- Open Access
Nearly n-homomorphisms and n-derivations in fuzzy ternary Banach algebras
© Hassani et al.; licensee Springer 2013
- Received: 2 March 2012
- Accepted: 7 February 2013
- Published: 27 February 2013
Let for some . We investigate the generalized Hyers-Ulam stability of n-homomorphisms and n-derivations on fuzzy ternary Banach algebras related to the generalized Cauchy-Jensen additive functional equation.
MSC:39B52, 46S40, 26E50.
- Hyers-Ulam-Rassias stability
- fixed point theorem
- fuzzy ternary Banach algebra
We say a functional equation (ξ) is stable if any function g satisfying the equation (ξ) approximately is near to a true solution of (ξ). We say that a functional equation (ξ) is superstable if every approximately solution of (ξ) is an exact solution of it (see ).
for all . Moreover, if is continuous in t for each fixed , then T is linear. It gave rise to the Hyers-Ulam type stability of functional equations. Hyers’ theorem was generalized by Rassias  for linear mappings by considering an unbounded Cauchy difference.
Theorem 1.1 (Th.M. Rassias)
for all . Also, if for each , the function is continuous in , 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 [5–45]).
Some mathematicians have defined fuzzy norms on a vector space from various points of view (see  and ). 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 .
for all , where are fixed integers with , . Especially, we observe that in the case , equation (1.1) yields the Cauchy additive equation . We observe that in the case , equation (1.1) yields the Jensen additive equation . 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 general -Cauchy-Jensen additive. For the case , 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 , a mapping satisfies equation (1.1) for all if and only if is Cauchy additive, where if . In particular, we have and for all .
Definition 1.1 Let X be a real vector space. A function is called a fuzzy norm on X if for all and all ,
(N1) for ;
(N2) if and only if for all ;
(N3) if ;
(N5) is a non-decreasing function of ℝ and ;
(N6) for , is continuous on ℝ.
is a fuzzy norm on X.
Definition 1.2 Let be a fuzzy normed vector space. A sequence in X is said to be convergent or converge if there exists an such that for all . In this case, x is called the limit of the sequence in X, and we denote it by .
Definition 1.3 Let be a fuzzy normed vector space. A sequence in X is called Cauchy if for each and each , there exists an such that for all and all , we have .
It is well known that every convergent sequence in a fuzzy normed vector space is Cauchy. 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 between fuzzy normed vector spaces X and Y is continuous at a point if for each sequence converging to , the sequence converges to . If is continuous at each , then is said to be continuous on X (see ).
- (1)The fuzzy normed space is called a fuzzy ternary normed algebra if
A complete ternary fuzzy normed algebra is called a ternary fuzzy Banach algebra.
Then is a fuzzy norm on X and is a ternary fuzzy normed (Banach) algebra.
From now on, we suppose that is a fixed positive integer and . Also, we assume that is a fixed positive integer.
- (1)a ℂ-linear mapping is called an -homomorphism if
- (2)a ℂ-linear mapping is called an -derivation if
for all .
We apply the following theorem on weighted spaces.
for all ;
the sequence converges to a fixed point of J;
is the unique fixed point of J in the set ;
for all .
There are several recent works on stability of functional equations on Banach algebras (see [10–28]). We investigate the stability of n-homomorphisms and n-derivations on fuzzy ternary Banach algebras.
In this section, by using the idea of Park et al. , we prove the generalized Hyers-Ulam-Rassias stability of 5-homomorphisms and 5-derivations related to the functional equation (1.1) on fuzzy ternary Banach algebras (see also ). We start our main results by the stability of 5-homomorphisms.
for all and all .
for all .
for all . This means that H is an -homomorphism. This completes the proof. □
for all and all .
for all . This implies that H is a unique mapping satisfying (2.8) such that there exists satisfying for all and .
The rest of the proof is similar to the proof of Theorem 2.1. □
Now, we investigate the Hyers-Ulam-Rassias stability of -derivations in ternary fuzzy Banach algebras.
for all and all .
for all and all . It follows that D is an -derivation. □
for all and all .
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