathbb{E}a\mathbb{U}$ outcome through explicit iterations solution and various forms of Ulam stability.

2. Preliminaries

In this section, we introduce several essential backgrounds for this research. Consider $\phi \in {\mathfrak{C}}^n(\mathcal{T},{\mathbb{R}}^{+})$, which denotes a positive increasing function, and satisfies ${\phi }^{\prime }\left(\mathsf{t}\right)\ne 0,$ $\mathrm{for}\mathrm{each}\mathsf{t}\in \mathcal{T}.$ Now, for $\delta >0,$ the $\phi$-$\mathbb{RL}$ fractional integral of order $\delta$ is given as follows [30]:

$$\left({\mathcal{I}}_{c^{+}}^{\delta ;\phi }\mathfrak{g}\right)\left(\mathsf{t}\right)={\displaystyle \frac{1}{\Gamma \left(\delta \right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{\delta -1}(\mathsf{t};\mathsf{v})\mathfrak{g}\left(\mathsf{v}\right)d\mathsf{v},\mathsf{t}>c.$$

Also, for $\delta \in (n-1,n]$, the $\phi$-$\mathbb{RL}$ fractional derivative of order $\delta$ is given by the following [30]:

$$\begin{array}{ccc}\hfill \left({\mathcal{D}}_{c^{+}}^{\delta ;\phi }\mathfrak{g}\right)\left(\mathsf{t}\right)& =& {\left({\displaystyle \frac{1}{{\phi }^{\prime }\left(\mathsf{t}\right)}}{\displaystyle \frac{d}{d\mathsf{t}}}\right)}^n\left({\mathcal{I}}_{c^{+}}^{n-\delta ;\phi }\mathfrak{g}\right)\left(\mathsf{t}\right),\mathsf{t}>c,\hfill \end{array}$$

In addition, the following identities act as examples for the above fractional operators: ${\mathcal{I}}_{c^{+}}^{\delta ;\phi }{\phi }^{\gamma -1}(\mathsf{t};c)={\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma +\delta )}}{\phi }^{\gamma +\delta -1}(\mathsf{t};c),\delta >0,$ and ${\mathcal{D}}_{c^{+}}^{\delta ;\phi }{\phi }^{\gamma -1}(\mathsf{t};c)={\displaystyle \frac{\Gamma \left(\gamma \right)}{\Gamma (\gamma -\delta )}}{\phi }^{\gamma -\delta -1}(\mathsf{t};c),n-1<\delta \le n,\gamma >0.$

Moreover, the fundamental fractional calculus of $\phi$-$\mathbb{RL}$ fractional derivative and integral are given as follows [31]:

$${\mathcal{I}}_{c^{+}}^{\delta ;\phi }{\mathcal{D}}_{c^{+}}^{\delta ;\phi }\mathfrak{g}\left(\mathsf{t}\right)=\mathfrak{g}\left(\mathsf{t}\right)-\sum _{k=1}^n{a}_k{\phi }^{\delta -k}(\mathsf{t};c),\forall \mathsf{t}\in \mathcal{T},$$

(2)

for $a_k\in \mathbb{R},$ $\delta \in (n-1,n]$, and ${\mathfrak{g}}^n\in \mathfrak{C}(\mathcal{T},\mathbb{R})$.

Theorem 1

([32]). Let $(\mathcal{O},\mathbb{B})$ denote the generalized completed metric space, and Ξ be a contractive operator that maps $\mathcal{O}$ into itself with the Lipschitz constant $z<1$. If there is $\ell \in \mathbb{N}$, where $\mathbb{B}({\Xi }^{\ell +1}\mathsf{t},{\Xi }^{\ell }\mathsf{t})<\infty ,$ for at least $\mathsf{t}\in \mathcal{O}$, then the following identities are fulfilled:

(i)

$\left\{{\Xi }^{\ell }\mathsf{t}\right\}$ be a convergence sequence tending to a fixed point ${\mathsf{t}}_0\in \mathcal{O}$;

(ii)

Ξ owns one fixed point ${\mathsf{t}}_0\in {\mathcal{O}}^{*}=\{\mathsf{s}\in \mathcal{O}|\mathbb{B}({\Xi }^{\ell }\mathsf{t},\mathsf{s})<\infty \}$;

(iii)

If $\mathsf{s}\in {\mathcal{O}}^{*}$, then $\mathbb{B}(\mathsf{s},{\mathsf{t}}_0)\le {\displaystyle \frac{1}{1-z}}\mathbb{B}(\Xi \mathsf{s},\mathsf{s}).$

Lemma 1

([33]). Consider ${\mathsf{\psi }}_{\mathfrak{p}}$ denotes the $\mathfrak{p}$-Laplacian function, so

(i)

If $|{\mathcal{U}}_1|,|{\mathcal{U}}_2|\ge \varrho >0$ and ${\mathcal{U}}_1,{\mathcal{U}}_2>0$, for every $1<\mathfrak{p}\le 2$, then

$$|{\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{U}}_1\right)-{\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{U}}_2\right)|\le (\mathfrak{p}-1){\varrho }^{\mathfrak{p}-2}|{\mathcal{U}}_1-{\mathcal{U}}_2|.$$

(ii)

If $|{\mathcal{U}}_1|,|{\mathcal{U}}_2|\le \rho$, for every $\mathfrak{p}>2$, then

$$|{\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{U}}_1\right)-{\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{U}}_2\right)|\le (\mathfrak{p}-1){\rho }^{\mathfrak{p}-2}|{\mathcal{U}}_1-{\mathcal{U}}_2|.$$

3. Main Results

This section discusses the $\mathbb{E}a\mathbb{U}$ of an unbounded explicit iterative solution and stability of Ulam–Hyers–Rassias ($\mathbb{UHR}$). To obtain a suitable analysis, we need to define the following spaces:

$$\begin{array}{cc}\hfill & {{\rm Y}}_1=\left\{\mathcal{U}|\mathcal{U}\left(\mathsf{t}\right)\in \mathfrak{C}(\mathcal{T},\mathbb{K}),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)\in {\mathfrak{C}}^1(\mathcal{T},\mathbb{K}),\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥\mathcal{U}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}<\infty ,\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}<\infty \right\},\hfill \\ \hfill & {{\rm Y}}_2=\left\{\mathcal{V}|\mathcal{V}\left(\mathsf{t}\right)\in \mathfrak{C}(\mathcal{T},\mathbb{K}),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)\in {\mathfrak{C}}^1(\mathcal{T},\mathbb{K}),\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥\mathcal{V}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}<\infty ,\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}<\infty \right\},\hfill \end{array}$$

which is endowed with the following norms:

$$\begin{array}{cc}\hfill & {\parallel \mathcal{U}\parallel }_{{{\rm Y}}_1}=\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥\mathcal{U}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}+\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}},\hfill \\ \hfill & {\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}=\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥\mathcal{V}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}+\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}.\hfill \end{array}$$

Similar to Lemma 2.4, [25], we can prove that $({{\rm Y}}_1,\parallel ·{\parallel }_{{{\rm Y}}_1})$ and $({{\rm Y}}_2,\parallel ·{\parallel }_{{{\rm Y}}_2})$ are Banach spaces.

Next, we also define the space $\mathbb{M}={{\rm Y}}_1\times {{\rm Y}}_2=\left\{(\mathcal{U},\mathcal{V})\right|\mathcal{U}\in {{\rm Y}}_1,\mathcal{V}\in {{\rm Y}}_2\}$ gifted by the maximum norm ${\parallel (\mathcal{U},\mathcal{V})\parallel }_{\mathbb{M}}=max\{{\parallel \mathcal{U}\parallel }_{{{\rm Y}}_1},{\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}\},(\mathcal{U},\mathcal{V})\in \mathbb{M},$ which is representing the Banach space as in the works [34,35,36].

Remark 1.

The proposed problem (1) can be reduced to the $\mathbb{RL}$-fractional derivatives sense when $\phi \left(\mathsf{t}\right)=\mathsf{t},$ and returns to the Hadamard fractional derivatives sense when $\phi \left(\mathsf{t}\right)=ln\left(\mathsf{t}\right),$ and reduces to the Katugampola fractional derivatives sense for $\phi \left(\mathsf{t}\right)={\mathsf{t}}^{\rho },\rho >0$.

Now, we derive an integral equation that is analogous to the essential coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system (1). Regarding this, we state the following hypothesis:

$\left({\mathbf{AS}}_1\right)$ The following are satisfied:

$$\begin{array}{cc}\hfill & {\int }_{c^{+}}^{\infty }{\phi }^{\prime }\left(\mathsf{v}\right){\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{v},\mathcal{V}\left(\mathsf{v}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{v}\right)\right)\right)d\mathsf{v}<\infty ,{\int }_{c^{+}}^{\infty }{\phi }^{\prime }\left(\mathsf{v}\right){\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{v},\mathcal{U}\left(\mathsf{v}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{v}\right)\right)\right)d\mathsf{v}<\infty ,\hfill \end{array}$$

and $\underset{\mathsf{t}\to \infty }{lim}\phi \left(\mathsf{t}\right)⟶\infty .$

Lemma 2.

Assume that ${\vartheta }_1,{\vartheta }_2\in (1,2),and{\theta }_1,{\theta }_2\in (0,1]$. If $\left({\mathbf{AS}}_1\right)$ holds. Then, the coupled φ-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) possesses parallel integral equations given by the following:

$$\begin{array}{cc}\hfill \mathcal{U}\left(\mathsf{t}\right)=& {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ & +{\Lambda }_1{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v},\hfill \end{array}$$

(3)

$$\begin{array}{cc}\hfill \mathcal{V}\left(\mathsf{t}\right)=& {\displaystyle \frac{1}{\Gamma \left({\vartheta }_2\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},\mathcal{U}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ & +{\Lambda }_2{\phi }^{{\vartheta }_2-1}(\mathsf{t};c)\sum _{i=1}^m{\zeta }_i{\displaystyle \frac{1}{\Gamma ({\vartheta }_2-1)}}{\int }_c^{{\epsilon }_i}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-2}({\epsilon }_i;\mathsf{v})\hfill \\ & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},\mathcal{U}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v},\hfill \end{array}$$

(4)

where ${\Lambda }_1={\displaystyle \frac{-1}{({\vartheta }_1-1){\sum }_{j=1}^n{\eta }_j{\phi }^{{\vartheta }_1-2}({ϵ}_j;c)}},{\Lambda }_2={\displaystyle \frac{-1}{({\vartheta }_2-1){\sum }_{i=1}^m{\zeta }_i{\phi }^{{\vartheta }_2-2}({\epsilon }_i;c)}}.$

Proof. 

