Existence–Uniqueness and Wright Stability Results of the Riemann–Liouville Fractional Equations by Random Controllers in MB-Spaces

Abstract

We apply the random controllers to stabilize pseudo Riemann–Liouville fractional equations in MB-spaces and investigate existence and uniqueness of their solutions. Next, we compute the optimum error of the estimate. The mentioned process i.e., stabilization by a control function and finding an approximation for a pseudo functional equation is called random HUR stability. We use a fixed point technique derived from the alternative fixed point theorem (FPT) to investigate random HUR stability of the Riemann–Liouville fractional equations in MB-spaces. As an application, we introduce a class of random Wright control function and investigate existence–uniqueness and Wright stability of these equations in MB-spaces.

1. Introduction and Preliminaries

We introduce the random control functions that help us to investigate existence, uniqueness, and random Wright stability of integro-differential equations in MB-spaces. Some good references for the theory and application of fractional analysis are [1,2,3].

We set $\mathbb{I}=[0,1]$, ${\mathbb{I}}^{\circ }=(0,1)$, ${\mathbb{R}}^{•}=[-\infty ,+\infty ]$, ${\mathbb{J}}^{•}=[0,+\infty ]$ and ${\mathbb{J}}^{\circ }=(0,+\infty )$. We denote the set of distribution distance mappings (DDM) by ${\Sigma }^{+}$. Now, $\sigma \in {\Sigma }^{+}$ means that $\sigma$ is a mapping from ${\mathbb{R}}^{•}$ to 𝕀, written by ${\sigma }_{\tau }$ for $\sigma \left(\tau \right)$, and is left continuous and non-decreasing on ℝ and also ${\sigma }_0$ is zero and ${\sigma }_{+\infty }=1$. Now, $S^{+}\subset {\Sigma }^{+}$ consists of all (proper) mappings $\sigma \in {\Sigma }^{+}$ for which ${\ell }^{-}{\sigma }_{+\infty }=1$, and ${\ell }^{-}{\sigma }_{\tau }$ means the left limit at the point $\tau$; for some more details, see [4,5,6]. Note that proper DDM’s are the DDM’s of non-negative random variables (i.e., of those random variables g that a.s. take non-negative real values, ($P\left(\right\{g<0\}\bigcup \{g=\infty \left\}\right)=0$)).

The maximal element in $({\Sigma }^{+},\le )$ is ${\nabla }_{\tau }^0$, which is defined by

$${\nabla }_{\tau }^0=\left\{\begin{array}{c}\begin{array}{cc}0,& \mathrm{if}\tau \in \mathbb{R}-{\mathbb{J}}^{\circ },\\ 1,& \mathrm{if}\tau \in {\mathbb{J}}^{\circ }.\end{array}\hfill \end{array}\right.$$

Definition 1

([5,7,8]). A continuous binary operation $\ast :\mathbb{I}\times \mathbb{I}\to \mathbb{I}$ is a CTN (continuous t-norm) whenever

(a) 

$\zeta \ast \xi =\xi \ast \zeta$ and $\zeta \ast (\kappa \ast \xi )=(\zeta \ast \kappa )\ast \xi$ for all $\zeta ,\xi ,\kappa \in \mathbb{I}$;

(b) 

$\zeta \ast 1=\zeta$ for all $\zeta \in \mathbb{I}$;

(c) 

$\zeta \ast \kappa \le {\zeta }^{\prime }\ast {\kappa }^{\prime }$ when $\zeta \le {\zeta }^{\prime }$ and $\kappa \le {\kappa }^{\prime }$ for every $\zeta ,{\zeta }^{\prime },\kappa ,{\kappa }^{\prime }\in \mathbb{I}$.

For instance,

(1)

$\vartheta {\ast }_P\kappa =\vartheta \kappa$ (: the product CTN);

(2)

$\vartheta {\ast }_M\kappa =\wedge \{\vartheta ,\kappa \}$ (: the minimum CTN);

(3)

$\vartheta {\ast }_L\kappa =\vee \{\vartheta +\kappa -1,0\}$ (: the Lukasiewicz CTN).

Definition 2

([6]). Let ∗ be a CTN, W be a linear space and $\zeta :W\to S^{+}$ be a DDM-valued mapping. Now, $(W,\zeta ,\ast )$ is called an MN-space if:

(MN1) ${\zeta }_{\tau }^w={\nabla }_{\tau }^0$ for every $\tau \in {\mathbb{J}}^{\circ }$ if and only if $w=0$;

(MN2) ${\zeta }_{\tau }^{\alpha w}={\zeta }_{\frac{\tau }{\left|\alpha \right|}}^w$ for each $w\in W$ and $\alpha \ne 0$ in ℂ

(MN3) ${\zeta }_{\tau +\varsigma }^{w+w^{\prime }}\ge {\zeta }_{\tau }^w\ast {\zeta }_{\varsigma }^{w^{\prime }}$ for every w and $w^{\prime }$ in W and $\tau ,\varsigma \in {\mathbb{J}}^{\circ }$.

Here, MB-space represents a complete MN-space [9,10]. In the following, we suppose that $\ast ={\ast }_M$.

Theorem 1

([11,12]). Consider the complete ${\mathbb{J}}^{•}$-valued metric space $(S,\delta )$ and also consider the self-map Λ on S such that

$$\delta (\Lambda s,\Lambda t)\le \kappa \delta (t,s),\kappa <1,\mathit{where}\kappa \mathit{is}a\mathit{Lipschitz}\mathit{constant}.$$

Assume that $s\in S$, so there are two options:

$$\left(I\right)\delta ({\Lambda }^{m}s,{\Lambda }^{m+1}s)=\infty ,\forall m\in \mathbb{N},$$

or

(II) there is $m_0\in \mathbb{N}$ such that

(1) 

$\delta ({\Lambda }^{m}s,{\Lambda }^{m+1}s)<\infty ,\forall m\ge m_0$;

(2) 

the sequence ${\Lambda }^{m}s$ converges to a fixed point $t^{\ast }$ of Λ;

(3) 

$t^{\ast }\in V=\{t\in S\mid \delta ({\Lambda }^{m_0}s,t)<\infty \}$ is the unique fixed point of Λ in V;

(4) 

$(1-\kappa )\delta (t,t^{\ast })\le \delta (t,\Lambda t)$ for every $t\in V$.

2. Riemann–Liouville Fractional Equations

Let $u:[p,q]\to \mathbb{R}$$(0<p<q<\infty )$ be a continuous function and $\varrho >0$ a real number. We define the Riemann–Liouville fractional integrals of order $\varrho$, by

$$\begin{array}{c}\hfill {}_p{\mathcal{I}}_s^{\varrho }u\left(s\right)=\frac{1}{\Gamma \left(\varrho \right)}{\int }_p^s{(s-\sigma )}^{\varrho -1}u\left(\sigma \right)d\sigma ,p<s.\end{array}$$

(1)

Using the definition of Riemann–Liouville fractional integrals, we define the Riemann–Liouville derivatives as follows:

$$\begin{array}{ccc}\hfill {}_p{\mathcal{D}}_s^{\varrho }u\left(s\right)& =& {\left(\frac{d}{ds}\right)}^k{}_p{\mathcal{I}}_s^{k-\varrho }u\left(s\right)\hfill \\ & =& \frac{1}{\Gamma (k-\varrho )}{\left(\frac{d}{ds}\right)}^k{\int }_p^s{(s-\sigma )}^{k-\varrho -1}u\left(\sigma \right)d\sigma ,p<s,k-1<\varrho <k.\hfill \end{array}$$

(2)

Let T be a real positive number. Consider the Riemann–Liouville fractional Volterra integro-differential equation, defined by

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_s^{\varrho }u\left(s\right)=\alpha (s,u\left(s\right))+{\int }_0^s\mathcal{K}(s,\sigma ,u\left(\sigma \right))d\sigma \end{array}$$

(3)

where $\varrho \in {\mathbb{I}}^{\circ }$, $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$.

