Radu–Miheţ Method for the Existence, Uniqueness, and Approximation of the ψ-Hilfer Fractional Equations by Matrix-Valued Fuzzy Controllers

Abstract

We apply the Radu–Miheţ method derived from an alternative fixed-point theorem with a class of matrix-valued fuzzy controllers to approximate a fractional Volterra integro-differential equation with the $\psi$-Hilfer fractional derivative in matrix-valued fuzzy k-normed spaces to obtain an approximation for this type of fractional equation.

1. Introduction

Fractional calculus is considered as a branch of mathematical analysis that deals with the investigation and applications of integrals and derivatives of arbitrary order. Therefore, fractional calculus is an extension of the integer-order calculus that considers integrals and derivatives of any real or complex order [1], i.e., unifying and generalizing the notions of integer-order differentiation and n-fold integration. Various forms of fractional operators have been introduced over time, such as the Riemann–Liouville, Grünwald–Letnikov, Weyl, Caputo, Marchaud, or Hadamard fractional derivatives. The first approach is the Riemann–Liouville one, which is based on the iteration of the classical integral operator for n times and then considering Cauchy’s formula where $n!$ is replaced by the Gamma function; hence, the fractional integral of non-integer order is defined. Results on the existence and stability of solutions of implicit fractional differential equations can be found in [2,3,4]. In this article, we study the fuzzy sets’ matrix valued with the generalized t-norms, to define a matrix-valued fuzzy k-Banach space (in short MV-k-FB-space) and introduce a new class of matrix-valued fuzzy controllers. We apply the Radu–Miheţ method to get an approximation for a class of $\psi$-Hilfer fractional Volterra integro-differential equations [5] in the matrix-valued fuzzy k-normed spaces (MVF-k-N-spaces).

The paper is organized as follows. In the next section, we present the definition of the generalized t-norm and define the matrix-valued fuzzy k-normed space. Next, we introduce the matrix-valued fuzzy controllers and the concept of Hyers–Ulam–Rassias stability. In Section 3, we apply the Radu–Miheţ method derived from the alternative fixed point theorem to study the Hyers–Ulam–Rassias stability of fractional Volterra integro-differential equations in MVF-k-B-spaces. In Section 4, we investigate the Hyers–Ulam–Rassias stability of fractional Volterra integral equations in MVF-k-B-spaces. In Section 5, we present some examples to illustrate our main results.

2. Preliminaries

Here, we let $E_1=[0,p]$, $E_2=(0,\infty )$, $E_3=(0,1]$, $E_4=[0,\infty ]$, $E_5=[0,1]$ (note that $E_5^{\circ }=(0,1)$ denotes the interior of $E_5$), and $E_6=[0,\infty )$.

Let:

$$\mathrm{diag}{M}_n\left(E_5\right)=\left\{\left[\begin{array}{ccc}{q}_1& & \\ & \ddots & \\ & & q_n\end{array}\right]=\mathrm{diag}[q_1,\cdots ,q_n],q_1,\dots ,q_n\in E_5\right\},$$

where $\mathrm{diag}{M}_n\left(E_5\right)$ is equipped with the partial order relation:

$$\begin{array}{cc}& \mathit{q}=\mathrm{diag}[q_1,\cdots ,q_n],\mathit{b}=\mathrm{diag}[b_1,\cdots ,b_n]\in \mathrm{diag}{M}_n\left(E_5\right),\\ & \mathit{q}⪯\mathit{b}⟺q_i\le b_i\mathrm{for}\mathrm{every}i=1,\dots ,n.\end{array}$$

Furthermore, $\mathit{q}\prec \mathit{b}$ denotes that $\mathit{q}⪯\mathit{b}$ and $\mathit{q}\ne \mathit{b}$; $\mathit{q}\ll \mathit{b}$ and $a_i<b_i$ for every $i=1,\dots ,n$. We define $\mathbf{e}=\mathrm{diag}[e,\dots ,e]$ in $\mathrm{diag}{M}_n\left(E_5\right)$ where $e\in E_5$. For instance, $\mathbf{1}=\mathrm{diag}[1,\dots ,1]$ and $\mathbf{0}=\mathrm{diag}[0,\dots ,0]$.

Now, we define a class of t-norms [6,7] on $\mathrm{diag}{M}_n\left(E_5\right)$.

Definition 1.

([6,7,8]) Consider the generalized t-norm (GTN) $⊛:\mathrm{diag}{M}_n\left(E_5\right)\times \mathrm{diag}{M}_n\left(E_5\right)\to \mathrm{diag}{M}_n\left(E_5\right)$, which satisfies the following conditions:

(a) 

$(\forall \mathit{q}\in \mathrm{diag}{M}_n\left(E_5\right))(\mathit{q}⊛\mathbf{1})=\mathit{q})$ (boundary condition);

(b) 

$(\forall (\mathit{q},\mathit{b})\in {\left(\mathrm{diag}{M}_n\left(E_5\right)\right)}^2)(\mathit{q}⊛\mathit{b}=\mathit{b}⊛\mathit{q})$ (commutativity);

(c) 

$(\forall (\mathit{q},\mathit{b},\mathit{c})\in {\left(\mathrm{diag}{M}_n\left(E_5\right)\right)}^3)(\mathit{q}⊛(\mathit{b}⊛\mathit{c})=(\mathit{q}⊛\mathit{b})⊛\mathit{c})$ (associativity);

(d) 

$(\forall ({\mathit{q}}_1,{\mathit{q}}_2,{\mathit{b}}_1,{\mathit{b}}_2)\in {\left(\mathrm{diag}{M}_n\left(E_5\right)\right)}^4)({\mathit{q}}_1⪯{\mathit{q}}_2\mathrm{and}{\mathit{b}}_1⪯{\mathit{b}}_2⟹{\mathit{q}}_1⊛{\mathit{b}}_1⪯{\mathit{q}}_2⊛{\mathit{b}}_2)$

(monotonicity).

For every $\mathit{q},\mathit{b}\in \mathrm{diag}{M}_n\left(E_5\right)$ and every sequences $\left\{{\mathit{q}}_k\right\}$ and $\left\{{\mathit{b}}_k\right\}$ converging to $\mathit{q}$ and $\mathit{b}$, suppose we have:

$$\underset{k}{lim}({\mathit{q}}_k⊛{\mathit{b}}_k)=\mathit{q}⊛\mathit{b},$$

then ⊛ on $\mathrm{diag}{M}_n\left(E_5\right)$ is the continuous GTN (CGTN). Now, we present some examples of the CGTN.

(1) Define ${⊛}_M:\mathrm{diag}{M}_n\left(E_5\right)\times \mathrm{diag}{M}_n\left(E_5\right)\to \mathrm{diag}{M}_n\left(E_5\right)$, such that,

$$\mathit{q}{⊛}_M\mathit{b}=\mathrm{diag}[q_1,\cdots ,q_n]{⊛}_M\mathrm{diag}[b_1,\cdots ,b_n]=\mathrm{diag}[min\{q_1,b_1\},\cdots ,min\{q_n,b_n\}],$$

then ${⊛}_M$ is the CGTN (minimum CGTN);

(2) Define ${⊛}_P:\mathrm{diag}{M}_n\left(E_5\right)\times \mathrm{diag}{M}_n\left(E_5\right)\to \mathrm{diag}{M}_n\left(E_5\right)$, such that,

$$\mathit{q}{⊛}_P\mathit{b}=\mathrm{diag}[q_1,\cdots ,q_n]{⊛}_P\mathrm{diag}[b_1,\cdots ,b_n]=\mathrm{diag}[q_1.b_1,\cdots ,q_n.b_n],$$

then ${⊛}_P$ is the CGTN (product CGTN);

(3) Define ${⊛}_L:\mathrm{diag}{M}_n\left(E_5\right)\times \mathrm{diag}{M}_n\left(E_5\right)\to \mathrm{diag}{M}_n\left(E_5\right)$, such that,

$$\mathit{q}{⊛}_L\mathit{b}=\mathrm{diag}[q_1,\cdots ,q_n]{⊛}_L\mathrm{diag}[b_1,\cdots ,b_n]=\mathrm{diag}[max\{q_1+b_1-1,0\},\cdots ,max\{q_n+b_n-1,0\}],$$

then ${⊛}_P$ is the CGTN (Lukasiewicz CGTN).

Now, we present some numerical examples:

$$\begin{array}{cc}\hfill & \mathrm{diag}\left[\frac{1}{2},\frac{3}{4},1,\frac{2}{3},\frac{1}{4}\right]{⊛}_M\mathrm{diag}\left[0,\frac{1}{3},\frac{2}{5},\frac{6}{7},1\right]=\hfill \\ \hfill & \left[\begin{array}{ccc}\frac{1}{2}& & \\ & \frac{3}{4}& \\ & & 1\\ & & & \frac{2}{3}\\ & & & & \frac{1}{4}\end{array}\right]{⊛}_M\left[\begin{array}{ccc}0& & \\ & \frac{1}{3}& \\ & & \frac{2}{5}\\ & & & \frac{6}{7}\\ & & & & 1\end{array}\right]=\left[\begin{array}{ccc}0& & \\ & \frac{1}{3}& \\ & & \frac{2}{5}\\ & & & \frac{2}{3}\\ & & & & \frac{1}{4}\end{array}\right]=\mathrm{diag}\left[0,\frac{1}{3},\frac{2}{5},\frac{2}{3},\frac{1}{4}\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathrm{diag}\left[\frac{1}{2},\frac{3}{4},1,\frac{2}{3},\frac{1}{4}\right]{⊛}_P\mathrm{diag}\left[0,\frac{1}{3},\frac{2}{5},\frac{6}{7},1\right]=\hfill \\ \hfill & \left[\begin{array}{ccc}\frac{1}{2}& & \\ & \frac{3}{4}& \\ & & 1\\ & & & \frac{2}{3}\\ & & & & \frac{1}{4}\end{array}\right]{⊛}_P\left[\begin{array}{ccc}0& & \\ & \frac{1}{3}& \\ & & \frac{2}{5}\\ & & & \frac{6}{7}\\ & & & & 1\end{array}\right]=\left[\begin{array}{ccc}0& & \\ & \frac{1}{4}& \\ & & \frac{2}{5}\\ & & & \frac{4}{7}\\ & & & & \frac{1}{4}\end{array}\right]=\mathrm{diag}\left[0,\frac{1}{4},\frac{2}{5},\frac{4}{7},\frac{1}{4}\right]\hfill \end{array}$$

$$\begin{array}{cc}\hfill & \mathrm{diag}\left[\frac{1}{2},\frac{3}{4},1,\frac{2}{3},\frac{1}{4}\right]{⊛}_L\mathrm{diag}\left[0,\frac{1}{3},\frac{2}{5},\frac{6}{7},1\right]=\hfill \\ \hfill & \left[\begin{array}{ccc}\frac{1}{2}& & \\ & \frac{3}{4}& \\ & & 1\\ & & & \frac{2}{3}\\ & & & & \frac{1}{4}\end{array}\right]{⊛}_L\left[\begin{array}{ccc}0& & \\ & \frac{1}{3}& \\ & & \frac{2}{5}\\ & & & \frac{6}{7}\\ & & & & 1\end{array}\right]=\left[\begin{array}{ccc}0& & \\ & \frac{1}{12}& \\ & & \frac{2}{5}\\ & & & \frac{11}{21}\\ & & & & \frac{1}{4}\end{array}\right]=\mathrm{diag}\left[0,\frac{1}{12},\frac{2}{5},\frac{11}{21},\frac{1}{4}\right].\hfill \end{array}$$

