$C^{*}$-Algebra valued fuzzy normed spaces with application of Hyers–Ulam stability of a random integral equation

Abstract

In this paper, we consider $C^{*}$-algebra valued fuzzy normed spaces. We study the random integral equation $(\frac{1}{2c})\int _{x-cd}^{x+cd}u(\gamma ,\tau ,d_{0})\,d\tau =u( \gamma ,x,d)$ which is related to the stochastic wave equation. In addition, using a $C^{*}$-algebra valued fuzzy controller function, we consider its $C^{*}$-algebra valued fuzzy Hyers–Ulam stability.

1 Introduction

The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms. In 1941, Hyers [2] gave the first affirmative answer to the question of Ulam for additive groups in Banach spaces. Hyers’ theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. A generalization of the Rassias theorem was obtained by Găvruta [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias’ approach. The stability problems for several functional equations or inequalities have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [6–11]).

Let A be a $C^{*}$-algebra and x be a self-adjoint element in A. Then if x is of the form $yy^{*}$ for some $y \in A$, then x is called a positive element. Denote by $A^{+}$ the cone of positive elements of A. We will denote $z \preceq w$ when $w-z \in A^{+}$ (see [12]).

Using random normed spaces introduced by S̆erstnev [13] and studied by Mus̆tari [14] and Radu [15], Cheng and Mordeson [16] defined fuzzy normed spaces.

In this paper, we generalize a recent paper of Saadati [17] using $C^{*}$-algebra valued fuzzy sets and applying t-norms on $C^{*}$-algebras (see [18, 19]).

2 $C^{*}$-Algebra valued fuzzy normed spaces

In this section, we discuss $C^{*}$-algebra. For more details, we refer the reader to [20–22].

Definition 1

Let $\mathcal{A}$ be an order commutative $C^{*}$-algebra and $\mathcal{A}^{+}$ be the positive section of $\mathcal{A}$. Let $U\neq \emptyset$. A $C^{*}$-algebra valued fuzzy set $\mathcal{C}$ on U is a function $\mathcal{C}:U \longrightarrow \mathcal{A}^{+}$. For each u in U, $\mathcal{C}(u)$ represents the degree (in $\mathcal{A}^{+}$) to which u satisfies $\mathcal{A}^{+}$.

We put $\mathbf{0}= \inf \mathcal{A}^{+}$ and $\mathbf{1}= \sup \mathcal{A}^{+}$. Now, we define the triangular norm (t-norm) on $\mathcal{A}^{+}$.

Definition 2

A function $\mathcal{T} : \mathcal{A}^{+}\times \mathcal{A}^{+} \to \mathcal{A}^{+}$ which satisfies

• (i) $(\forall u\in \mathcal{A}^{+})\ (\mathcal{T}(u,\mathbf{1}) = u)$; (boundary condition)

• (ii) $(\forall (u,v) \in \mathcal{A}^{+}\times \mathcal{A}^{+})\ ( \mathcal{T}(u,v) = \mathcal{T}(v,u))$; (commutativity)

• (iii) $(\forall (u,v,w) \in \mathcal{A}^{+}\times \mathcal{A}^{+}\times \mathcal{A}^{+})\ (\mathcal{T}(u, \mathcal{T}(v,w)) = \mathcal{T}( \mathcal{T}(u,v), w))$; (associativity)

• (iv) $(\forall (u,u',v,v') \in \mathcal{A}^{+}\times \mathcal{A}^{+} \times \mathcal{A}^{+}\times \mathcal{A}^{+})\ (u \preceq u'\text{ and } v \preceq v' \Rightarrow \mathcal{T}(u,v) \preceq \mathcal{T}(u',v'))$, (monotonicity)

is called a t-norm.

If, for every $u, v \in \mathcal{A}^{+}$ and sequences $\{u_{n}\}$ and $\{v_{n}\}$ converging to u and v, we have

$$ \lim_{n}\mathcal{T}(u_{n},v_{n})= \mathcal{T}(u,v), $$

then we say $\mathcal{T}$ on $\mathcal{A}^{+}$ is continuous (in short, a ct-norm).

Definition 3

Assume that $\mathcal{F}: \mathcal{A}^{+} \to \mathcal{A}^{+}$ satisfies $\mathcal{F}(\mathbf{0}) = \mathbf{1}$ and $\mathcal{F}(\mathbf{1}) = \mathbf{0}$ and is decreasing. Then $\mathcal{F}$ is called a negation on $\mathcal{A}^{+}$.

