Laplace inverse and MR approach to existence of a unique solution and the Hyers–Ulam–Wright stability analysis of the nonhomogeneous fractional delay oscillation equation by matrix-valued fuzzy controllers

Abstract

In this paper, we consider the nonhomogeneous fractional delay oscillation equation with order κ and investigate the existence of a unique solution in matrix-valued fuzzy Banach spaces for this equation using the alternative fixed point theorem. In a fuzzy environment, we introduce a class of the matrix-valued fuzzy Wright controller to investigate the Hyers–Ulam–Wright stability for the NH-FD-O equation with order κ. Finally, an illustrative example to demonstrate the application of the main theorem is also considered.

1 Introduction

One of the important topics in mathematics, and especially in mathematical analysis, is fractional calculus. Khusainov and Shuklin [1–3] applied the delayed exponential function and next matrix sine and a delayed matrix cosine to get the exact solution of a nonhomogeneous fractional delay oscillation equation (in short NH-FDO-E). In [4, 5], Li and Wang used a delayed Mittag-Leffler type matrix to solve a generalization of the mentioned problem.

In this paper, we consider the MVFB-space introduced in [6] and a modern class of MVF control functions based on the Wright functions. Our goal is to obtain an approximation for the NH-FDO-E by a new combined method powered by the Laplace inverse transform and MR approach in the MVFB-space [7–11]. Next, in Sect. 2, we present the basic definitions and concepts that are necessary to investigate the main results and introduce the matrix-valued fuzzy Wright function as a control function. In the third section, we prove the existence of a unique solution and the Hyers–Ulam–Wright stability for the NH-FDO-E in MVFN-spaces using the alternative FPT. Finally, we provide a numerical example as an application of our main theorem.

2 Preliminaries

In this manuscript, we consider the following NH-FDO-E:

$$\begin{aligned} &{D_{0}^{\kappa} \mathrm{u}(\mathfrak{x})=-\rho \mathrm{u}( \mathfrak{x}-\xi )+\mathrm{k}(\mathfrak{x})+\mathcal{L}^{-1} \bigl( \mathrm{u}(\mathfrak{r})\bigr) (\mathfrak{x}),} \quad \mathfrak{x} \in \mathfrak{L}:=[0, L], \end{aligned}$$

(1)

$$\begin{aligned} &\mathrm{u}(\mathfrak{x})= \Upsilon (\mathfrak{x}), \qquad \mathrm{u}^{ \prime}(\mathfrak{x})=\Upsilon ^{\prime} (\mathfrak{x}), \quad - \xi \leq \mathfrak{x}\leq 0, \end{aligned}$$

(2)

where

• $\mathrm{k}: \mathfrak{L}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}$ is an integrable function, $\mathrm{u}(\mathfrak{x}) \in \mathbb{R}^{n}$, $\rho \in \mathbb{R}^{n \times n}$ denotes constant matrix, $\Upsilon \in C^{2}([-\xi ,0], \mathbb{R}^{n})$ for $\xi >0$ being a fixed time.

• $D_{0}^{\kappa}$ is the standard Caputo fractional derivative with $\kappa \in (1,2)$, defined by

  $$ D_{0}^{\kappa} \mathrm{u}(\mathfrak{x})= \int _{0}^{\mathfrak{x}} \frac{(\mathfrak{x}-\mathit{t})^{1-\kappa}}{\Gamma (2-\kappa )} \mathrm{u}^{\prime \prime}(\mathit{t}) \,\mathrm{d} \mathit{t}, $$

  if the integral exists.

Definition 2.1

([12, 13])

The Wright function is defined by the following series representation:

$$ W_{\kappa , \varsigma}(\mathrm{k})=\sum_{\mathfrak{p}=0}^{+\infty} \frac{\mathrm{k}^{\mathfrak{p}}}{\mathfrak{p} ! \Gamma (\kappa \mathfrak{p}+\varsigma )} $$

for $\kappa >-1$, $\varsigma >0$, $\mathrm{k} \in \mathbb{R}$. It is an entire function of order $1 /(1+\kappa )$, which has also been known as the generalized Bessel (or Bessel Maitland) function.

Now, we present the Laplace transforms.

Definition 2.2

([2])

The classical Laplace transform is defined by the integral formula

$$ (\mathcal{L} \mathrm{k}) (\mathfrak{r})= \int _{0}^{+\infty} \mathrm{k}( \mathfrak{x}) e^{-\mathfrak{r}\mathfrak{x}} \,d \mathfrak{x}, $$

where $\mathrm{k}(\mathfrak{x})$ is absolutely integrable on $[0,+\infty )$. Let $J_{0}^{\kappa}$ denote the Riemann–Liouville fractional integral operator of order $\kappa \in (1,2)$. Assume that the Laplace transforms of $J_{0}^{\kappa}\mathrm{k}(\mathfrak{x})$ and $D_{0}^{\kappa}\mathrm{k}(\mathfrak{x})$ exist for $\mathfrak{x} \leq 0$ and the Laplace transform of $\mathrm{k}(\mathfrak{x}-\mathrm{b})$ exists for $\mathfrak{x} \leq \mathrm{b}$. Then we have

• $\mathcal{L} (J_{0}^{\kappa} \mathrm{k}(\mathfrak{x}) )( \mathfrak{r})=\mathfrak{r}^{-\kappa} \mathcal{L}(\mathrm{k}( \mathfrak{x}))(\mathfrak{r})$,

• $\mathcal{L} (D_{0}^{\kappa} \mathrm{k}(\mathfrak{x}) )( \mathfrak{r})=\mathfrak{r}^{\kappa} \mathcal{L}(\mathrm{k}( \mathfrak{x}))(\mathfrak{r})-\mathfrak{r}^{\kappa -1} \mathrm{k}(0)- \mathfrak{r}^{\kappa -2} \mathrm{k}^{\prime}(0)$,

• $\mathcal{L} (\mathrm{k}_{1}(\mathfrak{x}) )(\mathfrak{r})=e^{- \mathrm{b} \mathfrak{r}} \int _{-\mathrm{b}}^{0} e^{-\mathfrak{x} \mathfrak{r}} \mathrm{k}(\mathfrak{x}) \,d \mathfrak{x}+e^{-\mathrm{b} \mathfrak{r}} \mathcal{L}(\mathrm{k}(\mathfrak{x}))(\mathfrak{r})$.

Assuming $\mathfrak{F}_{1}=[0,p]$, $\mathfrak{F}_{2}=(0,{+\infty})$, $\mathfrak{F}_{3}=(0,1]$, $\mathfrak{F}_{4}=[0,{+\infty}]$, $\mathfrak{F}_{5}=[0,1]$ ($\mathfrak{F}_{5}^{\circ}=(0,1)$), and $\mathfrak{F}_{6} =[0,{+\infty})$, we consider the set of all matrices $n \times n$ on $\mathfrak{F}_{5}$ as follows:

$$ \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})= \left \{ \begin{bmatrix} {\ell}_{1} & & \\ & \ddots & \\ & & {\ell}_{n} \end{bmatrix} =\operatorname{diag} [\ell _{1},\ldots ,\ell _{n}], \ell _{1},\ldots ,\ell _{n}{ \in}\mathfrak{F}_{5} \right\}.$$

For the above set, we have

• $\boldsymbol{\ell}=\operatorname{diag} [\ell _{1},\ldots , \ell _{n}]$, $\boldsymbol{\jmath}=\operatorname{diag} [\jmath _{1},\ldots , \jmath _{n}] \in \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})$;

• $\boldsymbol{\ell}\preceq \boldsymbol{\jmath}$ if and only if $\ell _{i} \leq \jmath _{i}$ for every $i=1, \ldots , n$;

• $\boldsymbol{\ell} \prec \boldsymbol{\jmath}$ denotes that $\boldsymbol{\ell} \preceq \boldsymbol{\jmath}$ and $\boldsymbol{\ell} \neq \boldsymbol{\jmath}$; ${\ell}_{1} < {\jmath}_{i}$ for every $i=1, \ldots ,n$;

• Define $\boldsymbol{\mathrm{b}}=\operatorname{diag} [\mathrm{b},\ldots ,\mathrm{b}]$ in $\operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})$, where $\mathrm{b} \in \mathfrak{F}_{5}$. Note that $\operatorname{diag} [1,\ldots ,1]=\boldsymbol{1}$ and $\operatorname{diag} [0,\ldots ,0]=\boldsymbol{0}$.

