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# 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:

$∑ 1 ≤ i 1 < ⋯ < i m ≤ n 1 ≤ k l ≤ n k l ≠ i j , ∀ j ∈ { 1 , … , m } f ( ∑ j = 1 m x i j m + ∑ l = 1 n − m x k l ) = ( n − m + 1 ) n ( n m ) ∑ i = 1 n f( x i ).$

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 must be close to an exact solution of ?

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

Theorem 1.1 (TM Rassias)

Let $f:E→ E ′$ be a mapping from a normed vector space E into a Banach space $E ′$ subject to the following inequality:

$∥ f ( x + y ) − f ( x ) − f ( y ) ∥ ≤ϵ ( ∥ x ∥ p + ∥ y ∥ p )$

for all $x,y∈E$, where ϵ and p are constants with $ϵ>0$ and $0≤p<1$. Then the limit $L(x)= lim n → ∞ f ( 2 n x ) 2 n$ exists for all $x∈E$, and $L:E→ E ′$ is the unique additive mapping which satisfies

$∥ f ( x ) − L ( x ) ∥ ≤ 2 ϵ 2 − 2 p ∥ x ∥ p$

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

Găvruta  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 ).

Katsaras  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 , following Cheng and Mordeson , gave an idea of a fuzzy norm in such a manner that the corresponding fuzzy metric is of Karmosil and Michalek type . They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces .

Now, we consider a mapping $f:X→Y$ satisfying the following functional equation, which is introduced by Rassias and Kim  (see also ):

$∑ 1 ≤ i 1 < ⋯ < i m ≤ n 1 ≤ k l ≤ n k l ≠ i j , ∀ j ∈ { 1 , … , m } f ( ∑ j = 1 m x i j m + ∑ l = 1 n − m x k l ) = ( n − m + 1 ) n ( n m ) ∑ i = 1 n f( x i )$
(1.1)

for all $x 1 ,…, x n ∈X$, where $m,n∈N$ are fixed integers with $n≥2$ and $1≤m≤n$. Especially, we observe that, in the case $m=1$, equation (1.1) yields the Cauchy additive equation

$f ( ∑ l = 1 n x k l ) = ∑ l = 1 n f( x i ).$

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

$f ( ∑ j = 1 n x j n ) = 1 n ∑ l = 1 n f( x i ).$

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 .

Let X and Y be linear spaces. For each m with $1≤m≤n$, a mapping $f:X→Y$ satisfies equation (1.1) for all $n≥2$ if and only if $f(x)−f(0)=A(x)$ is Cauchy additive, where $f(0)=0$ if $m. In particular, we have $f((n−m+1)x)=(n−m+1)f(x)$ and $f(mx)=mf(x)$ for all $x∈X$.

Definition 1.1 Let X be a real vector space. A function $N:X×R→[0,1]$ is called a fuzzy norm on X if for all $x,y∈X$ and $s,t∈R$,

(N1) $N(x,t)=0$ for $t≤0$;

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

(N3) $N(cx,t)=N(x, t | c | )$ if $c≠0$;

(N4) $N(x+y,s+t)≥min{N(x,s),N(y,t)}$;

(N5) $N(x,⋅)$ is a non-decreasing function of and $lim t → ∞ N(x,t)=1$;

(N6) for any $x≠0$, $N(x,⋅)$ is continuous on .

Example 1.1 Let $(X,∥⋅∥)$ be a normed linear space and $β>0$. Then

$N(x,t)={ t t + β ∥ x ∥ , t > 0 , x ∈ X , 0 , t ≤ 0 , x ∈ X$

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 $x∈X$ such that

$lim n → ∞ N( x n −x,t)=1$

for all $t>0$. In this case, x is called the limit of the sequence ${ x n }$ in X, which is denoted by $N− lim t → ∞ x n =x$.

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 $n 0 ∈N$ such that, for all $n≥ n 0$ 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 $f:X→Y$ between fuzzy normed vector spaces X and Y is continuous at a point $x∈X$ if for each sequence ${ x n }$ converging to $x 0 ∈X$, the sequence ${f( x n )}$ converges to $f( x 0 )$. If $f:X→Y$ is continuous at each $x∈X$, then $f:X→Y$ is said to be continuous on X (see ).

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

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

1. (1)

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

$N ( [ x y z ] , s t u ) ≥N(x,s)N(y,t)N(z,u)$

for all $x,y,z∈X$ and $s,t,u>0$;

1. (2)

A complete ternary fuzzy normed algebra is called a ternary fuzzy Banach algebra.

Example 1.2 Let $(X,∥⋅∥)$ be a ternary normed (Banach) algebra. Let

$N(x,t)={ t t + ∥ x ∥ , t > 0 , x ∈ X , 0 , t ≤ 0 , x ∈ X .$

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 $(Y, N ′ )$ be two ternary fuzzy normed algebras.

