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

Abstract

By using Diaz and Margolis fixed point theorem, we establish the generalized Hyers-Ulam-Rassias stability of the ternary homomorphisms and ternary derivations between fuzzy ternary Banach algebras associated to the following $(m,n)$-Cauchy-Jensen additive functional equation:

$$\underset{k_l\ne i_j,\mathrm{\forall }j\in \{1,\dots ,m\}}{\underset{1\le k_l\le n}{\sum _{1\le i_1<\cdots <i_m\le n}}}f(\frac{{\sum }_{j=1}^m{x}_{i_j}}{m}+\sum _{l=1}^{n-m}{x}_{k_l})=\frac{(n-m+1)}{n}\left(\genfrac{}{}{0ex}{}{n}{m}\right)\sum _{i=1}^{n}f(x_i).$$

MSC:39B52, 46S40, 26E50.

1 Introduction

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)

Let $f:E\to E^{\prime }$ be a mapping from a normed vector space E into a Banach space $E^{\prime }$ subject to the following inequality:

$$\parallel f(x+y)-f(x)-f(y)\parallel \le ϵ({\parallel x\parallel }^p+{\parallel y\parallel }^p)$$

for all $x,y\in E$, where ϵ and p are constants with $ϵ>0$ and $0\le p<1$. Then the limit $L(x)={lim}_{n\to \mathrm{\infty }}\frac{f(2^{n}x)}{2^n}$ exists for all $x\in E$, and $L:E\to E^{\prime }$ is the unique additive mapping which satisfies

$$\parallel f(x)-L(x)\parallel \le \frac{2ϵ}{2-2^p}{\parallel x\parallel }^p$$

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].

Now, we consider a mapping $f:X\to Y$ satisfying the following functional equation, which is introduced by Rassias and Kim [37] (see also [38]):

$$\underset{k_l\ne i_j,\mathrm{\forall }j\in \{1,\dots ,m\}}{\underset{1\le k_l\le n}{\sum _{1\le i_1<\cdots <i_m\le n}}}f(\frac{{\sum }_{j=1}^m{x}_{i_j}}{m}+\sum _{l=1}^{n-m}{x}_{k_l})=\frac{(n-m+1)}{n}\left(\genfrac{}{}{0ex}{}{n}{m}\right)\sum _{i=1}^{n}f(x_i)$$

(1.1)

for all $x_1,\dots ,x_n\in X$, where $m,n\in \mathbb{N}$ are fixed integers with $n\ge 2$ and $1\le m\le n$. Especially, we observe that, in the case $m=1$, equation (1.1) yields the Cauchy additive equation

$$f\left(\sum _{l=1}^n{x}_{k_l}\right)=\sum _{l=1}^{n}f(x_i).$$

Also, we observe that, in the case $m=n$, equation (1.1) yields the Jensen additive equation

$$f\left(\frac{{\sum }_{j=1}^n{x}_j}{n}\right)=\frac{1}{n}\sum _{l=1}^{n}f(x_i).$$

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.

Example 1.1 Let $(X,\parallel \cdot \parallel )$ be a normed linear space and $\beta >0$. Then

$$N(x,t)=\{\begin{array}{cc}\frac{t}{t+\beta \parallel x\parallel },\hfill & t>0,x\in X,\hfill \\ 0,\hfill & t\le 0,x\in X\hfill \end{array}$$

is a fuzzy norm on X.

Definition 1.2 Let $(X,N)$ be a fuzzy normed vector space. A sequence $\{x_n\}$ in X is said to be convergent if there exists $x\in X$ such that

$$\underset{n\to \mathrm{\infty }}{lim}N(x_n-x,t)=1$$

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

Definition 1.3 Let $(X,N)$ be a fuzzy normed vector space. A sequence $\{x_n\}$ in X is called a Cauchy sequence if for each $ϵ>0$ and each $t>0$, there exists $n_0\in \mathbb{N}$ such that, for all $n\ge n_0$ and $p>0$,

$$N(x_{n+p}-x_n,t)>1-ϵ.$$

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].

Definition 1.4 Let X be a ternary algebra and $(X,N)$ be a fuzzy normed space.

• (1) The fuzzy normed space $(X,N)$ is called a ternary fuzzy normed algebra if

  $$N([xyz],stu)\ge N(x,s)N(y,t)N(z,u)$$

for all $x,y,z\in X$ and $s,t,u>0$;

• (2) A complete ternary fuzzy normed algebra is called a ternary fuzzy Banach algebra.

Example 1.2 Let $(X,\parallel \cdot \parallel )$ be a ternary normed (Banach) algebra. Let

$$N(x,t)=\{\begin{array}{cc}\frac{t}{t+\parallel x\parallel },\hfill & t>0,x\in X,\hfill \\ 0,\hfill & t\le 0,x\in X.\hfill \end{array}$$

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

Definition 1.5 Let $(X,N)$ and $(Y,N^{\prime })$ be two ternary fuzzy normed algebras.

