# A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Functional Equation

- JungRye Lee
^{1}, - Ji-hye Kim
^{2}and - Choonkil Park
^{2}Email author

**2010**:185780

**DOI: **10.1155/2010/185780

© Jung Rye Lee et al. 2010

**Received: **24 August 2009

**Accepted: **10 January 2010

**Published: **21 January 2010

## Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation in Banach spaces.

## 1. Introduction and Preliminaries

The stability problem of functional equations is originated from a question of Ulam [1] concerning the stability of group homomorphisms. Hyers [2] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' Theorem was generalized byAoki [3] for additive mappings and by Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [4] has provided a lot of influence in the development of what we call *generalized Hyers-Ulam stability* or as *Hyers-Ulam-Rassias stability* of functional equations. A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

is called a *quadratic functional equation*. In particular, every solution of the quadratic functional equation is said to be a *quadratic mapping*. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [6] for mappings
, where
is a normed space and
is a Banach space. Cholewa [7] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. Czerwik [8] proved the generalized Hyers-Ulam stability of the quadratic functional equation. The stability problems of several functional equations have been extensively investigated by a number of authors, and there are many interesting results concerning this problem (see [9–19]).

which is called a *cubic functional equation,* and every solution of the cubic functional equation is said to be a *cubic mapping*.

which is called a *quartic functional equation* and every solution of the quartic functional equation is said to be a *quartic mapping*. Quartic functional equations have been investigated in [22, 23].

Let
be a set. A function
is called a *generalized metric* on
if
satisfies

(1) if and only if ;

(2) for all ;

(3) for all .

We recall a fundamental result in fixed point theory.

for all nonnegative integers or there exists a positive integer such that

(1) , for all ;

(2)the sequence converges to a fixed point of ;

(3) is the unique fixed point of in the set ;

(4) for all .

In 1996, Isac and Th. M. Rassias [26] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [27–32]).

in Banach spaces for an odd case. In Section 3, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation (1.5) in Banach spaces for an even case.

Throughout this paper, assume that is a vector space and that is a Banach space.

## 2. Generalized Hyers-Ulam Stability of the Functional Equation (1.5): An Odd Case

for all .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in Banach spaces: an odd case.

Note that the fundamental ideas in the proofs of the main results in Sections 2 and 3 are contained in [24, 27, 28].

Theorem 2.1.

for all .

Proof.

for all .

for all .

for all .

where, as usual, . It is easy to show that is complete (see the proof of Lemma of [33]).

for all .

for all .

for all . So .

- (1)is a fixed point of , that is,(2.16)

- (2)as . This implies the equality(2.19)

- (3), which implies the inequality(2.20)

This implies that the inequality (2.4) holds.

for all . Thus the mapping is cubic, as desired.

Corollary 2.2.

for all .

Proof.

for all . Then we can choose and we get the desired result.

Theorem 2.3.

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

for all .

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.4.

for all .

Proof.

for all . Then we can choose and we get the desired result.

Theorem 2.5.

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

for all .

for all .

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.6.

for all .

Theorem 2.7.

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

for all .

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 2.8.

for all .

## 3. Generalized Hyers-Ulam Stability of the Functional Equation (1.5): An Even Case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in Banach spaces: an even case.

Theorem 3.1.

for all .

Proof.

for all .

for all .

for all .

Let be the generalized metric space defined in the proof of Theorem 2.1.

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.2.

for all .

Theorem 3.3.

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

for all .

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.4.

for all .

Theorem 3.5.

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

for all .

for all .

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.6.

for all .

Theorem 3.7.

for all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 2.1.

for all .

for all . So .

The rest of the proof is similar to the proof of Theorem 2.1.

Corollary 3.8.

for all .

## 4. Generalized Hyers-Ulam Stability of the Functional Equation (1.5)

It was shown in of [34, Lemma ] that and are cubic and additive, respectively, and that .

It was shown in of [35, Lemma ] that and are quartic and quadratic, respectively, and that . Functional equations of mixed type have been investigated in [36, 37].

So we obtain the following results.

Theorem 4.1.

for all .

Proof.

Since , , and . The result follows from Theorems 2.1, 2.5, 3.1, and 3.5.

Corollary 4.2.

for all .

Theorem 4.3.

for all .

Proof.

Since , , and . The result follows from Theorems 2.3, 2.7, 3.3, and 3.7.

Corollary 4.4.

for all .

## Declarations

### Acknowledgments

The first and third authors were supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0071229) and (NRF-2009-0070788), respectively.

## Authors’ Affiliations

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