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Using fixed point methods, we prove the generalized Hyers-Ulam stability of homomorphisms in

The stability problem of functional equations
originated from a question of Ulam [

We recall a fundamental result in fixed point theory.

Let

Let

the sequence

By the using fixed point method, the stability
problems of several functional equations have been extensively investigated by
a number of authors (see [

This paper is organized as follows: in Sections

In Sections

Throughout this section, assume that

For a given mapping

Note that a

We prove the generalized Hyers-Ulam stability of
homomorphisms in

Let

It
follows from

Consider the set

Now we consider the linear mapping

By [

Letting

By Theorem

It follows from(

Letting

It follows from
(

It follows from
(

Thus

Let

The proof follows from Theorem

Let

We
consider the linear mapping

It follows from
(

By Theorem

The rest of the proof is similar to the proof of
Theorem

Let

The
proof follows from Theorem

Let

Consider
the set

Now we consider the linear mapping

By [

Letting

By Theorem

The rest of the proof is similar to the proof of
Theorem

Let

The
proof follows from Theorem

Let

We
consider the linear mapping

It follows from
(

By Theorem

The rest of the proof is similar to the proof of
Theorem

Let

The
proof follows from Theorem

Throughout this section, assume that

Note that a

We prove the generalized Hyers-Ulam stability of
derivations on

Let

By the
same reasoning as the proof of Theorem

It follows from
(

Let

The
proof follows from Theorem

Let

The
proof is similar to the proofs of Theorems

Let

The
proof follows from Theorem

For inequalities controlled by the product of powers of norms, one can obtain similar results to Theorems

A

Let

Throughout this section, assume that

We prove the generalized Hyers-Ulam stability of
homomorphisms in Lie

Let

By the same reasoning as the proof
of Theorem

It follows from
(

Thus

Let

The
proof follows from Theorem

Let

The proof is similar to the proofs
of Theorems

Let

The
proof follows from Theorem

For inequalities controlled by the
product of powers of norms, one can obtain similar results to Theorems

Let

Throughout this section, assume that

We prove the generalized Hyers-Ulam stability of
derivations on Lie

Let

By the same reasoning as the proof
of Theorem

It follows from
(

Let

The
proof follows from Theorem

Let

The
proof is similar to the proofs of Theorems

Let

The
proof follows from Theorem

For inequalities controlled by the
product of powers of norms, one can obtain similar results to Theorems

The first author was supported by the R & E program of KOSEF in 2008.