Example 4

Let

$$ \operatorname{diag} M_{n}\bigl([0,1]\bigr)= \left\{ \begin{bmatrix} u_{1} & & \\ & \ddots & \\ & & u_{n} \end{bmatrix} =\operatorname{diag}[u_{1}, \ldots,u_{n}], u_{1},\ldots,u_{n}\in [0,1] \right\} . $$

We say $\operatorname{diag}[u_{1},\ldots,u_{n}]\preceq \operatorname{diag}[b_{1},\ldots,b_{n}]$ if and only if $a_{i}\leq b_{i}$ for all $i=1,\ldots,n$ and also ${\mathbf{1}}=\operatorname{diag}[1,\ldots,1]$ and ${\mathbf{0}}=\operatorname{diag}[0,\ldots,0]$. Now, we see that if $\mathcal{A}=\operatorname{diag} M_{n}([0,1])$, then $\operatorname{diag} M_{n}([0,1])=\mathcal{A}^{+}$. Let $\mathcal{T}_{P} : \operatorname{diag} M_{n}([0,1])\times \operatorname{diag} M_{n}([0,1]) \to \operatorname{diag} M_{n}([0,1])$ be

$$ {\mathcal{T}}_{P}\bigl(\operatorname{diag}[u_{1}, \ldots,u_{n}], \operatorname{diag}[v_{1}, \ldots,v_{n}]\bigr)=\operatorname{diag}[u_{1} \cdot v_{1},\ldots,u_{n}\cdot v_{n}]. $$

Then $\mathcal{T}_{P}$ is a t-norm (product t-norm). Note that this t-norm is continuous.

Example 5

Let $\operatorname{diag} M_{n}([0,1])=\mathcal{A}^{+}$ and $\mathcal{T}_{M} : \operatorname{diag} M_{n}([0,1])\times \operatorname{diag} M_{n}([0,1]) \to \operatorname{diag} M_{n}([0,1])$ be

$$ {\mathcal{T}}_{M}\bigl(\operatorname{diag}[u_{1}, \ldots,u_{n}], \operatorname{diag}[v_{1}, \ldots,v_{n}]\bigr)=\operatorname{diag}\bigl[ \min (u_{1},v_{1}), \ldots,\min (u_{n},v_{n})\bigr]. $$

Then $\mathcal{T}_{M}$ is a t-norm (minimum t-norm). Note that this t-norm is continuous.

Definition 6

The triple $(S,\eta ,\mathcal{T})$ is called a $C^{*}$-algebra valued fuzzy normed space (in short, $C^{*}$AVFN-space) if $S\neq \emptyset$, $\mathcal{T}$ is a ct-norm on $\mathcal{A}^{+}$ and η is a $C^{*}$-algebra valued fuzzy set on $S^{2} \times \mathopen]0,+\infty \mathclose[$ such that, for each $t,s,p\in T$ and τ, ς in $\mathopen]0,+\infty \mathclose[$, we have

• (a) $\eta (s,\tau ) \succ {\mathbf{0}}$;

• (b) $\eta (s,\tau ) = {\mathbf{1}}$ for all $\tau > 0$ if and only if $s = 0$;

• (c) $\eta (a s,\tau ) = \eta (s,\frac{\tau }{ \vert a \vert } )$ for all $s\in S$ and $a \in {\mathbb{R}}$ with $a \neq 0$;

• (d) $\eta (t+s,\tau +\varsigma ) \succeq \mathcal{T}(\eta (t,\tau ), \eta (s,\varsigma ))$ for all $t,s\in S$ and $\tau ,\varsigma \geq 0$;

• (e) $\eta (s,\cdot ):(0,\infty )\to \mathcal{A}^{+} \setminus \{{\mathbf{0}}\}$ is left continuous;

• (f) $\lim_{t\longrightarrow \infty }\eta (s,\tau )=\mathbf{1}$.

Also, η is a $C^{*}$-algebra valued fuzzy norm.