In [13], Goleţ, defined the concept of differentiable functions in an MB-space $(W,\zeta ,\ast )$ and proved that, if the function $f:U\to (W,\zeta ,\ast )$ is differentiable in $u_0\in U$, it is therefore continuous in the point $u_0$.

The Wright function [14,15] is one of the special functions defined by the series representation, valid in the whole complex plane

$$W_{\alpha ,\beta }\left(z\right)=\sum _{n=0}^{\infty }\frac{z^n}{n!\Gamma (n\alpha +\beta ),}$$

where $\alpha >-1$ and $\beta \in \mathbb{C}$. In this paper, we define a distribution distance mapping (DDM) based on Wright functions.

Consider the DDM-valued ${\mathbf{W}}_{\alpha ,\beta }:[0,T]\to S^{+}$ ($\alpha ,\beta \in {\mathbb{J}}^{\circ }$), a random control mapping, which is defined as follows:

$${\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s=\left\{\begin{array}{c}\begin{array}{cc}0,& \mathrm{if}\tau \in \mathbb{R}-{\mathbb{J}}^{\circ },\\ \sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )},& \mathrm{if}\tau \in {\mathbb{J}}^{\circ }.\end{array}\hfill \end{array}\right.$$

Then, we have

• It is left continuous and always increasing for positive values, it means that, for $\tau >0,{\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s>0$
• ${lim}_{\tau ⟶\infty }{\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s=1$
• For $\tau \le 0$, we have ${\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s=0,$

And the following conditions also apply to the DDM-valued Wright function

(MN1)

We show that ${\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s=1$ iff $s=0$.

$$\begin{array}{cc}\hfill & {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s=\sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )}=1+\sum _{n=1}^{\infty }\frac{{\left(\frac{-\parallel s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )}=1\hfill \\ \hfill & ⟹\sum _{n=1}^{\infty }\frac{{\left(\frac{-\parallel s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )}=0⟹\parallel s\parallel =0⟹s=0\hfill \end{array}$$

and vice versa with $s=0$.

(MN2)

$$\begin{array}{c}\hfill {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^{\alpha s}=\sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel \alpha s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )}=\sum _{n=0}^{\infty }\frac{{\left(\frac{-\left|\alpha \right|\parallel s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )}=\sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel s\parallel }{\frac{\tau }{\left|\alpha \right|}}\right)}^n}{n!\Gamma (n\alpha +\beta )}={\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\frac{\tau }{\left|\alpha \right|}}^s\end{array}$$

(MN3)

We assume that

$$\sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )}\le \sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel s^{\prime }\parallel }{\varsigma }\right)}^n}{n!\Gamma (n\alpha +\beta )}.$$

Thus,

$$\begin{array}{cc}\hfill & \frac{-\parallel s\parallel }{\tau }\le \frac{-\parallel s^{\prime }\parallel }{\varsigma }⟹\frac{\parallel s\parallel }{\tau }\ge \frac{\parallel s^{\prime }\parallel }{\varsigma }⟹\frac{\varsigma \parallel s\parallel }{\tau }\ge \parallel s^{\prime }\parallel ⟹\frac{\varsigma \parallel s\parallel }{\tau }+\parallel s\parallel \hfill \\ \hfill & \ge \parallel s^{\prime }\parallel +\parallel s\parallel \ge \parallel s+s^{\prime }\parallel ⟹\parallel s\parallel \left(\frac{\varsigma }{\tau }\right)\ge \parallel s+s^{\prime }\parallel ⟹\frac{\varsigma +\tau }{\tau }\parallel s\parallel \ge \parallel s+s^{\prime }\parallel ⟹\frac{\parallel s\parallel }{\tau }\hfill \\ \hfill & \ge \frac{\parallel s+s^{\prime }\parallel }{\tau +\varsigma }⟹\frac{-\parallel s\parallel }{\tau }\le \frac{-\parallel s+s^{\prime }\parallel }{\tau +\varsigma }⟹\sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel s+s^{\prime }\parallel }{\tau +\varsigma }\right)}^n}{n!\Gamma (n\alpha +\beta )}\ge \sum _{n=0}^{\infty }\frac{{\left(\frac{-\parallel s\parallel }{\tau }\right)}^n}{n!\Gamma (n\alpha +\beta )}.\hfill \end{array}$$

Now, we conclude that

$${\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau +\varsigma }^{s+s^{\prime }}\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s{\ast }_M{\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\varsigma }^{s^{\prime }}.$$

Let DDM-valued $\psi :[0,T]\to S^{+}$ be a random control mapping. We say that (3) is random HUR stable, when, for a differentiable mapping $u\left(s\right)$, satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right)-\alpha (s,u\left(s\right))-{\int }_0^s\mathcal{K}(s,\sigma ,u\left(\sigma \right))d\sigma }\ge {\psi }_{\tau }^s,\end{array}$$

$s\in [0,T]$, there is a solution $v\left(s\right)$ of Equation (3) such that, for some $r>0$,

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\frac{\tau }{r}}^s.\end{array}$$

If we replace the control function $\psi$ with the DDM-valued Wright function $\left({\mathbf{W}}_{\alpha ,\beta }\right)$, we say that (3) is random Wright stable.

3. Riemann–Liouville Fractional Volterra Integro-Differential Equation

In 2008, Mihet and Radu [16,17] introduced a new method to investigate random stability in MB-spaces and then some authors used this method to get stability results for new equations [18,19,20,21,22,23,24,25,26,27,28,29,30]. Here, we use the Mihet and Radu method and Theorem 1 to investigate random Wright stability of (3) and improve recent results [31]; we can suggest [32,33,34] for more details. We set

$$B:=\{u:[0,T]\to W,u\mathrm{is}\mathrm{differentiable}\}$$

and consider $\delta$ from $B\times B$ to ${\mathbb{J}}^{•}$ by

$$\begin{array}{c}\hfill \delta (u,v)=inf\left\{\mu \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\frac{\tau }{\mu }}^s,\forall u,v\in B,s\in [0,T],\tau \in {\mathbb{J}}^{\circ }\right\}.\end{array}$$

Theorem 2.

${\mathbb{J}}^{•}$-valued metric space $(B,\delta )$ is complete.

Proof. 