Definition 2.3

([6])

A mapping $\circledast : \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})\times \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}) \to \operatorname{diag} \mathrm{M}_{n}( \mathfrak{F}_{5})$ is called a GTN if:

• (1) $\boldsymbol{\ell} {\circledast} \mathbf{1}=\boldsymbol{\ell}$ for all $\boldsymbol{\ell} \in \operatorname{diag} \mathrm{M}_{n} ( \mathfrak{F}_{5})$ (boundary condition);

• (2) $\boldsymbol{\ell} \circledast \boldsymbol{\jmath} = \boldsymbol{\jmath} \circledast \boldsymbol{\ell}$ for all $(\boldsymbol{\ell},\boldsymbol{\jmath}) \in (\operatorname{diag} \mathrm{M}_{n}( \mathfrak{F}_{5}))^{2}$ (commutativity);

• (3) $\boldsymbol{\ell} \circledast (\boldsymbol{\jmath} \circledast \boldsymbol{\imath}) = (\boldsymbol{\ell} \circledast \boldsymbol{\jmath})\circledast \boldsymbol{\imath}$ for all $(\boldsymbol{\ell},\boldsymbol{\jmath}, \boldsymbol{\imath})\in (\operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}))^{3}$ (associativity);

• (4) $\boldsymbol{\ell}_{1} \preceq \boldsymbol{\ell}_{2}$ and $\boldsymbol{\jmath}_{1} \preceq \boldsymbol{\jmath}_{2}$ imply that $\boldsymbol{\ell}_{1} \circledast \boldsymbol{\jmath}_{1} \preceq \boldsymbol{\ell}_{2}\circledast \boldsymbol{\jmath}_{2}$ for all $(\boldsymbol{\ell}_{1}, \boldsymbol{\jmath}_{2} ,\boldsymbol{\jmath}_{1},\boldsymbol{\jmath}_{2}) \in (\operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}))^{4}$ (monotonicity).

• (5) If for every $\boldsymbol{\ell}, \boldsymbol{\jmath} \in \operatorname{diag} \mathrm{M}_{n}( \mathfrak{F}_{5})$ and each of sequences $\{\boldsymbol{\ell}_{q}\}$ and $\{\boldsymbol{\jmath}_{q}\}$ converges to ℓ and ȷ respectively, we get

  $$ \lim_{q\rightarrow +\infty}(\boldsymbol{\ell}_{q}\circledast \boldsymbol{\jmath}_{q})= \boldsymbol{\ell} \circledast \boldsymbol{ \jmath},$$

  which implies the continuity of ⊛ on $\operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})$ (CGTN).

The following are numerical examples of CGTNs:

(i) Define $\circledast _{M} : \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}) \times \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}) \to \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})$ such that

$$ \boldsymbol{\ell}\circledast _{M} \boldsymbol{\jmath}= \operatorname{diag}i [ \ell _{1},\ldots ,\ell _{n}] \circledast _{M} \operatorname{diag}[\jmath _{1}, \ldots , \jmath _{n}] =\operatorname{diag} \bigl[\min \{\ell _{1}, \jmath _{1}\}, \ldots ,\min \{\ell _{n},\jmath _{n} \}\bigr],$$

then $\circledast _{M}$ is CGTN (minimum CGTN). Here is an example of minimum CGTN:

$$\begin{aligned} & \operatorname{diag} [0.3,0.4,0.2 ]\circledast _{M} \operatorname{diag} [0.7,0.5,0.8 ] =\operatorname{diag} [0.3,0.4,0.2 ] \end{aligned}$$

or

$$\begin{aligned} &\begin{bmatrix} 0.3 & & \\ & 0.4 & \\ && 0.2 \end{bmatrix}\circledast _{M} \begin{bmatrix} 0.7 & & \\ & 0.5 & \\ & &0.8 \end{bmatrix} = \begin{bmatrix} 0.3 & & \\ & 0.4 & \\ & & 0.2 \end{bmatrix}. \end{aligned}$$

(ii) Define $\circledast _{P} : \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}) \times \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}) \to \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})$ such that

$$ \boldsymbol{\ell} \circledast _{P} \boldsymbol{\jmath} = \operatorname{diag} [ \ell _{1},\ldots ,\ell _{n}]\circledast _{P} \operatorname{diag}[\jmath _{1}, \ldots ,\jmath _{n}] =\operatorname{diag} [\ell _{1}.\jmath _{1},\ldots ,\ell _{n}. \jmath _{n}],$$

then $\circledast _{P}$ is CGTN (product CGTN). As an example of product CGTN,

$$\begin{aligned} & \operatorname{diag} [0.1,0.6,0.3 ]\circledast _{P} \operatorname{diag} [0.9,0.4,0.7 ] = \operatorname{diag} [0.09,0.24,0.21 ] \end{aligned}$$

or

$$\begin{aligned} &\begin{bmatrix} 0.1 & & \\ & 0.6 & \\ && 0.3 \end{bmatrix}\circledast _{P} \begin{bmatrix} 0.9 & & \\ & 0.4 & \\ & & 0.7 \end{bmatrix}= \begin{bmatrix} 0.09 & & \\ & 0.24 & \\ & & 0.21 \end{bmatrix}. \end{aligned}$$

(iii) Define $\circledast _{L} : \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}) \times \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5}) \to \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{5})$ such that

$$ \begin{aligned} \boldsymbol{\ell} \circledast _{L} \boldsymbol{\jmath} &= \operatorname{diag} [ \ell _{1},\ldots , \ell _{n}] \circledast _{L} \operatorname{diag} [\jmath _{1}, \ldots , \jmath _{n}] \\ &=\operatorname{diag} \bigl[\max \{\ell _{1}+ \jmath _{1}-1,0\}, \ldots ,\max \{\ell _{n} + \jmath _{n}-1,0\}\bigr],\end{aligned} $$

then $\circledast _{P}$ is CGTN (Lukasiewicz CGTN). A simple example of Lukasiewicz CGTN is

$$\begin{aligned} &\operatorname{diag} [0.2,0.1,0.8 ]\circledast _{L} \operatorname{diag} [0.4,0.5,0.6 ]= \operatorname{diag} [0,0,0.4 ] \end{aligned}$$

or

$$\begin{aligned} &\begin{bmatrix} 0.2 & & \\ & 0.1 & \\ && 0.8 \end{bmatrix}\circledast _{L} \begin{bmatrix} 0.4 & & \\ & 0.5 & \\ & & 0.6 \end{bmatrix}= \begin{bmatrix} 0 & & \\ & 0 & \\ & & 0.4 \end{bmatrix}. \end{aligned}$$

For the CGTNs introduced above, we clearly have the following relation:

$$\begin{aligned} &\operatorname{diag} [\ell _{1},\ldots , \ell _{n} ] \circledast _{M} \operatorname{diag} [\jmath _{1},\ldots , \jmath _{n} ] \\ &\quad \succeq \operatorname{diag} [\ell _{1},\ldots , \ell _{n} ] \circledast _{P} \operatorname{diag} [\jmath _{1},\ldots , \jmath _{n} ] \\ &\quad \succeq \operatorname{diag} [\ell _{1},\ldots , \ell _{n} ] \circledast _{L} \operatorname{diag} [\jmath _{1},\ldots , \jmath _{n} ]. \end{aligned}$$

Consider the matrix-valued fuzzy function (MVFF) $\mathcal{A}: \mathfrak{F}_{1} \times \mathfrak{F}_{2} \to \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{3})$, then we have

• It is left continuous and an increasing function.