We get:

$$\begin{array}{ccc}& & \mathrm{diag}\left[\frac{1}{2},\frac{3}{4},1,\frac{2}{3},\frac{1}{4}\right]{⊛}_M\mathrm{diag}\left[0,\frac{1}{3},\frac{2}{5},\frac{6}{7},1\right]\hfill \\ & ⪰& \mathrm{diag}\left[\frac{1}{2},\frac{3}{4},1,\frac{2}{3},\frac{1}{4}\right]{⊛}_P\mathrm{diag}\left[0,\frac{1}{3},\frac{2}{5},\frac{6}{7},1\right]\hfill \\ & ⪰& \mathrm{diag}\left[\frac{1}{2},\frac{3}{4},1,\frac{2}{3},\frac{1}{4}\right]{⊛}_L\mathrm{diag}\left[0,\frac{1}{3},\frac{2}{5},\frac{6}{7},1\right].\hfill \end{array}$$

We consider the set of matrix-valued fuzzy functions (MVFFs) $\Phi$, which are left continuous and increasing functions $\phi :{\left(E_1\right)}^k\times {\left(E_2\right)}^n\to \mathrm{diag}{M}_n\left(E_3\right)$, where $\overrightarrow{t}=(t_1,\cdots ,t_n)\in {\left(E_2\right)}^n$. Furthermore, ${lim}_{t\to +\infty }\phi (p_1,\cdots ,p_k,\overrightarrow{t})=\mathbf{1}$ for any $p_1,\cdots ,p_k$ in $E_1$.

As an example, the matrix-valued fuzzy function $\phi :{\left(E_1\right)}^k\times {\left(E_2\right)}^3\to \mathrm{diag}{M}_n\left(E_3\right)$:

$$\phi (p_1,\cdots ,p_k,\overrightarrow{t})=\mathrm{diag}\left[exp\left(\frac{-\parallel p_1\cdots ,p_k\parallel }{t_1}\right),\frac{t_2}{t_2+\parallel p_1,\cdots ,p_k\parallel },exp\left(\frac{-\parallel p_1,\cdots ,p_k\parallel }{t_3}\right)\right],$$

for $\overrightarrow{t}\in {\left(E_2\right)}^3.$

In $\Phi$, we define $″{⪯}^{″}$ as follows:

$$\phi \precsim \varphi ⟺\phi (p_1,\cdots ,p_k,\overrightarrow{t})⪯\varphi (p_1,\cdots ,p_k,\overrightarrow{t}),\forall \overrightarrow{t}\in {\left(E_2\right)}^n\mathrm{and}{p}_1,\cdots ,p_k\in E_1.$$

Definition 2.

Consider the CGTN ⊛, a vector space V, and the matrix-valued fuzzy set (MVFS) $\Theta :V^k\times {\left(E_2\right)}^n\to \mathrm{diag}{M}_n\left(E_3\right)$. In this case, we define a matrix-valued fuzzy k-normed space (MVF-K-N-space) $(V,\Theta ,⊛)$ as:

(MVF-K-N1) $\Theta (v_1,\cdots ,v_k,\overrightarrow{t})=\mathbf{1}$ if and only if $v_1,\cdots ,v_k$ are linearly dependent and $\overrightarrow{t}\in {\left(E_2\right)}^n$;

(MVF-K-N2) $\Theta (\alpha v_1,\cdots ,v_k,\overrightarrow{t})=\Theta (v_1,\cdots ,v_k,\frac{\overrightarrow{t}}{\left|\alpha \right|})$ for all $v_1,\cdots ,v_k\in V$ and $\alpha \in \mathbb{C}$ with $\alpha \ne 0$;

(MVF-K-N3) $\Theta (v_0+v_1,v_2,\cdots ,v_k,\overrightarrow{t}+\overrightarrow{s})⪰\Theta (v_0,v_2,\cdots ,v_k,\overrightarrow{t})⊛\Theta (v_1,v_2,\cdots ,v_k,\overrightarrow{s})$ for all $v_1,\cdots ,v_k\in V$ and any $\overrightarrow{t}\in {\left(E_2\right)}^n$ and $\overrightarrow{s}\in {\left(E_2\right)}^n$;

(MVF-K-N4) ${lim}_{t\to +\infty }\Theta (p_1,\cdots ,p_k,\overrightarrow{t})=\mathbf{1}$ for any $\overrightarrow{t}\in {\left(E_2\right)}^n$.

A complete MVF-k-N-space is called a matrix-valued fuzzy Banach space (MVF-k-B-space).

As an example, the matrix-valued fuzzy k-norm (MVF-k-N) $\Theta$,

$$\begin{array}{ccc}& & \Theta (v_1,v_2,\cdots ,v_k,\overrightarrow{t})\hfill \\ & =& \mathrm{diag}\left[exp(-\frac{\parallel v_1,v_2,\cdots ,v_k\parallel }{t_1}),\frac{t_2}{t_2+\parallel v_1,v_2,\cdots ,v_k\parallel },exp(-\frac{\parallel v_1,v_2,\cdots ,v_k\parallel }{t_3})\right],\hfill \end{array}$$

for $\overrightarrow{t}\in {\left(E_2\right)}^3$.

Define a matrix-valued fuzzy k-norm, and $(V,\Theta ,{⊛}_M)$ is an MVF-k-N-space; here, $(V,\parallel ·\parallel )$ is a k-normed vector space. In this paper, we assume that $⊛={⊛}_M$.

Theorem 1

([9,10]). Let $(W,d)$ be a complete $E_4$-valued metric space, and let $\Lambda :W\to W$ be a strictly contractive function with Lipschitz constant $Ł<1$. Thus, for a given element $\rho \in W$, either:

$$\begin{array}{c}\hfill d\left({\Lambda }^n\rho ,{\Lambda }^{n+1}\rho \right)=\infty ,\end{array}$$

for each $n\in \mathbb{N}$ or there is $n_0\in \mathbb{N}$ such that:

(i) 

$d\left({\Lambda }^n\rho ,{\Lambda }^{n+1}\rho \right)<\infty$ for every $n\ge n_0$;

(ii) 

the fixed point ${\varrho }^{*}$ of Λ is the convergent point of sequence $\left\{{\Lambda }^n\rho \right\}$;

(iii) 

in the set $U=\left\{\varrho \in W\mid d({\Lambda }^{n_0}\rho ,\varrho )<\infty \right\}$, ${\varrho }^{*}$ is the unique fixed point of Λ;

(iv) 

$(1-Ł)d\left(\varrho ,{\varrho }^{*}\right)\le d(\varrho ,\Lambda \varrho )$ for every $\varrho \in U$.

Definition 3

([5]). Let $\Omega :(a,b)\to \mathbb{R}$$(0<a<b<\infty )$ be a finite interval and $\kappa >0$. Furthermore, let $\psi \left(p\right)$ be an increasing and positive monotone function on $(a,b]$, having a continuous derivative ${\psi }^{\prime }\left(p\right)$ (we denote the first derivative as $\frac{d}{dt}\psi \left(p\right)={\psi }^{\prime }\left(p\right)$ on $(a,b)$). The left-sided fractional integral of a function f with respect to a function $\psi \left(p\right)$ on $(a,b)$ is defined by:

$$\begin{array}{c}\hfill {\mathcal{I}}_{a^{+}}^{\kappa ,\psi }\Omega \left(p\right)=\frac{1}{\Gamma \left(\kappa \right)}{\int }_a^p{\mathcal{A}}_{\psi }^{\kappa }(p,s)\Omega \left(s\right)ds,a<p,\end{array}$$

(1)

where ${\mathcal{A}}_{\psi }^{\kappa }(p,s):={\psi }^{\prime }\left(p\right){(\psi \left(p\right)-\psi \left(s\right))}^{\kappa -1}$. The right-sided fractional integral is defined in an analogous form.

As the aim of this paper is to present some types of stabilities involving a class of fractional integro-differential equations by means of a $\psi$-Hilfer fractional operator, we introduce such a fractional operator.

Definition 4.

[5] Let $\kappa \in \mathring{E_5}$, Ω be an integrable function on $E_1\mathrm{and}\psi \in C^1\left(E_1\right)$ be an increasing function with ${\psi }^{\prime }\left(p\right)\ne 0$, for each $p\in E_1$. Define the ψ-Hilfer fractional derivative as:

$$\begin{array}{c}\hfill {}^H{\mathcal{D}}_{0+}^{\kappa ,\tau ;\psi }\Omega \left(p\right)={\mathcal{I}}_{0+}^{\tau (1-\kappa );\psi }\left(\frac{1}{{\psi }^{\prime }\left(p\right)}\frac{d}{dp}\right){\mathcal{I}}_{0+}^{(1-\tau )(1-\kappa );\psi }\Omega \left(p\right).\end{array}$$

(2)

Consider the $\psi$-Hilfer fractional Volterra integro-differential equation, defined by:

$$\begin{array}{c}\hfill {}^H{\mathcal{D}}_{0^{+}}^{\kappa ,\tau ,\psi }\Omega \left(p\right)=\mathit{S}(p,\omega \left(p\right))+{\int }_0^p\mathcal{M}(p,\sigma ,\Omega \left(\sigma \right))d\sigma ,\end{array}$$

(3)

where $\mathit{S}(p,\omega (p\left)\right)$ is a continuous function (CF) with respect to the variables $p,\omega$, and also, $\mathcal{M}(p,\sigma ,\Omega (\sigma \left)\right)$ is a CF with respect to $p,\sigma$ and $\Omega$ on $E_1$, $\kappa \in E_5^{\circ }$, $\tau \in E_5$, $\mathit{S}:E_1\times V\to V$, and $\mathcal{M}:E_1\times E_1\times V\to V$.

Let function $\varphi :{\left(E_1\right)}^k\times {\left(E_2\right)}^n\to \mathrm{diag}{M}_n\left(E_3\right)$ be a matrix-valued fuzzy function. Equation (3) is said to be Hyers–Ulam–Rassias stable if $\Omega \left(p_1\right),\cdots ,\Omega \left(p_k\right)$ is a given differentiable function, satisfying:

$$\begin{array}{cc}\hfill \Theta ({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-\mathit{S}(p_1,\Omega \left(p_1\right))& -{\int }_0^{p_1}\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))d{\sigma }_1,\cdots ,\hfill \\ \hfill {}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-\mathit{S}(p_k,\Omega \left(p_k\right))& -{\int }_0^{p_k}\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))d{\sigma }_k,\overrightarrow{\mu })⪰\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu }),\hfill \end{array}$$

for $p_1,\cdots ,p_k\in E_1$, and we can find a solution ${\rm Y}\left(p_1\right)\cdots ,{\rm Y}\left(p_k\right)$ of Equation (3) such that for some $r>0$,

$$\begin{array}{c}\hfill \Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu })⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{r}\right).\end{array}$$

Using the Radu–Miheţ method, we study the Hyers–Ulam–Rassias stability of the $\psi$-Hilfer fractional Volterra integro-differential Equation (3) in MVF-k-B-space $(V,\Theta ,⊛)$. Our results can be applied to improve recent results [5], and by the methods used in this paper, we can extend some fractional Volterra integro-differential equations in MVF-k-B-spaces [11,12,13,14].

