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Theory and Modern Applications

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


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 [611]).

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 [2022].

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

  1. (i)

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

  2. (ii)

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

  3. (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)

  4. (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


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

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

  1. (a)

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

  2. (b)

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

  3. (c)

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

  4. (d)

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

  5. (e)

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

  6. (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


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 [3134].

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. (1)

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

  2. (2)

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

  3. (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. (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

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


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

we have that

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

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

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

in which

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

Consider the random integral equation

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

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

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

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

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

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

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

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:

  1. (a)

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

  2. (b)

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

  3. (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

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

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

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

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


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

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

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

Consider the operator \(\varLambda : T\rightarrow T\) given by

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

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

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

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

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

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

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

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

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

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

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


  1. (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. (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. (3)

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

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

    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.


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This work was supported by the Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science, and Technology (NRF-2017R1D1A1B04032937).

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The authors equally conceived of the study, participated in its design and coordination, drafted the manuscript, participated in the sequence alignment, and read and approved the final manuscript.

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Correspondence to Choonkil Park.

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Chaharpashlou, R., O’Regan, D., Park, C. et al. \(C^{*}\)-Algebra valued fuzzy normed spaces with application of Hyers–Ulam stability of a random integral equation. Adv Differ Equ 2020, 326 (2020).

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  • 54E50
  • 39B52
  • 39B62
  • 46L05
  • 47H10
  • 39B82


  • Nonlinear random integral equation
  • Stochastic wave equation
  • Iterative method
  • Random operator
  • Fuzzy normed space
  • \(C^{*}\)-algebra valued fuzzy set