Approximation of homomorphisms and derivations on non-Archimedean random Lie -algebras via fixed point method
© Kang and Saadati; licensee Springer 2012
Received: 18 April 2012
Accepted: 15 October 2012
Published: 29 October 2012
In this paper, using fixed point methods, we prove the generalized Hyers-Ulam stability of random homomorphisms in random -algebras and random Lie -algebras and of derivations on non-Archimedean random C∗-algebras and non-Archimedean random Lie C∗-algebras for an m-variable additive functional equation.
MSC:39A10, 39B52, 39B72, 46L05, 47H10, 46B03.
1 Introduction and preliminaries
if and only if ;
for any , , ;
- (iii)the strong triangle inequality (ultrametric) holds; namely
holds, a sequence is Cauchy if and only if converges to zero in a non-Archimedean normed space. By a complete non-Archimedean normed space, we mean the one in which every Cauchy sequence is convergent.
For any nonzero rational number x, there exists a unique integer such that , where a and b are integers not divisible by p. Then defines a non-Archimedean norm on ℚ. The completion of ℚ with respect to the metric is denoted by , which is called the p-adic number field.
If, in addition, for , then is a non-Archimedean -algebra.
The stability problem of functional equations was originated from a question of Ulam  concerning the stability of group homomorphisms. Let be a group and let be a metric group (a metric which is defined on a set with a group property) 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 a homomorphism is stable (see also [4–6]).
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 .
2 Random spaces
Definition 2.1 
T is commutative and associative;
T is continuous;
for all ;
whenever and for all .
Typical examples of continuous t-norms are , and (the Lukasiewicz t-norm).
Definition 2.2 
A non-Archimedean random normed space (briefly, NA-RN-space) is a triple , where X is a vector space, T is a continuous t-norm, and μ is a mapping from X into such that the following conditions hold:
(RN1) for all if and only if ;
(RN2) for all , ;
(RN3) for all and all .
for all , and is the minimum t-norm. This space is called the induced random normed space.
Definition 2.3 
A non-Archimedean random normed algebra is a non-Archimedean random normed space with an algebraic structure such that
(RN-4) for all and all , in which is a continuous t-norm.
This space is called an induced non-Archimedean random normed algebra.
Let and be non-Archimedean random normed algebras. An ℝ-linear mapping is called a homomorphism if for all .
An ℝ-linear mapping is called a derivation if for all .
If, in addition, for and , then is a non-Archimedean random -algebra.
A sequence in X is said to be convergent to x in X if, for every and , there exists a positive integer N such that whenever .
A sequence in X is called a Cauchy sequence if, for every and , there exists a positive integer N such that whenever .
An RN-space is said to be complete if and only if every Cauchy sequence in X is convergent to a point in X.
3 Stability of homomorphisms and derivations in non-Archimedean random -algebras
Throughout this section, assume that is a non-Archimedean random -algebra with the norm and that ℬ is a non-Archimedean random -algebra with the norm .
for all and all .
Note that a ℂ-linear mapping is called a homomorphism in non-Archimedean random -algebras if H satisfies and for all .
We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean random -algebras for the functional equation .
for all and .
for all and .
It is easy to show that is a generalized complete metric space (see ).
From this it is easy to see that for all , that is, J is a self-function of Ω with the Lipschitz constant L.
for all since .
By a similar method to the 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 (3.7) as desired.
Also by (3.3), (3.10), (3.11) and by a similar method, we have . □
for all and .
for all , and , we get the desired result. □
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean random -algebras for the functional equation .
for all and .
4 Stability of homomorphisms and derivations in non-Archimedean Lie -algebras
on , is called a Lie non-Archimedean random -algebra.
Definition 4.1 Let and ℬ be random Lie -algebras. A ℂ-linear mapping is called a non-Archimedean Lie -algebra homomorphism if for all .
Throughout this section, assume that is a non-Archimedean random Lie -algebra with the norm and that ℬ is a non-Archimedean random Lie -algebra with the norm .
We prove the generalized Hyers-Ulam stability of homomorphisms in non-Archimedean random Lie -algebras for the functional equation .
for all , all and . If there exists an and (3.4), (3.5), and (3.6) hold, then there exists a unique homomorphism such that (3.7) holds.
for all . Thus, is a Lie -algebra homomorphism satisfying (3.7), as desired. □
for all and .
Proof The proof follows from Theorem 4.2 and a method similar to Corollary 3.2. □
Definition 4.4 Let be a non-Archimedean random Lie -algebra. A ℂ-linear mapping is called a Lie derivation if for all .
We prove the generalized Hyers-Ulam stability of derivations on non-Archimedean random Lie -algebras for the functional equation .
for all . If there exists an and (3.4), (3.5), and (3.6) hold, then there exists a unique Lie derivation such that (3.7) holds.
for all .
for all . Thus, is a Lie derivation satisfying (3.7). □
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