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Stability of homomorphisms on fuzzy Lie -algebras via fixed point method
Journal of Inequalities and Applications volume 2014, Article number: 33 (2014)
Abstract
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.
1 Introduction and preliminaries
The stability problem of functional equations originated with a question of Ulam [1] concerning the stability of group homomorphisms: let be a group and let be a metric group with the metric . Given , does there exist a such that if a mapping satisfies the inequality for all , then there is a homomorphism with for all ? If the answer is affirmative, we would say that the equation of homomorphism is stable. We recall a fundamental result in fixed-point theory. Let Ω be a set. A function is called a generalized metric on Ω if d satisfies
-
(1)
if and only if ;
-
(2)
for all ;
-
(3)
for all .
Theorem 1.1 [2]
Let be a complete generalized metric space and let be a contractive mapping with Lipschitz constant . Then for each given element , either for all nonnegative integers n or there exists a positive integer such that
-
(1)
, ;
-
(2)
the sequence converges to a fixed point of J;
-
(3)
is the unique fixed point of J in the set ;
-
(4)
for all .
In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of homomorphisms and derivations in fuzzy Lie -algebras for the following additive functional equation [3]:
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 [4]
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.
Definition 1.3 [4]
-
(1)
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 .
-
(2)
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 [12]
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.
Every normed algebra defines a fuzzy normed algebra , where
for all iff
This space is called the induced fuzzy normed algebra.
Definition 1.5 (1) Let and be fuzzy normed algebras. An ℝ-linear mapping is called a homomorphism if for all .
-
(2)
An ℝ-linear mapping is called a derivation if for all .
Definition 1.6 Let be a fuzzy Banach algebra, then an involution on is a mapping from into which satisfies
-
(i)
for ;
-
(ii)
;
-
(iii)
for .
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 a given mapping , we define
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 .
Theorem 2.1 Let be a mapping for which there are functions , and such that
for all , all and . If there exists an such that
for all and , then there exists a unique homomorphism such that
for all and .
Proof Consider the set and introduce the generalized metric on X:
It is easy to show that is complete. Now, we consider the linear mapping such that for all . By Theorem 3.1 of [17], for all . Letting , and in equation (2.1), we get
for all and . Therefore
for all and . Hence . By Theorem 1.1, there exists a mapping such that
-
(1)
H is a fixed point of J, i.e.,
(2.10)
for all . The mapping H is a unique fixed point of J in the set
This implies that H is a unique mapping satisfying equation (2.10) such that there exists satisfying
for all and .
-
(2)
as . This implies the equality
(2.11)
for all .
-
(3)
, which implies the inequality . This implies that the inequality (2.8) holds.
It follows from equations (2.1), (2.2), and (2.11) that
for all and . So
for all .
By a similar method to above, we get for all and all . Thus one can show that the mapping is ℂ-linear.
It follows from equations (2.3), (2.4), and (2.11) that
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
A fuzzy -algebra , endowed with the Lie product
on , is called a fuzzy Lie -algebra (see [18–20]).
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 .
Theorem 3.2 Let be a mapping for which there are functions and such that
for all , all and . If there exists an such that
for all and , then there exists a unique homomorphism such that
for all and .
Proof By the same reasoning as the proof of Theorem 2.1, we can find that the mapping is given by
for all .
It follows from equation (3.3) that
for all and . So
for all .
Thus is a fuzzy Lie -algebra homomorphism satisfying equation (3.5), as desired. □
Corollary 3.3 Let and θ be nonnegative real numbers, and let be a mapping such that
for all , all and . Then there exists a unique homomorphism such that
for all and .
Proof The proof follows from Theorem 3.2 by taking
for all and . Putting , we get the desired result. □
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Vahidi, J., Lee, S.J. Stability of homomorphisms on fuzzy Lie -algebras via fixed point method. J Inequal Appl 2014, 33 (2014). https://doi.org/10.1186/1029-242X-2014-33
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DOI: https://doi.org/10.1186/1029-242X-2014-33