We start our proof by performing ${\mathcal{I}}_{c^{+}}^{{\theta }_1;\phi },{\mathcal{I}}_{c^{+}}^{{\theta }_2;\phi }$, on both sides of coupled system (1); using the identity (2), we obtain

$$\begin{array}{cc}\hfill & {\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{D}}_{c^{+}}^{{\vartheta }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)={\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)+a_0{\phi }^{{\theta }_1-1}(\mathsf{t};c),\hfill \\ \hfill {\displaystyle }& {\mathsf{\psi }}_{\mathfrak{p}}\left({\mathcal{D}}_{c^{+}}^{{\vartheta }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)={\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)+b_0{\phi }^{{\theta }_2-1}(\mathsf{t};c).\hfill \end{array}$$

such that $a_0,b_0\in \mathbb{R}.$ Thus, by applying the conditions ${\mathcal{D}}_{c^{+}}^{{\vartheta }_1,\phi }\mathcal{U}\left(c\right)=0$, and ${\mathcal{D}}_{c^{+}}^{{\vartheta }_2,\phi }\mathcal{V}\left(c\right)=0,$ one finds that $a_0=0,$ and $b_0=0.$

Therefore,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\mathcal{D}}_{c^{+}}^{{\vartheta }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)={\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right),\hfill \\ \hfill & {\mathcal{D}}_{c^{+}}^{{\vartheta }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)={\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right).\hfill \end{array}\right.\end{array}$$

(5)

Next, again performing ${\mathcal{I}}_{c^{+}}^{{\vartheta }_1;\phi },{\mathcal{I}}_{c^{+}}^{{\vartheta }_2;\phi }$, on both sides of coupled system (5), we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \mathcal{U}\left(\mathsf{t}\right)={\mathcal{I}}_{c^{+}}^{{\vartheta }_1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right)\right)+a_1{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)+a_2{\phi }^{{\vartheta }_1-2}(\mathsf{t};c),\hfill \\ \hfill & \mathcal{V}\left(\mathsf{t}\right)={\mathcal{I}}_{c^{+}}^{{\vartheta }_2,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right)\right)+b_1{\phi }^{{\vartheta }_2-1}(\mathsf{t};c)+b_2{\phi }^{{\vartheta }_2-2}(\mathsf{t};c),\hfill \end{array}\right.\end{array}$$

where $a_1,a_2,b_1,b_2\in \mathbb{R}$. Then, by applying the conditions $\mathcal{U}\left(c\right)=0$, and $\mathcal{V}\left(c\right)=0,$ one has that $a_2=0,$ and $b_2=0.$ Therefore, we have

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \mathcal{U}\left(\mathsf{t}\right)={\mathcal{I}}_{c^{+}}^{{\vartheta }_1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right)\right)+a_1{\phi }^{{\vartheta }_1-1}(\mathsf{t};c),\hfill \\ \hfill & \mathcal{V}\left(\mathsf{t}\right)={\mathcal{I}}_{c^{+}}^{{\vartheta }_2,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right)\right)+b_1{\phi }^{{\vartheta }_2-1}(\mathsf{t};c).\hfill \end{array}\right.\end{array}$$

(6)

Then,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\mathcal{D}}_{c^{+}}^{1,\phi }\mathcal{U}\left(\mathsf{t}\right)\hfill & {\displaystyle ={\mathcal{I}}_{c^{+}}^{{\vartheta }_1-1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right)\right)+a_1({\vartheta }_1-1){\phi }^{{\vartheta }_1-2}(\mathsf{t};c),}\hfill \\ {\mathcal{D}}_{c^{+}}^{1,\phi }\mathcal{V}\left(\mathsf{t}\right)\hfill & {\displaystyle ={\mathcal{I}}_{c^{+}}^{{\vartheta }_2-1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right)\right)+b_1({\vartheta }_2-1){\phi }^{{\vartheta }_2-2}(\mathsf{t};c).}\hfill \end{array}\right.\end{array}$$

Also,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}{\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_1-1,\phi }\mathcal{U}\left(\mathsf{t}\right)\hfill & {\displaystyle ={\phi }^{-1}(\mathsf{t};c){\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}}\hfill \\ \hfill & +a_1\Gamma \left({\vartheta }_1\right){\phi }^{-1}(\mathsf{t};c),\hfill \\ {\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_2-1,\phi }\mathcal{V}\left(\mathsf{t}\right)\hfill & {\displaystyle ={\phi }^{-1}(\mathsf{t};c){\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}}\hfill \\ & {\displaystyle +b_1\Gamma \left({\vartheta }_2\right){\phi }^{-1}(\mathsf{t};c),}\hfill \end{array}\right.\end{array}$$

and in view of $\left({\mathbf{AS}}_1\right)$, it follows that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & \underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_1-1,\phi }\mathcal{U}\left(\mathsf{t}\right)=0,\hfill \\ \hfill & \underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_2-1,\phi }\mathcal{V}\left(\mathsf{t}\right)=0.\hfill \end{array}\right.\end{array}$$

In what follows, by applying the conditions $\underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_1-1,\phi }\mathcal{U}\left(\mathsf{t}\right)={\sum }_{j=1}^n{\eta }_j{\mathcal{D}}_{c^{+}}^{1,\phi }\mathcal{U}\left({ϵ}_j\right),$ and $\underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};c){\mathcal{D}}_{c^{+}}^{{\vartheta }_2-1,\phi }\mathcal{V}\left(\mathsf{t}\right)={\sum }_{i=1}^m{\zeta }_i{\mathcal{D}}_{c^{+}}^{1,\phi }\mathcal{V}\left({\epsilon }_i\right),$ we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & 0=\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left({ϵ}_j,\mathcal{V}\left({ϵ}_j\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left({ϵ}_j\right)\right)\right)\right)+a_1({\vartheta }_1-1)\sum _{j=1}^n{\eta }_j{\phi }^{{\vartheta }_1-2}({ϵ}_j;c),\hfill \\ \hfill & 0=\sum _{i=1}^m{\zeta }_i{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left({\epsilon }_i,\mathcal{U}\left({\epsilon }_i\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left({\epsilon }_i\right)\right)\right)\right)+b_1({\vartheta }_2-1)\sum _{i=1}^m{\zeta }_i{\phi }^{{\vartheta }_2-2}({\epsilon }_i;c),\hfill \end{array}\right.\end{array}$$

which implies that

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & a_1={\displaystyle \frac{-1}{({\vartheta }_1-1){\sum }_{j=1}^n{\eta }_j{\phi }^{{\vartheta }_1-2}({ϵ}_j;c)}}\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left({ϵ}_j,\mathcal{V}\left({ϵ}_j\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left({ϵ}_j\right)\right)\right)\right),\hfill \\ \hfill & b_1={\displaystyle \frac{-1}{({\vartheta }_2-1){\sum }_{i=1}^m{\zeta }_i{\phi }^{{\vartheta }_2-2}({\epsilon }_i;c)}}\sum _{i=1}^m{\zeta }_i{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-1,\phi }\left({\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left({\epsilon }_i,\mathcal{U}\left({\epsilon }_i\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left({\epsilon }_i\right)\right)\right)\right).\hfill \end{array}\right.\end{array}$$

(7)

Hence, by substituting the values of $a_1,b_1$ into system (6), the required result is proved.  □

3.1. Existences and Uniqueness Result

In this subsection, we study the $\mathbb{E}a\mathbb{U}$ theorem to the coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system (1).

Now, we introduce the following sufficient assumptions:

• (AS₂) Suppose there exist positive functions ${\mathcal{N}}_1\left(\mathsf{t}\right),{\mathcal{N}}_2\left(\mathsf{t}\right)$, and continuous functions ${\mathcal{W}}_1:\mathcal{T}\times {{\rm Y}}_1\times {{\rm Y}}_1\to {{\rm Y}}_1$, and ${\mathcal{W}}_2:\mathcal{T}\times {{\rm Y}}_2\times {{\rm Y}}_2\to {{\rm Y}}_2$, where

  $$\begin{array}{cc}\hfill & \parallel {\mathcal{W}}_1\left(\mathsf{t},{\mathcal{V}}_1,{\mathcal{V}}_2\right)-{\mathcal{W}}_1\left(\mathsf{t},{\tilde{\mathcal{V}}}_1,{\tilde{\mathcal{V}}}_2\right)\parallel \le {\mathcal{N}}_1\left(\mathsf{t}\right)\left(\parallel {\mathcal{V}}_1-{\tilde{\mathcal{V}}}_1\parallel +\parallel {\mathcal{V}}_2-{\tilde{\mathcal{V}}}_2\parallel \right),\hfill \\ \hfill & \parallel {\mathcal{W}}_2\left(\mathsf{t},{\mathcal{U}}_1,{\mathcal{U}}_2\right)-{\mathcal{W}}_2\left(\mathsf{t},{\tilde{\mathcal{U}}}_1,{\tilde{\mathcal{U}}}_2\right)\parallel \le {\mathcal{N}}_2\left(\mathsf{t}\right)\left(\parallel {\mathcal{U}}_1-{\tilde{\mathcal{U}}}_1\parallel +\parallel {\mathcal{U}}_2-{\tilde{\mathcal{U}}}_2\parallel \right).\hfill \end{array}$$
• (AS₃) Suppose there exist positive functions ${\mathtt{f}}_1\left(\mathsf{t}\right),{\mathtt{f}}_2\left(\mathsf{t}\right)$, and continuous functions ${\mathcal{W}}_1:\mathcal{T}\times {{\rm Y}}_1\times {{\rm Y}}_1\to {{\rm Y}}_1$, and ${\mathcal{W}}_2:\mathcal{T}\times {{\rm Y}}_2\times {{\rm Y}}_2\to {{\rm Y}}_2$, where

  $$\parallel {\mathcal{W}}_1\left(\mathsf{t},{\mathcal{V}}_1,{\mathcal{V}}_2\right)\parallel \le {\mathtt{f}}_1\left(\mathsf{t}\right),\parallel {\mathcal{W}}_2\left(\mathsf{t},{\mathcal{U}}_1,{\mathcal{U}}_2\right)\parallel \le {\mathtt{f}}_2\left(\mathsf{t}\right).$$
• (AS₄) Suppose there are constants ${\mathcal{A}}_1,{\mathcal{B}}_1\in [0,1)$, and ${\mathcal{A}}_2,{\mathcal{B}}_2,{\mathcal{A}}_3,{\mathcal{B}}_3>0,$ where

