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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


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 [13] 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 [711]. 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:

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


  • \(\mathrm{k}: \mathfrak{L}\times \mathbb{R}^{n} \rightarrow \mathbb{R}^{n}\) 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


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


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. (1)

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

  2. (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. (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. (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. (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 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:

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


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

(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,

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


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

(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

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


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

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

  1. (N1)

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

  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 \(\varkappa \neq 0\);

  3. (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 \(\eta ,\mathfrak{z} \in \mathfrak{F}_{2}\);

  4. (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

  1. (a)

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

  2. (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

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


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

    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

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


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

    We shall deduce

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


    $$\begin{aligned} &\boldsymbol{ W_{\kappa ,\varsigma}}\biggl(-\frac{ \vert \mathfrak{x}) \vert }{\eta}\biggr) \\ &\quad =\operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )} \Biggr] \\ &\quad =\operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha \vert \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha \vert \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert \alpha \vert \vert (\mathfrak{x}) \vert }{\eta}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )} \Biggr] \\ &\quad =\operatorname{diag} \Biggl[\sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }})^{\mathfrak{p}}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }})^{\mathfrak{p}}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\tau +\omega )}, \sum_{\mathfrak{p}=0}^{+\infty} \frac{({\frac{- \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }}})^{\mathfrak{p}}}{\mathfrak{p}! \Gamma (\mathfrak{p}\kappa +\varsigma )} \Biggr] \\ &\quad=\boldsymbol{ W_{\kappa ,\varsigma}}\biggl(- \frac{ \vert (\mathfrak{x}) \vert }{\frac{\eta}{ \vert \alpha \vert }}\biggr). \end{aligned}$$
  4. (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

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


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


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


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

    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


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

  1. (i)

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


  1. (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. (1)

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

  2. (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. (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

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

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.


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

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

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

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


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

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

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


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

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

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

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

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

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 \(\eta \in{\mathfrak{F}_{2}}\).


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:

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

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}$$

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

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


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

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. (1)

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

  2. (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

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

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

  1. (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:


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



4 Example

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

Example 4.1

Consider the following the NH-FDO-E:

$$\begin{aligned} &D_{0}^{\frac{1}{2}}\mathrm{u}(\mathfrak{x})=-\rho \mathrm{u}\biggl( \mathfrak{x}-\frac{1}{7}\biggr)+ \frac{(\mathfrak{x}-1)^{-\frac{1}{4}} \sin (\mathfrak{x}-1)}{64(1+\sqrt{\mathfrak{x}-1})(1+ \vert \mathfrak{x} \vert )}+ \mathcal{L}^{-1}\bigl( \mathrm{u}(\mathfrak{r})\bigr) (\mathfrak{x}), \end{aligned}$$
$$\begin{aligned} & \mathrm{u}(\mathfrak{x})=\bigl[\mathrm{u}_{1}(\mathfrak{x}), \mathrm{u}_{2}( \mathfrak{x})\bigr]^{T}, \qquad \Upsilon ( \mathfrak{x})=\bigl[3 \mathfrak{x}, 4 \mathfrak{x}^{2} \bigr]^{T}, \quad -\xi \leq \mathfrak{x} \leq 0, \end{aligned}$$

where \(\kappa =\frac{1}{2}\), \(\xi =\frac{1}{7}\), and ρ= [ 3 5 0 0 2 5 ] . 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

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

then v is a solution of the inequality

A ( v ( x ) ( r κ + ρ e r ξ ) L ( v ( x ) ) ( r ) r κ 1 ϒ ( 0 ) s κ 2 ϒ ( 0 ) + ρ e r ξ ξ 0 e r ξ ϒ ( x ) d x L ( ( x 1 ) 1 4 sin ( x 1 ) 64 ( 1 + x 1 ) ( 1 + | x | ) ) ( r ) , η ) = diag [ W 1 2 , ς ( ( | v ( x ) ( r κ + ρ e r ξ ) L ( v ( x ) ) ( r ) r κ 1 ϒ ( 0 ) r κ 2 ϒ ( 0 ) + η e r ξ ξ 0 e r ξ ϒ ( x ) d x £ ( ( x 1 ) 1 4 sin ( x 1 ) 64 ( 1 + x 1 ) ( 1 + | x | ) ) ( r ) | ) / η ) , W 1 2 , ς ( ( | v ( x ) ( r κ + ρ e r ξ ) L ( v ( x ) ) ( r ) r κ 1 ϒ ( 0 ) r κ 2 ϒ ( 0 ) + ρ e r ξ ξ 0 e r ξ ϒ ( x ) d x L ( ( x 1 ) 1 4 sin ( x 1 ) 64 ( 1 + x 1 ) ( 1 + | x | ) ) ( r ) | ) / η ) ] W 1 2 , ς ( | L ( r ) ( x ) | η ϖ ) W κ , ς ( | x | 4 η ϖ ) ,

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

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


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


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

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

Figure 1
figure 1

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

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The authors are thankful to the area editor and referees for giving valuable comments and suggestions.


Chenkuan Li is supported by NSERC Discovery Grant number 2019-03907.

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Eidinejad, Z., Saadati, R. & Li, C. 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. J Inequal Appl 2022, 129 (2022).

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