• (1) A ℂ-linear mapping $H:(X,N)\to (Y,N^{\prime })$ is called a ternary homomorphism if

  $$H([xyz])=[H(x)H(y)H(z)]$$

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

• (2) A ℂ-linear mapping $D:(X,N)\to (X,N)$ is called a ternary fuzzy derivation if

  $$D([xyz])=[D(x)yz]+[xD(y)z]+[xyD(z)]$$

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)

Let $(X,d)$ be a complete metric space and $T:X\to X$ be a contraction, i.e., there exists $\alpha \in [0,1)$ such that

$$d(Tx,Ty)\le \alpha d(x,y)$$

for all $x,y\in X$. Then there exists a unique $a\in X$ such that $Ta=a$. Moreover, $a={lim}_{n\to \mathrm{\infty }}{T}^{n}x$ and

$$d(a,x)\le \frac{1}{1-\alpha }d(x,Tx)$$

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 $f:X\to Y$:

2 Main results

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]).

Theorem 2.1 Let $n\ge 3$ and $\phi :X^n\to [0,\mathrm{\infty })$ be a mapping such that there exists $L<\frac{1}{{(n-m+1)}^{n-2}}$ such that

$$\phi (\frac{x_1}{n-m+1},\dots ,\frac{x_n}{n-m+1})\le \frac{L\phi (x_1,x_2,\dots ,x_n)}{n-m+1}$$

for all $x_1,\dots ,x_n\in X$. Let $f:X\to Y$ with $f(0)=0$ be a mapping satisfying

$$N(\mathrm{\Delta }f(x_1,\dots ,x_n),t)\ge \frac{t}{t+\phi (x_1,\dots ,x_n)}$$

(2.1)

and

$$N(f([abc])-[f(a)f(b)f(c)],t)\ge \frac{t}{t+\phi (a,b,c,0,\dots ,0)}$$

(2.2)

for all $\mu \in {\mathbb{T}}_{\frac{1}{n_o}}^1$, $x_1,\dots ,x_n,a,b,c\in X$ and $t>0$. Then there exists a unique ternary homomorphism $H:X\to Y$ such that

$$N(f(x)-H(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+L\phi (x,\dots ,x)}$$

(2.3)

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

Proof Letting $\mu =1$ and putting $x_1=x_2=\cdots =x_n=x$ in (2.1), we have

$$N(\left(\genfrac{}{}{0ex}{}{n}{m}\right)f((n-m+1)x)-\left(\genfrac{}{}{0ex}{}{n}{m}\right)(n-m+1)f(x),t)\ge \frac{t}{t+\phi (x,\dots ,x)}$$

(2.4)

for all $x\in X$ and $t>0$. Set $S_0:=\{h:X\to Y:h(0)=0\}$ and define a mapping $d_0:S_0\times S_0\to [0,\mathrm{\infty }]$ by

$$d_0(f,g)=inf\{\mu \in {\mathbb{R}}^{+}:N(g(x)-h(x),\mu t)\ge \frac{t}{t+\phi (x,\dots ,x)},\mathrm{\forall }x\in X,t>0\},$$

where $inf\mathrm{\varnothing }=+\mathrm{\infty }$. Also, put $S:=\{h\in S_0:d_0(h,f)<\mathrm{\infty }\}$. Suppose that d is the restriction of $d_0$ on $S\times S$. By using the same technique in the proof of Theorem 3.2 [50], we can show that $(S,d)$ is a complete metric space. Now, we define a mapping $J:S\to S$ by

$$Jg(x):=(n-m+1)g\left(\frac{x}{n-m+1}\right)$$

for all $x\in X$. It is easy to see that $d(Jg,Jh)\le Ld(g,h)$ for all $g,h\in S$. This implies that

$$d(f,Jf)\le \frac{L}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)}.$$

Thus, by Banach’s fixed point theorem (Theorem 1.2), J has a unique fixed point $H:X\to Y$ in S satisfying

$$H\left(\frac{x}{n-m+1}\right)=\frac{H(x)}{n-m+1}$$

(2.5)

for all $x\in X$. This implies that H is a unique mapping with (2.5) 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 $t>0$.