• $\lim_{\eta \to {+\infty}}\mathcal{A}(\mathfrak{x},\eta )= \boldsymbol{1}$ for any $\mathfrak{x} \in \mathfrak{F}_{1}$ and $\eta \in \mathfrak{F}_{2}$.

• For MVFFs $\mathcal{A}$ and $\mathcal{W}$, the relation “⪯” is defined as follows:

  $$\begin{aligned} &\mathcal{A} \precsim \mathcal{W}\quad \text{if and only if} \quad \mathcal{A}( \mathfrak{x},\eta ) \preceq \mathcal{W}(\mathfrak{x}, \eta ) \quad \text{for all } \eta \in \mathfrak{F}_{2} \text{ and } \mathfrak{x} \in \mathfrak{F}_{1}. \end{aligned}$$

Definition 2.4

Let ⊛ be a CGTN, $\mathcal{J}$ be a vector space, and $\mathcal{A} :\mathcal{J} \times \mathfrak{F}_{2}\to \operatorname{diag} \mathrm{M}_{n}(\mathfrak{F}_{3})$ be a matrix-valued fuzzy set (MVFS). Triple $(\mathcal{J},\mathcal{A},\circledast )$ is called an MVFN-space if

• (N1) $\mathcal{A}(\epsilon ,\eta )=\boldsymbol{1}$ if and only if $\epsilon =0$ and $\eta \in \mathfrak{F}_{2}$;

• (N2) $\mathcal{A}({\varkappa} \epsilon ,\eta )=\mathcal{A}(\epsilon , \frac{\eta}{|\varkappa |})$ for all $\epsilon \in \mathcal{J}$ and $\varkappa \in \mathbb{C}$ with $\varkappa \neq 0$;

• (N3) $\mathcal{A}(\epsilon +w,\eta +\mathfrak{z})\succeq \mathcal{A}( \epsilon ,\eta )\circledast \mathcal{A}(w,\mathfrak{z})$ for all $\epsilon , w \in \mathcal{J}$ and any $\eta ,\mathfrak{z} \in \mathfrak{F}_{2}$;

• (N4) $\lim_{\eta \to {+\infty}}\mathcal{A}(\mathfrak{x},\eta )= \boldsymbol{1}$ for any $\eta \in \mathfrak{F}_{2}$.

When the MVFN-space is complete, we denote it by MVFB-space.

Using the concept of Wright function [13], we define an MVF Wright function $\boldsymbol{ W_{\kappa ,\varsigma}}$ ($\boldsymbol{\kappa ,\varsigma}\in \mathfrak{F}_{2}$) as a control function in the MVFN-spaces by

$$\begin{aligned} &\boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr)= \operatorname{diag} \biggl[W_{\kappa , \varsigma} \biggl(-\frac{ \vert \mathfrak{x} \vert }{\eta} \biggr),W_{\kappa , \varsigma} \biggl(-\frac{ \vert \mathfrak{x} \vert }{\eta} \biggr), W_{\kappa , \varsigma} \biggl(-\frac{ \vert \mathfrak{x} \vert }{\eta} \biggr) \biggr], \end{aligned}$$

where the Wright function $W_{\kappa ,\varsigma}$ is defined as follows:

$$ W_{\kappa ,\varsigma} \biggl(-\frac{ \vert \mathfrak{x} \vert }{\eta} \biggr)= \sum _{\mathfrak{p}=0}^{+\infty} \frac{ ({\frac{- \vert \mathfrak{x} \vert }{\eta}} )^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}.$$

Then, for the MVF Wright function $\boldsymbol{ W_{\kappa ,\varsigma}}$, we have

• It is left continuous and an increasing function for positive values.

• $\lim_{\eta \to {+\infty}}\boldsymbol{ W_{\kappa ,\varsigma}}(- \frac{|\mathfrak{x}|}{\eta}) =\boldsymbol{1}$.

• For $\boldsymbol{ W_{\kappa ,\varsigma}}$ and the matrix-valued fuzzy function $\boldsymbol{ \Xi _{\kappa ,\varsigma}}$, we have

  $$ \boldsymbol{ \Xi _{\kappa ,\varsigma}}\precsim \boldsymbol{ W_{\kappa ,\varsigma}} \quad \text{if and only if}\quad \boldsymbol{ \Xi _{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr) \preceq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr),$$

and we further get

• (a) $\boldsymbol{ W_{\kappa ,\varsigma}} (- \frac{|\mathfrak{x}|}{\eta} ) >0$.

• (b) For $\eta \in \mathfrak{F}_{2}$, $\boldsymbol{ W_{\kappa ,\varsigma}} (- \frac{|\mathfrak{x}|}{\eta} )=\boldsymbol{1}$ if and only if $\mathfrak{x} =0$.

  First, we assume that $\boldsymbol{ W_{\kappa ,\varsigma}} (- \frac{|\mathfrak{x}|}{\eta} )=\boldsymbol{1}$. Then

  $$\begin{aligned} &\operatorname{diag} \biggl[W_{\kappa ,\varsigma} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr),W_{\kappa ,\varsigma} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr), W_{\kappa ,\varsigma} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr) \biggr] \\ &\quad = \operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{ ({\frac{- \vert \mathfrak{x} \vert }{\eta}} )^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{ ({\frac{- \vert \mathfrak{x} \vert }{\eta}} )^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{ ({\frac{- \vert \mathfrak{x} \vert }{\eta}} )^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )} \Biggr] = \boldsymbol{1}, \end{aligned}$$

  then $\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{-|\mathfrak{x}|}{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}=1$. We have

  $$\begin{aligned} 1+ \sum_{\mathfrak{p}=1}^{+\infty} \frac{({\frac{- \vert \mathfrak{x} \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}=1 \quad \text{implies}\quad \sum_{\mathfrak{p}=1}^{+\infty} \frac{({\frac{- \vert \mathfrak{x} \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}=0, \end{aligned}$$

  therefore

  $$ \vert \mathfrak{x} \vert =0 \quad \text{and}\quad \mathfrak{x}=0.$$

  Conversely, suppose that $\mathfrak{x} =0$, then $\sum_{\mathfrak{p}=1}^{+\infty} \frac{({\frac{-|\mathfrak{x}}{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}=0$. As a result

  $$\begin{aligned} \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \mathfrak{x} \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}= 1+ \sum_{\mathfrak{p}=1}^{+\infty} \frac{({\frac{- \vert \mathfrak{x} \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}=1 \end{aligned}$$

  yields

  $$ \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \mathfrak{x} \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}=1.$$

• (c) We shall deduce

  $$ \boldsymbol{ W_{\kappa ,\varsigma}}\biggl(- \frac{ \vert \alpha (\mathfrak{x}) \vert }{\eta}\biggr) = \boldsymbol{ W_{\kappa ,\varsigma}}\biggl(- \frac{ \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }}\biggr).$$

