Research Article | Open Access

Fridoun Moradlou, Hamid Vaezi, Choonkil Park, "Fixed Points and Stability of an Additive Functional Equation of -Apollonius Type in -Algebras", *Abstract and Applied Analysis*, vol. 2008, Article ID 672618, 13 pages, 2008. https://doi.org/10.1155/2008/672618

# Fixed Points and Stability of an Additive Functional Equation of -Apollonius Type in -Algebras

**Academic Editor:**John Rassias

#### Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of -algebra homomorphisms and of generalized derivations on -algebras for the following functional equation of Apollonius type .

#### 1. Introduction and Preliminaries

A classical question in the theory of functional equations is the following: βwhen is it true that a function, which approximately satisfies a functional equation , must be close to an exact solution of ?β If the problem accepts a solution, we say that the equation is stable. Such a problem was formulated by Ulam [1] in 1940 and solved in the next year for the Cauchy functional equation by Hyers [2]. It gave rise to the stability theory for functional equations. The result of Hyers was extended by Aoki [3] in 1950 by considering the unbounded Cauchy differences. In 1978, Rassias [4] proved that the additive mapping , obtained by Hyers or Aoki, is linear if, in addition, for each , the mapping is continuous in . GΔvruΕ£ a [5] generalized the Rassias' result. Following the techniques of the proof of the corollary of Hyers [2], we observed that Hyers introduced (in 1941) the following Hyers continuity condition about the continuity of the mapping for each fixed point and then he proved homogeneity of degree one and, therefore, the famous linearity. This condition has been assumed further till now, through the complete Hyers direct method, in order to prove linearity for generalized Hyers-Ulam stability problem forms (see [6]). Beginning around 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 [7β21]).

Rassias [22], following the spirit of the innovative approach of Hyers [2], Aoki [3], and Rassias [4] for the unbounded Cauchy difference, proved a similar stability theorem in which he replaced the factor by for with (see also [23, 24] for a number of other new results).

In 2003, CΔdariu and Radu applied the fixed-point method
to the investigation of the Jensen functional equation [25] (see also [8, 26β30]). They could present a short and a simple proof
(different of the β*direct method*,β initiated by Hyers in 1941) for
the generalized Hyers-Ulam stability of Jensen functional equation [25], for Cauchy functional
equation [8], and for
quadratic functional equation [26].

The following
functional equation:is called a *quadratic functional equation,* and every
solution of (1.1) is said to be a *quadratic mapping*. Skof [31] proved the Hyers-Ulam
stability of the quadratic functional equation (1.1) for mappings ,
where is a normed space and is a Banach space. In [32], Czerwik proved the
Hyers-Ulam stability of the quadratic functional equation (1.1). Borelli
and Forti [33]
generalized the stability result of the quadratic functional equation (1.1).
Jun and Lee [34] proved
the Hyers-Ulam stability of the Pexiderized quadratic equationfor mappings ,
and The stability problem of the quadratic
equation has been extensively investigated by some mathematicians [35].

In an inner product space, the
equalityholds, then it is called the *Apollonius' identity.* The following
functional equation, which was motivated by this equation,holds, then it is called *quadratic* (see [36]). For this reason, the
functional equation (1.4) is called a *quadratic functional equation of Apollonius
type,* and each solution of the functional equation (1.4) is said to be a *quadratic mapping of Apollonius type.* The
quadratic functional equation and several other functional equations are useful
to characterize inner product spaces [37].

In [36], Park and Rassias
introduced and investigated a functional equation, which is called a *generalized Apollonius type quadratic
functional equation.* In [38], Najati introduced and investigated a functional
equation, which is called a *quadratic
functional equation of -Apollonius type.* Recently in [39], Park and Rassias
introduced and investigated the following functional equation:which is called an *Apollonius type additive functional equation,* and whose solution is called an *Apollonius type additive mapping.* In
[40], Park
introduced and investigated a functional equation, which is called a *generalized Apollonius-Jensen type additive
functional equation* and whose solution is said to be a *generalized Apollonius-Jensen type additive
mapping*.

In this paper, employing the above equality (1.5), for a fixed positive integer we introduce a new functional equation, which
is called an *additive functional
equation of -Apollonius type* and whose solution is
said to be an *additive mapping of -Apollonius type; *

We will adopt the idea of CΔdariu and Radu [8, 25, 28] to prove the generalized Hyers-Ulam stability results of -algebra homomorphisms as well as to prove the generalized Ulam-Hyers stability of generalized derivations on -algebra for additive functional equation of -Apollonius type.

We recall two fundamental results in fixed-point theory.

Theorem 1.1 (see [25]). * Let be a complete metric space and let be strictly contractive, that is, **for some Lipschitz constant .
Then, the following hold: *

(1)* the mapping has a unique fixed point ;*(2)

*(3)*

*the fixed point**is globally attractive, that is,**for any starting point*;

*one has the following estimation inequalities:**for all nonnegative integers*and all .Let be a set. A function is called a *generalized metric* on if satisfies the following:

(1) if and only if ;(2) for all ;(3) for all .

Theorem 1.2 (see [41]). * Let be a complete generalized metric space and let be a strictly contractive mapping with
Lipschitz constant .
Then for each given element ,
either **for all nonnegative integers or there exists a positive integer such that the following hold: *

(1)* for all ;*(2)

*(3)*

*he sequence**converges to a fixed point**of*;*(4)*

*is the unique fixed point of**in the set*;

*for all*.Throughout this paper, assume that is a -algebra with norm and that is a -algebra with norm .