Letting $\delta (u,v)=0$, we have

$$inf\left\{\mu \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\frac{\tau }{\mu }}^s,\forall u,v\in B,s\in [0,T],\tau \in {\mathbb{J}}^{\circ }\right\}=0$$

and so

$${\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\frac{\tau }{\mu }}^s,$$

for all $\mu \in {\mathbb{J}}^{\circ }$. Tend $\mu$ to zero in the above inequality, we get

$${\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}={\nabla }_{\tau }^0$$

and so

$${\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}={\nabla }_{\tau }^0;$$

thus, $u\left(s\right)=v\left(s\right)$ for every $s\in [0,T]$, and vice versa. In addition, we have $\delta (u,v)=\delta (v,u)$ for every $u,v\in B$. Now, let $\delta (u,v)=e_1\in {\mathbb{J}}^{\circ }$ and $\delta (v,w)=e_2\in {\mathbb{J}}^{\circ }$. Thus, we have

$${\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\frac{\tau }{e_1}}^s,$$

and

$${\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }v\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)}\ast {\zeta }_{\tau }^{v\left(s\right)-w\left(s\right)}\ge {\psi }_{\frac{\tau }{e_2}}^s,$$

for every $\tau \in {\mathbb{J}}^{\circ }$. Thus, we have

$$\begin{array}{ccc}& & {\zeta }_{(e_1+e_2)\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)}\ast {\zeta }_{(e_1+e_2)\tau }^{u\left(s\right)-w\left(s\right)}\hfill \\ & \ge & [{\zeta }_{e_1\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{e_2\tau }^{{}_0{\mathcal{D}}_s^{\varrho }v\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)}]\ast [{\zeta }_{e_1\tau }^{u\left(s\right)-v\left(s\right)}\ast {\zeta }_{e_2\tau }^{v\left(s\right)-w\left(s\right)}]\hfill \\ & \ge & {\psi }_{\tau }^s\ast {\psi }_{\tau }^s\hfill \\ & \ge & {\psi }_{\tau }^s,\hfill \end{array}$$

and so $\delta (u,w)\le e_1+e_2$. Thus, $\delta (u,w)\le \delta (u,v)+\delta (v,w)$. To show completeness of $(B,\delta )$, we consider the Cauchy sequence ${\left\{u_k\right\}}_k$ in $(B,\delta )$. Suppose that $s\in [0,T]$ is fixed. Assume that $\omega \in {\mathbb{J}}^{\circ }$ and $\lambda \in {\mathbb{I}}^{\circ }$ are arbitrary and consider $\tau \in {\mathbb{J}}^{\circ }$ such that ${\theta }_{\tau }^s>1-\lambda$. For $e\tau <\omega$, choose $k_0\in \mathbb{N}$ such that

$$\delta (u_k-u_{\ell })<e\forall k,\ell \ge k_0.$$

Consequently,

$$\begin{array}{ccc}& & {\zeta }_{\omega }^{{}_0{\mathcal{D}}_s^{\varrho }{u}_k\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{u}_{\ell }\left(s\right)}\ast {\zeta }_{\omega }^{u_k\left(s\right)-u_{\ell }\left(s\right)}\hfill \\ & \ge & {\zeta }_{e\tau }^{{}_0{\mathcal{D}}_s^{\varrho }{u}_k\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{u}_{\ell }\left(s\right)}\ast {\zeta }_{e\tau }^{u_k\left(s\right)-u_{\ell }\left(s\right)}\hfill \\ & \ge & {\psi }_{\tau }^s\hfill \\ & >& 1-\lambda .\hfill \end{array}$$

Hence, ${\zeta }_{\omega }^{{}_0{\mathcal{D}}_s^{\varrho }{u}_k\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{u}_{\ell }\left(s\right)}>1-\lambda$ and ${\zeta }_{\omega }^{u_k\left(s\right)-u_{\ell }\left(s\right)}>1-\lambda$ and i.e., the sequence both ${\left\{u_k\left(s\right)\right\}}_k$ and ${{\{}_0{\mathcal{D}}_s^{\varrho }{u}_k\left(s\right)\}}_k$ are Cauchy in complete space $(W,\zeta ,\ast )$ on compact set $[0,T]$, so they are uniformly convergent to the mapping $u:[0,T]\to W$ and ${}_0{\mathcal{D}}_s^{\varrho }u$, respectively. Now, if we apply the uniform convergence, we conclude that $u\in B$ and is differentiable; thus, $(B,\delta )$ is complete. □

Theorem 3.

Let $(W,\zeta ,\ast )$ be an MB-space and ${\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4$ and T be a positive constant such that $\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}<0.5$. Assume that the continuous mappings $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$ with DDM-valued $\psi :[0,T]\to S^{+}$ satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{\alpha (s,u(s\left)\right)-\alpha (s,v(s\left)\right)}\ge {\zeta }_{\frac{\tau }{{\ell }_1}}^{u\left(s\right)-v\left(s\right)},\end{array}$$

(4)

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{\mathcal{K}(s,\sigma ,u(\sigma \left)\right)-\mathcal{K}(s,\sigma ,v(\sigma \left)\right)}\ge {\zeta }_{\frac{\tau }{{\ell }_2}}^{u\left(\sigma \right)-v\left(\sigma \right)},\sigma \le s,\end{array}$$

(5)

$$\begin{array}{c}\hfill \underset{\rho \in [0,T]}{inf}{\psi }_{\tau }^{\rho }\ge {\psi }_{\frac{T\tau }{{\ell }_3}}^s,\end{array}$$

(6)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{u\left(s\right)}\ge {\psi }_{\tau }^s,\mathit{implies}\mathit{that}{\zeta }_{\tau }^{{}_0{\mathcal{I}}_s^{\varrho }u\left(\sigma \right)d\sigma }\ge {\psi }_{\frac{\tau }{{\ell }_4}}^s,\end{array}$$

(7)

for every $s\in [0,T]$, $u,v:[0,T]\to W$ and $\tau \in {\mathbb{J}}^{\circ }$. Let $w:[0,T]\to W$ be a differentiable function satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)-\alpha (s,w\left(s\right))-{\int }_0^s\mathcal{K}(s,\sigma ,w\left(\sigma \right))d\sigma }\ge {\psi }_{\tau }^s,\end{array}$$

(8)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. Thus, there is a unique differentiable function $w_0:[0,T]\to W$ such that

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)=\alpha (s,w_0\left(s\right))+{\int }_0^s\mathcal{K}(s,\sigma ,w_0\left(\sigma \right))d\sigma ,\end{array}$$

(9)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)}\ast {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\psi }_{\frac{(1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\})\tau }{\bigvee \{1,{\ell }_4\}}}^s,\end{array}$$

(10)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

Proof. 

We set

$$B:=\{u:[0,T]\to W,u\mathrm{is}\mathrm{differentiable}\}$$

and define

$$\begin{array}{c}\hfill \delta (u,v)=inf\left\{\mu \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\frac{\tau }{\mu }}^s,\forall u,v\in B,s\in [0,T],\tau \in {\mathbb{J}}^{\circ }\right\}.\end{array}$$

Theorem 2 guarantees that $(B,\delta )$ is a complete ${\mathbb{J}}^{•}$-valued metric space.