Let $(S,\eta ,\mathcal{T})$ be a $C^{*}$AVFN-space. For $\tau >0$, define the open ball$B(t,\varrho ,\tau )$ as

$$ B(s,\varrho ,\tau ) = \bigl\{ t \in S : \eta (t-s,\tau ) \succeq \mathcal{F}( \varrho )\bigr\} , $$

in which $s \in S$ is the center and $\varrho \in \mathcal{A}^{+} \setminus \{{\mathbf{0}}, {\mathbf{1}}\}$ is the radius. We say that $A \subseteq S$ is open if, for each $s \in A$, there exist $\tau > 0$ and $\varrho \in \mathcal{A}^{+} \setminus \{{\mathbf{0}}, {\mathbf{1}}\}$ such that $B(s,\varrho ,\tau ) \subseteq A$. We denote the family of all open subsets of S by $\tau _{\eta }$, and so $\tau _{\eta }$ is the ${C}^{*}$-fuzzy topology induced by the$C^{*}$-algebra valued fuzzy normη.

Example 7

Consider the linear normed space $(S, \Vert \cdot \Vert )$. Let $\mathcal{T}=\mathcal{T}_{M}$ and define the fuzzy set η on $S^{2}\times (0,\infty )$ as follows:

$$\begin{aligned} \eta (s,\tau ) =\operatorname{diag} \biggl[\frac{ \tau }{ \tau + \Vert s \Vert },\exp \biggl(- \frac{ \Vert s \Vert }{\tau } \biggr) \biggr] \end{aligned}$$

for all $\tau \in \mathbf{R}^{+}$. Then $(S,\eta ,\mathcal{T}_{M})$ is a $C^{*}$AVFN-space.

Lemma 8

([23])

Let$(S,\eta ,\mathcal{T})$be a$C^{*}$AVFN-space. Then$\eta (s,\tau )$is nondecreasing with respect toτfor all$s\in S$.

Definition 9

Let $\{s_{n}\}_{n \in \mathbf{{N}}}$ be a sequence in a $C^{*}$AVFN-space $(S,\eta ,\mathcal{T})$. If, for all $\varepsilon \in \mathcal{A}^{+} \setminus \{{\mathbf{0}}\}$ and $\tau > 0$, there exists $n_{0} \in {\mathbf{N}}$ such that, for all $m\geq n \ge n_{0}$,

$$ \eta (s_{m}-s_{n},\tau ) \succeq \mathcal{F}(\varepsilon ), $$

then $\{s_{n}\}_{n \in \mathbf{{N}}}$ is said to be Cauchy.

Also $\{s_{n}\}_{n \in \mathbf{{N}}}$ is said to be convergent to $s \in S$ ($s_{n} \stackrel{\eta }{\longrightarrow } s$) if $\eta (s_{n}-s,\tau ) = \eta (s-s_{n},\tau ) \to \mathbf{1}$ as $n \to +\infty$ for every $\tau > 0$. If every Cauchy sequence is convergent in a $C^{*}$AVFN-space, then the space is said to be complete. A complete $C^{*}$AVFN-space is called a $C^{*}$-algebra valued fuzzy Banach space (in short, a $C^{*}$AVFB-space).

3 Random operators in $C^{*}$AVFB-spaces

Let $(\varGamma , \varSigma , \xi )$ be a probability measure space. Assume that $(T,{\mathcal{B}}_{T})$ and $(S,{\mathcal{B}}_{S})$ are Borel measurable spaces, in which T and S are $C^{*}$AVFB-spaces. A mapping $F:\varGamma \times T\to S$ is said to be a random operator if $\{\gamma : F(\gamma ,t)\in B\}\in \varSigma$ for all t in T and $B\in {\mathcal{B}}_{S}$. Also, F is a random operator if $F(\gamma ,t)=s(\gamma )$ is an S-valued random variable for every t in T. A random operator $F:\varGamma \times T\to S$ is called linear if $F(\gamma ,a t_{1}+b t_{2})=a F(\gamma ,t_{1})+ b F(\gamma , t_{2})$ for almost every γ for each $t_{1}$, $t_{2}$ in T and scalars a, b and bounded if there exists a nonnegative real-valued random variable $M(\gamma )$ such that

$$ \eta \bigl(F(\gamma ,t_{1})-F(\gamma ,t_{2}),M(\gamma )\tau \bigr)\succeq \eta (t_{1}-t_{2}, \tau ), $$

almost every γ for each $t_{1}$, $t_{2}$ in T and $\tau >0$.

Recently, some authors discussed the approximation of functional equations in several spaces by using a direct technique and a fixed point technique; for fuzzy Menger normed algebras, see [24]; for fuzzy metric spaces, see [25, 26]; for FN spaces, see [27]; for non-Archimedean random Lie $C^{*}$-algebras, see [28]; for non-Archimedean random normed spaces, see [29]; for random multi-normed space, see [30]; and we also refer the reader to [31–34].