#### 2. Stability of -Algebra Homomorphisms

Lemma 2.1. *Let and be real-vector spaces. A mapping satisfies (1.6) for all if and only if the mapping is additive.*

*Proof. *Letting in (1.6), we get that .
Let and be fixed integers with Setting for all in (1.6), we havefor all Replacing by and by in (2.1), respectively, we getfor all Putting in (2.2), we conclude that for all This means that is an odd function. Letting in (2.1) and using the oddness of ,
we obtain thatfor all Letting in (2.1), using the oddness of and (2.3), we havefor all Therefore, is an additive mapping.

The converse is
obviously true.

For a given mapping and for a fixed positive integer , we definefor all and all .

We prove the generalized Hyers-Ulam stability of -algebra homomorphisms for the functional equation .

Theorem 2.2. *Let be a mapping satisfying for which there exists a function such that ** for all and all .
If for some there exists a Lipschitz constant such that ** for all ,
then there exists a unique -algebra homomorphism such that **for all .*

*Proof. *Consider the setand introduce the *generalized metric* on :It is easy to show that is complete.

For convenience, setfor all and all

Now we consider
the linear mapping such thatfor all ,
where

For any ,
we have

Therefore, we
see thatThis means is a strictly contractive self-mapping of ,
with the Lipschitz constant .

Letting ,
and for each with in (2.7), we getfor all .
Sofor all .
Hence .

By Theorem 1.2, there exists a mapping such that the following hold:

(1) is a fixed point of ,
that is,for all ;
the mapping is a unique fixed point of in the setand this implies that is a unique mapping satisfying (2.20) such
that there exists satisfyingfor all .

(2) as ;
and this implies the equalityfor all ;

(3) ,
which implies the inequalityand this implies that the
inequality (2.11) holds.

It follows from (2.6), (2.7), and (2.23)
thatfor all .
Sofor all .
By Lemma 2.1, the mapping is Cauchy additive, that is, for all .

By a similar method to the proof of [14], one can show that the
mapping is -linear.

It follows from (2.8) thatfor all .
Sofor all .

It follows from (2.9) thatfor all .
Sofor all .

Thus is a -algebra homomorphism satisfying (2.11) as
desired.

Corollary 2.3. *Let and be nonnegative real numbers, and let be a mapping such that ** for all and all .
Then there exists a unique -algebra homomorphism such that **for all .*

*Proof. *The
proof follows from Theorem 2.2 by takingfor all .
It follows from (2.31) that We can choose to get the desired result.

Theorem 2.4. *Let be a mapping satisfying for which there exists a function satisfying (2.7), (2.8),
and (2.9) such
that **for all .
If for some there exists a Lipschitz constant such that **for all ,
then there exists a unique -algebra homomorphism such that **for all .*

*Proof. *Similar to proof of Theorem (2.2),
we consider the linear mapping such thatfor all where .
We can conclude that is a strictly contractive self mapping of with the Lipschitz constant .

It follows from (2.18) thatfor all .
Hence, .

By Theorem 1.2, there exists a mapping such that the following hold:

(1) is a fixed point of ,
that is,for all ;
the mapping is a unique fixed point of in the setand this implies that is a unique mapping satisfying (2.41) such
that there exists satisfyingfor all ;

(2) as ;
and this implies the equalityfor all ;

(3) ,
which implies the inequalitywhich implies that the
inequality (2.38) holds.

The rest of the proof is similar to the proof of
Theorem 2.2.

Corollary 2.5. *Let and be nonnegative real numbers, and let be a mapping satisfying (2.31), (2.32), and
(2.33). Then there exists a unique -algebra homomorphism such that **for all and .*

*Proof. *The
proof follows from Theorem 2.4 by takingfor all .
It follows from (2.31) that We can choose to get the desired result.

#### 3. Stability of Generalized Derivations on -Algebras

For a given mapping and for a fixed positive integer , we definefor all and all .

*Definition 3.1 * (see [42]). A generalized derivation is involutive -linear and fulfills

for all .

We prove the generalized Hyers-Ulam stability of
derivations on -algebras for the functional equation .

Theorem 3.2. *Let be a mapping satisfying for which there exists a function such that ** for all and all .
If for some there exists a Lipschitz constant such that **for all ,
then there exists a unique generalized derivation such that **for all .*

*Proof. *By the same reasoning as in the
proof of Theorem 2.2, there exists a unique involutive -linear mapping satisfying (3.7). The mapping is given byfor all .

It follows from (3.4) thatfor all .
So for all .
Thus is a generalized derivation satisfying (3.7).

Theorem 3.3. *Let be a mapping satisfying for which there exists a function satisfying (2.36),(3.3),
(3.4) and (3.5) for
all .
If for some there exists a Lipschitz constant such that **for all ,
then there exists a unique generalized derivation such that **for all .*

*Proof. *The proof is similar to the proofs
of Theorems 2.4 and 3.2.

#### Acknowledgments

This paper is based on final report of the research project of the Ph.D. thesis in University of Tabriz and the third author was supported by Grant no. F01-2006-000-10111-0 from the Korea Science and Engineering Foundation. The authors would like to thank the referees for a number of valuable suggestions regarding a previous version of this paper.

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Copyright Β© 2008 Fridoun Moradlou et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.