Consider the self-map ${\rm Y}$ on B by

$$\begin{array}{c}\hfill {\rm Y}\left(u\left(s\right)\right)={}_0{\mathcal{I}}_s^{\varrho }\left(\alpha (\sigma ,u\left(\sigma \right))\right)+{}_0{\mathcal{I}}_s^{\varrho }\left({\int }_0^{\sigma }\mathcal{K}(\sigma ,\varsigma ,u\left(\varsigma \right))d\varsigma \right),\end{array}$$

(11)

where $\varrho \in {\mathbb{I}}^{\circ }$, $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$. We prove that ${\rm Y}$ is a strictly contractive mapping. Let $u,v\in B$, $\mu \in {\mathbb{J}}^{\circ }$ and $\delta (u,v)<\nu$; thus, we have

$$\begin{array}{c}\hfill {\zeta }_{\nu \tau }^{{}_0{\mathcal{D}}_s^{\varrho }u\left(s\right)-{}_0{\mathcal{D}}_s^{\varrho }v\left(s\right)}\ast {\zeta }_{\nu \tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\tau }^s,\forall u,v\in B,s\in [0,T],\tau \in {\mathbb{J}}^{\circ }.\end{array}$$

Using properties (MN2) and (MN3) of Definition 2 and (11), imply that

$$\begin{array}{ccc}\hfill & & {\zeta }_{2\nu \tau }^{{}_0{\mathcal{D}}_s^{\varrho }{\rm Y}\left(u\left(s\right)\right)-{}_0{\mathcal{D}}_s^{\varrho }{\rm Y}\left(v\left(s\right)\right)}\ast {\zeta }_{2\nu \tau }^{{\rm Y}\left(u\right(s\left)\right)-{\rm Y}\left(v\right(s\left)\right)}\hfill \\ & =& {\zeta }_{2\nu \tau }^{[\alpha (s,u\left(s\right))-\alpha (s,v\left(s\right))]+{\int }_0^s[\mathcal{K}(s,\sigma ,u\left(\sigma \right))-\mathcal{K}(s,\sigma ,v\left(\sigma \right))]d\sigma }\hfill \\ & \ast & {\zeta }_{2\nu \tau }^{{}_0{\mathcal{I}}_s^{\varrho }(\alpha (s,u\left(s\right))-\alpha (s,v\left(s\right)))+{}_0{\mathcal{I}}_s^{\varrho }\left({\int }_0^s[\mathcal{K}(s,\sigma ,u\left(\sigma \right))-\mathcal{K}(s,\sigma ,v\left(\sigma \right))]d\sigma \right)}\hfill \\ & \ge & {\zeta }_{\nu \tau }^{\alpha (s,u(s\left)\right)-\alpha (s,v(s\left)\right)}\ast {\zeta }_{\nu \tau }^{{\int }_0^s[\mathcal{K}(s,\sigma ,u\left(\sigma \right))-\mathcal{K}(s,\sigma ,v\left(\sigma \right))]d\sigma }\hfill \\ & \ast & {\zeta }_{\nu \tau }^{{}_0{\mathcal{I}}_s^{\varrho }(\alpha (\sigma ,u\left(\sigma \right))-\alpha (\sigma ,v\left(\sigma \right)))}\ast {\zeta }_{\nu \tau }^{{}_0{\mathcal{I}}_s^{\varrho }\left({\int }_0^{\sigma }[\mathcal{K}(\sigma ,\varsigma ,u\left(\varsigma \right))-\mathcal{K}(\sigma ,\varsigma ,v\left(\varsigma \right))]d\varsigma \right)}.\hfill \end{array}$$

(12)

Now, we connect (12) and control function $\psi$. Assume that $0={\varpi }_1<{\varpi }_2<\cdots <{\varpi }_k=s$, $\Delta {\varpi }_i={\varpi }_i-{\varpi }_{i-1}=\frac{s}{k}$, $i=1,2,\cdots ,k$ and $\parallel \Delta \varpi \parallel ={\bigvee }_{1\le i\le k}\left(\Delta {\varpi }_i\right)$.

Step 1. From (35), we have

$$\begin{array}{ccc}\hfill {\zeta }_{\nu \tau }^{\alpha (s,u(s\left)\right)-\alpha (s,v(s\left)\right)}& \ge & {\zeta }_{\frac{\nu \tau }{{\ell }_1}}^{u\left(s\right)-v\left(s\right)}\hfill \\ & \ge & {\psi }_{\frac{\tau }{{\ell }_1}}^s.\hfill \end{array}$$

(13)

Step 2. Using (MN2) and (MN3) of Definition 2, continuity property of DDM-valued $\zeta$, (36), and (37), we get

$$\begin{array}{ccc}\hfill {\zeta }_{\nu \tau }^{{\int }_0^s[\mathcal{K}(s,\sigma ,u\left(\sigma \right))-\mathcal{K}(s,\sigma ,v\left(\sigma \right))]d\sigma }& =& {\zeta }_{\nu \tau }^{\underset{\parallel \Delta \varpi \parallel \to 0}{lim}{\sum }_{j=1}^k[\mathcal{K}(s,{\varpi }_j,u\left({\varpi }_j\right))-\mathcal{K}(s,{\varpi }_j,v\left({\varpi }_j\right))]\Delta {\varpi }_i}\hfill \\ & =& \underset{\parallel \Delta \varpi \parallel \to 0}{lim}{\zeta }_{\nu \tau }^{{\sum }_{j=1}^k[\mathcal{K}(s,{\varpi }_j,u\left({\varpi }_j\right))-\mathcal{K}(s,{\varpi }_j,v\left({\varpi }_j\right))]\Delta {\varpi }_i}\hfill \\ & \ge & \underset{\parallel \Delta \varpi \parallel \to 0}{lim}\underset{j=1}{\overset{k}{\bigwedge }}{\zeta }_{\frac{\nu \tau }{k}}^{[\mathcal{K}(s,{\varpi }_j,u\left({\varpi }_j\right))-\mathcal{K}(s,{\varpi }_j,v\left({\varpi }_j\right))]\Delta {\varpi }_i}\hfill \\ & \ge & \underset{\rho \in [0,T]}{inf}{\zeta }_{\frac{k\nu \tau }{kT}}^{\mathcal{K}(s,\rho ,u(\rho \left)\right)-\mathcal{K}(s,\rho ,v(\rho \left)\right)}\hfill \\ & \ge & \underset{\rho \in [0,T]}{inf}{\zeta }_{\frac{k\nu \tau }{kT{\ell }_2}}^{u\left(\rho \right)-v\left(\rho \right)}\hfill \\ & \ge & \underset{\rho \in [0,T]}{inf}{\psi }_{\frac{\tau }{{\ell }_{2}T}}^{\rho }\hfill \\ & \ge & {\psi }_{\frac{\tau }{{\ell }_2{\ell }_3}}^s.\hfill \end{array}$$

(14)

Step 3. Using (38) and (13), we get

$$\begin{array}{ccc}\hfill {\zeta }_{\nu \tau }^{{}_0{\mathcal{I}}_s^{\varrho }(\alpha (\sigma ,u\left(\sigma \right))-\alpha (\sigma ,v\left(\sigma \right)))}& \ge & {\psi }_{\frac{\tau }{{\ell }_1{\ell }_4}}^s.\hfill \end{array}$$

(15)

Step 4. Using (38) and (14), we get

$$\begin{array}{ccc}\hfill {\zeta }_{\nu \tau }^{{}_0{\mathcal{I}}_s^{\varrho }\left({\int }_0^{\sigma }[\mathcal{K}(\sigma ,\varsigma ,u\left(\varsigma \right))-\mathcal{K}(\sigma ,\varsigma ,v\left(\varsigma \right))]d\varsigma \right)}& \ge & {\psi }_{\frac{\tau }{{\ell }_2{\ell }_3{\ell }_4}}^s.\hfill \end{array}$$

(16)

Form (12)–(16), we have

$$\begin{array}{ccc}\hfill & & {\zeta }_{2\nu \tau }^{{}_0{\mathcal{D}}_s^{\varrho }{\rm Y}\left(u\left(s\right)\right)-{}_0{\mathcal{D}}_s^{\varrho }{\rm Y}\left(v\left(s\right)\right)}\ast {\zeta }_{2\nu \tau }^{{\rm Y}\left(u\right(s\left)\right)-{\rm Y}\left(v\right(s\left)\right)}\hfill \\ & \ge & {\psi }_{\frac{\tau }{{\ell }_1}}^s\ast {\psi }_{\frac{\tau }{{\ell }_2{\ell }_3}}^s\ast {\psi }_{\frac{\tau }{{\ell }_1{\ell }_4}}^s\ast {\psi }_{\frac{\tau }{{\ell }_2{\ell }_3{\ell }_4}}^s\hfill \\ & \ge & {\psi }_{\frac{\tau }{\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}}}^s\hfill \end{array}$$