  $$\begin{array}{cc}\hfill \underset{\mathsf{t}\in \mathcal{T}}{sup}& ({\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\mathcal{I}}_{c^{+}}^{{\alpha }_1+{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))\right)\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\alpha }_1-1}(\mathsf{t};c)}{\Gamma \left({\alpha }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1-1;\phi }\left({\mathcal{N}}_1\left({ϵ}_j\right)(1+{\phi }^{{\vartheta }_1}({ϵ}_j;c))\right))\le {\mathcal{A}}_1<1,\hfill \\ \hfill \underset{\mathsf{t}\in \mathcal{T}}{sup}& ({\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{B}}_2\right)}^{\mathfrak{q}-2}}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}{\mathcal{I}}_{c^{+}}^{{\alpha }_2+{\theta }_2;\phi }\left({\mathcal{N}}_2\left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))\right)\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_2|\Gamma \left({\vartheta }_2\right){\phi }^{{\alpha }_2-1}(\mathsf{t};c)}{\Gamma \left({\alpha }_2\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}(\mathfrak{q}-1){\left({\mathcal{B}}_2\right)}^{\mathfrak{q}-2}\sum _{i=1}^m{\zeta }_i{\mathcal{I}}_{c^{+}}^{{\vartheta }_2+{\theta }_2-1;\phi }\left({\mathcal{N}}_2\left({\epsilon }_i\right)(1+{\phi }^{{\vartheta }_2}({\epsilon }_i;c))\right))\le {\mathcal{B}}_1<1,\hfill \\ \hfill \underset{\mathsf{t}\in \mathcal{T}}{sup}& \left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\theta }_1-1}(\mathsf{t};\mathsf{v}){\mathtt{f}}_1\left(\mathsf{v}\right)d\mathsf{v}\right)\le {\mathcal{A}}_2<\infty ,\hfill \\ \hfill \underset{\mathsf{t}\in \mathcal{T}}{sup}& \left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\theta }_2-1}(\mathsf{t};\mathsf{v}){\mathtt{f}}_2\left(\mathsf{v}\right)d\mathsf{v}\right)\le {\mathcal{B}}_2<\infty ,\hfill \\ \hfill \underset{\mathsf{t}\in \mathcal{T}}{sup}& ({\displaystyle \frac{1}{\Gamma \left({\alpha }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\alpha }_1-1}(\mathsf{t};\mathsf{v}){\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{\Gamma \left({\vartheta }_1\right)\left|{\Lambda }_1\right|{\phi }^{{\alpha }_1-1}(\mathsf{t};c)}{\Gamma \left({\alpha }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v})\le {\mathcal{A}}_3<\infty ,\hfill \\ \hfill \underset{\mathsf{t}\in \mathcal{T}}{sup}& ({\displaystyle \frac{1}{\Gamma \left({\alpha }_2\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\alpha }_2-1}(\mathsf{t};\mathsf{v}){\left({\mathcal{I}}_c^{{\theta }_2,\phi }\left({\mathcal{N}}_2\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}+{\mathcal{W}}_2^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{\Gamma \left({\vartheta }_2\right)\left|{\Lambda }_2\right|{\phi }^{{\alpha }_2-1}(\mathsf{t};c)}{\Gamma \left({\alpha }_2\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\sum _{i=1}^m{\zeta }_i{\displaystyle \frac{1}{\Gamma ({\vartheta }_2-1)}}{\int }_c^{{\epsilon }_i}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-2}({\epsilon }_i;\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_2,\phi }\left({\mathcal{N}}_2\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}+{\mathcal{W}}_2^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v})\le {\mathcal{B}}_3<\infty ,\hfill \\ \hfill & \mathrm{where}{\alpha }_1={\vartheta }_1\mathrm{or}{\vartheta }_1-{\gamma }_1,{\alpha }_2={\vartheta }_2\mathrm{or}{\vartheta }_2-{\gamma }_2,\mathrm{and}\hfill \end{array}$$

  $${\mathcal{I}}_c^{{\theta }_i,\phi }\left({\mathcal{N}}_i\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_i}(\mathsf{v};c))\right)<\infty ,{\mathcal{W}}_i^{*}=\underset{\mathsf{t}\in \mathcal{T}}{sup}\parallel {\mathcal{W}}_i(\mathsf{t},0,0)\parallel <\infty ,i=1,2.$$

Next, in the light of Lemma 2, we introduce the operator $\mathbb{Y}:\mathbb{M}\to \mathbb{M}$ as follows:

$$\mathbb{Y}(\mathcal{U},\mathcal{V})\left(\mathsf{t}\right)=\left(\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right),\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)\right),$$

(8)

where

$$\begin{array}{cc}\hfill \left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)=& {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ & +{\Lambda }_1{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v},\hfill \end{array}$$

(9)

$$\begin{array}{cc}\hfill \left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)=& {\displaystyle \frac{1}{\Gamma \left({\vartheta }_2\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},\mathcal{U}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ & +{\Lambda }_2{\phi }^{{\vartheta }_2-1}(\mathsf{t};c)\sum _{i=1}^m{\zeta }_i{\displaystyle \frac{1}{\Gamma ({\vartheta }_2-1)}}{\int }_c^{{\epsilon }_i}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-2}({\epsilon }_i;\mathsf{v})\hfill \\ & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},\mathcal{U}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}.\hfill \end{array}$$

(10)

Theorem 2.

Consider the assumptions $\left({\mathbf{AS}}_1\right),\left({\mathbf{AS}}_3\right)$ and $\left({\mathbf{AS}}_4\right)$ hold, and $\mathfrak{q}>2.$ Then, $\mathbb{Y}:\mathbb{M}\to \mathbb{M}.$

Proof. 

According to the assumptions $\left({\mathbf{AS}}_2\right)$ and $\left({\mathbf{AS}}_4\right)$, for each $(\mathcal{U},\mathcal{V})\in \mathbb{M},$ and $\mathsf{t}\in \mathcal{T},$ one has

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\parallel {\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)\parallel d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\parallel {\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)\parallel d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(\parallel \mathcal{V}\left(\mathit{w}\right)\parallel +\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\parallel )+\parallel {\mathcal{W}}_1(\mathit{w},0,0)\parallel \right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(\parallel \mathcal{V}\left(\mathit{w}\right)\parallel +\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\parallel )+\parallel {\mathcal{W}}_1(\mathit{w},0,0)\parallel \right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\right.\hfill \\ \hfill & \left.\times \left({\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c))\left({\displaystyle \frac{\parallel \mathcal{V}\left(\mathit{w}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathit{w};c))}}+{\displaystyle \frac{\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathit{w};c))}}\right)+\parallel {\mathcal{W}}_1(\mathit{w},0,0)\parallel \right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\right.\hfill \\ \hfill & \left.\times \left({\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c))\left({\displaystyle \frac{\parallel \mathcal{V}\left(\mathit{w}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathit{w};c))}}+{\displaystyle \frac{\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathit{w};c))}}\right)+\parallel {\mathcal{W}}_1(\mathit{w},0,0)\parallel \right)d\mathit{w}\right)d\mathsf{v}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)d\mathsf{v}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & \le {\mathcal{A}}_3<\infty .\hfill \end{array}$$

Furthermore,

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\parallel {\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)\parallel d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\parallel {\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)\parallel d\mathit{w}\right)d\mathsf{v}\hfill \\ & \le {\displaystyle \frac{1}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(\parallel \mathcal{V}\left(\mathit{w}\right)\parallel +\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\parallel )+\parallel {\mathcal{W}}_1(\mathit{w},0,0)\parallel \right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(\parallel \mathcal{V}\left(\mathit{w}\right)\parallel +\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\parallel )+\parallel {\mathcal{W}}_1(\mathit{w},0,0)\parallel \right)d\mathit{w}\right)d\mathsf{v}\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \le {\displaystyle \frac{1}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})\left({\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{\Gamma \left({\vartheta }_1\right)\left|{\Lambda }_1\right|{\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)d\mathsf{v}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{\Gamma \left({\vartheta }_1\right)\left|{\Lambda }_1\right|{\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_1,\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_1}+{\mathcal{W}}_1^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & \le {\mathcal{A}}_3<\infty .\hfill \end{array}$$

Hence,

$$\begin{array}{cc}\hfill & {\parallel {\mathbb{Y}}_1\mathcal{V}\parallel }_{{{\rm Y}}_1}=\underset{\mathsf{t}\in \mathcal{T}}{sup}∥{\displaystyle \frac{\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}∥+\underset{\mathsf{t}\in \mathcal{T}}{sup}∥{\displaystyle \frac{{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}∥<\infty .\hfill \end{array}$$

Similarly, we obtain that

$$\begin{array}{cc}\hfill ∥{\displaystyle \frac{\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}∥& \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_2\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_2,\phi }\left({\mathcal{N}}_2\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}+{\mathcal{W}}_2^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_2|{\phi }^{{\vartheta }_2-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\sum _{i=1}^m{\zeta }_i{\displaystyle \frac{1}{\Gamma ({\vartheta }_2-1)}}{\int }_c^{{\epsilon }_i}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-2}({\epsilon }_i;\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_2,\phi }\left({\mathcal{N}}_2\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}+{\mathcal{W}}_2^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & \le {\mathcal{B}}_3<\infty ,\hfill \end{array}$$

and

$$\begin{array}{cc}\hfill ∥{\displaystyle \frac{{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}∥& \le {\displaystyle \frac{1}{\Gamma ({\vartheta }_2-{\gamma }_2)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-{\gamma }_2-1}(\mathsf{t};\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_2,\phi }\left({\mathcal{N}}_2\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}+{\mathcal{W}}_2^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{\Gamma \left({\vartheta }_2\right)\left|{\Lambda }_2\right|{\phi }^{{\vartheta }_2-{\gamma }_2-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_2-{\gamma }_2)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\sum _{i=1}^m{\zeta }_i{\displaystyle \frac{1}{\Gamma ({\vartheta }_2-1)}}{\int }_c^{{\epsilon }_i}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-2}({\epsilon }_i;\mathsf{v})\hfill \\ \hfill & \times {\left({\mathcal{I}}_c^{{\theta }_2,\phi }\left({\mathcal{N}}_2\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{v};c)){\parallel \mathcal{V}\parallel }_{{{\rm Y}}_2}+{\mathcal{W}}_2^{*}\right)\right)}^{\mathfrak{q}-1}d\mathsf{v}\hfill \\ \hfill & \le {\mathcal{B}}_3<\infty ,\hfill \end{array}$$

which proves that

$$∥{\displaystyle \frac{\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}∥+\underset{\mathsf{t}\in \mathcal{T}}{sup}∥{\displaystyle \frac{{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}∥<\infty .$$

Therefore, we infer the following:

$${\parallel \mathbb{Y}(\mathcal{U},\mathcal{V})\parallel }_{\mathbb{M}}={\parallel ({\mathbb{Y}}_1\mathcal{V},{\mathbb{Y}}_2\mathcal{U})\parallel }_{\mathbb{M}}=max\{{\parallel {\mathbb{Y}}_1\mathcal{V}\parallel }_{{{\rm Y}}_1},{\parallel {\mathbb{Y}}_2\mathcal{U}\parallel }_{{{\rm Y}}_2}\}<\infty ,$$

and ${{\rm Y}}_1,{{\rm Y}}_2$ are continuous on $\mathcal{T}$. Thus, $\mathbb{Y}:\mathbb{M}\to \mathbb{M}.$  □

Theorem 3.