3. Best Approximation $\psi$-Hilfer Fractional Volterra Integro-Differential Equation

In this section, we apply the Radu–Miheţ method derived from Theorem 1 to study the Hyers–Ulam–Rassias stability of functional Equation (3); for more details, we refer to [15,16]. Consider the MVF-k-B-space $(V,\Theta ,⊛)$ and matrix-valued fuzzy function (MVFF) $\varphi :{\left(E_1\right)}^k\times {\left(E_2\right)}^n\to \mathrm{diag}{M}_n\left(E_3\right)$. We set:

$$B:=\{\Omega :E_1\to V,\Omega \mathrm{is} \mathrm{differentiable}\}$$

and define a mapping d from $B\times B$ to $E_4$ by:

$$\begin{array}{cc}\hfill d(\Omega ,{\rm Y})=inf\{C\in E_6:& \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta \left(\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right),⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{C}\right),\hfill \\ \hfill & \forall \Omega ,{\rm Y}\in B,p_1,\cdots ,p_k\in E_1,\overrightarrow{\mu }\in {\left(E_2\right)}^n\}.\hfill \end{array}$$

Theorem 2.

$(B,d)$ is a complete $E_4$-valued metric fuzzy space.

Proof. 

First, we show that $(B,d)$ is an $E_4$-valued metric fuzzy space.

We show that $d(\Omega ,{\rm Y})=0$ if and only if $\Omega ={\rm Y}$. Let $d(\Omega ,{\rm Y})=0$; we have:

$$\begin{array}{cc}\hfill inf\{C\in E_6:& \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu })⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{C}\right),\hfill \\ \hfill & \forall \Omega ,{\rm Y}\in B,p_1,\cdots ,p_k\in E_1,\overrightarrow{\mu }\in {\left(E_2\right)}^n\}=0\hfill \end{array}$$

and so:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu })⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{C}\right),\hfill \end{array}$$

for all $C\in E_6$. C tends to zero in the above inequality, and we get:

$$\Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right)=\mathbf{1}$$

and so:

$$\Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu })=\mathbf{1}$$

Thus, $\Omega (p_1,\cdots ,p_k)={\rm Y}(p_1,\cdots ,p_k)$ for every $p_1,\cdots ,p_k\in E_1$, and vice versa. It is straight forward to show $d(\Omega ,{\rm Y})=d(\Omega ,{\rm Y})$ for every $\Omega ,{\rm Y}\in B$. Now, let $d(\Omega ,{\rm Y})=e_1\in E_2$ and $d(\Omega ,{\rm Y})=e_2\in E_2$. Then, we have:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu })⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{e_1}\right),\hfill \end{array}$$

and:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right),\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta ({\rm Y}\left(p_1\right)-\omega \left(p_1\right),\cdots ,{\rm Y}\left(p_k\right)-\omega \left(p_k\right),\overrightarrow{\mu })⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{e_2}\right),\hfill \end{array}$$

for every $\overrightarrow{\mu }\in {\left(E_2\right)}^n$. Then, we have:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right),(e_1+e_2)\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta \left(\Omega \left(p_1\right)-\omega \left(p_1\right),\cdots ,\Omega \left(p_k\right)-\omega \left(p_k\right),(e_1+e_2\right)\overrightarrow{\mu })\hfill \\ \hfill & ⪰[\Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),e_1\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right),e_2\overrightarrow{\mu }\right)]\hfill \\ \hfill & ⊛\left[\Theta \left(\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),e_1\overrightarrow{\mu }\right)⊛\Theta \left({\rm Y}\left(p_1\right)-\omega \left(p_1\right),\cdots ,{\rm Y}\left(p_k\right)-\omega \left(p_k\right),e_2\overrightarrow{\mu }\right)\right]\hfill \\ \hfill & ⪰\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu })⊛\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu })\hfill \\ \hfill & =\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu })\hfill \end{array}$$

and so, $d(\Omega ,\omega )\le e_1+e_2$. Thus, $d(\Omega ,\omega )\le d(\Omega ,{\rm Y})+d({\rm Y},\omega )$. Now, we are ready to prove $(B,d)$ is complete. Suppose that ${\left\{{\Omega }_k\right\}}_k$ is a Cauchy sequence in $(B,d)$. Let $p_1,\cdots ,p_k\in E_1$. Assume that $\overrightarrow{v}\in {\left(E_2\right)}^n$ and $\lambda \in {E_5}^{\circ }$ are arbitrary, and consider $\overrightarrow{\mu }\in {\left(E_2\right)}^n$ such that $\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu })\succ 1-\lambda$ for each $\lambda$ in $(0,1)$. For $e\overrightarrow{\mu }<\overrightarrow{v}$, choose $k_0\in N$ such that:

$$d({\Omega }_k,{\Omega }_{\ell })<e\forall k,\ell \ge k_0.$$

Then:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\Omega }_l\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\Omega }_l\left(p_k\right),\overrightarrow{v}\right)⊛\hfill \\ \hfill & \Theta \left({\Omega }_k\left(p_1\right)-{\Omega }_l\left(p_1\right),\cdots ,{\Omega }_k\left(p_k\right)-{\Omega }_l\left(p_k\right),\overrightarrow{v}\right)⪰\hfill \\ \hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\Omega }_l\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\Omega }_l\left(p_k\right),e\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta ({\Omega }_k\left(p_1\right)-{\Omega }_l\left(p_1\right),\cdots ,{\Omega }_k\left(p_k\right)-{\Omega }_l\left(p_k\right),e\overrightarrow{\mu })\hfill \\ \hfill & ⪰\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu })\hfill \\ \hfill & \succ 1-\lambda .\hfill \end{array}$$

Then:

$$\Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\Omega }_l\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\Omega }_l\left(p_k\right),\overrightarrow{v}\right)\succ 1-\lambda$$

and $\Theta ({\Omega }_k\left(p_1\right)-{\Omega }_l\left(p_1\right),\cdots ,{\Omega }_k\left(p_k\right)-{\Omega }_l\left(p_k\right),\overrightarrow{v})\succ 1-\lambda$, i.e., the sequences ${\{{\Omega }_k\left(p_1\right),{\Omega }_k\left(p_2\right),\cdots ,{\Omega }_k\left(p_k\right)\}}_k$ and ${{\{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_1\right){,}_0{\mathcal{D}}_{p_2}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_2\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\Omega }_k\left(p_k\right)\}}_k$ are Cauchy in complete space $(V,\Theta ,⊛)$ on compact set $E_1$, so they are uniformly convergent to the mapping $\Omega :E_1\to V$ and ${}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega {,}_0{\mathcal{D}}_{p_2}^{\kappa ,\tau ,\psi }\Omega ,\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega$, respectively. The uniform convergence leads us to the fact that $\Omega$ is differentiable, i.e., an element of B; then, $(B,d)$ is complete. □

Now, we are ready to study the Hyers–Ulam–Rassias stability of the $\psi$-Hilfer fractional Volterra integro-differential Equation (3) and get the best approximation with a better estimate for the $\psi$-Hilfer fractional Volterra integro-differential equation.

Theorem 3.

Let $(V,\Theta ,⊛)$ be an MVFB-space and $L_1,L_2,L_3,L_4$ and P be positive constant such that $N_1>2$ where $N_1=min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]$. Assume that the continuous mappings $\mathit{S}:E_1\times V\to V$, $\mathcal{M}:E_1\times E_1\times V\to V$ with matrix-valued fuzzy function $\varphi :{\left(E_1\right)}^k\times {\left(E_2\right)}^n\to \mathrm{diag}{M}_n\left(E_3\right)$ satisfying:

$$\begin{array}{c}\hfill \Theta (\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right)),\cdots ,\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right)),\overrightarrow{\mu })⪰\\ \hfill \Theta \left(\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),L_1\overrightarrow{\mu }\right),\end{array}$$

(4)

$$\begin{array}{c}\hfill \Theta (\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right)),\cdots ,\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right),\overrightarrow{\mu })⪰\\ \hfill \Theta \left(\Omega \left({\sigma }_1\right)-{\rm Y}\left({\sigma }_1\right),\cdots ,\Omega \left({\sigma }_k\right)-{\rm Y}\left({\sigma }_k\right),L_2\overrightarrow{\mu }\right),\end{array}$$

(5)

${\sigma }_i⪯p_i(i=1,2,\cdots ,k)$.

$$\begin{array}{c}\hfill \underset{{\xi }_1,\cdots ,{\xi }_k\in E_1}{inf}\varphi ({\xi }_1,\cdots ,{\xi }_k,\overrightarrow{\mu })⪰\varphi \left(p_1,\cdots ,p_k,L_{3}P\overrightarrow{\mu }\right),\end{array}$$

(6)

and:

$$\begin{array}{c}\hfill \Theta (\Omega \left(p_1\right),\Omega \left(p_2\right),\cdots ,\Omega \left(p_k\right),\overrightarrow{\mu })⪰\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu }),\mathrm{implies}\mathrm{that}\\ \hfill \Theta \left({}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\Omega \left({\sigma }_1\right)d{\sigma }_1,\cdots {,}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\Omega \left({\sigma }_k\right)d{\sigma }_K,\overrightarrow{\mu }\right)⪰\varphi \left(p_1,\cdots ,p_k,L_4\overrightarrow{\mu }\right),\end{array}$$

(7)

for every $p_1,\cdots ,p_k\in E_1$, $\Omega ,{\rm Y}:E_1\to V$, and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$.

Let $\omega :E_1\to V$ be a differentiable function satisfying:

$$\begin{array}{cc}\hfill \Theta ({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)& -\mathit{S}(p_1,\omega \left(p_1\right))-{\int }_0^{p_1}\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))d{\sigma }_1,\cdots ,\hfill \\ \hfill {}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)& -\mathit{S}(p_k,\omega \left(p_k\right))-{\int }_0^{p_k}\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))d{\sigma }_k,\overrightarrow{\mu })\hfill \\ \hfill ⪰\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu }),\end{array}$$

(8)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$. Therefore, we are able find a unique differentiable function ${\omega }_0:E_1\to V$ such that:

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_p^{\kappa ,\tau ,\psi }{\omega }_0\left(p\right)=\mathit{S}(p,{\omega }_0\left(p\right))+{\int }_0^p\mathcal{M}(p,\sigma ,{\omega }_0\left(\sigma \right))d\sigma ,\end{array}$$

(9)

and:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_k\right),\overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta (\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right),\overrightarrow{\mu })⪰\hfill \\ \hfill & \varphi \left(p_1,\cdots ,p_k,\frac{N_2}{N_3}\overrightarrow{\mu }\right),\hfill \end{array}$$

(10)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$, and $N_2=min[1,L_4]$ and $N_3=\frac{min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]}{min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]-2}$.

Proof. 

