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.


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][7][8][9][10][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 ∈ A, then x is called a positive element. Denote by A + the cone of positive elements of A. We will denote z w when wz ∈ A + (see [12]).
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 [20][21][22].
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Definition 1 Let
A be an order commutative C * -algebra and A + be the positive section of A. Let U = ∅. A C * -algebra valued fuzzy set C on U is a function C : U − → A + . For each u in U, C(u) represents the degree (in A + ) to which u satisfies A + .
We put 0 = inf A + and 1 = sup A + . Now, we define the triangular norm (t-norm) on A + .
If, for every u, v ∈ A + and sequences {u n } and {v n } converging to u and v, we have then we say T on A + is continuous (in short, a ct-norm).
Definition 3 Assume that F : A + → A + satisfies F(0) = 1 and F(1) = 0 and is decreasing. Then F is called a negation on A + . We Then T P is a t-norm (product t-norm). Note that this t-norm is continuous.
Then T M is a t-norm (minimum t-norm). Note that this t-norm is continuous.

Definition 6
The triple (S, η, T ) is called a C * -algebra valued fuzzy normed space (in short, C * AVFN-space) if S = ∅, T is a ct-norm on A + and η is a C * -algebra valued fuzzy set on S 2 × ]0, +∞[ such that, for each t, s, p ∈ T and τ , ς in ]0, +∞[, we have We denote the family of all open subsets of S by τ η , and so τ η is the C * -fuzzy topology induced by the C * -algebra valued fuzzy norm η.
Example 7 Consider the linear normed space (S, · ). Let T = T M and define the fuzzy set η on S 2 × (0, ∞) as follows: for all τ ∈ R + . Then (S, η, T M ) is a C * AVFN-space. Also {s n } n∈N is said to be convergent to s ∈ S (s n as n → +∞ for every τ > 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
almost every γ for each t 1 , t 2 in T and τ > 0.
Note that a [0, ∞]-valued metric is called a generalized metric.

Random integral equation related to the stochastic wave equation
Let (Γ , Σ, ξ ) be a probability space and (S, η, T M ) be a C * AVFB-space. Assume that the real numbers c > 0 and d 0 are fixed, and suppose that γ ∈ Γ . Consider the stochastic wave equation we have that is a solution of (4.1) for any random differentiable S-valued function H on Γ × R.
On the other hand, Jung [37] showed that if the S-valued functions F and G on Γ ×R 2 are twice differentiable, then the S-valued solution u on Γ × R 2 of (4.1) has a representation of the form (4.5) Consider the random integral equation which is controlled by the continuous fuzzy set ϕ(x, d, t) as We say that the random integral equation (4.6) has fuzzy Hyers-Ulam stability if there are u 0 (γ , x, d) and λ > 0 such that (4.8)

Theorem 11 Suppose that a random operator u ∈ T satisfies the random integral inequality
for all x ∈ α 0 , d ∈ β, t > 0, and γ ∈ Γ . Then there is a unique random operator u 0 ∈ T which satisfies for all x ∈ α 0 , d ∈ β, t > 0, and γ ∈ Γ .