Note that a $[0,\infty ]$-valued metric is called a generalized metric.

Theorem 10

([35, 36])

Consider a complete generalized metric space$(T, \delta )$and a strictly contractive function$\varLambda : T \rightarrow T$with Lipschitz constant$L <1$. For every given element$t\in T$, either

$$ \delta \bigl(\varLambda ^{n}t,\varLambda ^{n+1}t\bigr) = \infty $$

for each$n\in \mathbb{N}$or there is$n_{0}\in \mathbb{N}$such that

• (1) $\delta (\varLambda ^{n}t,\varLambda ^{n+1}t)<\infty$, $\forall n\ge n_{0}$;

• (2) the fixed point$s^{*}$ofΛis the convergent point of sequence$\{\varLambda ^{n} t\}$;

• (3) in the set$V = \{s\in T \mid \delta (\varLambda ^{n_{0}}t,s) <\infty \}$, $s^{*}$is the unique fixed point of Λ;

• (4) $(1-L)\delta (s,s^{\ast }) \le \delta (s,\varLambda s)$for every$s \in V$.

4 Random integral equation related to the stochastic wave equation

Let $(\varGamma , \varSigma ,\xi )$ be a probability space and $(S,\eta , \mathcal{T}_{M})$ be a $C^{*}$AVFB-space. Assume that the real numbers $c>0$ and $d_{0}$ are fixed, and suppose that $\gamma \in \varGamma$. Consider the stochastic wave equation

$$\begin{aligned} u_{dd}(\gamma ,x,d)=c^{2}u_{xx}( \gamma ,x,d). \end{aligned}$$

(4.1)

Since

$$\begin{aligned} \begin{gathered} \begin{aligned} u_{d}( \gamma ,x,d) &=\frac{1}{2c}\frac{\partial }{\partial d} \int _{x-cd}^{x+cd}H( \gamma ,\tau ,d_{0})\,d \tau \\ &=\frac{1}{2}H(\gamma ,x+cd,d_{0})+\frac{1}{2}H(\gamma ,x-cd,d_{0}), \end{aligned} \\ u_{dd}(\gamma ,x,d) =\frac{c}{2}H_{d}(\gamma ,x+cd,d_{0})-\frac{c}{2}H_{d}( \gamma ,x-cd,d_{0}), \\ u_{x}(\gamma ,x,d) =\frac{1}{2c}H(\gamma ,x+cd,d_{0})- \frac{1}{2c}H( \gamma ,x-cd,d_{0}), \\ u_{xx}(\gamma ,x,d) =\frac{1}{2c}H_{x}(\gamma ,x+cd,d_{0})- \frac{1}{2c}H_{x}(\gamma ,x-cd,d_{0}), \end{gathered} \end{aligned}$$

(4.2)

we have that

$$\begin{aligned} u(\gamma ,x,d):=\frac{1}{2c} \int _{x-cd}^{x+cd}H(\gamma ,\tau ,d_{0})\,d \tau \end{aligned}$$

(4.3)

is a solution of (4.1) for any random differentiable S-valued function H on $\varGamma \times \mathbb{R}$.

On the other hand, Jung [37] showed that if the S-valued functions F and G on $\varGamma \times \mathbb{R}^{2}$ are twice differentiable, then the S-valued solution u on $\varGamma \times \mathbb{R}^{2}$ of (4.1) has a representation of the form

$$\begin{aligned} u(\gamma ,x,d)=F(\gamma ,x+cd)+G(\gamma ,x-cd), \end{aligned}$$

(4.4)

in which

$$\begin{aligned} \begin{gathered} \frac{1}{2c} \int _{x-cd}^{x+cd}F(\gamma ,\tau )\,d\tau =F(\gamma ,x+cd), \\ \frac{1}{2c} \int _{x-cd}^{x+cd}G(\gamma ,\tau )\,d\tau =G(\gamma ,x-cd). \end{gathered} \end{aligned}$$

(4.5)

Consider the random integral equation

$$\begin{aligned} \frac{1}{2c} \int _{x-cd}^{x+cd}u(\gamma ,\tau ,d_{0})\,d \tau =u(\gamma ,x,d), \end{aligned}$$