(17)

and so

$$\begin{array}{c}\hfill {\zeta }_{\nu \tau }^{{}_0{\mathcal{D}}_s^{\varrho }{\rm Y}\left(u\left(s\right)\right)-{}_0{\mathcal{D}}_s^{\varrho }{\rm Y}\left(v\left(s\right)\right)}\ast {\zeta }_{\nu \tau }^{{\rm Y}\left(u\right(s\left)\right)-{\rm Y}\left(v\right(s\left)\right)}\ge {\psi }_{\frac{\tau }{2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}}}^s,\end{array}$$

(18)

which implies that

$$\begin{array}{c}\hfill \delta ({\rm Y}\left(u\right),{\rm Y}\left(v\right))\le 2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}\nu ,\end{array}$$

(19)

and so

$$\begin{array}{c}\hfill \delta ({\rm Y}\left(u\right),{\rm Y}\left(v\right))\le 2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}\delta (u,v).\end{array}$$

(20)

Consequently, ${\rm Y}$ is a strictly contractive mapping with Lipschitz constant

$$2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}.$$

Letting $w\in B$, we show that $\delta ({\rm Y}(w),w)<\infty$. Using (38) and (39), we get

$$\begin{array}{ccc}\hfill & & {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }[{\rm Y}\left(w\left(s\right)\right)-w\left(s\right)]}\ast {\zeta }_{\tau }^{{\rm Y}\left(w\right(s\left)\right)-w\left(s\right)}\hfill \\ & =& {\zeta }_{\tau }^{\alpha (s,w\left(s\right))+{\int }_0^s\mathcal{K}(s,\sigma ,w\left(\sigma \right))d\sigma -{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)}\ast {\zeta }_{\tau }^{{}_0{\mathcal{I}}_s^{\varrho }\left(\alpha (\sigma ,w\left(\sigma \right))\right)+{}_0{\mathcal{I}}_s^{\varrho }\left({\int }_0^{\sigma }\mathcal{K}(\sigma ,\varsigma ,w\left(\varsigma \right))d\varsigma \right)-{}_0{\mathcal{I}}_s^{\varrho }{}_0{\mathcal{D}}_{\sigma }^{\varrho }w\left(\sigma \right)}\hfill \\ & =& {\zeta }_{\tau }^{\alpha (s,w\left(s\right))+{\int }_0^s\mathcal{K}(s,\sigma ,w\left(\sigma \right))d\sigma -{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)}\ast {\zeta }_{\tau }^{{}_0{\mathcal{I}}_s^{\varrho }\left[\left(\alpha (\sigma ,w\left(\sigma \right))\right)+\left({\int }_0^{\sigma }\mathcal{K}(\sigma ,\varsigma ,w\left(\varsigma \right))d\varsigma \right)-{}_0{\mathcal{D}}_{\sigma }^{\varrho }w\left(\sigma \right)\right]}\hfill \\ & \ge & {\psi }_{\tau }^s\ast {\psi }_{\frac{\tau }{{\ell }_4}}^s\hfill \\ & \ge & {\psi }_{\frac{\tau }{\bigvee \{1,{\ell }_4\}}}^s,\hfill \end{array}$$

(21)

for every $\tau \in {\mathbb{J}}^{\circ }$. Then, we have $\delta ({\rm Y}\left(w\right),w)<\bigvee \{1,{\ell }_4\}<\infty$.

By Theorem 1, there is $w_0$ in B such that

(1) $w_0$ is a fixed point of ${\rm Y}$, i.e.,

$$\begin{array}{ccc}\hfill w_0\left(s\right)& =& {\rm Y}\left(w_0\left(s\right)\right)\hfill \\ \hfill & =& {}_0{\mathcal{I}}_s^{\varrho }\left(\alpha (\sigma ,w_0\left(\sigma \right))\right)+{}_0{\mathcal{I}}_s^{\varrho }\left({\int }_0^{\sigma }\mathcal{K}(\sigma ,\varsigma ,w_0\left(\varsigma \right))d\varsigma \right),\hfill \end{array}$$

(22)

which is unique in

$$B^{\ast }=\{u\in B:\delta ({\rm Y}\left(w\right),u)<\infty \}.$$

Taking ${}_0{\mathcal{D}}_s^{\varrho }$ from (22), we get

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)=\alpha (s,w_0\left(s\right))+{\int }_0^s\mathcal{K}(s,\sigma ,w_0\left(\sigma \right))d\sigma ,\end{array}$$

(23)

where $\varrho \in {\mathbb{I}}^{\circ }$, $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$.

(2) $\delta ({{\rm Y}}^k\left(w\right),w_0)\to 0$ as $n\to \infty$;

(3) $\delta (w,w_0)\le \frac{1}{1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}}\delta ({\rm Y}\left(w\right),w)\le \frac{\bigvee \{1,{\ell }_4\}}{1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}}$, which implies that

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)}\ast {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\psi }_{\frac{(1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\})\tau }{\bigvee \{1,{\ell }_4\}}}^s,\end{array}$$

(24)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

Now, we prove that $B^{\ast }=B$. Considering $z_0$ in B satisfying (40) and (41), we show that $z_0=w_0$ and $z_0\in B^{\ast }$. From (40), we get

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_s^{\varrho }{z}_0\left(s\right)=\alpha (s,z_0\left(s\right))+{\int }_0^s\mathcal{K}(s,\sigma ,z_0\left(\sigma \right))d\sigma ,\end{array}$$

(25)

and so

$$\begin{array}{cc}\hfill z_0\left(s\right)& ={}_0{\mathcal{I}}_s^{\varrho }\alpha (\sigma ,z_0\left(\sigma \right))+{}_0{\mathcal{I}}_s^{\varrho }{\int }_0^{\sigma }\mathcal{K}(\sigma ,\varsigma ,z_0\left(\varsigma \right))d\varsigma \hfill \\ \hfill & ={\rm Y}\left(z_0\left(s\right)\right),\hfill \end{array}$$

(26)

where $\varrho \in {\mathbb{I}}^{\circ }$, $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$.