  Indeed,

  $$\begin{aligned} &\boldsymbol{ W_{\kappa ,\varsigma}}\biggl(-\frac{ \vert \mathfrak{x}) \vert }{\eta}\biggr) \\ &\quad =\operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )} \Biggr] \\ &\quad =\operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha \vert \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha \vert \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha \vert \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )} \Biggr] \\ &\quad =\operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }})^{\mathfrak{p}}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }})^{\mathfrak{p}}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\tau +\omega )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )} \Biggr] \\ &\quad=\boldsymbol{ W_{\kappa ,\varsigma}}\biggl(- \frac{ \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }}\biggr). \end{aligned}$$

• (d) Finally, we prove

  $$\begin{aligned} &\boldsymbol{ W_{\kappa ,\varsigma}}\biggl(- \frac{ \vert (\mathfrak{x}+ \epsilon ) \vert }{\eta +\zeta}\biggr) \succeq \boldsymbol{ W_{\kappa ,\varsigma}}\biggl(-\frac{ \vert \mathfrak{x} \vert }{\eta}\biggr) \circledast \boldsymbol{ W_{\kappa ,\varsigma}}\biggl(- \frac{ \vert \epsilon \vert }{\zeta}\biggr) . \end{aligned}$$

  Suppose that $\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{|(\mathfrak{x})|}{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )} \leq \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{|\epsilon |}{\zeta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}$, then

  $$ \frac{- \vert (\mathfrak{x}) \vert }{\eta} \leq \frac{- \vert \epsilon \vert }{\zeta}$$

  implies

  $$\begin{aligned} &\frac{ \vert (\mathfrak{x}) \vert }{\eta} \geq \frac{ \vert \epsilon \vert }{\zeta}, \\ &\frac{\zeta \vert \mathfrak{x} \vert }{\eta}\geq \vert \epsilon \vert , \\ &\frac{\zeta \vert (\mathfrak{x}) \vert }{\eta} + \bigl\vert (\mathfrak{x}) \bigr\vert \geq \bigl\vert ( \epsilon ) \bigr\vert + \vert \mathfrak{x} \vert , \end{aligned}$$

  then

  $$\begin{aligned} &\frac{\zeta \vert (\mathfrak{x}) \vert }{\eta} + \bigl\vert (\mathfrak{x}) \bigr\vert \geq \bigl\vert ( \mathfrak{x}) +(\epsilon ) \bigr\vert , \\ &\bigl\vert (\mathfrak{x}) \bigr\vert \biggl(\frac{\zeta}{\eta}+1\biggr) \geq \bigl\vert (\mathfrak{x} ) +( \epsilon ) \bigr\vert , \\ &\bigl\vert (\mathfrak{x}) \bigr\vert \biggl(\frac{\zeta +\eta}{\eta}\biggr)\geq \bigl\vert (\mathfrak{x}) +( \epsilon ) \bigr\vert , \\ &\frac{ \vert (\mathfrak{x}) \vert }{\eta}\geq \frac{ \vert (\mathfrak{x}) +(\epsilon ) \vert }{\zeta +\eta}, \\ &\frac{- \vert (\mathfrak{x}) \vert }{\eta}\leq \frac{- \vert (\mathfrak{x} ) +(\epsilon ) \vert }{\zeta +\eta} \end{aligned}$$

  yields

  $$ \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x} ) +(\epsilon ) \vert }{\zeta +\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )} \geq \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}.$$

  Therefore, we have

  $$\begin{aligned} & \operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) +(\epsilon ) \vert }{\eta +\zeta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) +(\epsilon ) \vert }{\eta +\zeta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{ \vert \alpha \vert (- \vert (\mathfrak{x} ) +(\epsilon ) \vert )}{\eta +\zeta})^{\mathfrak{p}}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )} \Biggr] \\ &\quad {} \succeq\operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}, \sum_{\mathfrak{p}=0}^{\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa ,\varsigma )} \Biggr] \\ &\qquad {}\circledast\operatorname{diag} \Biggl[\sum _{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \epsilon \vert }{\zeta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}, \sum _{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \epsilon \vert }{\zeta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )}, \sum _{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\epsilon ) \vert }{\zeta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa + \varsigma )} \Biggr]. \end{aligned}$$

Consequently, if

$$\begin{aligned} &\mathcal{A}(\epsilon ,\eta ) \\ &\quad =\operatorname{diag} \biggl[W_{\kappa ,\varsigma}\biggl(-\frac{ \vert \epsilon \vert }{\eta} \biggr),W_{ \kappa ,\varsigma}\biggl(-\frac{ \vert \epsilon \vert }{\eta}\biggr), W_{\kappa ,\varsigma} \biggl(- \frac{ \vert \epsilon \vert }{\eta}\biggr) \biggr] \end{aligned}$$

for $\eta \in \mathfrak{F}_{2}$, then $(\mathcal{J},\mathcal{A},\circledast _{M})$ is an MVFN-space. From now on, we assume $\circledast =\circledast _{M}$.

Theorem 2.1

([14])

Consider the $\mathfrak{F}_{4}$-valued metric space $(\mathcal{Z},d)$. For $\mathrm{u}, \mathrm{v} \in \mathcal{Z}$, let Δ be the self mapping on $\mathcal{Z}$ such that

$$ d(\Delta \mathrm{u}, \Delta \mathrm{v}) \leq \lambda d(\mathrm{v}, \mathrm{u}), $$

where $0 < \lambda <1$ is a Lipschitz constant. Let $\mathrm{u} \in \mathcal{Z}$, then we have either

• (i) $d (\Delta ^{\mathbf{q}} \mathrm{u}, \Delta ^{\mathbf{q}+1} \mathrm{u} )=+\infty$ for all $\mathbf{q} \in \mathbb{N}$

or

• (ii) we can find $\mathbf{q}_{0} \in \mathbb{N}$ such that $d (\Delta ^{\mathbf{q}} \mathrm{u}, \Delta ^{\mathbf{q}+1} \mathrm{u} )<+\infty$ for all $\mathbf{q} \geq \mathbf{q}_{0}$.

If condition (ii) holds, then we always have

• (1) the fixed point $\mathrm{v}^{*}$ of Δ is the convergent point of the sequence $\{\Delta ^{\mathbf{q}} \mathrm{u} \}$;

• (2) in the set $\mathcal{Z}^{*}= \{\mathrm{v} \in \mathcal{Z} \mid d ( \Delta ^{\mathbf{q}_{0}} \mathrm{u}, \mathrm{v} )<+\infty \}, \mathrm{v}^{*}$ is the unique fixed point of Δ;

• (3) $(1-\lambda ) \,d (\mathrm{v}, \mathrm{v}^{*} ) \leq d( \mathrm{v}, \Delta \mathrm{v})$ for every $\mathrm{v} \in \mathcal{Z}$.