Consider the assumptions $\left({\mathbf{AS}}_1\right)$–$\left({\mathbf{AS}}_4\right)$ hold, and $\mathfrak{q}>2$. If $0\le \mathcal{G}=max\{{\mathcal{A}}_1,{\mathcal{B}}_1\}<1,$ then the coupled φ-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) possesses exactly one solution on the infinite domain $\mathcal{T}\times \mathcal{T}$. Additionally, there is an iterative monotone sequence $({\left\{{\mathcal{U}}_n\right\}}_{n\in \mathbb{N}},{\left\{{\mathcal{V}}_n\right\}}_{n\in \mathbb{N}})$ which converges uniformly to $(\tilde{\mathcal{U}},\tilde{\mathcal{V}})$ on $\mathcal{T}\times \mathcal{T}$, where

$$\begin{array}{cc}\hfill {\mathcal{U}}_n\left(\mathsf{t}\right)=& {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},{\mathcal{V}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathcal{V}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ & +{\Lambda }_1{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})\hfill \\ & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},{\mathcal{V}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathcal{V}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v},\hfill \end{array}$$

(11)

$$\begin{array}{cc}\hfill {\mathcal{V}}_n\left(\mathsf{t}\right)=& {\displaystyle \frac{1}{\Gamma \left({\vartheta }_2\right)}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},{\mathcal{U}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathcal{U}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ & +{\Lambda }_2{\phi }^{{\vartheta }_2-1}(\mathsf{t};c)\sum _{i=1}^m{\zeta }_i{\displaystyle \frac{1}{\Gamma ({\vartheta }_2-1)}}{\int }_c^{{\epsilon }_i}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_2-2}({\epsilon }_i;\mathsf{v})\hfill \\ & \times {\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},{\mathcal{U}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathcal{U}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}.\hfill \end{array}$$

(12)

Moreover, an estimate of the error for the approximation sequence is given as follows:

$${\parallel (\tilde{\mathcal{U}},\tilde{\mathcal{V}})-({\mathcal{U}}_n,{\mathcal{V}}_n)\parallel }_{\mathbb{M}}\le {\displaystyle \frac{{\mathcal{G}}^n}{1-\mathcal{G}}}{\parallel ({\mathcal{U}}_1,{\mathcal{V}}_1)-({\mathcal{U}}_0,{\mathcal{V}}_0)\parallel }_{\mathbb{M}},n\in \mathbb{N}.$$

Proof. 

First, let us take the operator $\mathbb{Y}$ as defined in (8). Next, we aim to prove the mapping $\mathbb{Y}$ is contractive. Regarding this, by using $\left({\mathbf{AS}}_2\right)$, for each $(\mathcal{U},\mathcal{V}),(\overline{\mathcal{U}},\overline{\mathcal{V}})\in \mathbb{M},$ and $\mathit{w}\in \mathcal{T},$ we have

$$\begin{array}{cc}\hfill & \parallel {\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)-{\mathcal{W}}_1\left(\mathit{w},\overline{\mathcal{V}}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\overline{\mathcal{V}}\left(\mathit{w}\right)\right)\parallel \hfill \\ \hfill & \le {\mathcal{N}}_1\left(\mathit{w}\right)\left(\parallel \mathcal{V}\left(\mathit{w}\right)-\overline{\mathcal{V}}\left(\mathit{w}\right)\parallel +\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\overline{\mathcal{V}}\left(\mathit{w}\right)\parallel \right)\hfill \\ \hfill & \le {\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c)){\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}.\hfill \end{array}$$

(13)

Likewise, one has

$$\begin{array}{cc}\hfill & \parallel {\mathcal{W}}_2\left(\mathit{w},\mathcal{U}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathit{w}\right)\right)-{\mathcal{W}}_2\left(\mathit{w},\overline{\mathcal{U}}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\overline{\mathcal{U}}\left(\mathit{w}\right)\right)\parallel \hfill \\ \hfill & \le {\mathcal{N}}_2\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_2}(\mathit{w};c)){\parallel \mathcal{U}-\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2}.\hfill \end{array}$$

(14)

Additionally, in view of $\left({\mathbf{AS}}_3\right)$ and $\left({\mathbf{AS}}_4\right)$, we establish the boundedness of ${\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{v},\mathcal{V}\left(\mathsf{v}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{v}\right)\right)$, and ${\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{v},\mathcal{U}\left(\mathsf{v}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{v}\right)\right)$ as below:

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w})∥{\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)∥d\mathit{w}\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathtt{f}}_1\left(\mathit{w}\right)d\mathit{w}\le {\mathcal{A}}_2<\infty ,\hfill \end{array}$$

(15)

and similarly, we find

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},\mathcal{U}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathit{w}\right)\right)d\mathit{w}∥\hfill \\ \hfill & \le {\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathtt{f}}_2\left(\mathit{w}\right)d\mathit{w}\le {\mathcal{B}}_2<\infty .\hfill \end{array}$$

(16)

Next, for simplicity analysis, we take

$$\begin{array}{cc}\hfill & \left({\mathbb{Y}}_{10}\mathcal{V}\right)\left(\mathsf{v}\right)={\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},\mathcal{V}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathit{w}\right)\right)d\mathit{w}\right),\hfill \\ \hfill & \left({\mathbb{Y}}_{20}\mathcal{U}\right)\left(\mathsf{v}\right)={\mathsf{\psi }}_{\mathfrak{q}}\left({\displaystyle \frac{1}{\Gamma \left({\theta }_2\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_2-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},\mathcal{U}\left(\mathit{w}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathit{w}\right)\right)d\mathit{w}\right),\hfill \end{array}$$

So, by utilizing inequalities (13) and (15), one has

$$\begin{array}{cc}\hfill & ∥\left({\mathbb{Y}}_{10}\mathcal{V}\right)\left(\mathsf{v}\right)-\left({\mathbb{Y}}_{10}\overline{\mathcal{V}}\right)\left(\mathsf{v}\right)∥\hfill \\ \hfill & \le {\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}}{\Gamma \left({\theta }_1\right)}}{\int }_c^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{{\theta }_1-1}(\mathsf{v};\mathit{w}){\mathcal{N}}_1\left(\mathit{w}\right)(1+{\phi }^{{\vartheta }_1}(\mathit{w};c))\parallel \mathcal{V}-\overline{\mathcal{V}}{\parallel }_{{{\rm Y}}_1}d\mathit{w}\hfill \\ \hfill & \le (\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}\parallel \mathcal{V}-\overline{\mathcal{V}}{\parallel }_{{{\rm Y}}_1}{\mathcal{I}}_{c^{+}}^{{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c))\right),\hfill \end{array}$$

(17)

and according to inequalities (14) and (16), one finds

$$\begin{array}{cc}\hfill & ∥\left({\mathbb{Y}}_{20}\mathcal{U}\right)\left(\mathsf{v}\right)-\left({\mathbb{Y}}_{20}\overline{\mathcal{U}}\right)\left(\mathsf{v}\right)∥\hfill \\ \hfill & \le (\mathfrak{q}-1){\left({\mathcal{B}}_2\right)}^{\mathfrak{q}-2}\parallel \mathcal{U}-\overline{\mathcal{U}}{\parallel }_{{{\rm Y}}_2}{\mathcal{I}}_{c^{+}}^{{\theta }_2;\phi }\left({\mathcal{N}}_2\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{v};c))\right).\hfill \end{array}$$

(18)

Then, due to inequality (17), and $\left({\mathbf{AS}}_2\right)$, we obtain

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}-{\displaystyle \frac{\left({\mathbb{Y}}_1\overline{\mathcal{V}}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}∥\hfill \\ & \le {\displaystyle \frac{1}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v})∥\left({\mathbb{Y}}_{10}\mathcal{V}\right)\left(\mathsf{v}\right)-\left({\mathbb{Y}}_{10}\overline{\mathcal{V}}\right)\left(\mathsf{v}\right)∥d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})∥\left({\mathbb{Y}}_{10}\mathcal{V}\right)\left(\mathsf{v}\right)-\left({\mathbb{Y}}_{10}\overline{\mathcal{V}}\right)\left(\mathsf{v}\right)∥d\mathsf{v}\hfill \\ & \le {\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}}{\Gamma \left({\vartheta }_1\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-1}(\mathsf{t};\mathsf{v}){\mathcal{I}}_{c^{+}}^{{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c))\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}}{\Gamma ({\vartheta }_1-1)}}\hfill \\ \hfill & \times {\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v}){\mathcal{I}}_{c^{+}}^{{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c))\right)d\mathsf{v}\hfill \\ & \le {\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))\right)\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1-1;\phi }\left({\mathcal{N}}_1\left({ϵ}_j\right)(1+{\phi }^{{\vartheta }_1}({ϵ}_j;c))\right)\hfill \\ & \le [{\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))\right)\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1-1;\phi }\left({\mathcal{N}}_1\left({ϵ}_j\right)(1+{\phi }^{{\vartheta }_1}({ϵ}_j;c))\right)]{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}\hfill \\ & \le {\mathcal{A}}_1\parallel \mathcal{V}-\overline{\mathcal{V}}{\parallel }_{{{\rm Y}}_1}.\hfill \end{array}$$

Also,

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{\left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}-{\displaystyle \frac{\left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\overline{\mathcal{V}}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}∥\hfill \\ & \le {\displaystyle \frac{1}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};\mathsf{v})∥\left({\mathbb{Y}}_{10}\mathcal{V}\right)\left(\mathsf{v}\right)-\left({\mathbb{Y}}_{10}\overline{\mathcal{V}}\right)\left(\mathsf{v}\right)∥d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{1}{\Gamma ({\vartheta }_1-1)}}{\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v})∥\left({\mathbb{Y}}_{10}\mathcal{V}\right)\left(\mathsf{v}\right)-\left({\mathbb{Y}}_{10}\overline{\mathcal{V}}\right)\left(\mathsf{v}\right)∥d\mathsf{v}\hfill \\ & \le {\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\int }_c^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};\mathsf{v}){\mathcal{I}}_{c^{+}}^{{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c))\right)d\mathsf{v}\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\sum _{j=1}^n{\eta }_j{\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}}{\Gamma ({\vartheta }_1-1)}}\hfill \\ \hfill & \times {\int }_c^{{ϵ}_j}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{{\vartheta }_1-2}({ϵ}_j;\mathsf{v}){\mathcal{I}}_{c^{+}}^{{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{v}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{v};c))\right)d\mathsf{v}\hfill \\ & \le {\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-{\gamma }_1+{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))\right)\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1-1;\phi }\left({\mathcal{N}}_1\left({ϵ}_j\right)(1+{\phi }^{{\vartheta }_1}({ϵ}_j;c))\right)\hfill \\ \hfill & \le {\mathcal{A}}_1{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}.\hfill \end{array}$$

Hence, we infer that

$${\parallel {\mathbb{Y}}_1\mathcal{V}-{\mathbb{Y}}_1\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}\le {\mathcal{A}}_1{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}.$$

(19)