We set:

$$B:=\{\Omega :E_1\to V,\Omega \mathrm{is} \mathrm{differenriable}\}$$

and introduce the $E_4$-valued metric on B as,

$$\begin{array}{c}d(\Omega ,{\rm Y})=inf\{C\in E_6:\Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right)\\ ⊛\Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu })⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{C}\right),\\ \forall \Omega ,{\rm Y}\in B,p_1,\cdots ,p_k\in E_1,\overrightarrow{\mu }\in {\left(E_2\right)}^n\}.\end{array}$$

By Theorem 2, we have that $(B,d)$ is a complete $E_4$-valued metric space.

Now, we define the mapping $\Lambda$ from B to B by:

$$\begin{array}{c}\hfill \Lambda (\Omega \left(p_i\right))={}_0{\mathcal{I}}_{p_i}^{\kappa ,\psi }\left(\mathit{S}(\sigma ,\Omega \left(\sigma \right))\right)+{}_0{\mathcal{I}}_{p_i}^{\kappa ,\psi }\left({\int }_0^{\sigma }\mathcal{M}(\sigma ,\epsilon ,\Omega \left(\epsilon \right))d\epsilon \right),\end{array}$$

(11)

where $\kappa \in {E_5}^{\circ }$, $\mathit{S}:E_1\times V\to V$, $\mathcal{M}:E_1\times E_1\times V\to V$, and $p_i\in E_1(i=1,2,\cdots ,k)$. We prove $\Lambda$ is a strictly contractive mapping. Let $\Omega ,{\rm Y}\in B$, $C\in E_2$, and $d(\Omega ,{\rm Y})<\vartheta$, then we have:

$$\begin{array}{ccc}& & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\rm Y}\left(p_k\right),\vartheta \overrightarrow{\mu }\right)⊛\hfill \\ & & \Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\vartheta \overrightarrow{\mu })⪰\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu }),\hfill \\ & & \forall \Omega ,{\rm Y}\in B,p_1,\cdots ,p_k\in E_1,\overrightarrow{\mu }\in {\left(E_2\right)}^n.\hfill \end{array}$$

Using the properties (MVF-K-N2) and (MVF-K-N3) of Definition 2 and (11), we have:

$$\begin{array}{ccc}\hfill & & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Lambda (\Omega \left(p_1\right))-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Lambda ({\rm Y}\left(p_1\right)),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Lambda (\Omega \left(p_k\right))-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Lambda ({\rm Y}\left(p_k\right)),2\vartheta \overrightarrow{\mu }\right)\hfill \\ \hfill & ⊛& \Theta \left(\Lambda (\Omega \left(p_1\right))-\Lambda ({\rm Y}\left(p_1\right)),\cdots ,\Lambda (\Omega \left(p_k\right))-\Lambda ({\rm Y}\left(p_k\right)),2\vartheta \overrightarrow{\mu }\right)\hfill \\ & =& \Theta ([\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right))]+{\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right))]d{\sigma }_1,\hfill \\ \hfill & \cdots ,& [\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right))]+{\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right))]d{\sigma }_k,2\vartheta \overrightarrow{\mu })\hfill \\ & ⊛& \Theta ({}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left(\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right))\right)+{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left({\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right))]d{\sigma }_1\right),\hfill \\ \hfill & \cdots ,& {}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left(\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right))\right)+\hfill \\ \hfill & & {}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left({\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right))]d{\sigma }_k\right),2\vartheta \overrightarrow{\mu })\hfill \\ & ⪰& \Theta \left(\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right)),\cdots ,\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right)),\vartheta \overrightarrow{\mu }\right)\hfill \\ & ⊛& \Theta ({\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right))]d{\sigma }_1,\hfill \\ \hfill & \cdots ,& {\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right))]d{\sigma }_k,\vartheta \overrightarrow{\mu })\hfill \\ \hfill & ⊛& \Theta \left({}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }[\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right))],\cdots {,}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }[\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right))],\vartheta \overrightarrow{\mu }\right)\hfill \\ \hfill & ⊛& \Theta ({}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }{\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right))]d{\sigma }_1,\hfill \\ & \cdots ,& {}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }{\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right))]d{\sigma }_k,\vartheta \overrightarrow{\mu }).\hfill \end{array}$$

(12)

In the last part of (12), there are four formulas, in the next steps, we work on them to get new formulas derived from the control function $\varphi$. Let $0={\overline{y}}_1<{\overline{y}}_2<\cdots <{\overline{y}}_k=p_i$, $\Delta {\overline{y}}_i={\overline{y}}_i-{\overline{y}}_{i-1}=\frac{p_i}{k}$, $i=1,2,\cdots ,k$, and $\parallel \Delta \overline{y}\parallel ={max}_{1\le i\le k}\left(\Delta {\overline{y}}_i\right)$.

Step 1. From (4), we have:

$$\begin{array}{cc}\hfill & \Theta \left(\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right)),\cdots ,\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right)),\vartheta \overrightarrow{\mu }\right)⪰\hfill \\ \hfill & \Theta \left(\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),L_1\vartheta \overrightarrow{\mu }\right)⪰\varphi \left(p_1,\cdots ,p_k,L_1\overrightarrow{\mu }\right).\hfill \end{array}$$

(13)

Step 2. Using (MVF-K-N2) and (MVF-K-N3) of Definition 2, the continuity property of MVFF $\Theta$, (5), and (6), we get:

$$\begin{array}{cc}\hfill & \Theta ({\int }_0^{p_1}\left[\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right))\right]d{\sigma }_1,\hfill \\ \hfill & \cdots ,{\int }_0^{p_k}\left[\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right))\right]d{\sigma }_k,\vartheta \overrightarrow{\mu })\hfill \\ \hfill & =\Theta (\underset{\parallel \Delta \overline{y}\parallel \to 0}{lim}\sum _{j=1}^k[\mathcal{M}(p_1,{\overline{y}}_j,\Omega \left({\overline{y}}_j\right))-\mathcal{M}(p_1,{\overline{y}}_j,{\rm Y}\left({\overline{y}}_j\right))]\Delta {\overline{y}}_i,\hfill \\ \hfill & \cdots ,\underset{\parallel \Delta \overline{y}\parallel \to 0}{lim}\sum _{j=1}^k[\mathcal{M}(p_k,{\overline{y}}_j,\Omega \left({\overline{y}}_j\right))-\mathcal{M}(p_k,{\overline{y}}_j,{\rm Y}\left({\overline{y}}_j\right))]\Delta {\overline{y}}_i,\vartheta \overrightarrow{\mu })\hfill \\ \hfill & =\underset{\parallel \Delta \overline{y}\parallel \to 0}{lim}\Theta (\sum _{j=1}^k[\mathcal{M}(p_1,{\overline{y}}_j,\omega \left({\overline{y}}_j\right))-\mathcal{M}(p_1,{\overline{y}}_j,{\rm Y}\left({\overline{y}}_j\right))]\Delta {\overline{y}}_i,\hfill \\ \hfill & \cdots ,\sum _{j=1}^k[\mathcal{M}(p_k,{\overline{y}}_j,\omega \left({\overline{y}}_j\right))-\mathcal{M}(p_k,{\overline{y}}_j,{\rm Y}\left({\overline{y}}_j\right))]\Delta {\overline{y}}_i,\vartheta \overrightarrow{\mu })\hfill \\ \hfill & ⪰\underset{\parallel \Delta \overline{y}\parallel \to 0}{lim}{⊛}_M\Theta ([\mathcal{M}(p_1,{\overline{y}}_j,\Omega \left({\overline{y}}_j\right))-\mathcal{M}(p_1,{\overline{y}}_j,{\rm Y}\left({\overline{y}}_j\right))]\Delta {\overline{y}}_i,\hfill \\ \hfill & \cdots ,[\mathcal{M}(p_k,{\overline{y}}_j,\Omega \left({\overline{y}}_j\right))-\mathcal{M}(p_k,{\overline{y}}_j,{\rm Y}\left({\overline{y}}_j\right))]\Delta {\overline{y}}_i,\frac{\vartheta \overrightarrow{\mu }}{k})\hfill \\ \hfill & ⪰\underset{{\xi }_1,\cdots ,{\xi }_k\in E_1}{inf}\Theta (\mathcal{M}(p_1,{\xi }_1,\Omega \left({\xi }_1\right))-\mathcal{M}(p_1,{\xi }_1,{\rm Y}\left({\xi }_1\right)),\hfill \\ \hfill & \cdots ,\mathcal{M}(p_k,{\xi }_k,\Omega \left({\xi }_k\right))-\mathcal{M}(p_k,{\xi }_k,{\rm Y}\left({\xi }_k\right)),\frac{k\vartheta \overrightarrow{\mu }}{kP})\hfill \\ \hfill & ⪰\underset{{\xi }_1,\cdots ,{\xi }_k\in E_1}{inf}\Theta \left(\Omega \left({\xi }_1\right)-{\rm Y}\left({\xi }_1\right),\cdots ,\Omega \left({\xi }_k\right)-{\rm Y}\left({\xi }_k\right),\frac{k{L}_2\vartheta \overrightarrow{\mu }}{kP}\right)\hfill \\ \hfill & ⪰\underset{{\xi }_1,\cdots ,{\xi }_k\in E_1}{inf}\varphi \left({\xi }_1,\cdots ,{\xi }_k,\frac{P\overrightarrow{\mu }}{L_2}\right)\hfill \\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,L_2{L}_3\overrightarrow{\mu }\right).\hfill \end{array}$$

(14)

Step 3. Using (7) and (13), we get:

$$\begin{array}{cc}\hfill & \Theta {(}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left(\mathit{S}({\sigma }_1,\Omega \left({\sigma }_1\right))-\mathit{S}({\sigma }_1,{\rm Y}\left({\sigma }_1\right))\right),\\ \hfill & \cdots {,}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left(\mathit{S}({\sigma }_k,\Omega \left({\sigma }_k\right))-\mathit{S}({\sigma }_k,{\rm Y}\left({\sigma }_k\right))\right),\vartheta \overrightarrow{\mu })& ⪰\varphi \left(p_1,\cdots ,p_k,L_1{L}_4\overrightarrow{\mu }\right).\hfill \end{array}$$

(15)

Step 4. Using (7) and (14), we get:

$$\begin{array}{cc}\hfill & \Theta {(}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left({\int }_0^{{\sigma }_1}[\mathcal{M}({\sigma }_1,{\epsilon }_1,\Omega \left({\epsilon }_1\right))-\mathcal{M}({\sigma }_1,{\epsilon }_1,{\rm Y}\left({\epsilon }_1\right))]d{\epsilon }_1\right),\\ \hfill & \cdots {,}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left({\int }_0^{{\sigma }_k}[\mathcal{M}({\sigma }_k,{\epsilon }_k,\Omega \left({\epsilon }_k\right))-\mathcal{M}({\sigma }_k,{\epsilon }_k,{\rm Y}\left({\epsilon }_k\right))]d{\epsilon }_k\right),\vartheta \overrightarrow{\mu })& ⪰\varphi \left(p,L_2{L}_3{L}_4\overrightarrow{\mu }\right).\hfill \\ \hfill \end{array}$$

(16)

From (12)–(16), we have:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Lambda (\Omega \left(p_1\right))-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Lambda ({\rm Y}\left(p_1\right)),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Lambda (\Omega \left(p_k\right))-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Lambda ({\rm Y}\left(p_k\right)),2\vartheta \overrightarrow{\mu }\right)⊛\hfill \\ \hfill & \Theta \left(\Lambda (\Omega \left(p_1\right))-\Lambda ({\rm Y}\left(p_1\right)),\cdots ,\Lambda (\Omega \left(p_k\right))-\Lambda ({\rm Y}\left(p_k\right)),2\vartheta \overrightarrow{\mu }\right)\hfill \\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,L_1\overrightarrow{\mu }\right)⊛\varphi \left(p_1,\cdots ,p_k,L_2{L}_3\overrightarrow{\mu }\right)⊛\varphi \left(p_1,\cdots ,p_k,L_1{L}_4\overrightarrow{\mu }\right)⊛\varphi \left(p_1,\cdots ,p_k,L_2{L}_3{L}_4\overrightarrow{\mu }\right)\hfill \\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,N_1\overrightarrow{\mu }\right)\hfill \end{array}$$

(17)

where in $N_1=min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]$.