Definition 2.5

Consider the MVF $\boldsymbol{ W_{\kappa ,\varsigma}}$. We say (1) has Hyers–Ulam–Wright stability when a given differentiable map $\mathrm{v}(\mathfrak{x})$ satisfies

$$\begin{aligned} &\mathcal{A} \bigl(D_{0}^{\kappa}\mathrm{v}( \mathfrak{x})+\rho \mathrm{v}(\mathfrak{x}-\xi )-\mathrm{k}(\mathfrak{x})- \mathcal{L}^{-1} \bigl(\mathrm{v}(\mathfrak{r})\bigr) (\mathfrak{x}),\eta \bigr)\succeq \boldsymbol{ W_{\kappa ,\varsigma}}\biggl(-\frac{ \vert \mathfrak{x} \vert }{\eta}\biggr) \end{aligned}$$

(3)

for $\mathfrak{x}\in \mathfrak{F}_{1}$ and there is a solution $\mathrm{u}(\mathfrak{x})$ of (1) such that, for some $\lambda >0$,

$$\begin{aligned} \mathcal{A} \bigl(\mathrm{u}(\mathfrak{x}) -\mathrm{v}(\mathfrak{x}), \eta \bigr) \succeq\boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{\lambda}} \biggr). \end{aligned}$$

3 Hyers–Ulam–Wright stability for nonhomogeneous fractional delay oscillation equation

Now, we use a new method, the Laplace–Mihet–Radu method, to show (1) is Hyers–Ulam–Wright stable [12] in MVFB-space $(\mathcal{J},\mathcal{A},\circledast )$ with MVFF $\boldsymbol{W_{\kappa ,\varsigma}}$.

We choose the set $\mathcal{Z}$ as follows:

$$ \mathcal{Q}=\bigl\{ \mathrm{u}:\mathfrak{L} \rightarrow \mathbb{R}^{n}, \mathrm{u} \text{ is differentiable}\bigr\} $$

and define the mapping $d:\mathcal{Q}\times \mathcal{Q}\rightarrow [0,+\infty ]$ as

$$\begin{aligned} d(\mathrm{u},\mathrm{v})=\inf \biggl\{ \wp \in {\mathfrak{F}_{6}} : & \mathcal{A} \bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}(\mathfrak{x}), \eta \bigr) \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{\wp}} \biggr), \forall\ \mathrm{u},\mathrm{v}\in \mathcal{Z}, \mathfrak{x} \in \mathfrak{F}_{1}, \eta \in \mathfrak{F}_{2} \biggr\} . \end{aligned}$$

Theorem 3.1

The $\mathfrak{F}_{4}$-valued metric space $(\mathcal{Q}, d)$ is complete.

Proof

We have $d(\mathrm{u},\mathrm{v})=0$ iff $\mathrm{u}=\mathrm{v}$. Assuming that $d(\mathrm{u},\mathrm{v})=0$, then we come to

$$\begin{aligned} \inf \biggl\{ \wp \in {\mathfrak{F}_{6}} :&\mathcal{A} \bigl( \mathrm{u}( \mathfrak{x})-\mathrm{v}(\mathfrak{x}),\eta \bigr)\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{ \wp}} \biggr), \forall \ \mathrm{u}, \mathrm{v} \in \mathcal{Q}, \mathfrak{x} \in \mathfrak{F}_{1}, \eta \in \mathfrak{F}_{2} \biggr\} =0, \end{aligned}$$

and so

$$\begin{aligned} \mathcal{A} \bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}( \mathfrak{x}), \eta \bigr)\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{ \wp}} \biggr) \end{aligned}$$

(4)

for all $\wp \in \mathfrak{F}_{6}$. Let $\wp \to 0$ in (4), we get

$$ \mathcal{A} \bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}(\mathfrak{x}), \eta \bigr)= \boldsymbol{1}.$$

Thus $\mathrm{u}(\mathfrak{x})=\mathrm{v}(\mathfrak{x})$ for every $\mathfrak{x} \in \mathfrak{F}_{1}$ and vice versa. Also we have $d(\mathrm{u},\mathrm{v})=d(\mathrm{v},\mathrm{u})$ for every $\mathrm{u},\mathrm{v}\in \mathcal{Q}$. Assuming $d(\mathrm{u},\mathrm{v})=\alpha _{1}\in{\mathfrak{F}_{2}}$ and $d(\mathrm{v},\mathrm{w})=\alpha _{2}\in{\mathfrak{F}_{2}}$, then we have

$$\begin{aligned} \mathcal{A} \bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}(\mathfrak{x}), \eta \bigr) \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{ \alpha _{1}}} \biggr) \end{aligned}$$

and

$$\begin{aligned} \mathcal{A} \bigl(\mathrm{v}(\mathfrak{x})-\mathrm{w}(\mathfrak{x}), \eta \bigr) \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{ \alpha _{2}}} \biggr) \end{aligned}$$

for every $\eta \in \mathfrak{F}_{2}$. Therefore, we get

$$\begin{aligned} \mathcal{A} (\bigl(\mathrm{u}(\mathfrak{x})-\mathrm{w}(\mathfrak{x}), ( \alpha _{1}+\alpha _{2})\eta \bigr)&\succeq \bigl[ \mathcal{A} ( \bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}(\mathfrak{x}), (\alpha _{1}) \eta \bigr)\circledast \mathcal{A} (\bigl(\mathrm{v}(\mathfrak{x})- \mathrm{w}( \mathfrak{x}), (\alpha _{2})\eta \bigr) \bigr] \\ & \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr) \circledast \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr) \\ &= \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr). \end{aligned}$$

This infers that $d(\mathrm{v},\mathrm{w})\le d(\mathrm{u},\mathrm{v})+d(\mathrm{v}, \mathrm{w})$. To show the completeness of $(\mathcal{Q}, d)$, we suppose that $\{\mathrm{v}_{\mathbf{q}}\}_{k}$ is a Cauchy sequence in $(\mathcal{Q}, d)$. Assume that $\mathfrak{x} \in \mathfrak{F}_{1}$, $\tau \in{\mathfrak{F}_{2}}$, $\Im \in{\mathfrak{F}_{5}}^{\circ}$, and $\eta \in{\mathfrak{F}_{2}}$ in which $\boldsymbol{ W_{\kappa ,\varsigma}} (- \frac{|\mathfrak{x}|}{\eta} ) \succ \boldsymbol{ 1-\Im}$. For $\alpha \eta <\tau$, choose $\mathbf{q}_{0}\in {\mathbb{N}}$ such that

$$ d(\mathrm{v}_{\mathbf{q}},\mathrm{v}_{\mathbf{p}})< \alpha \quad \text{for all } \mathbf{q},\mathbf{p}\ge \mathbf{q}_{0}.$$

Then

$$\begin{aligned} \mathcal{A} \bigl(\mathrm{v}_{\mathbf{q}}(\mathfrak{x})-\mathrm{v}_{ \mathbf{p}}( \mathfrak{x}), \tau \bigr)&\succeq \mathcal{A} \bigl( \mathrm{v}_{\mathbf{q}}( \mathfrak{x})-\mathrm{v}_{\mathbf{p}}( \mathfrak{x}), \alpha \eta \bigr)\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\eta} \biggr)\succ \boldsymbol{1-\Im}, \end{aligned}$$

and so

$$ \mathcal{A} \bigl( \mathrm{v}_{\mathbf{q}}(\mathfrak{x})-\mathrm{v}( \mathfrak{x})_{\mathbf{p}}, \tau \bigr) \succ \boldsymbol{1-\Im},$$

which implies that the sequence $\{ \mathrm{v}_{\mathbf{q}}(\mathfrak{x})\}_{k}$ is Cauchy in the complete space $(\mathcal{J},\mathcal{A},\circledast )$ on a compact set $\mathfrak{F}_{1}$. Then it is uniformly convergent to the mapping $\mathrm{v}:\mathfrak{F}_{1}\to \mathcal{J}$. By the uniform convergence property we conclude that $\mathrm{v}\in \mathcal{Q}$, and then $(\mathcal{Q}, d)$ is complete. □

Now, we can investigate Hyers–Ulam–Wright stability and get an approximation for the nonhomogeneous fractional delay oscillation equation (1).