1. (1)

A -linear mapping $H:(X,N)→(Y, N ′ )$ is called a ternary homomorphism if

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

for all $x,y,z∈X$;

1. (2)

A -linear mapping $D:(X,N)→(X,N)$ is called a ternary fuzzy derivation if

$D ( [ x y z ] ) = [ D ( x ) y z ] + [ x D ( y ) z ] + [ x y D ( z ) ]$

for all $x,y,z∈X$.

We apply the following theorem on weighted spaces (see ).

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

Let $(X,d)$ be a complete metric space and $T:X→X$ be a contraction, i.e., there exists $α∈[0,1)$ such that

$d(Tx,Ty)≤αd(x,y)$

for all $x,y∈X$. Then there exists a unique $a∈X$ such that $Ta=a$. Moreover, $a= lim n → ∞ T n x$ and

$d(a,x)≤ 1 1 − α d(x,Tx)$

for all $x∈X$.

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 $n 0 ∈N$ is a positive integer and $T 1 n o 1 :={ e i θ :0≤θ≤ 2 π n o }$. For the convenience, we use the following abbreviation for a given mapping $f:X→Y$: ## 2 Main results

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

Theorem 2.1 Let $n≥3$ and $φ: X n →[0,∞)$ be a mapping such that there exists $L< 1 ( n − m + 1 ) n − 2$ such that

$φ ( x 1 n − m + 1 , … , x n n − m + 1 ) ≤ L φ ( x 1 , x 2 , … , x n ) n − m + 1$

for all $x 1 ,…, x n ∈X$. Let $f:X→Y$ with $f(0)=0$ be a mapping satisfying

$N ( Δ f ( x 1 , … , x n ) , t ) ≥ t t + φ ( x 1 , … , x n )$
(2.1)

and

$N ( f ( [ a b c ] ) − [ f ( a ) f ( b ) f ( c ) ] , t ) ≥ t t + φ ( a , b , c , 0 , … , 0 )$
(2.2)

for all $μ∈ T 1 n o 1$, $x 1 ,…, x n ,a,b,c∈X$ and $t>0$. Then there exists a unique ternary homomorphism $H:X→Y$ such that

$N ( f ( x ) − H ( x ) , t ) ≥ ( n − m + 1 ) ( n m ) ( 1 − L ) t ( n − m + 1 ) ( n m ) ( 1 − L ) t + L φ ( x , … , x )$
(2.3)

for all $x∈X$ and $t>0$.

Proof Letting $μ=1$ and putting $x 1 = x 2 =⋯= x n =x$ in (2.1), we have

$N ( ( n m ) f ( ( n − m + 1 ) x ) − ( n m ) ( n − m + 1 ) f ( x ) , t ) ≥ t t + φ ( x , … , x )$
(2.4)

for all $x∈X$ and $t>0$. Set $S 0 :={h:X→Y:h(0)=0}$ and define a mapping $d 0 : S 0 × S 0 →[0,∞]$ by

$d 0 (f,g)=inf { μ ∈ R + : N ( g ( x ) − h ( x ) , μ t ) ≥ t t + φ ( x , … , x ) , ∀ x ∈ X , t > 0 } ,$

where $inf∅=+∞$. Also, put $S:={h∈ S 0 : d 0 (h,f)<∞}$. Suppose that d is the restriction of $d 0$ on $S×S$. By using the same technique in the proof of Theorem 3.2 , we can show that $(S,d)$ is a complete metric space. Now, we define a mapping $J:S→S$ by

$Jg(x):=(n−m+1)g ( x n − m + 1 )$

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

$d(f,Jf)≤ L ( n − m + 1 ) ( n m ) .$

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

$H ( x n − m + 1 ) = H ( x ) n − m + 1$
(2.5)

for all $x∈X$. This implies that H is a unique mapping with (2.5) such that there exists $μ∈(0,∞)$ satisfying

$N ( f ( x ) − H ( x ) , μ t ) ≥ t t + φ ( x , … , x )$

for all $x∈X$ and $t>0$.