Therefore,

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Lambda (\Omega \left(p_1\right))-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\Lambda ({\rm Y}\left(p_1\right)),\cdots ,{}_0\mathcal{D}{p_k}^{\kappa ,\tau ,\psi }\Lambda (\Omega \left(p_k\right))-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\Lambda ({\rm Y}\left(p_k\right)),2\vartheta \overrightarrow{\mu }\right)\\ \hfill & ⊛\Theta \left(\Lambda (\Omega \left(p_1\right))-\Lambda ({\rm Y}\left(p_1\right)),\cdots ,\Lambda (\Omega \left(p_k\right))-\Lambda ({\rm Y}\left(p_k\right)),2\vartheta \overrightarrow{\mu }\right)\\ & ⪰\varphi \left(p_1,\cdots ,p_k,N_4\overrightarrow{\mu }\right),\end{array}$$

(18)

where $N_4=\frac{1}{2}min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]$ and implying that:

$$\begin{array}{c}\hfill d(\Lambda (\Omega ),\Lambda ({\rm Y}))\le \frac{\vartheta }{N_4}.\end{array}$$

(19)

Therefore,

$$\begin{array}{c}\hfill d(\Lambda (\Omega ),\Lambda ({\rm Y}))\le \frac{1}{N_4}d(\Omega ,{\rm Y}).\end{array}$$

(20)

Thus, $\Lambda$ with Lipschitz constant $\frac{1}{N_4}$ is a strictly contractive mapping.

Let $\omega \in B$. We show that $d(\Lambda (\omega ),\omega )<\infty$. Using (7) and (8), we get:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }[\Lambda \left(\omega \left(p_1\right)\right)-\omega \left(p_1\right)],\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }[\Lambda \left(\omega \left(p_k\right)\right)-\omega \left(p_k\right)],\overrightarrow{\mu }\right)\hfill \\ \hfill & ⊛\Theta \left(\Lambda \left(\omega \left(p_1\right)\right)-\omega \left(p_1\right),\cdots ,\Lambda \left(\omega \left(p_k\right)\right)-\omega \left(p_k\right),\overrightarrow{\mu }\right)\hfill \\ \hfill & =\Theta (\mathit{S}(p_1,\omega \left(p_1\right))+{\int }_0^{p_1}\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))d{\sigma }_1-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right),\hfill \\ \hfill & \cdots ,\mathit{S}(p_k,\omega \left(p_k\right))+{\int }_0^{p_k}\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))d{\sigma }_k-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right),\overrightarrow{\mu })\hfill \\ \hfill & ⊛\Theta {(}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left(\mathit{S}(p_1,\omega \left(p_1\right))\right)+{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left({\int }_0^{p_1}\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))d{\sigma }_1\right)-{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right),\hfill \\ \hfill & \cdots {,}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left(\mathit{S}(p_k,\omega \left(p_k\right))\right)+{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left({\int }_0^{p_k}\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))d{\sigma }_k\right)-{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right),\overrightarrow{\mu })\hfill \\ \hfill & =\Theta (\mathit{S}(p_1,\omega \left(p_1\right))+{\int }_0^{p_1}\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))d{\sigma }_1-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right),\hfill \\ \hfill & \cdots ,\mathit{S}(p_k,\omega \left(p_k\right))+{\int }_0^{p_k}\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))d{\sigma }_k-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right),\overrightarrow{\mu })\hfill \\ \hfill & ⊛\Theta {(}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left[\mathit{S}(p_1,\omega \left(p_1\right))+\left({\int }_0^{p_1}\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))d{\sigma }_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)\right],\hfill \\ \hfill & \cdots {,}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left[\mathit{S}(p_k,\omega \left(p_k\right))+\left({\int }_0^{p_k}\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))d{\sigma }_k)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)\right],\overrightarrow{\mu }\right)\hfill \\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,\overrightarrow{\mu }\right)⊛\varphi \left(p_1,\cdots ,p_k,L_4\overrightarrow{\mu }\right)\hfill \\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,N_2\overrightarrow{\mu }\right),\hfill \end{array}$$

(21)

for every $\overrightarrow{\mu }\in {\left(E_2\right)}^n$ and $N_2=min[1,L_4]$. Then, we have $d(\Lambda \left(\omega \right),\omega )<\frac{1}{N_2}<\infty$.

Now, Theorem 1 enables us to find an element ${\omega }_0$ in B satisfying the following:

(1) ${\omega }_0$ is a fixed point of $\Lambda$, i.e.,

$$\begin{array}{ccc}\hfill {\omega }_0\left(p\right)& =& \Lambda \left({\omega }_0\left(p\right)\right)\hfill \\ \hfill & =& {}_0{\mathcal{I}}_p^{\kappa ,\psi }\left(\mathit{S}(\sigma ,{\omega }_0\left(\sigma \right))\right){+}_0{\mathcal{I}}_p^{\kappa ,\psi }\left({\int }_0^{\sigma }\mathcal{M}(\sigma ,\epsilon ,{\omega }_0\left(\epsilon \right))d\epsilon \right),\hfill \end{array}$$

(22)

which is unique in the set:

$$B^{*}=\{\Omega \in B:d(\Lambda \left(\omega \right),\Omega )<\infty \}.$$

Take ${}_0{\mathcal{D}}_p^{\kappa ,\tau ,\psi }$ from (22). We get:

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_p^{\kappa ,\tau ,\psi }{\omega }_0\left(p\right)=\mathit{S}(p,{\omega }_0\left(p\right))+{\int }_0^p\mathcal{M}(p,\sigma ,{\omega }_0\left(\sigma \right))d\sigma ,\end{array}$$

(23)

where $\kappa \in {E_5}^{\circ }$, $\mathit{S}:E_1\times V\to V$, $\mathcal{M}:E_1\times E_1\times V\to V$.

(2) $d({\Lambda }^k\left(\omega \right),{\omega }_0)\to 0$ as $n\to \infty$;

(3) $d(\omega ,{\omega }_0)\le N_{3}d(\Lambda \left(\omega \right),w)\le \frac{N_3}{N_2}$,

where $N_2=min[1,L_4]$ and $N_3=\frac{min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]}{min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]-2}$ and implying that:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_k\right),\overrightarrow{\mu }\right)⊛\\ \hfill & \Theta \left(\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right),\overrightarrow{\mu }\right)\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,\frac{N_2}{N_3}\overrightarrow{\mu }\right),\end{array}$$

(24)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$.

Now, we prove that the fixed point in $B^{*}$ is unique. Suppose that ${\varsigma }_0$ is an element of B satisfying (9) and (10). We prove that ${\varsigma }_0={\omega }_0$ and ${\varsigma }_0\in B^{*}$. From (9), we get:

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_p^{\kappa ,\tau ,\psi }{\varsigma }_0\left(p\right)=\mathit{S}(p,{\varsigma }_0\left(p\right))+{\int }_0^p\mathcal{M}(p,\sigma ,{\varsigma }_0\left(\sigma \right))d\sigma ,\end{array}$$

(25)

and so:

$$\begin{array}{cc}\hfill {\varsigma }_0\left(p\right)& {=}_0{\mathcal{I}}_p^{\kappa ,\psi }\mathit{S}(\sigma ,{\varsigma }_0\left(\sigma \right)){+}_0{\mathcal{I}}_p^{\kappa ,\psi }{\int }_0^{\sigma }\mathcal{M}(\sigma ,\epsilon ,{\varsigma }_0\left(\epsilon \right))d\epsilon \hfill \\ \hfill & =\Lambda \left({\varsigma }_0\left(p\right)\right),\hfill \end{array}$$

(26)

where $\kappa \in {E_5}^{\circ }$, $\mathit{S}:E_1\times V\to V$, $\mathcal{M}:E_1\times E_1\times V\to V$.

Now, we show that:

$${\varsigma }_0\in \{\Omega \in B:d(\Lambda \left(\omega \right),\Omega )<\infty \},$$

i.e., $d(\Lambda \left(\omega \right),{\varsigma }_0)<\infty$. We set $j=\frac{min[1,L_4](min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]-2)}{min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]}$. From (10), we get:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\varsigma }_0\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\varsigma }_0\left(p_k\right),\overrightarrow{\mu }\right)⊛\\ \hfill & \Theta \left(\omega \left(p_1\right)-{\varsigma }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\varsigma }_0\left(p_k\right),\overrightarrow{\mu }\right)⪰\varphi \left(p_1,\cdots ,p_k,j\overrightarrow{\mu }\right),\end{array}$$

(27)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$.

From (4) and (27), we get:

$$\begin{array}{cc}\hfill & \Theta \left(\mathit{S}(p_1,\omega \left(p_1\right))-\mathit{S}(p_1,{\varsigma }_0\left(p_1\right)),\cdots ,\mathit{S}(p_k,\omega \left(p_k\right))-\mathit{S}(p_k,{\varsigma }_0\left(p_k\right)),\overrightarrow{\mu }\right)\\ \hfill & ⪰\Theta \left(\omega \left(p_1\right)-{\varsigma }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\varsigma }_0\left(p_k\right),L_1\overrightarrow{\mu }\right)\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,L_{1}j\overrightarrow{\mu }\right).\end{array}$$

(28)

Furthermore, from (5) and (27), we get:

$$\begin{array}{cc}\hfill & \Theta (\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\varsigma }_0\left({\sigma }_1\right)),\cdots ,\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\varsigma }_0\left({\sigma }_k\right)),\overrightarrow{\mu })\\ \hfill & ⪰\Theta \left(\omega \left({\sigma }_1\right)-{\varsigma }_0\left({\sigma }_1\right),\cdots ,\omega \left({\sigma }_k\right)-{\varsigma }_0\left({\sigma }_k\right),L_2\overrightarrow{\mu }\right)\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,L_{2}j\overrightarrow{\mu }\right),\end{array}$$

(29)

for every $p_1,\cdots ,p_k\in E_1$, ${\sigma }_i⪯p_i(i=1,2,\cdots ,k)$, and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$.