Theorem 3.2

Consider the MVFB-space $(\mathcal{J},\mathcal{A},\circledast )$ and the constant θ such that $0<\theta <1$. Suppose that the following conditions hold:

▶:

By considering the MVFF as the control function, we have

$$\begin{aligned} & \boldsymbol{ W_{\kappa ,\varsigma}} \biggl( \frac{- \vert \mathcal{L} ((\mathfrak{r}))(\mathfrak{x}) \vert }{\eta} \biggr){ \succeq} \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{ \theta}} \biggr). \end{aligned}$$

(5)

Let $\mathrm{v}:\mathfrak{F}_{1}\rightarrow \mathcal{J}$ be a differentiable function satisfying

$$\begin{aligned} &\mathcal{A} \bigl(D_{0}^{\kappa} \mathrm{v}( \mathfrak{x})+\rho \mathrm{v}(\mathfrak{x}-\xi )-\mathfrak{k}(\mathfrak{x})- \mathcal{L}^{-1} \bigl(\mathrm{v}(\mathfrak{r})\bigr) (\mathfrak{x}), \eta \bigr) \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{\varpi}} \biggr). \end{aligned}$$

(6)

Then we can find a unique solution $\mathrm{u}:\mathfrak{F}_{1}\rightarrow \mathcal{J}$ for equation (1) such that

$$\begin{aligned} &\mathcal{A} \bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}( \mathfrak{x}), \eta \bigr) \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta }{\digamma \varpi}} \biggr), \end{aligned}$$

where $\digamma =\frac{\theta}{1-\theta}$, and for every $\mathfrak{x} \in \mathfrak{F}_{1}$ and $\eta \in{\mathfrak{F}_{2}}$.

Proof

Consider complete ${\mathfrak{F}_{4}}$-valued metric space $(\mathcal{Q}, d)$ defined in Theorem 3.1.

Step 1. According to the main equation, we define the mapping $\Delta : \mathcal{Q}\rightarrow \mathcal{Q}$ as follows:

$$\begin{aligned} \Delta \bigl(\mathrm{u}(\mathfrak{x})\bigr)&= \mathcal{L} \bigl(D_{0}^{\kappa} \mathrm{u}(\mathfrak{x})+\rho \mathrm{u}(\mathfrak{x}-\xi )- \mathrm{k}( \mathfrak{x})\bigr) (\mathfrak{r})= \bigl(\mathfrak{r}^{\kappa}+\rho e^{- \mathfrak{r}\xi}\bigr)\mathcal{L}\mathrm{u}(\mathfrak{x})) (\mathfrak{r})- \mathfrak{r}^{\kappa -1} \Upsilon (0) \\ &\quad{}-\mathfrak{r}^{\kappa -2}\Upsilon ^{\prime}(0)+\rho e^{-\mathfrak{r} \xi} \int _{-\xi}^{0} e^{-\mathfrak{r}\xi}\Upsilon ( \mathfrak{x}) \,\mathrm{d}\mathfrak{x} - \mathcal{L}\bigl(\mathrm{k}(\mathfrak{x}) \bigr) ( \mathfrak{r}) \end{aligned}$$

(7)

for $\mathfrak{x} \in \mathfrak{F}_{1}$, and we show that Δ is a strictly contractive mapping.

Let $\mathrm{u},\mathrm{v}\in \mathcal{Q}$ and consider the coefficient $B_{\mathrm{u}\mathrm{v}}\in \mathfrak{F}_{4}$ with $d(\mathrm{u},\mathrm{v})\leq {B}_{\mathrm{u}\mathrm{v}}$, thus

$$\begin{aligned} &\mathcal{A} \bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}(\mathfrak{x}),B_{ \mathrm{u}\mathrm{v}} \eta \bigr) \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl(-\frac{ \vert \mathfrak{x} \vert }{\eta} \biggr) \end{aligned}$$

for all $\mathrm{u},\mathrm{v} \in \mathcal{Q}$, $\mathfrak{x} \in \mathfrak{F}_{1}$, and $\eta \in {\mathfrak{F}_{2}}$. Applying (N2) and (N3), we have

$$\begin{aligned} \mathcal{A} \bigl(\Delta \mathrm{u}(\mathfrak{x})-\Delta \mathrm{v}( \mathfrak{x}), {B}_{\mathrm{u}\mathrm{v}} \eta \bigr) &=\mathcal{A} ( \bigl( \mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}\bigr)\mathcal{L}\biggl( \mathrm{u}(\mathfrak{x}) (\mathfrak{r})-\mathfrak{r}^{\kappa -1} \Upsilon (0) \\ &\quad{}-\mathfrak{r}^{\kappa -2}\Upsilon ^{\prime}(0)+\rho e^{-\mathfrak{r} \xi} \int _{-\xi}^{0} e^{-\mathfrak{r}\xi}\Upsilon ( \mathfrak{x}) \,\mathrm{d}\mathfrak{x} - \mathcal{L}\bigl(\mathrm{k}(\mathfrak{x}) \bigr) ( \mathfrak{r}) \\ &\quad{}- \bigl(\mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}\bigr) \mathcal{L}\bigl( \mathrm{v}(\mathfrak{x})\bigr) (\mathfrak{r})+ \mathfrak{r}^{\kappa -1} \Upsilon (0) \\ &\quad{}+\mathfrak{r}^{\kappa -2}\Upsilon ^{\prime}(0)_{\rho }e^{- \mathfrak{r}\xi} \int _{-\xi}^{0} e^{-\mathfrak{r}\xi}\Upsilon ( \mathfrak{x})\,\mathrm{d}\mathfrak{x} + \mathcal{L}\bigl(\mathrm{k}( \mathfrak{x}) \bigr) (\mathfrak{r}), \eta \biggr) \\ &=\mathcal{A} \bigl(\bigl(\mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}\bigr) \mathcal{L}\bigl(\mathrm{u}(\mathfrak{x})\bigr) ( \mathfrak{r})- \bigl(\mathfrak{r}^{ \kappa}+\rho e^{-\mathfrak{r}\xi}\bigr) \mathcal{L}\bigl(\mathrm{v}( \mathfrak{x})\bigr) (\mathfrak{r}), \eta \bigr) \\ &=\mathcal{A} \bigl(\bigl(\mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}\bigr) \mathcal{L}\bigl(\mathrm{u}(\mathfrak{x})-\mathrm{v}( \mathfrak{x})\bigr) ( \mathfrak{r}), \eta \bigr) \\ &=\mathcal{A} \biggl( \mathcal{L}\bigl( \bigl\vert \mathrm{u}( \mathfrak{x})-\mathrm{v}( \mathfrak{x}) \bigr\vert \bigr) (\mathfrak{r}), \frac{\eta}{ \vert (\mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}) \vert } \biggr) \\ &\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl( \frac{-\mathcal{L}(\mathfrak{x})(\mathfrak{r})|}{\frac{\eta}{{B}_{\mathrm{u}\mathrm{v}} \vert (\mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}) \vert }} \biggr) \\ &\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl( \frac{- \vert \mathfrak{x} \vert }{\frac{\eta}{{B}_{\mathrm{u}\mathrm{v}}\theta \vert (\mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}) \vert }} \biggr) \\ &\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl( \frac{- \vert \mathfrak{x} \vert }{\frac{\eta}{{B}_{\mathrm{u}\mathrm{v}}\theta}} \biggr), \end{aligned}$$

(8)

which implies that

$$ d(\Delta \mathrm{u},\Delta \mathrm{v}) \leq \theta {B}_{\mathrm{u} \mathrm{v}},$$

and so

$$ d(\Delta \mathrm{u},\Delta \mathrm{v})\leq \theta d(\mathrm{u}, \mathrm{v}),$$

where $0< \theta <1$, therefore Δ is a contractive mapping.

