Fixed points and stability of functional equations in fuzzy ternary Banach algebras

A classical question in the theory of functional equations is the following:

When is it true that a function which approximately satisfies a functional equation ℰ must be close to an exact solution of ℰ?

If the problem admits a solution, we say that the equation ℰ is stable. Such a problem was formulated by Ulam [1] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [2]. Since Hyers, many authors have studied the stability theory for functional equations. The result of Hyers was extended by Aoki [3] in 1950, by considering the unbounded Cauchy differences. Also, Hyers’ theorem was generalized by Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (TM Rassias)

for all $x\in E$. Also, if for each $x\in E$, the function $f(tx)$ is continuous in $t\in \mathbb{R}$, then L is linear.

Găvruta [5] generalized the Rassias’ result. Beginning around the year 1980, the stability problems of several functional equations and approximate homomorphisms have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [6–29]).

Katsaras [30] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view (see [31, 32]). In particular, Bag and Samanta [33], following Cheng and Mordeson [34], gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type [35]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [36].

Therefore, equation (1.1) is a generalized form of the Cauchy-Jensen additive equation and thus every solution of equation (1.1) may be analogously called the general $(m,n)$-Cauchy-Jensen additive. For the case $m=2$, the authors have established new theorems about the Ulam-Hyers-Rassias stability in quasi-β-normed spaces [37].

Let X and Y be linear spaces. For each m with $1\le m\le n$, a mapping $f:X\to Y$ satisfies equation (1.1) for all $n\ge 2$ if and only if $f(x)-f(0)=A(x)$ is Cauchy additive, where $f(0)=0$ if $m<n$. In particular, we have $f((n-m+1)x)=(n-m+1)f(x)$ and $f(mx)=mf(x)$ for all $x\in X$.

Definition 1.1 Let X be a real vector space. A function $N:X\times \mathbb{R}\to [0,1]$ is called a fuzzy norm on X if for all $x,y\in X$ and $s,t\in \mathbb{R}$,

(N1) $N(x,t)=0$ for $t\le 0$;

(N2) $x=0$ if and only if $N(x,t)=1$ for all $t>0$;

(N3) $N(cx,t)=N(x,\frac{t}{|c|})$ if $c\ne 0$;

(N4) $N(x+y,s+t)\ge min\{N(x,s),N(y,t)\}$;

(N5) $N(x,\cdot )$ is a non-decreasing function of ℝ and ${lim}_{t\to \mathrm{\infty }}N(x,t)=1$;

(N6) for any $x\ne 0$, $N(x,\cdot )$ is continuous on ℝ.

is a fuzzy norm on X.

for all $t>0$. In this case, x is called the limit of the sequence $\{x_n\}$ in X, which is denoted by $N-{lim}_{t\to \mathrm{\infty }}{x}_n=x$.

It is well known that every convergent sequence in a fuzzy normed vector space is a Cauchy sequence. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping $f:X\to Y$ between fuzzy normed vector spaces X and Y is continuous at a point $x\in X$ if for each sequence $\{x_n\}$ converging to $x_0\in X$, the sequence $\{f(x_n)\}$ converges to $f(x_0)$. If $f:X\to Y$ is continuous at each $x\in X$, then $f:X\to Y$ is said to be continuous on X (see [36]).

Ternary algebraic operations were considered in the nineteenth century by several mathematicians such as Cayley [39] who introduced the notion of cubic matrix which in turn was generalized by Kapranov, Gelfand and Zelevinskii in 1990 [40]. The comments on physical applications of ternary structures can be found in [41–45].

Then $N(x,t)$ is a fuzzy norm on X and $(X,N)$ is a ternary fuzzy normed (Banach) algebra.

for all $x,y,z\in X$.

We apply the following theorem on weighted spaces (see [46–49]).

Theorem 1.2 (The generalized fixed point theorem of Diaz and Margolis)

for all $x\in X$.

Throughout this paper, we suppose that X is a ternary fuzzy normed algebra and Y is a ternary fuzzy Banach algebra. Moreover, we assume that $n_0\in \mathbb{N}$ is a positive integer and ${\mathbb{T}}_{\frac{1}{n_o}}^1:=\{e^{i\theta }:0\le \theta \le \frac{2\pi }{n_o}\}$. For the convenience, we use the following abbreviation for a given mapping $f:X\to Y$:

In this section, by using the idea of Gavruta and Gavruta [14], we prove the generalized Hyers-Ulam-Rassias stability of ternary homomorphisms related to functional equation (1.1) on ternary fuzzy Banach algebras (see also [50]).

for all $x\in X$ and $t>0$.

for all $x\in X$ and $t>0$.

for all $a,b,c\in X$ and $t>0$, where

Then we have, as $p\to +\mathrm{\infty }$,

for all $a,b,c\in X$ and $t>0$. Hence we have $H([abc])=[H(a)H(b)H(c)]$ for all $a,b,c\in X$. This means that H is a ternary homomorphism. This completes the proof. □

for all $x\in X$ and $t>0$.

for all $x\in X$. This implies that H is a unique mapping satisfying (2.8) such that there exists $\mu \in (0,\mathrm{\infty })$ satisfying $N(f(x)-H(x),\mu t)\ge \frac{t}{t+\phi (x,\dots ,x)}$ for all $x\in X$ and $t>0$.

The rest of the proof is similar to the proof of Theorem 2.1. This completes the proof. □

Now, we investigate the Hyers-Ulam-Rassias stability of ternary derivations in ternary fuzzy Banach algebras.

for all $x\in X$ and $t>0$.

Proof By the same reasoning as that in the proof of Theorem 2.1, the mapping $D:X\to X$ is a unique ℂ-linear mapping which satisfies (2.10).

Now, we show that D is a ternary derivation. By (2.9), we have

(2.11)

for all $a,b,c\in X$ and $t>0$. Then we have

for all $a,b,c\in X$ and $t>0$. Hence we have $D([abc])=[D(a)bc]+[aD(b)c]+[abD(c)]$ for all $a,b,c\in X$ . This means that D is a ternary derivation. This completes the proof. □

for all $x\in X$ and $t>0$.