(4.6)

which is controlled by the continuous fuzzy set $\varphi (x,d,t)$ as

$$\begin{aligned} \eta \biggl(\frac{1}{2c} \int _{x-cd}^{x+cd}u(\gamma ,\tau ,d_{0})\,d \tau -u( \gamma ,x,d),t\biggr)\geq \varphi (x,d,t). \end{aligned}$$

(4.7)

We say that the random integral equation (4.6) has fuzzy Hyers–Ulam stability if there are $u_{0}(\gamma ,x,d)$ and $\lambda >0$ such that

$$\begin{aligned} \begin{gathered} \frac{1}{2c} \int _{x-cd}^{x+cd}u_{0}(\gamma ,\tau ,d_{0})\,d\tau =u_{0}( \gamma ,x,d), \\ \eta \bigl(u(\gamma ,x,d)-u_{0}(\gamma ,x,d),t\bigr) \geq \varphi \biggl(x,d, \frac{t}{\lambda }\biggr). \end{gathered} \end{aligned}$$

(4.8)

5 $C^{*}$-Algebra-valued fuzzy Hyers–Ulam stability

Let $c>0$, $d_{0}>0$, and $a+cd_{0}< b-cd_{0}$. Let $(\varGamma , \varSigma , \xi )$ be a probability measure space, $(S,\eta ,\mathcal{T}_{M})$ be a $C^{*}$AVFB-space, $\alpha :=[a,b]$, $\beta :=\mathopen(0,d_{0}\mathclose]$, and $\alpha _{0}:=[a+cd_{0},b-cd_{0}]$. Let $M>0$ and $0< L<1$. Consider a continuous $C^{*}$-algebra-valued fuzzy set $\varphi : \alpha \times \beta \times (0,\infty )\rightarrow J$ which is increasing in the second and third components and satisfies

$$\begin{aligned} \inf_{\tau \in [x-cd, x+cd]}\varphi \biggl(\tau ,d, \frac{t}{d} \biggr) \succeq \varphi \biggl( x,d,\frac{t}{L} \biggr) \end{aligned}$$

(5.1)

for all $x\in \alpha _{0}$, $d\in \beta$, and $t>0$.

The set T consists of all random operators $F:\varGamma \times \alpha \times \beta \rightarrow S$ which satisfy the following:

• (a) $F(\gamma ,x,d)$ is continuous for each $x\in \alpha _{0}$, $d\in \beta$, and $\gamma \in \varGamma$;

• (b) $F(\gamma ,x,d)=0_{S}$ for all $x\in \alpha \setminus \alpha _{0}$, $d\in \beta$, and $\gamma \in \varGamma$;

• (c) $\eta (F(\gamma ,x,d),t) \succeq \varphi (x,d,\frac{t}{M})$ for all $x\in \alpha _{0}$, $d\in \beta$, $t>0$, and $\gamma \in \varGamma$.

Theorem 11

Suppose that a random operator$u\in T$satisfies the random integral inequality

$$\begin{aligned} \eta \biggl(\frac{1}{2c} \int _{x-cd}^{x+cd}u(\gamma ,\tau ,d_{0})\,d \tau -u(\gamma ,x,d),t \biggr)\succeq \varphi (x,d,t) \end{aligned}$$

(5.2)

for all$x\in \alpha _{0}$, $d\in \beta$, $t>0$, and$\gamma \in \varGamma$. Then there is a unique random operator$u_{0}\in T$which satisfies

$$\begin{aligned}& \frac{1}{2c} \int _{x-cd}^{x+cd}u_{0}(\gamma ,\tau ,d_{0})\,d\tau =u_{0}( \gamma ,x,d), \end{aligned}$$

(5.3)

$$\begin{aligned}& \eta \bigl(u(\gamma ,x,d)-u_{0}(\gamma ,x,d)\bigr) \succeq \varphi \bigl(x,d,(1-L)t\bigr) \end{aligned}$$

(5.4)

for all$x\in \alpha _{0}$, $d\in \beta$, $t>0$, and$\gamma \in \varGamma$.

Proof

We consider the $[0,\infty ]$-valued metric δ on T defined by

$$\begin{aligned}& \delta (F,G) \\& \quad := \inf \biggl\{ \lambda \in [0,\infty ]\Bigm| \eta \bigl(F(\gamma , x,d)-G( \gamma ,x,d),t\bigr)\succeq \varphi \biggl(x,d,\frac{t}{\lambda }\biggr) \\& \quad\quad \ \forall x\in \alpha _{0}, d\in \beta , \gamma \in \varGamma , t>0 \biggr\} . \end{aligned}$$

(5.5)

In [38], Miheţ and Radu proved that $(B, \delta )$ is complete (see also [39]).