Step 2. We will show that $d(\Delta (\mathrm{v}),\mathrm{v})<+\infty$.

Let $\mathrm{v}\in \mathfrak{L}$, we have

$$\begin{aligned} &\mathcal{A} \bigl(\Delta \bigl(\mathrm{v}(\mathfrak{x})\bigr)- \mathrm{v}( \mathfrak{x}),\eta \bigr) \\ &\quad=\mathcal{A} (\mathcal{L}\bigl( D_{0}^{\kappa} \mathrm{v}( \mathfrak{x})+\rho \mathrm{v}(\mathfrak{x}-\xi )-\mathrm{k}( \mathfrak{x})-\mathcal{L}^{-1}\bigl(\mathrm{v}(\mathfrak{r})\bigr) ( \mathfrak{x}), \eta \bigr) \\ &\quad=\mathcal{A} \biggl( \bigl(\mathfrak{r}^{\kappa}+\rho e^{-\mathfrak{r}\xi}\bigr) \mathcal{L}\bigl(\mathrm{v}(\mathfrak{x})\bigr) ( \mathfrak{r})-\mathfrak{r}^{ \kappa -1} \Upsilon (0) -\mathfrak{r}^{\kappa -2} \Upsilon ^{\prime}(0)\\ & \qquad {}+ \rho e^{-\mathfrak{r}\xi} \int _{-\xi}^{0} e^{-\mathfrak{r}\xi} \Upsilon ( \mathfrak{x})\,\mathrm{d}\mathfrak{x} - \mathcal{L}\bigl( \mathrm{k}(\mathfrak{x}) \bigr) (\mathfrak{r}) -\mathrm{v}(\mathfrak{x}), \eta \biggr) \\ &\quad\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl( - \frac{\mathcal{L}( \vert \mathfrak{r} \vert )(\mathfrak{x})}{\frac{\eta}{\varpi}} \biggr) \\ &\quad\succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl( - \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{\varpi \theta}} \biggr). \end{aligned}$$

(9)

Consequently,

$$\begin{aligned} d(\Delta \mathrm{v},\mathrm{v}) \leq \varpi \theta < {+\infty }, \quad \theta < 1, \end{aligned}$$

(10)

for every $\eta \in{\mathfrak{F}_{2}}$. Then we have $d(\Delta \mathrm{v},\mathrm{v})<{+\infty}$.

Therefore, all the conditions in (ii) of Theorem 2.1 hold. Then we have

• (1) The sequence $\{\Delta ^{\mathbf{q}} \mathrm{v}\}$ converges to a fixed point such as v.

• (2) The unique element v is in the set $\mathcal{Q}^{\ast}=\{ \mathrm{v}\in \mathcal{Q}: d(\Delta \mathrm{v}, \mathrm{v})<+\infty \}$ and is the unique fixed point of Δ, it means that $\Delta \mathrm{v}=\mathrm{v}$ or equivalently

  $$\begin{aligned} \mathrm{v}(\mathfrak{x})&= \bigl(\mathfrak{r}^{\kappa}+\rho e^{- \mathfrak{r}\xi}\bigr)\mathcal{L}\bigl(\mathrm{u}(\mathfrak{x})\bigr) ( \mathfrak{r})- \mathfrak{r}^{\kappa -1} \Upsilon (0) \\ &\quad{}-\mathfrak{r}^{\kappa -2}\Upsilon ^{\prime}(0)+\rho e^{-\mathfrak{r} \xi} \int _{-\xi}^{0} e^{-\mathfrak{r}\xi}\Upsilon ( \mathfrak{x}) \,\mathrm{d}\mathfrak{x} - \mathcal{L}\bigl(\mathrm{k}(\mathfrak{x}) \bigr) ( \mathfrak{r}). \end{aligned}$$

  (11)

Since u is a differentiable function, by the NH-FDO-E and according to equation (11), we have

(12)

• (3) Using inequality (10), we get

  $$\begin{aligned} d(\mathrm{u},\mathrm{v}) \leq \frac{1}{1-\theta}d(\Delta \mathrm{v}, \mathrm{v}) \leq \frac{\varpi \theta}{1-\theta}, \end{aligned}$$

  thus, equation (1) has the Hyers–Ulam–Wright stability property.

Now, we show the uniqueness of the obtained point. For convenience, we consider

and let h be another differentiable function satisfying equation (12), and this means that the following equation holds:

(13)

We are ready to prove that h is a fixed point of Δ and $\mathrm{h}\in \mathcal{Q}^{\ast}$. Using equation (13), we get $\Delta \mathrm{h}= \mathrm{h}$. Now, we show that $d(\Delta \mathrm{v}, \mathrm{h})<+\infty$. Let $\mathrm{v}\in \mathcal{Q}$, , and using equation (13), we get

then

 □

4 Example

Now, we provide a numerical example to demonstrate the main results obtained.

Example 4.1

Consider the following the NH-FDO-E:

$$\begin{aligned} &D_{0}^{\frac{1}{2}}\mathrm{u}(\mathfrak{x})=-\rho \mathrm{u}\biggl( \mathfrak{x}-\frac{1}{7}\biggr)+ \frac{(\mathfrak{x}-1)^{-\frac{1}{4}} \sin (\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+ \vert \mathfrak{x} \vert )}+ \mathcal{L}^{-1}\bigl( \mathrm{u}(\mathfrak{r})\bigr) (\mathfrak{x}), \end{aligned}$$

(14)

$$\begin{aligned} & \mathrm{u}(\mathfrak{x})=\bigl[\mathrm{u}_{1}(\mathfrak{x}), \mathrm{u}_{2}( \mathfrak{x})\bigr]^{T}, \qquad \Upsilon ( \mathfrak{x})=\bigl[3 \mathfrak{x}, 4 \mathfrak{x}^{2} \bigr]^{T}, \quad -\xi \leq \mathfrak{x} \leq 0, \end{aligned}$$

(15)

where $\kappa =\frac{1}{2}$, $\xi =\frac{1}{7}$, and $\rho =\left[\begin{array}{cc}\frac{3}{5}& 0\\ 0& \frac{2}{5}\end{array}\right]$. Assume that the following condition is true for the given continuous function:

▶:

$$\begin{aligned} & \boldsymbol{ W_{\frac{1}{2},\varsigma}} \biggl( - \frac{ \vert \mathcal{L} ((s))(\mathfrak{x}) \vert }{\eta} \biggr)\succeq \boldsymbol{ W_{\frac{1}{2},\varsigma}} \biggl(- \frac{ \vert \mathfrak{x} \vert }{4\eta} \biggr). \end{aligned}$$

Let be a differentiable function such that

$$\begin{aligned} &{{\mathcal{A}} { \biggl( D_{0}^{\frac{1}{2}} \mathrm{v}({\mathfrak{x}})+ \rho \mathrm{v}\biggl({\mathfrak{x}}- \frac{1}{7}\biggr)- \frac{({\mathfrak{x}}-1)^{-\frac{1}{4}} \sin ({\mathfrak{x}}-1)}{64(1+\sqrt{{\mathfrak{x}}-1})(1+ \vert {\mathfrak{x}} \vert )}-{ \mathcal{L}}^{-1}\bigl({ \mathrm{u}}(s)\bigr) ({\mathfrak{x}}), \eta \biggr)}} \end{aligned}$$