Now, using Step 2 and (29), we get:

$$\begin{array}{cc}\hfill & \Theta ({\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\varsigma }_0\left({\sigma }_1\right))]d{\sigma }_1,\cdots ,\\ \hfill & {\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\varsigma }_0\left({\sigma }_k\right))]d{\sigma }_k,\overrightarrow{\mu })\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,L_2{L}_3\overrightarrow{\mu }\right)\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,L_2{L}_{3}j\overrightarrow{\mu }\right).\end{array}$$

(30)

Using the triangular inequality (MVF-K-N3), (28), and (30), we get:

$$\begin{array}{cc}\hfill & \Theta (\mathit{S}(p_1,\omega \left(p_1\right))-\mathit{S}(p_1,{\varsigma }_0\left(p_1\right))+{\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\varsigma }_0\left({\sigma }_1\right))]d{\sigma }_1,\cdots ,\\ \hfill & \mathit{S}(p_k,\omega \left(p_k\right))-\mathit{S}(p_k,{\varsigma }_0\left(p_k\right))+{\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\varsigma }_0\left({\sigma }_k\right))]d{\sigma }_k,2\overrightarrow{\mu })\\ \hfill & ⪰\Theta \left(\mathit{S}(p_1,\omega \left(p_1\right))-\mathit{S}(p_1,{\varsigma }_0\left(p_1\right)),\cdots ,\mathit{S}(p_k,\omega \left(p_k\right))-\mathit{S}(p_k,{\varsigma }_0\left(p_k\right)),\overrightarrow{\mu }\right)\\ \hfill & ⊛\Theta ({\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\varsigma }_0\left({\sigma }_1\right))]d{\sigma }_1,\cdots ,\\ \hfill & {\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\varsigma }_0\left({\sigma }_k\right))]d{\sigma }_k,\overrightarrow{\mu })\\ & ⪰\varphi \left(p_1,\cdots ,p_k,L_{1}j\overrightarrow{\mu }\right)⊛\varphi \left(p_1,\cdots ,p_k,L_2{L}_{3}j\overrightarrow{\mu }\right)\\ & ⪰\varphi \left(p_1,\cdots ,p_k,N_{5}j\overrightarrow{\mu }\right),\end{array}$$

(31)

where in $N_5=min[L_1,L_2{L}_3]$, and so:

$$\begin{array}{cc}\hfill & \Theta (\mathit{S}(p_1,\omega \left(p_1\right))-\mathit{S}(p_1,{\varsigma }_0\left(p_1\right))+{\int }_0^{p_1}\left[\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\varsigma }_0\left({\sigma }_1\right))\right]d{\sigma }_1,\cdots ,\\ \hfill & \mathit{S}(p_k,\omega \left(p_k\right))-\mathit{S}(p_k,{\varsigma }_0\left(p_k\right))+{\int }_0^{p_k}\left[\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\varsigma }_0\left({\sigma }_k\right))\right]d{\sigma }_k,\overrightarrow{\mu })\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,N_{6}j\overrightarrow{\mu }\right),\end{array}$$

(32)

where $N_6=\frac{1}{2}min[L_1,L_2{L}_3]$.

We apply (7) and get:

$$\begin{array}{cc}\hfill & \Theta {(}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\left[\mathit{S}({\sigma }_1,\omega \left({\sigma }_1\right))-\mathit{S}({\sigma }_1,{\varsigma }_0\left({\sigma }_1\right))\right]+{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }{\int }_0^{{\sigma }_1}[\mathcal{M}({\sigma }_1,{\epsilon }_1,\omega \left({\epsilon }_1\right))-\mathcal{M}({\sigma }_1,{\epsilon }_1,{\varsigma }_0\left({\epsilon }_1\right))]d{\epsilon }_1,\\ \hfill & \cdots {,}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\left[\mathit{S}(\sigma ,\omega \left(\sigma \right))-\mathit{S}(\sigma ,{\varsigma }_0\left(\sigma \right))\right]+{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }{\int }_0^{\sigma }[\mathcal{M}(\sigma ,\epsilon ,\omega \left(\epsilon \right))-\mathcal{M}(\sigma ,\epsilon ,{\varsigma }_0\left(\epsilon \right))]d\epsilon ,\overrightarrow{\mu })\\ & ⪰\varphi \left(p_1,\cdots ,p_k,L_4{N}_{6}j\overrightarrow{\mu }\right),\end{array}$$

(33)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$, and $N_6=\frac{1}{2}min[L_1,L_2{L}_3]$.

Using (32) and (33), we get:

$$\begin{array}{cc}\hfill & \Theta \left({}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }[\Lambda \left(\omega \left(p_1\right)\right)-{\varsigma }_0\left(p_1\right)],\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }[\Lambda \left(\omega \left(p_k\right)\right)-{\varsigma }_0\left(p_k\right)],\overrightarrow{\mu }\right)\\ \hfill & ⊛\Theta \left(\Lambda \left(\omega \left(p_1\right)\right)-{\varsigma }_0\left(p_1\right),\cdots ,\Lambda \left(\omega \left(p_k\right)\right)-{\varsigma }_0\left(p_k\right),\overrightarrow{\mu }\right)\\ \hfill & =\Theta (\mathit{S}(p_1,\omega \left(p_1\right))-\mathit{S}(p_1,{\varsigma }_0\left(p_1\right))+{\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\varsigma }_0\left({\sigma }_1\right))]d{\sigma }_1,\cdots ,\\ \hfill & \mathit{S}(p_k,\omega \left(p_k\right))-\mathit{S}(p_k,{\varsigma }_0\left(p_k\right))+{\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\varsigma }_0\left({\sigma }_k\right))]d{\sigma }_k,\overrightarrow{\mu })\\ \hfill & ⊛\Theta {(}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }[\mathit{S}({\sigma }_1,\omega \left({\sigma }_1\right))-\mathit{S}({\sigma }_1,{\varsigma }_0\left({\sigma }_1\right))]+{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }{\int }_0^{{\sigma }_1}[\mathcal{M}({\sigma }_1,{\epsilon }_1,\omega \left({\epsilon }_1\right))-\mathcal{M}({\sigma }_1,{\epsilon }_1,{\varsigma }_0\left({\epsilon }_1\right))]d{\epsilon }_1,\cdots ,\\ \hfill & {}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }[\mathit{S}({\sigma }_k,\omega \left({\sigma }_k\right))-\mathit{S}({\sigma }_k,{\varsigma }_0\left({\sigma }_k\right))]+{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }{\int }_0^{{\sigma }_k}[\mathcal{M}({\sigma }_k,{\epsilon }_k,\omega \left({\epsilon }_k\right))-\mathcal{M}({\sigma }_k,{\epsilon }_k,{\varsigma }_0\left({\epsilon }_k\right))]d{\epsilon }_k,\overrightarrow{\mu })\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,N_{6}j\overrightarrow{\mu }\right)⊛\varphi \left(p_1,\cdots ,p_k,L_4{N}_{6}j\overrightarrow{\mu }\right)\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,\frac{(L_4+1)N_{6}j\overrightarrow{\mu }}{L_4}\right),\end{array}$$

(34)

which $N_6=\frac{1}{2}min[L_1,L_2{L}_3]$, which implies that:

$$d(\Lambda \left(\omega \right),{\varsigma }_0)\le \frac{L_4}{j(L_4+1)N_6}<\infty$$

then ${\varsigma }_0\in B^{*}$. □

4. Best Approximation of $\psi$-Hilfer Fractional Volterra Integral Equation

Now, we are ready to study the Hyers–Ulam–Rassias stability of the $\psi$-Hilfer fractional Volterra integral equation:

$$\begin{array}{c}\hfill \Omega \left(p\right)=\mathit{S}(p,\Omega \left(p\right))+{}_0{\mathcal{I}}_p^{\kappa ,\psi }\mathcal{M}(p,\sigma ,\Omega \left(\sigma \right)),\end{array}$$

(35)

where $\kappa \in {E_5}^{\circ }$, $\mathit{S}:E_1\times V\to V$, $\mathcal{M}:E_1\times E_1\times V\to V$ and get the best approximation with a better estimate for the pseudo $\psi$-Hilfer fractional Volterra integral equation. Our method can be used for new problems; for more problems and details, we refer to [17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,34].

Theorem 4.

Let $(V,\Theta ,⊛)$ be an MVFB-space and $L_1,L_2,L_3,L_4$ and T be positive constant such that $N_1>2$ where $N_1=min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]$. Assume that the continuous mappings $\mathit{S}:E_1\times V\to V$, $\mathcal{M}:E_1\times E_1\times V\to V$ with MVFF $\varphi :{\left(E_1\right)}^k\times {\left(E_2\right)}^n\to \mathrm{diag}\left(M_n\left(E_3\right)\right)$ satisfying (4)–(7).

Let $\omega :E_1\to V$ be a differentiable function satisfying:

$$\begin{array}{cc}\hfill & \Theta (\omega \left(p_1\right)-\mathit{S}(p_1,\omega \left(p_1\right))-{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right)),\cdots ,\\ \hfill & \omega \left(p_k\right)-\mathit{S}(p_k,\omega \left(p_k\right))-{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right)),\overrightarrow{\mu })\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,\overrightarrow{\mu }\right),\end{array}$$

(36)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$. Then, we are to be able find a unique differentiable function ${\omega }_0:E_1\to V$ such that:

$$\begin{array}{c}\hfill {\omega }_0\left(p\right)=\mathit{S}(p,{\omega }_0\left(p\right))+{}_0{\mathcal{I}}_p^{\kappa ,\psi }\mathcal{M}(p,\sigma ,{\omega }_0\left(\sigma \right)),\end{array}$$

(37)

and:

$$\begin{array}{c}\hfill \Theta (\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right),\overrightarrow{\mu })⪰\\ \hfill \varphi \left(p_1,\cdots ,p_k,\frac{N_2}{N_3}\overrightarrow{\mu }\right),\end{array}$$

(38)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$, and $N_2=min[1,L_4]$ and $N_3=\frac{min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]}{min[L_1,L_2{L}_3,L_1{L}_4,L_2{L}_3{L}_4]-2}$.

Proof. 

We set

$$B:=\{\Omega :E_1\to V,\Omega \mathrm{is} \mathrm{differentiable}\}$$

and introduce the $E_4$-valued metric on B as,

$$\begin{array}{c}\hfill d(\Omega ,{\rm Y})=inf\{C\in E_6:\Theta \left(\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\overrightarrow{\mu }\right)⪰\varphi \left(p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{C}\right),\\ \hfill \forall \Omega ,{\rm Y}\in B,p_1,\cdots ,p_k\in E_1,\overrightarrow{\mu }\in {\left(E_2\right)}^n\}.\end{array}$$

By Theorem 2, we have that $(B,d)$ is a complete $E_4$-valued metric space.