Likewise, using inequality (18) and $\left({\mathbf{AS}}_2\right)$, we can also deduce that

$$\begin{array}{cc}\hfill & ∥{\displaystyle \frac{\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}-{\displaystyle \frac{\left({\mathbb{Y}}_2\overline{\mathcal{U}}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}∥\le {\mathcal{B}}_1{\parallel \mathcal{U}-\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2},\hfill \\ \hfill & ∥{\displaystyle \frac{\left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}-{\displaystyle \frac{\left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\overline{\mathcal{U}}\right)\left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}∥\le {\mathcal{B}}_1{\parallel \mathcal{U}-\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2},\hfill \end{array}$$

which implies that

$${\parallel {\mathbb{Y}}_2\mathcal{U}-{\mathbb{Y}}_2\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2}\le {\mathcal{B}}_1{\parallel \mathcal{U}-\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2}.$$

(20)

Hence, using inequalities (19) and (20), we find

$$\begin{array}{cc}\hfill {\parallel \mathbb{Y}(\mathcal{U},\mathcal{V})-\mathbb{Y}(\overline{\mathcal{U}},\overline{\mathcal{V}})\parallel }_{\mathbb{M}}& ={\parallel ({\mathbb{Y}}_1\mathcal{V},{\mathbb{Y}}_2\mathcal{U})-({\mathbb{Y}}_1\overline{\mathcal{V}},{\mathbb{Y}}_2\overline{\mathcal{U}})\parallel }_{\mathbb{M}}\hfill \\ \hfill & ={\parallel \left({\mathbb{Y}}_1\mathcal{V}-{\mathbb{Y}}_1\overline{\mathcal{V}},{\mathbb{Y}}_2\mathcal{U}-{\mathbb{Y}}_2\overline{\mathcal{U}}\right)\parallel }_{\mathbb{M}}\hfill \\ \hfill & =max\{{\parallel {\mathbb{Y}}_1\mathcal{V}-{\mathbb{Y}}_1\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1},{\parallel {\mathbb{Y}}_2\mathcal{U}-{\mathbb{Y}}_2\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2}\}\hfill \\ \hfill & \le max\{{\mathcal{A}}_1{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1},{\mathcal{B}}_1{\parallel \mathcal{U}-\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2}\}\hfill \\ \hfill & =max\{{\mathcal{B}}_1{\parallel \mathcal{U}-\overline{\mathcal{U}}\parallel }_{{{\rm Y}}_2},{\mathcal{A}}_1{\parallel \mathcal{V}-\overline{\mathcal{V}}\parallel }_{{{\rm Y}}_1}\}\hfill \\ \hfill & \le \mathcal{G}{\parallel (\mathcal{U},\mathcal{V})-(\overline{\mathcal{U}},\overline{\mathcal{V}})\parallel }_{\mathbb{M}},\hfill \end{array}$$

(21)

where $\mathcal{G}=max\{{\mathcal{A}}_1,{\mathcal{B}}_1\},$ and since $\mathcal{G}\in [0,1)$, then the operator $\mathbb{Y}$ bis contractive. Hence, in view of the Banach fixed point result, we infer that $\mathbb{Y}$ has only one fixed point $(\tilde{\mathcal{U}},\tilde{\mathcal{V}})$ in $\mathbb{M}$ and acts a solution of the coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) on an infinite domain $\mathcal{T}\times \mathcal{T}$.

Moreover, for any $({\mathcal{U}}_0,{\mathcal{V}}_0)\in \mathbb{M}$, ${\parallel ({\mathcal{U}}_n,{\mathcal{V}}_n)-(\tilde{\mathcal{U}},\tilde{\mathcal{V}})\parallel }_{\mathbb{M}}\to 0,$ as $n\to \infty$, where $({\mathcal{U}}_n,{\mathcal{V}}_n)=\mathbb{Y}({\mathcal{U}}_{n-1},{\mathcal{V}}_{n-1}),n\in \mathbb{N}$. Furthermore, in view of (21), we obtain

$${\parallel ({\mathcal{U}}_n,{\mathcal{V}}_n)-({\mathcal{U}}_{n-1},{\mathcal{V}}_{n-1})\parallel }_{\mathbb{M}}\le {\mathcal{G}}^{n-1}{\parallel ({\mathcal{U}}_1,{\mathcal{V}}_1)-({\mathcal{U}}_0,{\mathcal{V}}_0)\parallel }_{\mathbb{M}},n\in \mathbb{N}.$$

Then, for any k, and $k>n$, one finds

$$\begin{array}{cc}\hfill \parallel ({\mathcal{U}}_k,{\mathcal{V}}_k)-({\mathcal{U}}_n,{\mathcal{V}}_n){\parallel }_{\mathbb{M}}& \le \parallel ({\mathcal{U}}_k,{\mathcal{V}}_k)-({\mathcal{U}}_{k-1},{\mathcal{V}}_{k-1}){\parallel }_{\mathbb{M}}+{\parallel ({\mathcal{U}}_{k-1},{\mathcal{V}}_{k-1})-({\mathcal{U}}_{k-2},{\mathcal{V}}_{k-2})\parallel }_{\mathbb{M}}+\cdots \hfill \\ \hfill & +\parallel ({\mathcal{U}}_{n+1},{\mathcal{V}}_{n+1})-({\mathcal{U}}_n,{\mathcal{V}}_n){\parallel }_{\mathbb{M}}\hfill \\ & \le \left({\mathcal{G}}^{k-1}+{\mathcal{G}}^{k-2}+\cdots +{\mathcal{G}}^k\right){\parallel ({\mathcal{U}}_1,{\mathcal{V}}_1)-({\mathcal{U}}_0,{\mathcal{V}}_0)\parallel }_{\mathbb{M}}\hfill \\ & \le {\mathcal{G}}^n{\parallel ({\mathcal{U}}_1,{\mathcal{V}}_1)-({\mathcal{U}}_0,{\mathcal{V}}_0)\parallel }_{\mathbb{M}}\sum _{r=0}^{k-n-1}{\mathcal{G}}^r.\hfill \end{array}$$

This shows that, for $k\to \infty$,

$${\parallel (\tilde{\mathcal{U}},\tilde{\mathcal{V}})-({\mathcal{U}}_n,{\mathcal{V}}_n)\parallel }_{\mathbb{M}}\le {\displaystyle \frac{{\mathcal{G}}^n}{1-\mathcal{G}}}{\parallel ({\mathcal{U}}_1,{\mathcal{V}}_1)-({\mathcal{U}}_0,{\mathcal{V}}_0)\parallel }_{\mathbb{M}},n\in \mathbb{N}.$$

Hence, the proof is completed.  □

3.2. Stability Results

In this section, we are devoting our attention to investigating the stabilities of $\mathbb{UHR}$, and semi-$\mathbb{UHR}$ types. For more detail, we refer readers to [37]. We present the beneficial metrics ${\mathrm{d}}_1(·,·)$ and ${\mathrm{d}}_2(·,·)$ on the Banach space $\mathbb{M}.$ Furthermore, for non-decreasing continuous mapping $\beta \left(\mathsf{t}\right)>0$, the metric ${\mathrm{d}}_1(·,·)$ is defined by the following:

$$\begin{array}{cc}\hfill {\mathrm{d}}_1((\mathcal{U},\mathcal{V}),(\overline{\mathcal{U}},\overline{\mathcal{V}}))=inf& \left\{\mathcal{H}\in [c,\infty )|{\displaystyle \frac{\parallel \mathcal{U}\left(\mathsf{t}\right)-\overline{\mathcal{U}}\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le \mathcal{H}\beta \left(\mathsf{t}\right),{\displaystyle \frac{\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\overline{\mathcal{U}}\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le \mathcal{H}\beta \left(\mathsf{t}\right),\right.\hfill \\ \hfill & \left.{\displaystyle \frac{\parallel \mathcal{V}\left(\mathsf{t}\right)-\overline{\mathcal{V}}\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le \mathcal{H}\beta \left(\mathsf{t}\right),{\displaystyle \frac{\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\overline{\mathcal{V}}\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le \mathcal{H}\beta \left(\mathsf{t}\right),\mathsf{t}\in \mathcal{T}\right\}.\hfill \end{array}$$

Also, for non-increasing continuous mapping $\beta \left(\mathsf{t}\right)>0$, the metric ${\mathrm{d}}_2(·,·)$ is defined by:

$$\begin{array}{cc}\hfill {\mathrm{d}}_2((\mathcal{U},\mathcal{V}),(\overline{\mathcal{U}},\overline{\mathcal{V}}))=sup& \left\{\mathcal{H}\in [c,\infty )|{\displaystyle \frac{\parallel \mathcal{U}\left(\mathsf{t}\right)-\overline{\mathcal{U}}\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le \mathcal{H},{\displaystyle \frac{\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\overline{\mathcal{U}}\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le \mathcal{H},\right.\hfill \\ \hfill & \left.{\displaystyle \frac{\parallel \mathcal{V}\left(\mathsf{t}\right)-\overline{\mathcal{V}}\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le \mathcal{H},{\displaystyle \frac{\parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\overline{\mathcal{V}}\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le \mathcal{H},\mathsf{t}\in \mathcal{T}\right\}.\hfill \end{array}$$

Note that ${\mathrm{d}}_1(·,·)$ and ${\mathrm{d}}_2(·,·)$ are metrics on Banach space $\mathbb{M}$ [38].

In what follows, we investigate the stabilities of the $\mathbb{UHR}$ type.

Theorem 4.