(16)

$$\begin{aligned} &\quad =\operatorname{diag} \biggl[ {{W}_{\frac{1}{2},\varsigma}} \biggl(- \frac {- \vert D_{0}^{\frac{1}{2}} \mathrm{v}(\mathfrak{x})+\rho \mathrm{v}(\mathfrak{x}-\frac{1}{7})-\frac{(\mathfrak{x}-1)^{-\frac{1}{4}} \sin (\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+ \vert \mathfrak{x} \vert )}-\mathcal{L}^{-1}(\mathrm{u}(\mathfrak{r}))(\mathfrak{x}) \vert }{\eta} \biggr), \\ &\qquad W_{\frac{1}{2},\varsigma} \biggl(- \frac {- \vert D_{0}^{\frac{1}{2}} \mathrm{v}(\mathfrak{x})+\rho \mathrm{v}(\mathfrak{x}-\frac{1}{7})-\frac{(\mathfrak{x}-1)^{-\frac{1}{4}} \sin (\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+ \vert \mathfrak{x} \vert )}-\mathcal{L}^{-1}(\mathrm{v}(\mathfrak{r}))(\mathfrak{x}) \vert }{\eta} \biggr) \biggr] \\ &\quad \succeq W_{\frac{1}{2},\varsigma} \biggl(- \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{\varpi}} \biggr) , \end{aligned}$$

(17)

then v is a solution of the inequality

$$\begin{array}{rl}& \mathcal{A}(\mathrm{v}(\mathfrak{x})-({\mathfrak{r}}^{\kappa }+\rho e^{-\mathfrak{r}\xi })\mathcal{L}(\mathrm{v}(\mathfrak{x}))(\mathfrak{r})-{\mathfrak{r}}^{\kappa -1}\mathrm{\Upsilon }(0)\\ & -s^{\kappa -2}{\mathrm{\Upsilon }}^{\prime }(0)+\rho e^{-\mathfrak{r}\xi }{\int }_{-\xi }^0{e}^{-\mathfrak{r}\xi }\mathrm{\Upsilon }(\mathfrak{x})\mathrm{d}\mathfrak{x}-\mathcal{L}\left(\frac{{(\mathfrak{x}-1)}^{-\frac{1}{4}}sin(\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+|\mathfrak{x}|)}\right)(\mathfrak{r}),\eta )\\ & =diag\left[W_{\frac{1}{2},\varsigma }\right(-\left(\right|\mathrm{v}(\mathfrak{x})-({\mathfrak{r}}^{\kappa }+\rho e^{-\mathfrak{r}\xi })\mathcal{L}(\mathrm{v}(\mathfrak{x}))(\mathfrak{r})-{\mathfrak{r}}^{\kappa -1}\mathrm{\Upsilon }(0)-{\mathfrak{r}}^{\kappa -2}{\mathrm{\Upsilon }}^{\prime }(0)\\ & +\eta e^{-\mathfrak{r}\xi }{\int }_{-\xi }^0{e}^{-\mathfrak{r}\xi }\mathrm{\Upsilon }(\mathfrak{x})\mathrm{d}\mathfrak{x}-\pounds \left(\frac{{(\mathfrak{x}-1)}^{-\frac{1}{4}}sin(\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+|\mathfrak{x}|)}\right)(\mathfrak{r})\left|\right)/\eta ),\\ & W_{\frac{1}{2},\varsigma }(-(|\mathrm{v}(\mathfrak{x})-({\mathfrak{r}}^{\kappa }+\rho e^{-\mathfrak{r}\xi })\mathcal{L}(\mathrm{v}(\mathfrak{x}))(\mathfrak{r})-{\mathfrak{r}}^{\kappa -1}\mathrm{\Upsilon }(0)-{\mathfrak{r}}^{\kappa -2}{\mathrm{\Upsilon }}^{\prime }(0)\\ & +\rho e^{-\mathfrak{r}\xi }{\int }_{-\xi }^0{e}^{-\mathfrak{r}\xi }\mathrm{\Upsilon }(\mathfrak{x})\mathrm{d}\mathfrak{x}-\mathcal{L}\left(\frac{{(\mathfrak{x}-1)}^{-\frac{1}{4}}sin(\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+|\mathfrak{x}|)}\right)(\mathfrak{r})\left|\right)/\eta \left)\right]\\ & ⪰{\mathit{W}}_{\frac{\mathbf{1}}{\mathbf{2}}\mathbf{,}\mathit{\varsigma }}(-\frac{|\mathcal{L}(\mathfrak{r})(\mathfrak{x})|}{\frac{\eta }{\varpi }})\\ & ⪰{\mathit{W}}_{\mathit{\kappa }\mathbf{,}\mathit{\varsigma }}(-\frac{|\mathfrak{x}|}{\frac{4\eta }{\varpi }}),\end{array}$$

where $\Upsilon (\mathfrak{x})=[3 \mathfrak{x}, 4 \mathfrak{x}^{2}]^{T}$. Thus we can find a unique differentiable function from (14) such that for each $\varrho \in [1,2]$ we have

$$\begin{aligned} \Delta \bigl(\mathrm{u}(\mathfrak{x})\bigr)&= \bigl(\mathfrak{r}^{\kappa}+ \rho e^{- \mathfrak{r}\xi}\bigr)\mathcal{L}\bigl(\mathrm{u}(\mathfrak{x})\bigr) ( \mathfrak{r})- \mathfrak{r}^{\kappa -1} \Upsilon (0) \\ &\quad{}-\mathfrak{r}^{\kappa -2}\Upsilon ^{\prime}(0)+\rho e^{-\mathfrak{r} \xi} \int _{-\xi}^{0} e^{-\mathfrak{r}\xi}\Upsilon ( \mathfrak{x}) \,\mathrm{d}\mathfrak{x} - \mathcal{L}\biggl( \frac{(\mathfrak{x}-1)^{-\frac{1}{4}} \sin (\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+ \vert \mathfrak{x} \vert )} \biggr) ( \mathfrak{r}). \end{aligned}$$

Therefore,

$$\begin{aligned} d(\mathrm{u},\mathrm{v}) \leq \digamma \varpi \end{aligned}$$

and

$$\begin{aligned} & \mathcal{A} \bigl( \mathrm{v}(\mathfrak{x})-\mathrm{u}(\mathfrak{x}), \eta \bigr)\\ &\quad \succeq \operatorname{diag} \biggl[W_{\kappa ,\varsigma}\biggl(- \frac{ \vert \mathrm{v}(\mathfrak{x})-\mathrm{u}(\mathfrak{x}) \vert }{\eta}\biggr), W_{ \kappa ,\varsigma}\biggl(- \frac{ \vert \mathrm{v}(\mathfrak{x})-\mathrm{u}(\mathfrak{x}) \vert }{\eta}\biggr), W_{ \kappa ,\varsigma}\biggl(- \frac{ \vert \mathrm{v}(\mathfrak{x})-\mathrm{u}(\mathfrak{x}) \vert }{\eta}\biggr) \biggr] \\ &\quad \succeq \boldsymbol{ W_{\kappa ,\varsigma}} \biggl( - \frac{ \vert \mathfrak{x} \vert }{\frac{\eta}{\digamma \varpi}} \biggr), \end{aligned}$$

where in $\digamma =\frac{1}{3}$. See Fig. 1.

Figure 1

Graphs related to the exact solution of the NH-FDO-E (14) for $\xi =\frac{1}{7}$, $\kappa =\frac{1}{2}$