Stability of homomorphisms on fuzzy Lie -algebras via fixed point method
© Vahidi and Lee; licensee Springer. 2014
Received: 20 August 2013
Accepted: 17 December 2013
Published: 24 January 2014
In this paper, first, we define fuzzy -algebras and fuzzy Lie -algebras; then, using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy -algebras and fuzzy Lie -algebras for an m-variable additive functional equation.
MSC: 39A10, 39B52, 39B72, 46L05, 47H10, 46B03.
Keywordsfuzzy normed spaces additive functional equation fixed point homomorphism in -algebras and Lie -algebras generalized Hyers-Ulam stability
1 Introduction and preliminaries
if and only if ;
for all ;
for all .
Theorem 1.1 
the sequence converges to a fixed point of J;
is the unique fixed point of J in the set ;
for all .
We use the definition of fuzzy normed spaces given in [4–10] to investigate a fuzzy version of the Hyers-Ulam stability for the Cauchy-Jensen functional equation in the fuzzy normed algebra setting (see also [11–16]).
Definition 1.2 
Let X be a real vector space. A function is called a fuzzy norm on X if for all and all ,
() for ;
() if and only if for all ;
() if ;
() is a non-decreasing function of ℝ and ;
() for , is continuous on ℝ.
The pair is called a fuzzy normed vector space.
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 and we denote it by .
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 in X, then the sequence converges to . If is continuous at each , then is said to be continuous on X (see [4, 10]).
Definition 1.4 
A fuzzy normed algebra is a fuzzy normed space with algebraic structure such that
() for all and all , in which ∗′ is a continuous t-norm.
This space is called the induced fuzzy normed algebra.
An ℝ-linear mapping is called a derivation if for all .
If, in addition for and , then is a fuzzy -algebra.
2 Stability of homomorphisms in fuzzy -algebras
Throughout this section, assume that A is a fuzzy -algebra with norm and that B is a fuzzy -algebra with norm .
for all and all .
Note that a ℂ-linear mapping is called a homomorphism in fuzzy -algebras if H satisfies and for all .
We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy -algebras for the functional equation .
for all and .
- (1)H is a fixed point of J, i.e.,(2.10)
- (2)as . This implies the equality(2.11)
, which implies the inequality . This implies that the inequality (2.8) holds.
for all .
By a similar method to above, we get for all and all . Thus one can show that the mapping is ℂ-linear.
for all . So for all . Thus is a homomorphism satisfying equation (2.7), as desired.
Also by equations (2.5), (2.6), (2.11), and by a similar method we have . □
3 Stability of homomorphisms in fuzzy Lie -algebras
Definition 3.1 Let A and B be fuzzy Lie -algebras. A ℂ-linear mapping is called a fuzzy Lie -algebra homomorphism if for all .
Throughout this section, assume that A is a fuzzy Lie -algebra with norm and that B is a fuzzy Lie -algebra with norm .
We prove the generalized Hyers-Ulam stability of homomorphisms in fuzzy Lie -algebras for the functional equation .
for all and .
for all .
for all .
Thus is a fuzzy Lie -algebra homomorphism satisfying equation (3.5), as desired. □
for all and .
for all and . Putting , we get the desired result. □
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