Consider the operator $\varLambda : T\rightarrow T$ given by

$$ (\varLambda H) (\gamma ,x,d):= \textstyle\begin{cases} \frac{1}{2c}\int _{x-cd}^{x+cd}H(\gamma ,\tau ,d_{0})\,d\tau ,&(x\in \alpha _{0},d\in \beta ,\gamma \in \varGamma ), \\ 0,& (\text{otherwise}). \end{cases} $$

(5.6)

It is easy to show that ΛH is continuous on $\varGamma \times \alpha _{0}\times \beta$. Let $x-cd=\xi _{1}<\xi _{2}<\cdots<\xi _{k}=x+cd$, $\bigtriangleup s_{i}=\xi _{i}-\xi _{i-1}$, $i=1,2,\ldots,k$. Using (5.1), (c), and (5.6), we obtain

$$\begin{aligned} \eta (\varLambda H) (\gamma ,x,d),t) &= \eta \biggl(\frac{1}{2c} \int _{x-cd}^{x+cd}H( \gamma ,\tau ,d_{0})\,d \tau ,t\biggr) \\ &= \eta \Biggl(\frac{1}{2c}\lim_{ \Vert \Delta s \Vert \to 0}\sum _{i=1}^{k} H(\gamma ,\xi _{i},d_{0})\Delta s_{i},t\Biggr) \\ &= \eta \Biggl(\lim_{ \Vert \Delta s \Vert \to 0}\sum _{i=1}^{k} H( \gamma ,\xi _{i},d_{0}) \Delta s_{i},2ct\Biggr) \\ &= \lim_{ \Vert \Delta s \Vert \to 0}\eta \Biggl(\sum _{i=1}^{k} H( \gamma ,\xi _{i},d_{0}) \Delta s_{i},2ct\Biggr) \\ &\succeq \lim_{ \Vert \Delta s \Vert \to 0}\mathcal{T}_{M} \eta \biggl(H(\gamma ,\xi _{i},d_{0})\Delta s_{i}, \frac{2ct}{k}\biggr) \\ &\succeq \inf_{\tau \in [x-cd, x+cd]} \eta \biggl(H(\gamma ,\xi _{i},d_{0}), \frac{2ct}{ \vert \Delta s_{i} \vert k} \biggr) \\ &\succeq \inf_{\tau \in [x-cd, x+cd]} \eta \biggl(H(\gamma ,\tau ,d_{0}), \frac{2ctk}{2cdk} \biggr) \\ &\succeq \inf_{\tau \in [x-cd, x+cd]} \varphi \biggl(\tau ,d_{0}, \frac{t}{dM} \biggr) \\ &\succeq \varphi \biggl(x,d,\frac{t}{LM} \biggr) \\ &\succ \varphi \biggl(x,d,\frac{t}{M} \biggr) \end{aligned}$$

(5.7)

for any given $x \in \alpha _{0}$, $d \in \beta$, $t>0$, and $\gamma \in \varGamma$, and then $\varLambda H\in T$. Let $F, G \in T$ and $\lambda _{FG}\in [0,\infty ]$ such that $\delta (F,G)\leq \lambda _{FG}$. Then we have

$$\begin{aligned} \eta \bigl(F(\gamma ,x,d)-G(\gamma ,x,d),t\bigr)\succeq \varphi \biggl(x,d, \frac{t}{\lambda _{FG}} \biggr) \end{aligned}$$

(5.8)

for all $x\in \alpha _{0}$, $d\in \beta$, $t>0$, and $\gamma \in \varGamma$, i.e., Λ is strictly contractive on T. From (5.1), (5.6), and (5.8), we get