Now, we define the mapping $\Lambda$ from B to B by:

$$\begin{array}{c}\hfill \Lambda (\Omega \left(p_i\right))=\mathit{S}(\sigma ,\Omega \left(\sigma \right))+{}_0{\mathcal{I}}_{p_i}^{\kappa ,\psi }\left({\int }_0^{\sigma }\mathcal{M}(\sigma ,\epsilon ,\Omega \left(\epsilon \right))d\epsilon \right),\end{array}$$

(39)

where $\kappa \in {\left(E_5\right)}^{\circ }$, $\mathit{S}:E_1\times V\to V$, $\mathcal{M}:E_1\times E_1\times V\to V$, and $p_i\in E_1(i=1,2,\cdots ,k)$. We prove that $\Lambda$ is a strictly contractive mapping. Let $\Omega ,{\rm Y}\in B$, $C\in E_6$ and $d(\Omega ,{\rm Y})<\vartheta$, then we have:

$$\begin{array}{c}\hfill \Theta (\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),\vartheta \overrightarrow{\mu })⪰\varphi (p_1,\cdots ,p_k,\overrightarrow{\mu }),\\ \hfill \forall \Omega ,{\rm Y}\in B,p_1,\cdots ,p_k\in E_1,\overrightarrow{\mu }\in {\left(E_2\right)}^n.\end{array}$$

Using the properties (MVF-K-N2) and (MVF-K-N3) of Definition 2, (4)–(7), and (39), we have:

$$\begin{array}{cc}\hfill & \Theta \left(\Lambda (\Omega \left(p_1\right))-\Lambda ({\rm Y}\left(p_1\right)),\cdots ,\Lambda (\Omega \left(p_k\right))-\Lambda ({\rm Y}\left(p_k\right)),2\vartheta \overrightarrow{\mu }\right)\\ & =\Theta ([\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right))]+{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }[\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right))]d{\sigma }_1,\cdots ,\\ \hfill & [\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right))]+{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }[\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right))]d{\sigma }_k,2\vartheta \overrightarrow{\mu })\\ & ⪰\Theta \left(\mathit{S}(p_1,\Omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right)),\cdots ,\mathit{S}(p_k,\Omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right)),\vartheta \overrightarrow{\mu }\right)\end{array}$$

(40)

$$\begin{array}{cc}& ⊛\Theta ({\int }_0^{p_1}[\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right))]d{\sigma }_1,\cdots ,\\ \hfill & {\int }_0^{p_k}[\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right))]d{\sigma }_k,\vartheta \overrightarrow{\mu })\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,L_1\overrightarrow{\mu }\right)⊛\varphi \left(p_1,\cdots ,p_k,L_2{L}_4\overrightarrow{\mu }\right)\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,N_7\overrightarrow{\mu }\right),\end{array}$$

(41)

where in $N_7=min[L_1,L_2{L}_4]$, and so:

$$\begin{array}{cc}\hfill & \Theta \left(\Lambda (\Omega \left(p_1\right))-\Lambda ({\rm Y}\left(p_1\right)),\cdots ,\Lambda (\Omega \left(p_k\right))-\Lambda ({\rm Y}\left(p_k\right)),\vartheta \overrightarrow{\mu }\right)\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,N_8\overrightarrow{\mu }\right),\end{array}$$

(42)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$, and $N_8=\frac{1}{2}min[L_1,L_2{L}_4]$.

Then:

$$\begin{array}{c}\hfill d(\Lambda (\Omega ),\Lambda ({\rm Y}))\le \frac{\vartheta }{N_8},\end{array}$$

(43)

and so:

$$\begin{array}{c}\hfill d(\Lambda (\Omega ),\Lambda ({\rm Y}))\le \frac{1}{N_8}d(\Omega ,{\rm Y}).\end{array}$$

(44)

Then, $\Lambda$ is a strictly contractive mapping with Lipschitz constant $\frac{1}{N_8}$.

Let $\omega \in B$. We show that $d(\Lambda (\omega ),\omega )<\infty$. Using (36), we get:

$$\begin{array}{cc}\hfill & \Theta \left(\Lambda \left(\omega \left(p_1\right)\right)-\omega \left(p_1\right),\cdots ,\Lambda \left(\omega \left(p_k\right)\right)-\omega \left(p_k\right),\overrightarrow{\mu }\right)\\ & =\Theta (\mathit{S}(p_1,\omega \left(p_1\right))+{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }\mathcal{M}(p_1,{\sigma }_1,\omega \left({\sigma }_1\right))d{\sigma }_1-\omega \left(p_1\right),\\ \hfill & \cdots ,\mathit{S}(p_k,\omega \left(p_k\right))+{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }\mathcal{M}(p_k,{\sigma }_k,\omega \left({\sigma }_k\right))d{\sigma }_k-\omega \left(p_k\right),\overrightarrow{\mu })\\ \hfill & ⪰\varphi \left(p_1,\cdots ,p_k,\overrightarrow{\mu }\right)\end{array}$$

(45)

for every $\overrightarrow{\mu }\in {\left(E_2\right)}^n$. Then, we have $d(\Lambda (\omega ),\omega )<1$.

Now, Theorem 1 enables us to find an element ${\omega }_0$ in B satisfying the following:

(1) ${\omega }_0$ is a fixed point of $\Lambda$, i.e.,

$$\begin{array}{ccc}\hfill {\omega }_0\left(p\right)& =& \Lambda \left({\omega }_0\left(p\right)\right)\hfill \\ \hfill & =& \mathit{S}(\sigma ,{\omega }_0\left(\sigma \right))+{}_0{\mathcal{I}}_p^{\kappa ,\psi }\left(\mathcal{M}(\sigma ,\epsilon ,{\omega }_0\left(\epsilon \right))d\epsilon \right),\hfill \end{array}$$

(46)

which is unique in the set:

$$B^{*}=\{\Omega \in B:d(\Lambda \left(\omega \right),\Omega )<\infty \}.$$

(2) $d({\Lambda }^k\left(\omega \right),{\omega }_0)\to 0$ as $n\to \infty$;

(3) $d(\omega ,{\omega }_0)\le N_{9}d(\Lambda \left(\omega \right),\omega )\le N_9$,

where $N_9=\frac{min[L_1,L_2{L}_4]}{min[L_1,L_2{L}_4]-2}$ and implying that:

$$\begin{array}{c}\hfill \Theta (\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right),\overrightarrow{\mu })⪰\varphi (p_1,\cdots ,p_k,\frac{\overrightarrow{\mu }}{N_9}),\end{array}$$

(47)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^n$. We use the same method of proving Theorem 3 and show that the fixed point in $B^{*}$ is unique. □

5. Examples

In this section, we apply the main results to solve some examples.

Example 1.

Let $(\mathbb{R},\Theta ,⊛)$ be an MVF-k-B-space. Let $\Omega ,{\rm Y}:E_1\to \mathbb{R}$ such that $\Omega \left(p_i\right)={p_i}^4$ and ${\rm Y}\left(p_i\right)={p_i}^2$, and define $\mathit{S}(p_i,\Omega \left(p_i\right))=\frac{1}{L_1}\Omega \left(p_i\right)$. Define $\mathcal{M}:E_1\times E_1\times \mathbb{R}\to \mathbb{R}$ as $\mathcal{M}(p_i,{\sigma }_i,\Omega \left({\sigma }_i\right))=exp({p_i}^2-{{\sigma }_i}^2)\Omega \left({\sigma }_i\right)$ for every $p_i\in E_1$ and ${\sigma }_i⪯p_i(i=1,2,\cdots ,k)$.

Then, we have:

$$\begin{array}{cc}\hfill & \Theta \left(\mathit{S}(p_1,\omega \left(p_1\right))-\mathit{S}(p_1,{\rm Y}\left(p_1\right)),\cdots ,\mathit{S}(p_1,\omega \left(p_k\right))-\mathit{S}(p_k,{\rm Y}\left(p_k\right)),\overrightarrow{\mu }\right)\hfill \\ \hfill & =\Theta (\frac{1}{L_1}\Omega \left(p_1\right)-\frac{1}{L_1}{\rm Y}\left(p_1\right),\cdots ,\frac{1}{L_1}\Omega \left(p_k\right)-\frac{1}{L_1}{\rm Y}\left(p_k\right),\overrightarrow{\mu })\hfill \\ \hfill & =\Theta (\frac{1}{L_1}{p}_1^4-\frac{1}{L_1}{p}_1^2,\cdots ,\frac{1}{L_1}{p}_k^4-\frac{1}{L_1}{p}_k^2,\overrightarrow{\mu })\hfill \\ \hfill & =\mathrm{diag}\left[exp\left(\frac{-\frac{1}{L_1}|p_1^4-p_1^2,\cdots ,p_k^4-p_k^2|}{{\mu }_1}\right),\frac{{\mu }_2}{{\mu }_2+\frac{1}{L_1}|p_1^4-p_1^2,\cdots ,p_k^4-p_k^2|},exp\left(\frac{-\frac{1}{L_1}|p_1^4-p_1^2,\cdots ,p_k^4-p_k^2|}{{\mu }_3}\right)\right]\hfill \\ \hfill & ⪰\mathrm{diag}\left[exp\left(\frac{-|p_1^4-p_1^2,\cdots ,p_k^4-p_k^2|}{L_1{\mu }_1}\right),\frac{L_1{\mu }_2}{L_1{\mu }_2+|p_1^4-p_1^2,\cdots ,p_k^4-p_k^2|},exp\left(\frac{-|p_1^4-p_1^2,\cdots ,p_k^4-p_k^2|}{L_1{\mu }_3}\right)\right]\hfill \\ \hfill & =\Theta \left(p_1^4-p_1^2,\cdots ,p_k^4-p_k^2,L_1\overrightarrow{\mu }\right)\hfill \\ \hfill & =\Theta \left(\Omega \left(p_1\right)-{\rm Y}\left(p_1\right),\cdots ,\Omega \left(p_k\right)-{\rm Y}\left(p_k\right),L_1\overrightarrow{\mu }\right),\hfill \end{array}$$

(48)

$$\begin{array}{c}\Theta \left(\mathcal{M}(p_1,{\sigma }_1,\Omega \left({\sigma }_1\right))-\mathcal{M}(p_1,{\sigma }_1,{\rm Y}\left({\sigma }_1\right)),\cdots ,\mathcal{M}(p_k,{\sigma }_k,\Omega \left({\sigma }_k\right))-\mathcal{M}(p_k,{\sigma }_k,{\rm Y}\left({\sigma }_k\right)),\overrightarrow{\mu }\right)\\ =\Theta (exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^4-exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^2,\cdots ,exp\left(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^4-exp(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^2,\overrightarrow{\mu }\right)\\ =\mathrm{diag}[exp\left(\frac{-|exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^4-exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^2,\cdots ,exp(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^4-exp(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^2|}{{\mu }_1}\right),\\ \frac{{\mu }_2}{{\mu }_2+|exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^4-exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^2,\cdots ,exp(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^4-exp(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^2|},\\ exp\left(\frac{-|exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^4-exp(p_1^2-{{\sigma }_1}^2){{\sigma }_1}^2,\cdots ,exp(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^4-exp(p_k^2-{{\sigma }_k}^2){{\sigma }_k}^2|}{{\mu }_3}\right)]\\ ⪰\mathrm{diag}[exp\left(\frac{-|{{\sigma }_1}^4-{{\sigma }_1}^2,\cdots ,{{\sigma }_k}^4-{{\sigma }_k}^2|}{\frac{{\mu }_1}{exp(p_1^2-{{\sigma }_1}^2),\cdots ,exp(p_k^2-{{\sigma }_k}^2)}}\right),\frac{\frac{{\mu }_2}{exp(p_1^2-{{\sigma }_1}^2),\cdots ,exp(p_k^2-{{\sigma }_k}^2)}}{\frac{{\mu }_2}{exp(p_1^2-{{\sigma }_1}^2),\cdots ,exp(p_k^2-{{\sigma }_k}^2)}+|{{\sigma }_1}^4-{{\sigma }_1}^2,\cdots ,{{\sigma }_k}^4-{{\sigma }_k}^2|},\\ exp\left(\frac{-|{{\sigma }_1}^4-{{\sigma }_1}^2,\cdots ,{{\sigma }_k}^4-{{\sigma }_k}^2|}{\frac{{\mu }_3}{exp(p_1^2-{{\sigma }_1}^2),\cdots ,exp(p_k^2-{{\sigma }_k}^2)}}\right)]\\ =\Theta \left(\Omega \left({\sigma }_1\right)-{\rm Y}\left({\sigma }_1\right),\cdots ,\Omega \left({\sigma }_k\right)-{\rm Y}\left({\sigma }_k\right),\frac{\overrightarrow{\mu }}{|exp(p_1^2-{{\sigma }_1}^2),\cdots ,exp(p_k^2-{{\sigma }_k}^2)|}\right)\\ ⪰\Theta \left(\Omega \left({\sigma }_1\right)-{\rm Y}\left({\sigma }_1\right),\cdots ,\Omega \left({\sigma }_k\right)-{\rm Y}\left({\sigma }_k\right),\frac{\overrightarrow{\mu }}{K}\right),\end{array}$$

(49)

for some $K\in E_6$.