Moreover, we have $d( J p f,H)→0$ as $p→∞$, which implies

$N- lim p → ∞ f ( x ( n − m + 1 ) p ) ( n − m + 1 ) − p =H(x)$
(2.6)

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

$∑ 1 ≤ i 1 < ⋯ < i m ≤ n 1 ≤ k l ≤ n k l ≠ i j , ∀ j ∈ { 1 , … , m } H ( ∑ j = 1 m μ x i j m + ∑ l = 1 n − m μ x k l ) = ( n − m + 1 ) n ( n m ) ∑ i = 1 n μH( x i )$

for all $μ∈ T 1 n o 1$ and $x 1 ,…, x n ∈X$. This means that $H:X→Y$ is additive. By using the same technique as in the proof of Theorem 2.1 , we can show that H is -linear. On the other hand, by (2.2), we have

$N(α,β)≥ t t + φ ( a ( n − m + 1 ) p , b ( n − m + 1 ) p , c ( n − m + 1 ) p , 0 , 0 , … , 0 )$

for all $a,b,c∈X$ and $t>0$, where Then we have, as $p→+∞$, for all $a,b,c∈X$ and $t>0$. So, it follows that

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

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

Theorem 2.2 Let $φ: X n →[0,∞)$ be a mapping such that there exists $L<1$ with

$φ( x 1 ,…, x n )≤(n−m+1)Lφ ( x 1 n − m + 1 , … , x n n − m + 1 )$

for all $x 1 , x 2 ,…, x n ∈X$. Let $f:X→Y$ be a mapping with $f(0)=0$ satisfying (2.1). Then the limit $H(x):=N- lim p → ∞ f ( ( n − m + 1 ) p x ) ( n − m + 1 ) p$ exists for all $x∈X$ and $H:X→Y$ is defined as a ternary homomorphism such that

$N ( f ( x ) − H ( x ) , t ) ≥ ( n − m + 1 ) ( n m ) ( 1 − L ) t ( n − m + 1 ) ( n m ) ( 1 − L ) t + φ ( x , … , x )$
(2.7)

for all $x∈X$ and $t>0$.

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

$H(x):=N- lim p → ∞ f ( ( n − m + 1 ) p x ) ( n − m + 1 ) p$

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 $x∈X$. This implies that H is a unique mapping satisfying (2.8) such that there exists $μ∈(0,∞)$ satisfying $N(f(x)−H(x),μt)≥ t t + φ ( x , … , x )$ for all $x∈X$ 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 $φ: X n →[0,∞)$ be a function such that there exists $L< 1 ( n − m + 1 ) n − 2$ with

$φ ( x 1 n − m + 1 , … , x n n − m + 1 ) ≤ L φ ( x 1 , x 2 , … , x n ) n − m + 1$

for all $x 1 ,…, x n ∈X$. Let $f:X→X$ be a mapping with $f(0)=0$ satisfying (2.1) and

$N ( f ( [ a b c ] ) − [ f ( a ) b c ] − [ a f ( b ) c ] − [ a b f ( c ) ] , t ) ≥ t t + φ ( a , b , c , 0 , 0 , … , 0 )$
(2.9)

for all $a,b,c∈X$ and $t>0$. Then $D(x):=N- lim p → ∞ f ( x ( n − m + 1 ) p ) ( n − m + 1 ) − p$ exists for all $x∈X$ and $D:X→X$ is defined as a unique ternary derivation such that

$N ( f ( x ) − D ( x ) , t ) ≥ ( n − m + 1 ) ( n m ) ( 1 − L ) t ( n − m + 1 ) ( n m ) ( 1 − L ) t + L φ ( x , … , x )$
(2.10)

for all $x∈X$ and $t>0$.

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

Now, we show that D is a ternary derivation. By (2.9), we have (2.11)

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

$N ( D ( [ a b c ] ) − [ D ( a ) b c ] − [ a D ( b ) c ] − [ a b H ( c ) ] , t ) =1$

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

Theorem 2.4 Let X be a fuzzy Banach algebra. Let $φ: X n →[0,∞)$ be a function such that there exists $L<1$ with

$φ( x 1 ,…, x n )≤(n−m+1)Lφ ( x 1 n − m + 1 , … , x n n − m + 1 )$

for all $x 1 , x 2 ,…, x n ∈X$. Let $f:X→X$ be a mapping with $f(0)=0$ satisfying (2.1) and (2.9). Then the limit $D(x):=N- lim p → ∞ f ( ( n − m + 1 ) p x ) ( n − m + 1 ) p$ exists for all $x∈X$ and $D:X→X$ is defined as a ternary derivation such that

$N ( f ( x ) − D ( x ) , t ) ≥ ( n − m + 1 ) ( n m ) ( 1 − L ) t ( n − m + 1 ) ( n m ) ( 1 − L ) t + φ ( x , … , x )$
(2.12)

for all $x∈X$ and $t>0$.

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## Acknowledgements

The second author of this work was partially supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (Grant No. 2012-0008170).

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Correspondence to YJ Cho or M Eshaghi Gordji.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

All authors read and approved the final manuscript.

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Asgari, G., Cho, Y., Lee, Y. et al. Fixed points and stability of functional equations in fuzzy ternary Banach algebras. J Inequal Appl 2013, 166 (2013). https://doi.org/10.1186/1029-242X-2013-166 