$$\begin{aligned} \eta \bigl((\varLambda F) (\gamma ,x,d)-(\varLambda G) (\gamma ,x,d),t \bigr) &= \eta \biggl(\frac{1}{2c} \int _{x-cd}^{x+cd}\bigl(F(\gamma ,\tau ,d_{0})-G( \gamma ,\tau ,d_{0})\bigr)\,d\tau ,t \biggr) \\ &= \eta \Biggl(\frac{1}{2c}\lim_{ \Vert \Delta s \Vert \to 0} \sum _{i=1}^{k} \bigl(F(\gamma ,\xi _{i},d_{0})-G(\gamma ,\xi _{i},d_{0}) \bigr) \Delta s_{i},t \Biggr) \\ &= \eta \Biggl(\lim_{ \Vert \Delta s \Vert \to 0}\sum _{i=1}^{k} \bigl(F(\gamma ,\xi _{i},d_{0})-G( \gamma ,\xi _{i},d_{0})\bigr)\Delta s_{i},2ct \Biggr) \\ &= \lim_{ \Vert \Delta s \Vert \to 0}\eta \Biggl(\sum _{i=1}^{k} \bigl(F(\gamma ,\xi _{i},d_{0})-G( \gamma ,\xi _{i},d_{0})\bigr)\Delta s_{i},2ct \Biggr) \\ &\succeq \lim_{ \Vert \Delta s \Vert \to 0}\mathcal{T}_{M} \eta \biggl(\bigl(F(\gamma ,\xi _{i},d_{0})-G(\gamma ,\xi _{i},d_{0})\bigr), \frac{2ct}{ \vert \Delta s_{i} \vert k} \biggr) \\ &\succeq \inf_{\tau \in [x-cd, x+cd]} \eta \biggl(\bigl(F(\gamma ,\tau ,d_{0})-G( \gamma ,\tau ,d_{0})\bigr),\frac{2ctk}{2cdk} \biggr) \\ &\succeq \inf_{\tau \in [x-cd, x+cd]} \varphi \biggl(\tau ,d_{0}, \frac{t}{d\lambda _{FG}} \biggr) \\ &\succeq \varphi \biggl(x,d_{0},\frac{t}{L\lambda _{FG}} \biggr) \\ &\succ \varphi \biggl(x,d,\frac{t}{L\lambda _{FG}} \biggr) \end{aligned}$$

(5.9)

for any given $x \in \alpha _{0}$, $d \in \beta$, $t>0$, and $\gamma \in \varGamma$, which implies that $\delta (\varLambda F,\varLambda G)\leq L\lambda _{FG}$, and so $\delta (\varLambda F,\varLambda G)\leq Ld(F,G)$. Suppose $H_{0}\in T$. Using (5.2) and (5.5), we get

$$\begin{aligned} \eta \bigl((\varLambda H_{0}) (\gamma ,x,d)-H_{0}( \gamma ,x,d),t\bigr) &\succeq \eta \biggl(\frac{1}{2c} \int _{x-cd}^{x+cd}H_{0}(\gamma ,\tau ,d_{0})\,d \tau -H_{0}(\gamma ,x,d),t \biggr) \\ &\succeq \varphi (x,d,t) \end{aligned}$$

(5.10)

for any $x\in \alpha _{0}$, $d\in \beta$, $t>0$, and $\gamma \in \varGamma$. Thus (5.5) implies that

$$\begin{aligned} \delta (\varLambda H_{0},H_{0})\leq 1< \infty . \end{aligned}$$

(5.11)

Now,

• (1) Theorem 10$(2)$ implies that there is $u_{0}\in T$ such that $\varLambda ^{n}H_{0}\rightarrow u_{0}$ in $(T,\delta )$ and $\varLambda u_{0}=u_{0}$.

• (2) Theorem 10$(3)$ implies that $u_{0}$ is the unique element of T which satisfies $(\varLambda u_{0})(\gamma ,x,d)=u_{0}(\gamma ,x,d)$ for any $x\in \alpha _{0}$, $d\in \beta$, $t>0$, and $\gamma \in \varGamma$.

• (3) Theorem 10$(3)$, together with (5.5) and (5.2), implies that

  $$\begin{aligned} \delta (u,u_{0})\leq \frac{1}{1-L}\delta ( \varLambda u,u)\leq \frac{1}{1-L}, \end{aligned}$$

  (5.12)

  since (5.2) means that $\delta (\varLambda u,u)\leq 1$. In view of (5.5), we can conclude that (5.4) holds for all $x\in \alpha _{0}$ and $d\in \beta$.

 □

6 Conclusion

In this paper, we modified and generalized fuzzy normed spaces and introduced the concept of a $C^{*}$AVFN-space. As an application, we studied the Hyers–Ulam stability of a random integral equation related to the stochastic wave equation in $C^{*}$AVFB-spaces.