Let MVFF $\varphi :{\left(E_1\right)}^k\times {\left(E_2\right)}^3\to \mathrm{diag}{M}_n\left(E_3\right)$, satisfying (6) and (7).

Let $\omega :E_1\to R$ be a differentiable function satisfying:

$$\begin{array}{c}\hfill \mathrm{diag}[exp\left(\frac{{-|}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-\frac{1}{L_1}\omega \left(p_1\right)-{\int }_0^{p_1}exp(p_1^2-{{\sigma }_1}^2)\omega \left({\sigma }_1\right)d{\sigma }_1,\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-\frac{1}{L_1}\omega \left(p_k\right)-{\int }_0^{p_k}exp(p_k^2-{{\sigma }_k}^2)\omega \left({\sigma }_k\right)d{\sigma }_k|}{{\mu }_1}\right),\\ \hfill \frac{{\mu }_2}{{\mu }_2+{|}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-\frac{1}{L_1}\omega \left(p_1\right)-{\int }_0^{p_1}exp(p_1^2-{{\sigma }_1}^2)\omega \left({\sigma }_1\right)d{\sigma }_1,\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-\frac{1}{L_1}\omega \left(p_k\right)-{\int }_0^{p_k}exp(p_k^2-{{\sigma }_k}^2)\omega \left({\sigma }_k\right)d{\sigma }_k|},\\ \hfill exp\left(\frac{{-|}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-\frac{1}{L_1}\omega \left(p_1\right)-{\int }_0^{p_1}exp(p_1^2-{{\sigma }_1}^2)\omega \left({\sigma }_1\right)d{\sigma }_1,\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-\frac{1}{L_1}\omega \left(p_k\right)-{\int }_0^{p_k}exp(p_k^2-{{\sigma }_k}^2)\omega \left({\sigma }_k\right)d{\sigma }_k|}{{\mu }_3}\right)]\\ \hfill ⪰\mathrm{diag}\left[exp\left(\frac{-|p_1,\cdots ,p_k|}{{\mu }_1}\right),\frac{{\mu }_2}{{\mu }_2+|p_1,\cdots ,p_k|},exp\left(\frac{-|p_1,\cdots ,p_k|}{{\mu }_3}\right)\right].\end{array}$$

(50)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^3$. Now, Theorem 3 implies that, if $H_1>2$, $H_1=min[L_1,\frac{L_3}{K},L_1{L}_4,\frac{L_3{L}_4}{K}]$. We are to be able find a unique differentiable function ${\omega }_0:E_1\to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_p^{\kappa ,\tau ,\psi }{\omega }_0\left(p\right)=\frac{1}{L_1}{\omega }_0\left(p\right)+{\int }_0^{p}exp(p^2-{\sigma }^2)\omega \left(\sigma \right)d\sigma ,\end{array}$$

(51)

and:

$$\begin{array}{c}\mathrm{diag}[exp\left(\frac{{-|}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_k\right)|}{{\mu }_1}\right),\\ \frac{{\mu }_2}{{\mu }_2+{|}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_k\right)|},\\ exp\left(\frac{{-|}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }\omega \left(p_1\right)-{}_0{\mathcal{D}}_{p_1}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_1\right),\cdots {,}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }\omega \left(p_k\right)-{}_0{\mathcal{D}}_{p_k}^{\kappa ,\tau ,\psi }{\omega }_0\left(p_k\right)|}{{\mu }_3}\right)]\\ ⊛\mathrm{diag}[exp\left(\frac{-|\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right)|}{{\mu }_1}\right),\frac{{\mu }_2}{{\mu }_2+|\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right)|},\\ exp\left(\frac{-|\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right)|}{{\mu }_3}\right)]\\ ⪰\mathrm{diag}[exp\left(\frac{(-|p_1,\cdots ,p_k\left|\right)}{\frac{N_1}{H_2}{\mu }_1}\right),\frac{\frac{N_1}{H_2}{\mu }_2}{\frac{N_1}{H_2}{\mu }_2+|p_1,\cdots ,p_k|},\\ exp\left(\frac{(-|p_1,\cdots ,p_k\left|\right)}{\frac{N_1}{H_2}{\mu }_3}\right)],\end{array}$$

(52)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^3$, and $N_1=min[1,L_4]$ and $H_2=\frac{min[L_1,\frac{L_3}{K},L_1{L}_4,\frac{L_3{L}_4}{K}]}{min[L_1,\frac{L_3}{K},L_1{L}_4,\frac{L_3{L}_4}{K}]-2}$.

Example 2.

Let $(\mathbb{R},\Theta ,⊛)$ be an MVF-k-B-space. Consider $\Omega ,{\rm Y}:E_1\to \mathbb{R}$ such that $\Omega \left(p_i\right)={p_i}^4$ and ${\rm Y}\left(p_i\right)={p_i}^2$, and define $\mathit{S}(p_i,\Omega \left(p_i\right))=\frac{1}{L_1}\Omega \left(p_i\right)$. Define $\mathcal{M}:E_1\times E_1\times \mathbb{R}\to \mathbb{R}$ as $\mathcal{M}(p_i,{\sigma }_i,\Omega \left({\sigma }_i\right))=exp({p_i}^2-{{\sigma }_i}^2)\Omega \left({\sigma }_i\right)$ for every $p_1,\cdots ,p_k\in E_1$ and ${\sigma }_i⪯p_i(i=1,2,\cdots ,k)$ satisfying (49).

Let MVFF $\varphi :{\left(E_1\right)}^k\times {\left(E_2\right)}^3\to \mathrm{diag}{M}_n\left(E_3\right)$, satisfying (6) and (7).

Let $\omega :E_1\to \mathbb{R}$ be a differentiable function satisfying:

$$\begin{array}{c}\mathrm{diag}[exp\left(\frac{-|\omega \left(p_1\right)-\frac{1}{L_1}\omega \left(p_1\right)-{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }exp(p_1^2-{{\sigma }_1}^2)\omega \left({\sigma }_1\right)d{\sigma }_1,\cdots ,\omega \left(p_k\right)-\frac{1}{L_1}\omega \left(p_k\right)-{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }exp(p_k^2-{{\sigma }_k}^2)\omega \left({\sigma }_k\right)d{\sigma }_k|}{{\mu }_1}\right)\\ ,\frac{{\mu }_2}{{\mu }_2+|\omega \left(p_1\right)-\frac{1}{L_1}\omega \left(p_1\right)-{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }exp(p_1^2-{{\sigma }_1}^2)\omega \left({\sigma }_1\right)d{\sigma }_1,\cdots ,\omega \left(p_k\right)-\frac{1}{L_1}\omega \left(p_k\right)-{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }exp(p_k^2-{{\sigma }_k}^2)\omega \left({\sigma }_k\right)d{\sigma }_k|}\\ ,exp\left(\frac{-|\omega \left(p_1\right)-\frac{1}{L_1}\omega \left(p_1\right)-{}_0{\mathcal{I}}_{p_1}^{\kappa ,\psi }exp(p_1^2-{{\sigma }_1}^2)\omega \left({\sigma }_1\right)d{\sigma }_1,\cdots ,\omega \left(p_k\right)-\frac{1}{L_1}\omega \left(p_k\right)-{}_0{\mathcal{I}}_{p_k}^{\kappa ,\psi }exp(p_k^2-{{\sigma }_k}^2)\omega \left({\sigma }_k\right)d{\sigma }_k|}{{\mu }_3}\right)]\\ ⪰\mathrm{diag}\left[exp\left(\frac{-|p_1,\cdots ,p_k|}{{\mu }_1}\right),\frac{{\mu }_2}{{\mu }_2+|p_1,\cdots ,p_k|},exp\left(\frac{-|p_1,\cdots ,p_k|}{{\mu }_3}\right)\right]\end{array}$$

(53)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^3$.

Now, Theorem 4 implies that, if $H_3>2$, $H_3=min[L_1,\frac{L_4}{K}]$. We are to be able find a unique differentiable function ${\omega }_0:E_1\to \mathbb{R}$ such that:

$$\begin{array}{c}\hfill {\omega }_0\left(p\right)=\frac{1}{L_1}{\omega }_0\left(p\right)+{}_0{\mathcal{I}}_p^{\kappa ,\psi }exp(p^2-{\sigma }^2)\omega \left(\sigma \right)d\sigma ,\end{array}$$

(54)

and:

$$\begin{array}{cc}\hfill & \mathrm{diag}[exp\left(\frac{-|\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right)|}{{\mu }_1}\right),\frac{{\mu }_2}{{\mu }_2+|\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right)|},\\ \hfill & exp\left(\frac{-|\omega \left(p_1\right)-{\omega }_0\left(p_1\right),\cdots ,\omega \left(p_k\right)-{\omega }_0\left(p_k\right)|}{{\mu }_3}\right)]\\ & ⪰\mathrm{diag}\left[exp\left(\frac{-|p_1,\cdots ,p_k|}{\frac{N_1}{H_4}{\mu }_1}\right),\frac{\frac{N_1}{H_4}{\mu }_2}{\frac{N_1}{H_4}{\mu }_2+|p_1,\cdots ,p_k|},exp\left(\frac{-|p_1,\cdots ,p_k|}{\frac{N_1}{H_4}{\mu }_3}\right)\right]\end{array}$$

(55)

for every $p_1,\cdots ,p_k\in E_1$ and $\overrightarrow{\mu }\in {\left(E_2\right)}^3$, and $N_2=min[1,L_4]$ and $H_4=\frac{min[L_1,\frac{L_4}{K}]}{min[L_1,\frac{L_4}{K}]-2}$.

6. Conclusions

In this paper, we studied the concept of matrix-valued fuzzy k-normed spaces (MVF-k-N-spaces), and we applied the alternative fixed-point theorem to investigate the Hyers–Ulam–Rassias stability of some fractional equations in these spaces. We defined a class of matrix-valued fuzzy control functions for stabilizing fractional Volterra integro-differential equations with $\psi$-Hilfer fractional derivative in the complete MVF-k-N-spaces, and we obtained the best approximation for this kind of fractional equations.