Now, we show that

$$z_0\in \{u\in B:\delta ({\rm Y}\left(w\right),u)<\infty \},$$

i.e., $\delta ({\rm Y}\left(w\right),z_0)<\infty$. We set $j=\frac{1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}}{\bigvee \{1,{\ell }_4\}}$, from (41), we get

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{z}_0\left(s\right)}\ast {\zeta }_{\tau }^{w\left(s\right)-z_0\left(s\right)}\ge {\psi }_{j\tau }^s,\end{array}$$

(27)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

Form (35) and (27), we get

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{\alpha (s,w\left(s\right))-\alpha (s,z_0\left(s\right))}& \ge & {\zeta }_{\frac{\tau }{{\ell }_1}}^{w\left(s\right)-z_0\left(s\right)}\hfill \\ & \ge & {\psi }_{j\frac{\tau }{{\ell }_1}}^s;\hfill \end{array}$$

(28)

in addition, from (36) and (27), we get

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{\mathcal{K}(s,\sigma ,w\left(\sigma \right))-\mathcal{K}(s,\sigma ,z_0\left(\sigma \right))}& \ge & {\zeta }_{\frac{\tau }{{\ell }_2}}^{w\left(\sigma \right)-z_0\left(\sigma \right)}\hfill \\ & \ge & {\psi }_{j\frac{\tau }{{\ell }_2}}^s,\hfill \end{array}$$

(29)

for every $s\in [0,T]$, $\sigma \le s$ and $\tau \in {\mathbb{J}}^{\circ }$. By step 2 and (29), we get

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{{\int }_0^s[\mathcal{K}(s,\sigma ,w\left(\sigma \right))-\mathcal{K}(s,\sigma ,z_0\left(\sigma \right))]d\sigma }& \ge & {\psi }_{\frac{\tau }{{\ell }_2{\ell }_3}}^s\hfill \\ & \ge & {\psi }_{j\frac{\tau }{{\ell }_2{\ell }_3}}^s.\hfill \end{array}$$

Using triangular inequality (MN3), (28) and (30), we get

$$\begin{array}{ccc}\hfill & & {\zeta }_{2\tau }^{\alpha (s,w\left(s\right))-\alpha (s,z_0\left(s\right))+{\int }_0^s[\mathcal{K}(s,\sigma ,w\left(\sigma \right))-\mathcal{K}(s,\sigma ,z_0\left(\sigma \right))]d\sigma }\hfill \\ \hfill & \ge & {\zeta }_{\tau }^{\alpha (s,w\left(s\right))-\alpha (s,z_0\left(s\right))}\ast {\zeta }_{\tau }^{{\int }_0^s[\mathcal{K}(s,\sigma ,w\left(\sigma \right))-\mathcal{K}(s,\sigma ,z_0\left(\sigma \right))]d\sigma }\hfill \\ & \ge & {\psi }_{j\frac{\tau }{{\ell }_1}}^s\ast {\psi }_{j\frac{\tau }{{\ell }_2{\ell }_3}}^s\hfill \end{array}$$

(30)

$$\begin{array}{ccc}& \ge & {\psi }_{j\frac{\tau }{\bigvee \{{\ell }_1,{\ell }_2{\ell }_3\}}}^s,\hfill \end{array}$$

(31)

and so

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{\alpha (s,w\left(s\right))-\alpha (s,z_0\left(s\right))+{\int }_0^s[\mathcal{K}(s,\sigma ,w\left(\sigma \right))-\mathcal{K}(s,\sigma ,z_0\left(\sigma \right))]d\sigma }\ge {\psi }_{j\frac{\tau }{2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3\}}}^s.\end{array}$$

(32)

We apply (38) and get

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{I}}_s^{\varrho }[\alpha (\sigma ,w\left(\sigma \right))-\alpha (\sigma ,z_0\left(\sigma \right))+{}_0{\mathcal{I}}_s^{\varrho }{\int }_0^{\sigma }[\mathcal{K}(\sigma ,\varsigma ,w\left(\varsigma \right))-\mathcal{K}(\sigma ,\varsigma ,z_0\left(\varsigma \right))]d\varsigma }\ge {\psi }_{j\frac{\tau }{2{\ell }_4\bigvee \{{\ell }_1,{\ell }_2{\ell }_3\}}}^s,\end{array}$$

(33)

for every $s\in [0,T]$, $\sigma \le s$ and $\tau \in {\mathbb{J}}^{\circ }$.

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

$$\begin{array}{ccc}\hfill & & {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }[{\rm Y}\left(w\left(s\right)\right)-z_0\left(s\right)]}\ast {\zeta }_{\tau }^{{\rm Y}\left(w\left(s\right)\right)-z_0\left(s\right)}\hfill \\ \hfill & =& {\zeta }_{\tau }^{\alpha (s,w\left(s\right))-\alpha (s,z_0\left(s\right))+{\int }_0^s[\mathcal{K}(s,\sigma ,w\left(\sigma \right))-\mathcal{K}(s,\sigma ,z_0\left(\sigma \right))]d\sigma }\hfill \\ & \ast & {\zeta }_{\tau }^{{}_0{\mathcal{I}}_s^{\varrho }[\alpha (\sigma ,w\left(\sigma \right))-\alpha (\sigma ,z_0\left(\sigma \right))+{}_0{\mathcal{I}}_s^{\varrho }{\int }_0^{\sigma }[\mathcal{K}(\sigma ,\varsigma ,w\left(\varsigma \right))-\mathcal{K}(\sigma ,\varsigma ,z_0\left(\varsigma \right))]d\varsigma }\hfill \\ & \ge & {\psi }_{j\frac{\tau }{2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3\}}}^s\ast {\psi }_{j\frac{\tau }{2{\ell }_4\bigvee \{{\ell }_1,{\ell }_2{\ell }_3\}}}^s\hfill \\ & \ge & {\psi }_{j\frac{\tau }{2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3\}(1+{\ell }_4)}}^s,\hfill \end{array}$$

(34)

which implies that $\delta ({\rm Y}\left(w\right),z_0)\le \frac{2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3\}(1+{\ell }_4)}{j}<\infty$, hence $z_0\in B^{\ast }$. □

Corollary 1.

Let $(\mathbb{R},\zeta ,\ast )$ be an MB-space and ${\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4$ and T be a positive constant such that $\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}<0.5$. Assume that the continuous mappings $\alpha :[0,T]\times \mathbb{R}\to \mathbb{R}$, $\mathcal{K}:[0,T]\times [0,T]\times \mathbb{R}\to \mathbb{R}$ with DDM-valued $\psi :[0,T]\to S^{+}$ satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{\alpha (s,u(s\left)\right)-\alpha (s,v(s\left)\right)}\ge {\zeta }_{\frac{\tau }{{\ell }_1}}^{u\left(s\right)-v\left(s\right)},\end{array}$$

(35)

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{\mathcal{K}(s,\sigma ,u(\sigma \left)\right)-\mathcal{K}(s,\sigma ,v(\sigma \left)\right)}\ge {\zeta }_{\frac{\tau }{{\ell }_2}}^{u\left(\sigma \right)-v\left(\sigma \right)},\sigma \le s,\end{array}$$

(36)

$$\begin{array}{c}\hfill \underset{\rho \in [0,T]}{inf}{\psi }_{\tau }^{\rho }\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\frac{T\tau }{{\ell }_3}}^s,\end{array}$$

(37)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{u\left(s\right)}\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s,\mathit{implies}\mathit{that}{\zeta }_{\tau }^{{}_0{\mathcal{I}}_s^{\varrho }u\left(\sigma \right)d\sigma }\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\frac{\tau }{{\ell }_4}}^s,\end{array}$$

(38)

for every $s\in [0,T]$, $u,v:[0,T]\to \mathbb{R}$ and $\tau \in {\mathbb{J}}^{\circ }$. Let $w:[0,T]\to \mathbb{R}$ be a differentiable function satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)-\alpha (s,w\left(s\right))-{\int }_0^s\mathcal{K}(s,\sigma ,w\left(\sigma \right))d\sigma }\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s,\end{array}$$

(39)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. Thus, there is a unique differentiable function $w_0:[0,T]\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)=\alpha (s,w_0\left(s\right))+{\int }_0^s\mathcal{K}(s,\sigma ,w_0\left(\sigma \right))d\sigma ,\end{array}$$

(40)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)}\ast {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\frac{(1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\})\tau }{\bigvee \{1,{\ell }_4\}}}^s,\end{array}$$

(41)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

Proof. 