Assume that all assumptions of Theorem 3 are verified. Also, let $\beta \left(\mathsf{t}\right)$ denote a continuous non-decreasing non-negative function on $\mathcal{T}$, verifying that

$$\begin{array}{cc}\hfill & \parallel \mathcal{U}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\Lambda }_1{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left({ϵ}_j,\mathcal{V}\left({ϵ}_j\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left({ϵ}_j\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }\beta \left(\mathsf{t}\right)\right),\hfill \\ \hfill & \parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-{\gamma }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{\Gamma \left({\vartheta }_1\right){\Lambda }_1{\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)}}\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left({ϵ}_j,\mathcal{V}\left({ϵ}_j\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left({ϵ}_j\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_1-{\gamma }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }\beta \left(\mathsf{t}\right)\right),\hfill \end{array}$$

(22)

and

$$\begin{array}{cc}\hfill & \parallel \mathcal{V}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\Lambda }_2{\phi }^{{\vartheta }_2-1}(\mathsf{t};c)\sum _{i=1}^m{\zeta }_i{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left({\epsilon }_i,\mathcal{U}\left({\epsilon }_i\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left({\epsilon }_i\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }\beta \left(\mathsf{t}\right)\right),\hfill \\ \hfill & \parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-{\gamma }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{\Gamma \left({\vartheta }_2\right){\Lambda }_2{\phi }^{{\vartheta }_2-{\gamma }_2-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_2-{\gamma }_2)}}\sum _{i=1}^m{\zeta }_i{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left({\epsilon }_i,\mathcal{U}\left({\epsilon }_i\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left({\epsilon }_i\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_2-{\gamma }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }\beta \left(\mathsf{t}\right)\right),\hfill \end{array}$$

(23)

then there is a solution $(\tilde{\mathcal{U}},\tilde{\mathcal{V}})\in \mathbb{M},$ satisfying

$$\begin{array}{cc}\hfill & {\displaystyle \frac{∥\mathcal{U}\left(\mathsf{t}\right)-\tilde{\mathcal{U}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le {\displaystyle \frac{{\mathcal{R}}_1}{1-{\mathcal{A}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\tilde{\mathcal{U}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le {\displaystyle \frac{{\mathcal{R}}_1}{1-{\mathcal{A}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{∥\mathcal{V}\left(\mathsf{t}\right)-\tilde{\mathcal{V}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\displaystyle \frac{{\mathcal{R}}_2}{1-{\mathcal{B}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\tilde{\mathcal{V}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\displaystyle \frac{{\mathcal{R}}_2}{1-{\mathcal{B}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1,\hfill \end{array}$$

which suggests that the coupled φ-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) allows for a stable $\mathbb{UHR}$ type, and, as a consequence $\mathbb{UH}$ is stable. Given that

$$\begin{array}{cc}\hfill & \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\left({\displaystyle \frac{\beta \left(\mathsf{t}\right)}{\Gamma ({\theta }_1+1)}}\right)}^{\mathfrak{q}-1}{\displaystyle \frac{\Gamma ({\theta }_1(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_1(\mathfrak{q}-1)+{\alpha }_1+1)}}{\phi }^{{\theta }_1(\mathfrak{q}-1)+{\alpha }_1}(\mathsf{t};c)\le {\mathcal{R}}_1<\infty ,\hfill \\ \hfill & \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}{\left({\displaystyle \frac{\beta \left(\mathsf{t}\right)}{\Gamma ({\theta }_2+1)}}\right)}^{\mathfrak{q}-1}{\displaystyle \frac{\Gamma ({\theta }_2(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_2(\mathfrak{q}-1)+{\alpha }_2+1)}}{\phi }^{{\theta }_2(\mathfrak{q}-1)+{\alpha }_2}(\mathsf{t};c)\le {\mathcal{R}}_2<\infty ,\hfill \end{array}$$

where ${\alpha }_1={\vartheta }_1\mathrm{or}{\vartheta }_1-{\gamma }_1$, ${\alpha }_2={\vartheta }_2\mathrm{or}{\vartheta }_2-{\gamma }_2$, and $\mathcal{R}=min\{{\mathcal{R}}_1,{\mathcal{R}}_2\}$.

Proof. 

We recall the contractive operator $\mathbb{Y}:\mathbb{M}\to \mathbb{M}$ defined in (8). In view of metric ${\mathrm{d}}_1(·,·)$, and the assumptions $\left({\mathbf{AS}}_1\right)$–$\left({\mathbf{AS}}_4\right)$, for any $(\mathcal{U},\mathcal{V}),(\overline{\mathcal{U}},\overline{\mathcal{V}})\in {{\rm Y}}_1\times {{\rm Y}}_2,$ one finds

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_1\overline{\mathcal{V}}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\hfill \\ \hfill & \le {\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}\mathcal{H}\beta \left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))\right)\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}\mathcal{H}\beta \left(\mathsf{t}\right)\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1-1;\phi }\left({\mathcal{N}}_1\left({ϵ}_j\right)(1+{\phi }^{{\vartheta }_1}({ϵ}_j;c))\right)\hfill \\ \hfill & \le {\mathcal{A}}_1\mathcal{H}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1.\hfill \end{array}$$

Also

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\overline{\mathcal{V}}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\hfill \\ \hfill & \le {\displaystyle \frac{(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}\mathcal{H}\beta \left(\mathsf{t}\right)}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-{\gamma }_1+{\theta }_1;\phi }\left({\mathcal{N}}_1\left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))\right)\hfill \\ \hfill & +{\displaystyle \frac{|{\Lambda }_1|\Gamma \left({\vartheta }_1\right){\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}(\mathfrak{q}-1){\left({\mathcal{A}}_2\right)}^{\mathfrak{q}-2}\mathcal{H}\beta \left(\mathsf{t}\right)\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1+{\theta }_1-1;\phi }\left({\mathcal{N}}_1\left({ϵ}_j\right)(1+{\phi }^{{\vartheta }_1}({ϵ}_j;c))\right)\hfill \\ \hfill & \le {\mathcal{A}}_1\mathcal{H}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1.\hfill \end{array}$$

In the same manner, one has

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_2\overline{\mathcal{U}}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{B}}_1\mathcal{H}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\overline{\mathcal{U}}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{B}}_1\mathcal{H}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1.\hfill \end{array}$$

Then, we obtain

$${\mathrm{d}}_1(\mathbb{Y}(\mathcal{U},\mathcal{V}),\mathbb{Y}(\overline{\mathcal{U}},\overline{\mathcal{V}}))\le \mathcal{G}\mathcal{H}=\mathcal{G}{\mathrm{d}}_1((\mathcal{U},\mathcal{V}),(\overline{\mathcal{U}},\overline{\mathcal{V}})),0\le \mathcal{G}<1.$$

Now, using inequalities (22)–(23), we obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left(\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\hfill \\ \hfill & \le \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\left({\displaystyle \frac{\beta \left(\mathsf{t}\right)}{\Gamma ({\theta }_1+1)}}\right)}^{\mathfrak{q}-1}{\displaystyle \frac{\Gamma ({\theta }_1(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_1(\mathfrak{q}-1)+{\vartheta }_1+1)}}{\phi }^{{\theta }_1(\mathfrak{q}-1)+{\vartheta }_1}(\mathsf{t};c)\hfill \\ & \le {\mathcal{R}}_1\beta \left(\mathsf{t}\right),\mathsf{t}\in \mathcal{T},\hfill \end{array}$$

(24)

and

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\hfill \\ \hfill & \le \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\left({\displaystyle \frac{\beta \left(\mathsf{t}\right)}{\Gamma ({\theta }_1+1)}}\right)}^{\mathfrak{q}-1}{\displaystyle \frac{\Gamma ({\theta }_1(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_1(\mathfrak{q}-1)+{\vartheta }_1-{\gamma }_1+1)}}{\phi }^{{\theta }_1(\mathfrak{q}-1)+{\vartheta }_1-{\gamma }_1}(\mathsf{t};c)\hfill \\ & \le {\mathcal{R}}_1\beta \left(\mathsf{t}\right),\mathsf{t}\in \mathcal{T},\hfill \end{array}$$

(25)

Analogously, we also obtain

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left(\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{R}}_2\beta \left(\mathsf{t}\right),\mathsf{t}\in \mathcal{T},\hfill \end{array}$$

(26)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)\parallel }{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{R}}_2\beta \left(\mathsf{t}\right),\mathsf{t}\in \mathcal{T}.\hfill \end{array}$$

(27)

Then, the above inequalities imply that

$${\mathrm{d}}_1((\mathcal{U},\mathcal{V}),\mathbb{Y}(\mathcal{U},\mathcal{V}))\le \mathcal{R}<\infty .$$

Thus, by employing Theorem 1, the mapping $\mathbb{Y}$ admits a fixed point $(\tilde{\mathcal{U}},\tilde{\mathcal{V}})$, and it follows that

$${\mathrm{d}}_1((\mathcal{U},\mathcal{V}),(\tilde{\mathcal{U}},\tilde{\mathcal{V}}))\le {\displaystyle \frac{1}{1-\mathcal{G}}}{\mathrm{d}}_1(\mathbb{Y}(\mathcal{U},\mathcal{V}),(\mathcal{U},\mathcal{V}))\le {\displaystyle \frac{\mathcal{R}}{1-\mathcal{G}}},0\le \mathcal{G}<1.$$

Hence, we infer that the coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) is $\mathbb{UHR}$ stable. Additionally, when $\beta \left(\mathsf{t}\right)=1$, the coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) is called, $\mathbb{UH}$ stable.  □

Theorem 5.

Assume that all assumptions of Theorem 3 are fulfilled. Also, let $\beta \left(\mathsf{t}\right)$ denote a continuous non-increasing non-negative function on $\mathcal{T}$, and for $\mu >0$ verifying

$$\begin{array}{cc}\hfill & \parallel \mathcal{U}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\Lambda }_1{\phi }^{{\vartheta }_1-1}(\mathsf{t};c)\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left({ϵ}_j,\mathcal{V}\left({ϵ}_j\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left({ϵ}_j\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }\mu \right),\hfill \\ \hfill & \parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-{\gamma }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{\Gamma \left({\vartheta }_1\right){\Lambda }_1{\phi }^{{\vartheta }_1-{\gamma }_1-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_1-{\gamma }_1)}}\sum _{j=1}^n{\eta }_j{\mathcal{I}}_{c^{+}}^{{\vartheta }_1-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }{\mathcal{W}}_1\left({ϵ}_j,\mathcal{V}\left({ϵ}_j\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\left({ϵ}_j\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_1-{\gamma }_1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_1,\phi }\mu \right),\hfill \end{array}$$

(28)

and

$$\begin{array}{cc}\hfill & \parallel \mathcal{V}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\Lambda }_2{\phi }^{{\vartheta }_2-1}(\mathsf{t};c)\sum _{i=1}^m{\zeta }_i{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left({\epsilon }_i,\mathcal{U}\left({\epsilon }_i\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left({\epsilon }_i\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }\mu \right),\hfill \\ \hfill & \parallel {\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)-{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-{\gamma }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)\right)\hfill \\ \hfill & -{\displaystyle \frac{\Gamma \left({\vartheta }_2\right){\Lambda }_2{\phi }^{{\vartheta }_2-{\gamma }_2-1}(\mathsf{t};c)}{\Gamma ({\vartheta }_2-{\gamma }_2)}}\sum _{i=1}^m{\zeta }_i{\mathcal{I}}_{c^{+}}^{{\vartheta }_2-1,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }{\mathcal{W}}_2\left({\epsilon }_i,\mathcal{U}\left({\epsilon }_i\right),{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{U}\left({\epsilon }_i\right)\right)\right)\parallel \le {\mathcal{I}}_{c^{+}}^{{\vartheta }_2-{\gamma }_2,\phi }{\mathsf{\psi }}_{\mathfrak{q}}\left({\mathcal{I}}_{c^{+}}^{{\theta }_2,\phi }\mu \right),\hfill \end{array}$$

(29)