Moreover, we have $d(J^{p}f,H)\to 0$ as $p\to \mathrm{\infty }$, which implies

$$N\text{-}\underset{p\to \mathrm{\infty }}{lim}\frac{f(\frac{x}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}=H(x)$$

(2.6)

for all $x\in X$. Thus it follows from (2.1) and (2.6) that

$$\underset{k_l\ne i_j,\mathrm{\forall }j\in \{1,\dots ,m\}}{\underset{1\le k_l\le n}{\sum _{1\le i_1<\cdots <i_m\le n}}}H(\frac{{\sum }_{j=1}^m\mu x_{i_j}}{m}+\sum _{l=1}^{n-m}\mu x_{k_l})=\frac{(n-m+1)}{n}\left(\genfrac{}{}{0ex}{}{n}{m}\right)\sum _{i=1}^n\mu H(x_i)$$

for all $\mu \in {\mathbb{T}}_{\frac{1}{n_o}}^1$ and $x_1,\dots ,x_n\in X$. This means that $H:X\to Y$ is additive. By using the same technique as in the proof of Theorem 2.1 [51], we can show that H is ℂ-linear. On the other hand, by (2.2), we have

$$N(\alpha ,\beta )\ge \frac{t}{t+\phi (\frac{a}{{(n-m+1)}^p},\frac{b}{{(n-m+1)}^p},\frac{c}{{(n-m+1)}^p},0,0,\dots ,0)}$$

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 $t>0$. So, it follows that

$$N(H([abc])-[H(a)H(b)H(c)],t)=1$$

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. □

Theorem 2.2 Let $\phi :X^n\to [0,\mathrm{\infty })$ be a mapping such that there exists $L<1$ with

$$\phi (x_1,\dots ,x_n)\le (n-m+1)L\phi (\frac{x_1}{n-m+1},\dots ,\frac{x_n}{n-m+1})$$

for all $x_1,x_2,\dots ,x_n\in X$. Let $f:X\to Y$ be a mapping with $f(0)=0$ satisfying (2.1). Then the limit $H(x):=N\text{-}{lim}_{p\to \mathrm{\infty }}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^p}$ exists for all $x\in X$ and $H:X\to Y$ is defined as a ternary homomorphism such that

$$N(f(x)-H(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+\phi (x,\dots ,x)}$$

(2.7)

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

Proof Let $(S,d)$ be the metric space defined as in the proof of Theorem 2.1. Consider the mapping $T:S\to S$ defined by $Tg(x):=\frac{g((n-m+1)x)}{n-m+1}$ for all $x\in X$. One can show that $d(g,h)=ϵ$ implies that $d(Tg,Th)\le Lϵ$ for all positive real numbers ϵ. This means that T is a contraction on $(S,d)$. The mapping

$$H(x):=N\text{-}\underset{p\to \mathrm{\infty }}{lim}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^p}$$

is the unique fixed point of T in S and H has the following property:

$$(n-m+1)H(x)=H((n-m+1)x)$$

(2.8)

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.

Theorem 2.3 Let X be a fuzzy Banach algebra. Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists $L<\frac{1}{{(n-m+1)}^{n-2}}$ with

$$\phi (\frac{x_1}{n-m+1},\dots ,\frac{x_n}{n-m+1})\le \frac{L\phi (x_1,x_2,\dots ,x_n)}{n-m+1}$$

for all $x_1,\dots ,x_n\in X$. Let $f:X\to X$ be a mapping with $f(0)=0$ satisfying (2.1) and

$$N(f([abc])-[f(a)bc]-[af(b)c]-[abf(c)],t)\ge \frac{t}{t+\phi (a,b,c,0,0,\dots ,0)}$$

(2.9)

for all $a,b,c\in X$ and $t>0$. Then $D(x):=N\text{-}{lim}_{p\to \mathrm{\infty }}\frac{f(\frac{x}{{(n-m+1)}^p})}{{(n-m+1)}^{-p}}$ exists for all $x\in X$ and $D:X\to X$ is defined as a unique ternary derivation such that

$$N(f(x)-D(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+L\phi (x,\dots ,x)}$$

(2.10)

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 $t>0$. So, we have

$$N(D([abc])-[D(a)bc]-[aD(b)c]-[abH(c)],t)=1$$

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. □

Theorem 2.4 Let X be a fuzzy Banach algebra. Let $\phi :X^n\to [0,\mathrm{\infty })$ be a function such that there exists $L<1$ with

$$\phi (x_1,\dots ,x_n)\le (n-m+1)L\phi (\frac{x_1}{n-m+1},\dots ,\frac{x_n}{n-m+1})$$

for all $x_1,x_2,\dots ,x_n\in X$. Let $f:X\to X$ be a mapping with $f(0)=0$ satisfying (2.1) and (2.9). Then the limit $D(x):=N\text{-}{lim}_{p\to \mathrm{\infty }}\frac{f({(n-m+1)}^{p}x)}{{(n-m+1)}^p}$ exists for all $x\in X$ and $D:X\to X$ is defined as a ternary derivation such that

$$N(f(x)-D(x),t)\ge \frac{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t}{(n-m+1)\left(\genfrac{}{}{0ex}{}{n}{m}\right)(1-L)t+\phi (x,\dots ,x)}$$

(2.12)

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