Put $\psi =\left({\mathbf{W}}_{\alpha ,\beta }\right)$ and apply Theorem 3. □

4. Random Stability of Riemann–Liouville Fractional Volterra Integral Equation

Consider the Riemann–Liouville fractional Volterra integral equation

$$\begin{array}{c}\hfill u\left(s\right)=\alpha (s,u\left(s\right))+{}_0{\mathcal{I}}_s^{\varrho }\mathcal{K}(s,\sigma ,u\left(\sigma \right)),\end{array}$$

(42)

where $\varrho \in {\mathbb{I}}^{\circ }$, $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$. In this section, we study random Wright stability of (42).

Theorem 4.

Suppose that $(W,\zeta ,\ast )$ is an MB-space and ${\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4$ and T are positive constants such that $\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}<0.5$. Assume the continuous mappings $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$ with DDM-valued $\psi :[0,T]\to S^{+}$ satisfying (35)–(38).

Let $w:[0,T]\to W$ be a differentiable function satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{w\left(s\right)-\alpha (s,w\left(s\right))-{}_0{\mathcal{I}}_s^{\varrho }\mathcal{K}(s,\sigma ,w\left(\sigma \right))}\ge {\psi }_{\tau }^s,\end{array}$$

(43)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. Thus, we can find a unique differentiable function $w_0:[0,T]\to W$ such that

$$\begin{array}{c}\hfill w_0\left(s\right)=\alpha (s,w_0\left(s\right))+{}_0{\mathcal{I}}_s^{\varrho }\mathcal{K}(s,\sigma ,w_0\left(\sigma \right)),\end{array}$$

(44)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\psi }_{\frac{(1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\})\tau }{\bigvee \{1,{\ell }_4\}}}^s,\end{array}$$

(45)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

Proof. 

We set

$$B:=\{u:[0,T]\to W,u\mathrm{as}\mathrm{differentiable}\}$$

and define

$$\begin{array}{c}\hfill \delta (u,v)=inf\left\{\mu \in {\mathbb{J}}^{\circ }:{\zeta }_{\tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\frac{\tau }{\mu }}^s,\forall u,v\in B,s\in [0,T],\tau \in {\mathbb{J}}^{\circ }\right\}.\end{array}$$

Theorem 2 guarantees that $(B,\delta )$ is a complete ${\mathbb{J}}^{•}$-valued metric space.

Consider ${\rm Y}$ from B to B by

$$\begin{array}{c}\hfill {\rm Y}\left(u\left(s\right)\right)=\alpha (\sigma ,u\left(\sigma \right))+{}_0{\mathcal{I}}_s^{\varrho }\left({\int }_0^{\sigma }\mathcal{K}(\sigma ,\varsigma ,u\left(\varsigma \right))d\varsigma \right),\end{array}$$

(46)

where $\varrho \in {\mathbb{I}}^{\circ }$, $\alpha :[0,T]\times W\to W$, $\mathcal{K}:[0,T]\times [0,T]\times W\to W$. We show that ${\rm Y}$ is a strictly contractive mapping. Let $u,v\in B$, $\mu \in {\mathbb{J}}^{\circ }$ and $\delta (u,v)<\nu$; then, we have

$$\begin{array}{c}\hfill {\zeta }_{\nu \tau }^{u\left(s\right)-v\left(s\right)}\ge {\psi }_{\tau }^s,\forall u,v\in B,s\in [0,T],\tau \in {\mathbb{J}}^{\circ }.\end{array}$$

From (MN2) and (MN3) of Definition 2, (35)–(38) and (46), we get

$$\begin{array}{ccc}\hfill & & {\zeta }_{2\nu \tau }^{{\rm Y}\left(u\right(s\left)\right)-{\rm Y}\left(v\right(s\left)\right)}\hfill \\ & =& {\zeta }_{2\nu \tau }^{[\alpha (s,u\left(s\right))-\alpha (s,v\left(s\right))]+{}_0{\mathcal{I}}_s^{\varrho }[\mathcal{K}(s,\sigma ,u\left(\sigma \right))-\mathcal{K}(s,\sigma ,v\left(\sigma \right))]d\sigma }\hfill \\ & \ge & {\zeta }_{\nu \tau }^{\alpha (s,u(s\left)\right)-\alpha (s,v(s\left)\right)}\ast {\zeta }_{\nu \tau }^{{\int }_0^s[\mathcal{K}(s,\sigma ,u\left(\sigma \right))-\mathcal{K}(s,\sigma ,v\left(\sigma \right))]d\sigma }\hfill \\ & \ge & {\psi }_{\frac{\tau }{{\ell }_1}}^s\ast {\psi }_{\frac{\tau }{{\ell }_2{\ell }_4}}^s\hfill \\ & \ge & {\psi }_{\frac{\tau }{\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\}}}^s,\hfill \end{array}$$

(47)

and so

$$\begin{array}{c}\hfill {\zeta }_{\nu \tau }^{{\rm Y}\left(u\right(s\left)\right)-{\rm Y}\left(v\right(s\left)\right)}\ge {\psi }_{\frac{\tau }{2\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\}}}^s,\end{array}$$

(48)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. Consequently,

$$\begin{array}{c}\hfill \delta ({\rm Y}\left(u\right),{\rm Y}\left(v\right))\le 2\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\}\nu ,\end{array}$$

(49)

and so

$$\begin{array}{c}\hfill \delta ({\rm Y}\left(u\right),{\rm Y}\left(v\right))\le 2\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\}\delta (u,v).\end{array}$$

(50)

Thus, ${\rm Y}$ is strictly contractive with Lipschitz constant $2\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\}$.

Letting $w\in B$, we prove that $\delta ({\rm Y}(w),w)<\infty$. In addition, (53) implies that

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{{\rm Y}\left(w\right(s\left)\right)-w\left(s\right)}& =& {\zeta }_{\tau }^{\alpha (s,w\left(s\right))+{}_0{\mathcal{I}}_s^{\varrho }\mathcal{K}(s,\sigma ,w\left(\sigma \right))d\sigma -w\left(s\right)}\hfill \\ & \ge & {\psi }_{\tau }^s\hfill \end{array}$$

for every $\tau \in {\mathbb{J}}^{\circ }$. Thus, $\delta ({\rm Y}(w),w)<1$.

By Theorem 1, there is $w_0$ in B such that

(1) $w_0$ is a fixed point of ${\rm Y}$, i.e.,

$$\begin{array}{ccc}\hfill w_0\left(s\right)& =& {\rm Y}\left(w_0\left(s\right)\right)\hfill \\ \hfill & =& \alpha (\sigma ,w_0\left(\sigma \right))+{}_0{\mathcal{I}}_s^{\varrho }\left(\mathcal{K}(\sigma ,\varsigma ,w_0\left(\varsigma \right))d\varsigma \right),\hfill \end{array}$$

(51)

which is unique in the set

$$B^{\ast }=\{u\in B:\delta ({\rm Y}\left(w\right),u)<\infty \}.$$

(2) $\delta ({{\rm Y}}^k\left(w\right),w_0)\to 0$ as $n\to \infty$;

(3) $\delta (w,w_0)\le \frac{1}{1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\}}\delta ({\rm Y}\left(w\right),w)\le \frac{1}{1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\}}$, which implies that

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\psi }_{(1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_4\})\tau }^s,\end{array}$$

(52)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. By the same method of the proof of Theorem 3, we can show that $B^{\ast }=B$. □

Corollary 2.