Then, one has a real number $\varrho >0$, and a solution $(\tilde{\mathcal{U}},\tilde{\mathcal{V}})\in \mathbb{M},$ satisfying

$$\begin{array}{cc}\hfill & {\displaystyle \frac{∥\mathcal{U}\left(\mathsf{t}\right)-\tilde{\mathcal{U}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le {\displaystyle \frac{\mu {\mathcal{F}}_1\varrho }{1-{\mathcal{A}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\tilde{\mathcal{U}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le {\displaystyle \frac{\mu {\mathcal{F}}_1\varrho }{1-{\mathcal{A}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{∥\mathcal{V}\left(\mathsf{t}\right)-\tilde{\mathcal{V}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\displaystyle \frac{\mu {\mathcal{F}}_2\varrho }{1-{\mathcal{B}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{∥{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\mathcal{V}\left(\mathsf{t}\right)-{\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }\tilde{\mathcal{V}}\left(\mathsf{t}\right)∥}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\displaystyle \frac{\mu {\mathcal{F}}_2\varrho }{1-{\mathcal{B}}_1}}\beta \left(\mathsf{t}\right),\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1,\hfill \end{array}$$

which suggests that the coupled φ-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) allows for a stable semi-$\mathbb{UHR}$ type.

$$\begin{array}{cc}\hfill & \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\displaystyle \frac{{\mu }^{\mathfrak{q}-2}}{{(\Gamma ({\theta }_1+1))}^{\mathfrak{q}-1}}}{\displaystyle \frac{\Gamma ({\theta }_1(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_1(\mathfrak{q}-1)+{\alpha }_1+1)}}{\phi }^{{\theta }_1(\mathfrak{q}-1)+{\alpha }_1}(\mathsf{t};c)\le {\mathcal{F}}_1<\infty ,\hfill \\ \hfill & \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}{\displaystyle \frac{{\mu }^{\mathfrak{q}-2}}{{(\Gamma ({\theta }_2+1))}^{\mathfrak{q}-1}}}{\displaystyle \frac{\Gamma ({\theta }_2(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_2(\mathfrak{q}-1)+{\alpha }_2+1)}}{\phi }^{{\theta }_2(\mathfrak{q}-1)+{\alpha }_2}(\mathsf{t};c)\le {\mathcal{F}}_2<\infty ,\hfill \end{array}$$

where ${\alpha }_1={\vartheta }_1\mathrm{or}{\vartheta }_1-{\gamma }_1$, ${\alpha }_2={\vartheta }_2\mathrm{or}{\vartheta }_2-{\gamma }_2$, and $\mathcal{F}=min\{{\mathcal{F}}_1,{\mathcal{F}}_2\}$.

Proof. 

The producers in Theorem 4 are similar. We define a contractive operator $\mathbb{Y}:\mathbb{M}\to \mathbb{M}$ as given in (8). In view of metric ${\mathrm{d}}_2(·,·)$, and the assumptions $\left({\mathbf{AS}}_1\right)$–$\left({\mathbf{AS}}_4\right)$, we find

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_1\overline{\mathcal{V}}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le {\mathcal{A}}_1\mathcal{H},\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\overline{\mathcal{V}}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\le {\mathcal{A}}_1\mathcal{H},\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{A}}_1<1.\hfill \end{array}$$

Also,

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_2\overline{\mathcal{U}}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{B}}_1\mathcal{H},\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1,\hfill \\ \hfill & {\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\overline{\mathcal{U}}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{B}}_1\mathcal{H},\forall \mathsf{t}\in \mathcal{T},0\le {\mathcal{B}}_1<1.\hfill \end{array}$$

Therefore, we deduce that

$${\mathrm{d}}_2(\mathbb{Y}(\mathcal{U},\mathcal{V}),\mathbb{Y}(\overline{\mathcal{U}},\overline{\mathcal{V}}))\le \mathcal{G}\mathcal{H}=\mathcal{G}{\mathrm{d}}_2((\mathcal{U},\mathcal{V}),(\overline{\mathcal{U}},\overline{\mathcal{V}})),0\le \mathcal{G}<1.$$

Next, based on the positiveness and continuity of a non-increasing function $\beta \left(\mathsf{t}\right)$, we obtain

$${\displaystyle \frac{1}{\beta \left(\mathsf{t}\right)}}\le \varrho ,\forall \mathsf{t}\in \mathcal{T},0<\varrho .$$

Moreover, through inequalities (28) and (29), one has

$$\begin{array}{c}{\displaystyle \frac{\parallel \left(\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\hfill \\ \le \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\left({\displaystyle \frac{\mu }{\Gamma ({\theta }_1+1)}}\right)}^{\mathfrak{q}-1}{\displaystyle \frac{\Gamma ({\theta }_1(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_1(\mathfrak{q}-1)+{\vartheta }_1+1)}}{\phi }^{{\theta }_1(\mathfrak{q}-1)+{\vartheta }_1}(\mathsf{t};c)\hfill \\ \le {\mathcal{F}}_1\varrho \mu ,\mathsf{t}\in \mathcal{T},\hfill \end{array}$$

(30)

$$\begin{array}{c}{\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{U}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }{\mathbb{Y}}_1\mathcal{V}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}\hfill \\ \le \underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{1}{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_1}(\mathsf{t};c))}}{\left({\displaystyle \frac{\mu }{\Gamma ({\theta }_1+1)}}\right)}^{\mathfrak{q}-1}{\displaystyle \frac{\Gamma ({\theta }_1(\mathfrak{q}-1)+1)}{\Gamma ({\theta }_1(\mathfrak{q}-1)+{\vartheta }_1-{\gamma }_1+1)}}{\phi }^{{\theta }_1(\mathfrak{q}-1)+{\vartheta }_1-{\gamma }_1}(\mathsf{t};c)\hfill \\ \le {\mathcal{F}}_1\varrho \mu ,\mathsf{t}\in \mathcal{T},\hfill \end{array}$$

(31)

Through analogue, we also have

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left(\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{F}}_2\varrho \mu ,\mathsf{t}\in \mathcal{T},\hfill \end{array}$$

(32)

$$\begin{array}{cc}\hfill & {\displaystyle \frac{\parallel \left({\mathcal{D}}_{c^{+}}^{{\gamma }_1,\phi }\mathcal{V}\right)\left(\mathsf{t}\right)-\left({\mathcal{D}}_{c^{+}}^{{\gamma }_2,\phi }{\mathbb{Y}}_2\mathcal{U}\right)\left(\mathsf{t}\right)\parallel }{\beta \left(\mathsf{t}\right)(1+{\phi }^{{\vartheta }_2}(\mathsf{t};c))}}\le {\mathcal{F}}_2\varrho \mu ,\mathsf{t}\in \mathcal{T},\hfill \end{array}$$

(33)

that yield

$${\mathrm{d}}_2((\mathcal{U},\mathcal{V}),\mathbb{Y}(\mathcal{U},\mathcal{V}))\le \mathcal{F}\varrho \mu <\infty .$$

Hence, according on Theorem 1, the mapping $\mathbb{Y}$ admits a fixed point $(\tilde{\mathcal{U}},\tilde{\mathcal{V}})$, and one infers that

$${\mathrm{d}}_2((\mathcal{U},\mathcal{V}),(\tilde{\mathcal{U}},\tilde{\mathcal{V}}))\le {\displaystyle \frac{1}{1-\mathcal{G}}}{\mathrm{d}}_2(\mathbb{Y}(\mathcal{U},\mathcal{V}),(\mathcal{U},\mathcal{V}))\le {\displaystyle \frac{\mathcal{F}\varrho \mu }{1-\mathcal{G}}},0\le \mathcal{G}<1.$$

Therefore, we deduce that the coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system (1) is semi-$\mathbb{UHR}$-stable.  □

4. Application to the Turbulent Flow System

Herein, we present a numerical application to test the strength of our results. Let us consider the following coupled $\phi$-$\mathbb{RL}$ fractional turbulent flow model:

$$\left\{\begin{array}{c}{\displaystyle {\mathcal{D}}_{0^{+}}^{12,\phi }{\mathsf{\psi }}_{97}\left({\mathcal{D}}_{0^{+}}^{32,\phi }\mathcal{U}\left(\mathsf{t}\right)\right)={\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{0^{+}}^{43,\phi }\mathcal{V}\left(\mathsf{t}\right)\right),\mathsf{t}\in \mathcal{T}=[0,\infty ),}\hfill \\ {\displaystyle {\mathcal{D}}_{0^{+}}^{14,\phi }{\mathsf{\psi }}_{97}\left({\mathcal{D}}_{0^{+}}^{54,\phi }\mathcal{V}\left(\mathsf{t}\right)\right)={\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{0^{+}}^{76,\phi }\mathcal{U}\left(\mathsf{t}\right)\right),\mathsf{t}\in \mathcal{T}=[0,\infty ),}\hfill \\ {\displaystyle \mathcal{U}\left(0\right)=0,{\mathcal{D}}_{0^{+}}^{32,\phi }\mathcal{U}\left(0\right)=0,\underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};0){\mathcal{D}}_{0^{+}}^{1,\phi }\mathcal{U}\left(\mathsf{t}\right)=0.2{\mathcal{D}}_{0^{+}}^{1,\phi }\mathcal{U}\left(0.01\right),}\hfill \\ {\displaystyle \mathcal{V}\left(0\right)=0,{\mathcal{D}}_{0^{+}}^{54,\phi }\mathcal{V}\left(0\right)=0,\underset{\mathsf{t}\to \infty }{lim}{\phi }^{-1}(\mathsf{t};0){\mathcal{D}}_{0^{+}}^{1,\phi }\mathcal{V}\left(\mathsf{t}\right)=0.3{\mathcal{D}}_{0^{+}}^{1,\phi }\mathcal{V}\left(0.02\right),}\hfill \end{array}\right.$$

(34)

Here, ${\theta }_1=12,$ ${\theta }_2=14,$ ${\vartheta }_1=32,$ ${\vartheta }_2=54,$ ${\gamma }_1=43,$ ${\gamma }_2=76,$ $\mathfrak{p}=97,$ $\mathfrak{q}=92,$ ${\eta }_1=0.2,$ ${\zeta }_1=0.3,m=n=1,$ ${ϵ}_1=0.01,$ ${\epsilon }_1=0.02$, and the appropriate Banach spaces are given by the following:

$$\begin{array}{cc}\hfill & {{\rm Y}}_1=\left\{\mathcal{U}|\mathcal{U}\left(\mathsf{t}\right)\in \mathfrak{C}(\mathcal{T},\mathbb{R}),{\mathcal{D}}_{0^{+}}^{43,\phi }\mathcal{U}\left(\mathsf{t}\right)\in {\mathfrak{C}}^1(\mathcal{T},\mathbb{R}),\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥\mathcal{U}\left(\mathsf{t}\right)∥}{(1+{\phi }^{32}(\mathsf{t};c))}}<\infty ,\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥{\mathcal{D}}_{0^{+}}^{43,\phi }\mathcal{U}\left(\mathsf{t}\right)∥}{(1+{\phi }^{32}(\mathsf{t};c))}}<\infty \right\},\hfill \end{array}$$

