Skip to main content

\(C^{*}\)-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.

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]).

\(C^{*}\)-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

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

  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

([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).

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\(s \in V\).

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)

\(C^{*}\)-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:

  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

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

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

 □

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.

References

  1. 1.

    Ulam, S.M.: Problems in Modern Mathematics. Wiley, New York (1960)

    Google Scholar 

  2. 2.

    Hyers, D.H.: On the stability of the linear functional equation. Proc. Natl. Acad. Sci. USA 27, 222–224 (1941)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Aoki, T.: On the stability of the linear transformation in Banach spaces. J. Math. Soc. Jpn. 2, 64–66 (1950)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Rassias, T.M.: On the stability of the linear mapping in Banach spaces. Proc. Am. Math. Soc. 72, 297–300 (1978)

    MathSciNet  Article  Google Scholar 

  5. 5.

    Gǎvruta, P.: A generalization of the Hyers–Ulam–Rassias stability of approximately additive mappings. J. Math. Anal. Appl. 184, 431–436 (1994)

    MathSciNet  Article  Google Scholar 

  6. 6.

    Jung, S., Popa, D., Rassias, M.T.: On the stability of the linear functional equation in a single variable on complete metric spaces. J. Glob. Optim. 59, 13–16 (2014)

    Article  Google Scholar 

  7. 7.

    Lee, Y., Jung, S., Rassias, M.T.: Uniqueness theorems on functional inequalities concerning cubic–quadratic-additive equation. J. Math. Inequal. 12, 43–61 (2018)

    MathSciNet  Article  Google Scholar 

  8. 8.

    Li, T., Zada, A.: Connections between Hyers–Ulam stability and uniform exponential stability of discrete evolution families of bounded linear operators over Banach spaces. Adv. Differ. Equ. 2016, 153 (2016)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Li, T., Zada, A., Faisal, S.: Hyers–Ulam stability of nth order linear differential equations. J. Nonlinear Sci. Appl. 9, 2070–2075 (2016)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Zada, A., Yar, M., Li, T.: Existence and stability analysis of nonlinear sequential coupled system of Caputo fractional differential equations with integral boundary conditions. Ann. Univ. Paedagog. Crac. Stud. Math. 17, 103–125 (2018)

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Zada, A., Ali, S., Li, T.: Analysis of a new class of impulsive implicit sequential fractional differential equations. Int. J. Nonlinear Sci. Numer. Simul. (in press). https://doi.org/10.1515/ijnsns-2019-0030

  12. 12.

    Dixmier, J.: \(C^{*}\)-Algebras. North-Holland, New York (1977)

    Google Scholar 

  13. 13.

    S̆erstnev, A.N.: On the notion of a random normed space. Dokl. Akad. Nauk USSR 149, 280–283 (1963)

    Google Scholar 

  14. 14.

    Mus̆tari, D.H.: Almost sure convergence in linear spaces of random variables. Teor. Veroâtn. Primen. 15, 351–357 (1970)

    MathSciNet  Google Scholar 

  15. 15.

    Radu, V.: Linear operators in random normed spaces. Bull. Math. Soc. Sci. Math. R. S. Roumanie (N.S.) 17(65), 217–220 (1975)

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Cheng, S.C., Mordeson, J.N.: Fuzzy linear operators and fuzzy normed linear spaces. Bull. Calcutta Math. Soc. 86(5), 429–436 (1994)

    MathSciNet  MATH  Google Scholar 

  17. 17.

    Saadati, R.: Nonlinear contraction and fuzzy compact operator in fuzzy Banach algebras. Fixed Point Theory 20, 289–297 (2019)

    MathSciNet  Article  Google Scholar 

  18. 18.

    Beg, I., Ahmed, M., Nafadi, H.: Fixed points of \(\mathcal{L}\)-fuzzy mappings in ordered b-metric spaces. J. Funct. Spaces 2018, Article ID 5650242 (2018)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Huang, H., Wu, C.: Characterizations of compact sets in fuzzy set spaces with \(L_{p}\) metric. Fuzzy Sets Syst. 330, 16–40 (2018)

    Article  Google Scholar 

  20. 20.

    Glück, J.: A note on lattice ordered \(C^{*}\)-algebra and Perron–Frobenius theory. Math. Nachr. 291(11–12), 1727–1732 (2020)

    MathSciNet  MATH  Google Scholar 

  21. 21.

    Green, M.D.: The lattice structure of \(C^{*}\)-algebras and their duals. Math. Proc. Camb. Philos. Soc. 81(2), 245–248 (1977)

    MathSciNet  Article  Google Scholar 

  22. 22.