Suppose that $(\mathbb{R},\zeta ,\ast )$ is an MB-space and ${\ell }_1,{\ell }_2,{\ell }_3,{\ell }_4$ and T are positive constants such that $\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\}<0.5$. Assume that the continuous mappings $\alpha :[0,T]\times \mathbb{R}\to \mathbb{R}$, $\mathcal{K}:[0,T]\times [0,T]\times \mathbb{R}\to \mathbb{R}$ with DDM-valued $\psi :[0,T]\to S^{+}$ satisfying (35)–(38).

Let $w:[0,T]\to \mathbb{R}$ be a differentiable function satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{w\left(s\right)-\alpha (s,w\left(s\right))-{}_0{\mathcal{I}}_s^{\varrho }\mathcal{K}(s,\sigma ,w\left(\sigma \right))}\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s,\end{array}$$

(53)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. Thus, we can find a unique differentiable function $w_0:[0,T]\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill w_0\left(s\right)=\alpha (s,w_0\left(s\right))+{}_0{\mathcal{I}}_s^{\varrho }\mathcal{K}(s,\sigma ,w_0\left(\sigma \right)),\end{array}$$

(54)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\frac{(1-2\bigvee \{{\ell }_1,{\ell }_2{\ell }_3,{\ell }_1{\ell }_4,{\ell }_2{\ell }_3{\ell }_4\})\tau }{\bigvee \{1,{\ell }_4\}}}^s,\end{array}$$

(55)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

Proof. 

Put $\psi =\left({\mathbf{W}}_{\alpha ,\beta }\right)$ and apply Theorem 4. □

5. Applications, Random Wright Stability

Now, as applications, we study the concept of random Wright stability for some fractional equations.

Example 1.

Assume that $(\mathbb{R},\zeta ,\ast )$ is an MB-space. Consider $u,v:[0,T]\to \mathbb{R}$ and define $\alpha (s,u\left(s\right))={\ell }_{1}u\left(s\right)$. Let $\theta \in {\mathcal{W}}^{1,1}({\mathbb{J}}^{\circ },\mathbb{R})$, in which ${\mathcal{W}}^{1,1}$ is the Sobolev space, define $\mathcal{K}:[0,T]\times [0,T]\times \mathbb{R}\to \mathbb{R}$ as $\mathcal{K}(s,\sigma ,u(\sigma \left)\right)=\theta (s-\sigma )u\left(\sigma \right)$ for every $s\in [0,T]$ and $\sigma \le s$.

Consequently, we have

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{\alpha (s,u(s\left)\right)-\alpha (s,v(s\left)\right)}& =& {\zeta }_{\tau }^{{\ell }_{1}u\left(s\right)-{\ell }_{1}v\left(s\right)}\hfill \\ & =& {\zeta }_{\frac{\tau }{{\ell }_1}}^{u\left(s\right)-v\left(s\right)},\hfill \end{array}$$

(56)

$$\begin{array}{ccc}\hfill {\zeta }_{\tau }^{\mathcal{K}(s,\sigma ,u(\sigma \left)\right)-\mathcal{K}(s,\sigma ,v(\sigma \left)\right)}& =& {\zeta }_{\tau }^{\theta (s-\sigma )\left[u\right(\sigma )-v(\sigma \left)\right]}\hfill \\ & \ge & {\zeta }_{\frac{\tau }{\left|\theta \right(s-\sigma \left)\right|}}^{u\left(\sigma \right)-v\left(\sigma \right)}\hfill \\ & \ge & {\zeta }_{\frac{\tau }{K}}^{u\left(\sigma \right)-v\left(\sigma \right)},\hfill \end{array}$$

(57)

for some $K\in {\mathbb{J}}^{\circ }$. Let DDM-valued ${\mathbf{W}}_{\alpha ,\beta }:[0,T]\to S^{+}$ satisfying (37) and (38).

Let $w:[0,T]\to \mathbb{R}$ be a differentiable function satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right)-{\ell }_{1}w\left(s\right)-{\int }_0^s\theta (s-\sigma )w\left(\sigma \right)d\sigma }\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s,\end{array}$$

(58)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. Now, Theorem 3 implies that, if $\bigvee \{{\ell }_1,K{\ell }_3,{\ell }_1{\ell }_4,K{\ell }_3{\ell }_4\}<0.5$, there is a unique differentiable function $w_0:[0,T]\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill {}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)={\ell }_1{w}_0\left(s\right)+{\int }_0^s\theta (s-\sigma )w\left(\sigma \right)d\sigma ,\end{array}$$

(59)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{{}_0{\mathcal{D}}_s^{\varrho }w\left(s\right){-}_0{\mathcal{D}}_s^{\varrho }{w}_0\left(s\right)}\ast {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\frac{(1-2\bigvee \{{\ell }_1,K{\ell }_3,{\ell }_1{\ell }_4,K{\ell }_3{\ell }_4\})\tau }{\bigvee \{1,{\ell }_4\}}}^s,\end{array}$$

(60)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

Example 2.

Assume that $(\mathbb{R},\zeta ,\ast )$ is an MB-space. Consider $u,v:[0,T]\to \mathbb{R}$ and define $\alpha (s,u\left(s\right))={\ell }_{1}u\left(s\right)$. Let $\theta \in {\mathcal{W}}^{1,1}({\mathbb{J}}^{\circ },\mathbb{R})$, in which ${\mathcal{W}}^{1,1}$ is the Sobolev space, and define $\mathcal{K}:[0,T]\times [0,T]\times \mathbb{R}\to \mathbb{R}$ as $\mathcal{K}(s,\sigma ,u(\sigma \left)\right)=\theta (s-\sigma )u\left(\sigma \right)$ for every $s\in [0,T]$ and $\sigma \le s$ satisfying (57).

Let DDM-valued ${\mathbf{W}}_{\alpha ,\beta }:[0,T]\to S^{+}$ satisfying (37) and (38).

Let $w:[0,T]\to \mathbb{R}$ be a differentiable function satisfying

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{w\left(s\right)-{\ell }_{1}w\left(s\right)-{}_0{\mathcal{I}}_s^{\varrho }\theta (s-\sigma )w\left(\sigma \right)d\sigma }\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\tau }^s,\end{array}$$

(61)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$. Now, Theorem 4 implies that, if $\bigvee \{{\ell }_1,K{\ell }_4\}<0.5$, there is a unique differentiable function $w_0:[0,T]\to \mathbb{R}$ such that

$$\begin{array}{c}\hfill w_0\left(s\right)={\ell }_1{w}_0\left(s\right)+{}_0{\mathcal{I}}_s^{\varrho }\theta (s-\sigma )w\left(\sigma \right)d\sigma ,\end{array}$$

(62)

and

$$\begin{array}{c}\hfill {\zeta }_{\tau }^{w\left(s\right)-w_0\left(s\right)}\ge {\left({\mathbf{W}}_{\alpha ,\beta }\right)}_{\frac{(1-2\bigvee \{{\ell }_1,K{\ell }_4\})\tau }{\bigvee \{1,{\ell }_4\}}}^s,\end{array}$$

(63)

for every $s\in [0,T]$ and $\tau \in {\mathbb{J}}^{\circ }$.

6. Conclusions

We considered a class of random control functions. By a method from Mihet and Radu emanating from the alternative fixed point theorem and random controllers, we stabilized some fractional equations in MB-spaces in the sense of Wright.