(35)

$$\begin{array}{cc}\hfill & {{\rm Y}}_2=\left\{\mathcal{V}|\mathcal{V}\left(\mathsf{t}\right)\in \mathfrak{C}(\mathcal{T},\mathbb{R}),{\mathcal{D}}_{0^{+}}^{76,\phi }\mathcal{V}\left(\mathsf{t}\right)\in {\mathfrak{C}}^1(\mathcal{T},\mathbb{R}),\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥\mathcal{V}\left(\mathsf{t}\right)∥}{(1+{\phi }^{54}(\mathsf{t};c))}}<\infty ,\underset{\mathsf{t}\in \mathcal{T}}{sup}{\displaystyle \frac{∥{\mathcal{D}}_{0^{+}}^{76,\phi }\mathcal{V}\left(\mathsf{t}\right)∥}{(1+{\phi }^{54}(\mathsf{t};c))}}<\infty \right\},\hfill \end{array}$$

(36)

Now, let us choose $\phi \left(\mathsf{t}\right)=\mathsf{t};$ we obtain

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}\hfill & {\mathcal{W}}_1\left(\mathsf{t},\mathcal{V}\left(\mathsf{t}\right),{\mathcal{D}}_{0^{+}}^{43,\mathsf{t}}\mathcal{V}\left(\mathsf{t}\right)\right)={\displaystyle \frac{1}{(1+{\mathsf{t}}^{32})}}\left[{\displaystyle \frac{1}{9e^{\mathsf{t}}}}+{\displaystyle \frac{1}{15+\mathsf{t}}}sin\left(\mathcal{V}\left(\mathsf{t}\right)\right)+{\displaystyle \frac{1}{9+\mathsf{t}}}{\displaystyle \frac{{\mathcal{D}}_{0^{+}}^{43,\mathsf{t}}\mathcal{V}\left(\mathsf{t}\right)}{(1+{\mathcal{D}}_{0^{+}}^{43,\mathsf{t}}\mathcal{V}\left(\mathsf{t}\right))}}\right],\hfill \\ & {\mathcal{W}}_2\left(\mathsf{t},\mathcal{U}\left(\mathsf{t}\right),{\mathcal{D}}_{0^{+}}^{76,\mathsf{t}}\mathcal{U}\left(\mathsf{t}\right)\right)={\displaystyle \frac{1}{(1+{\mathsf{t}}^{54})}}\left[{\displaystyle \frac{1}{5e^{2\mathsf{t}}}}+{\displaystyle \frac{1}{20+{\mathsf{t}}^2}}cos\left(\mathcal{U}\left(\mathsf{t}\right)\right)+{\displaystyle \frac{1}{16+{\mathsf{t}}^2}}{\displaystyle \frac{{\mathcal{D}}_{0^{+}}^{76,\mathsf{t}}\mathcal{U}\left(\mathsf{t}\right)}{(1+{\mathcal{D}}_{0^{+}}^{76,\mathsf{t}}\mathcal{U}\left(\mathsf{t}\right))}}\right].\hfill \end{array}\right.\end{array}$$

(37)

Then, one can see that

$$\begin{array}{cc}\hfill & \left|{\mathcal{W}}_1\left(\mathsf{t},{\mathcal{V}}_1,{\mathcal{V}}_2\right)\right|\le {\displaystyle \frac{1}{(1+{\mathsf{t}}^{32})}}\left[{\displaystyle \frac{1}{9e^{\mathsf{t}}}}+{\displaystyle \frac{1}{15+\mathsf{t}}}+{\displaystyle \frac{1}{9+\mathsf{t}}}\right],\hfill \\ \hfill & \left|{\mathcal{W}}_2\left(\mathsf{t},{\mathcal{U}}_1,{\mathcal{U}}_2\right)\right|\le {\displaystyle \frac{1}{(1+{\mathsf{t}}^{54})}}\left[{\displaystyle \frac{1}{5e^{2\mathsf{t}}}}+{\displaystyle \frac{1}{20+{\mathsf{t}}^2}}+{\displaystyle \frac{1}{16+{\mathsf{t}}^2}}\right],\hfill \end{array}$$

yielding ${\mathtt{f}}_1\left(\mathsf{t}\right)={\displaystyle \frac{1}{(1+{\mathsf{t}}^{32})}}\left[{\displaystyle \frac{1}{9e^{\mathsf{t}}}}+{\displaystyle \frac{1}{15+\mathsf{t}}}+{\displaystyle \frac{1}{9+\mathsf{t}}}\right],{\mathtt{f}}_2\left(\mathsf{t}\right)={\displaystyle \frac{1}{(1+{\mathsf{t}}^{54})}}\left[{\displaystyle \frac{1}{5e^{2\mathsf{t}}}}+{\displaystyle \frac{1}{20+{\mathsf{t}}^2}}+{\displaystyle \frac{1}{16+{\mathsf{t}}^2}}\right].$

We also obtain

$$\begin{array}{cc}\hfill & |{\mathcal{W}}_1\left(\mathsf{t},{\mathcal{V}}_1,{\mathcal{V}}_2\right)-{\mathcal{W}}_1\left(\mathsf{t},{\tilde{\mathcal{V}}}_1,{\tilde{\mathcal{V}}}_2\right)|\le {\displaystyle \frac{1}{(1+{\mathsf{t}}^{32})}}{\displaystyle \frac{1}{9+\mathsf{t}}}\left(|{\mathcal{V}}_1-{\tilde{\mathcal{V}}}_1|+|{\mathcal{V}}_2-{\tilde{\mathcal{V}}}_2|\right),\hfill \\ \hfill & |{\mathcal{W}}_2\left(\mathsf{t},{\mathcal{U}}_1,{\mathcal{U}}_2\right)-{\mathcal{W}}_2\left(\mathsf{t},{\tilde{\mathcal{U}}}_1,{\tilde{\mathcal{U}}}_2\right)|\le {\displaystyle \frac{1}{(1+{\mathsf{t}}^{54})}}{\displaystyle \frac{1}{16+{\mathsf{t}}^2}}\left(|{\mathcal{U}}_1-{\tilde{\mathcal{U}}}_1|+|{\mathcal{U}}_2-{\tilde{\mathcal{U}}}_2|\right).\hfill \end{array}$$

${\mathcal{N}}_1\left(\mathsf{t}\right)={\displaystyle \frac{1}{(1+{\mathsf{t}}^{32})}}{\displaystyle \frac{1}{9+\mathsf{t}}},{\mathcal{N}}_2\left(\mathsf{t}\right)={\displaystyle \frac{1}{(1+{\mathsf{t}}^{54})}}{\displaystyle \frac{1}{16+{\mathsf{t}}^2}},{\mathcal{W}}_1^{*}=19,$ and ${\mathcal{W}}_2^{*}=15.$ Therefore, we can infer that

$$\begin{array}{cc}\hfill & \left|{\Lambda }_1\right|=1,{\mathcal{A}}_1\approx 0.0377694<1,{\mathcal{A}}_2\approx 0.276993<\infty ,{\mathcal{A}}_3\approx 1.72682\times {10}^{106}<\infty ,\hfill \\ \hfill & \left|{\Lambda }_2\right|=0.709106,{\mathcal{B}}_1\approx 0.000171751<1,{\mathcal{B}}_2\approx 0.0553989<\infty ,{\mathcal{B}}_3\approx 1.7349\times {10}^{59}<\infty ,\hfill \\ \hfill & \mathrm{and}\mathcal{G}=max\{{\mathcal{A}}_1,{\mathcal{B}}_1\}<1.\hfill \end{array}$$

Hence, all conditions of Theorem 3 hold. Then, the coupled $\phi$-$\mathbb{RL}$ fractional Laplacian implicit differential system (34) possesses a unique solution $(\tilde{\mathcal{U}},\tilde{\mathcal{V}})$ on unbounded domain $[0,\infty )\times [0,\infty )$. Additionally, it has an explicit iterated unbounded solution, represented as follows:

$$\begin{array}{cc}\hfill & \left({\mathcal{U}}_n\left(\mathsf{t}\right),{\mathcal{V}}_n\left(\mathsf{t}\right)\right)\hfill \\ \hfill & =({\displaystyle \frac{1}{\Gamma \left(32\right)}}{\int }_0^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{32-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_3\left({\displaystyle \frac{1}{\Gamma \left(12\right)}}{\int }_0^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{12-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},{\mathcal{V}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{0^{+}}^{43,\mathsf{t}}{\mathcal{V}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\Lambda }_1{\phi }^{32-1}(\mathsf{t};c){\displaystyle \frac{0.2}{\Gamma (32-1)}}{\int }_0^{0.01}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{32-2}(0.01;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_3\left({\displaystyle \frac{1}{\Gamma \left(12\right)}}{\int }_0^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{12-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_1\left(\mathit{w},{\mathcal{V}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{0^{+}}^{43,\mathsf{t}}{\mathcal{V}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v},\hfill \\ \hfill & {\displaystyle \frac{1}{\Gamma \left(54\right)}}{\int }_0^{\mathsf{t}}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{54-1}(\mathsf{t};\mathsf{v}){\mathsf{\psi }}_3\left({\displaystyle \frac{1}{\Gamma \left(14\right)}}{\int }_0^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{14-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},{\mathcal{U}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{0^{+}}^{76,\mathsf{t}}{\mathcal{U}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}\hfill \\ \hfill & +{\Lambda }_2{\phi }^{54-1}(\mathsf{t};c){\displaystyle \frac{0.3}{\Gamma (54-1)}}{\int }_0^{0.02}{\phi }^{\prime }\left(\mathsf{v}\right){\phi }^{54-2}(0.02;\mathsf{v})\hfill \\ \hfill & \times {\mathsf{\psi }}_3\left({\displaystyle \frac{1}{\Gamma \left(14\right)}}{\int }_0^{\mathsf{v}}{\phi }^{\prime }\left(\mathit{w}\right){\phi }^{14-1}(\mathsf{v};\mathit{w}){\mathcal{W}}_2\left(\mathit{w},{\mathcal{U}}_{n-1}\left(\mathit{w}\right),{\mathcal{D}}_{0^{+}}^{76,\mathsf{t}}{\mathcal{U}}_{n-1}\left(\mathit{w}\right)\right)d\mathit{w}\right)d\mathsf{v}),\hfill \end{array}$$

where ${\mathcal{W}}_1$ and ${\mathcal{W}}_2$ are provided in Equation (37).

5. Conclusions

In this paper, we aimed to study a new coupled $\phi$-$\mathbb{RL}$ fractional implicit differential system involving a $\mathfrak{p}$-Laplacian operator (1) with multi-point strips boundary conditions on unbounded domains $[c,\infty )$. We investigated the sufficient criteria of the $\mathbb{E}a\mathbb{U}$ of the explicit iterative solution for the suggested system (1), by employing the Banach fixed point strategy. Also, we discussed the different types of stability, such as $\mathbb{UH}$, $\mathbb{UHR}$, and semi-$\mathbb{UHR}$. Finally, we provided a numerical example of a coupled $\phi$-$\mathbb{RL}$ fractional turbulent flow model to ensure the validity of our findings.