    Hussain, S.: Fixed point and common fixed point theorems on ordered cone b-metric space over Banach algebra. J. Nonlinear Sci. Appl. 13, 22–33 (2020)

    MathSciNet  Article  Google Scholar 

  23. 23.

    Saadati, R., Vaezpour, S.M.: Some results on fuzzy Banach spaces. J. Appl. Math. Comput. 17(1–2), 475–484 (2005)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Mirmostafaee, A.K.: Perturbation of generalized derivations in fuzzy Menger normed algebras. Fuzzy Sets Syst. 195, 109–117 (2012)

    MathSciNet  Article  Google Scholar 

  25. 25.

    Naeem, R., Anwar, M.: Jensen type functionals and exponential convexity. J. Math. Comput. Sci. 17, 429–436 (2017)

    Article  Google Scholar 

  26. 26.

    Park, C., Shin, D., Saadati, R., Lee, R.: A fixed point approach to the fuzzy stability of an AQCQ-functional equation. Filomat 30(7), 1833–1851 (2016)

    MathSciNet  Article  Google Scholar 

  27. 27.

    Naeem, R., Anwar, M.: Weighted Jensen’s functionals and exponential convexity. J. Math. Comput. Sci. 19, 171–180 (2019)

    Article  Google Scholar 

  28. 28.

    Shoaib, A., Azam, A., Arshad, M., Ameer, E.: Fixed point results for multivalued mappings on a sequence in a closed ball with applications. J. Math. Comput. Sci. 17, 308–316 (2017)

    Article  Google Scholar 

  29. 29.

    Ciepliski, K.: On a functional equation connected with bi-linear mappings and its Hyers–Ulam stability. J. Nonlinear Sci. Appl. 10(11), 5914–5921 (2017)

    MathSciNet  Article  Google Scholar 

  30. 30.

    Agarwal, R.P., Saadati, R., Salamati, A.: Approximation of the multiplicatives on random multi-normed space. J. Inequal. Appl. 2017, 204 (2017)

    MathSciNet  Article  Google Scholar 

  31. 31.

    EL-Fassi, I.: Solution and approximation of radical quintic functional equation related to quintic mapping in quasi-β-Banach spaces. Rev. R. Acad. Cienc. Exactas Fís. Nat., Ser. A Mat. 113, 675–687 (2019)

    MathSciNet  Article  Google Scholar 

  32. 32.

    Jang, S., Saadati, R.: Approximation of an additive \((\varrho _{1},\varrho _{2})\)-random operator inequality. J. Funct. Spaces 2020, Article ID 7540303 (2020)

    MathSciNet  MATH  Google Scholar 

  33. 33.

    Saadati, R., Park, C.: Approximation of derivations and the superstability in random Banach -algebras. Adv. Differ. Equ. 2018, 418 (2018)

    MathSciNet  Article  Google Scholar 

  34. 34.

    Wang, Z., Saadati, R.: Approximation of additive functional equations in NA Lie \(C^{*}\)-algebras. Demonstr. Math. 51, 37–44 (2018)

    MathSciNet  Article  Google Scholar 

  35. 35.

    Cădariu, L., Radu, V.: Fixed point methods for the generalized stability of functional equations in a single variable. Fixed Point Theory Appl. 2008, Article ID 749392 (2008)

    MathSciNet  Article  Google Scholar 

  36. 36.

    Diaz, J.B., Margolis, B.: A fixed point theorem of the alternative, for contractions on a generalized complete metric space. Bull. Am. Math. Soc. 74, 305–309 (1968)

    MathSciNet  Article  Google Scholar 

  37. 37.

    Jung, S.: A fixed point approach to the stability of an integral equation related to the wave equation. Abstr. Appl. Anal. 2013, Article ID 612576 (2013)

    MathSciNet  MATH  Google Scholar 

  38. 38.

    Miheţ, D., Radu, V.: On the stability of the additive Cauchy functional equation in random normed spaces. J. Math. Anal. Appl. 343, 567–572 (2008)

    MathSciNet  Article  Google Scholar 

  39. 39.

    Miheţ, D., Saadati, R.: On the stability of the additive Cauchy functional equation in random normed spaces. Appl. Math. Lett. 24, 2005–2009 (2011)

    MathSciNet  Article  Google Scholar 

Download references

Acknowledgements

Not applicable.

Availability of data and materials

Not applicable.

Funding

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).

Author information

Affiliations

Authors

Contributions

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.

Corresponding author

Correspondence to Choonkil Park.

Ethics declarations

Competing interests

The authors declare that they have no competing interests.

Rights and permissions

Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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). https://doi.org/10.1186/s13662-020-02780-0

Download citation

MSC

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

Keywords

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