- Research Article
- Open Access

# A Fixed Point Approach to the Stability of Quintic and Sextic Functional Equations in Quasi- -Normed Spaces

- TianZhou Xu
^{1}Email author, - JohnMichael Rassias
^{2}, - MatinaJohn Rassias
^{2}and - WanXin Xu
^{3}

**2010**:423231

https://doi.org/10.1155/2010/423231

© Tian Zhou Xu et al. 2011

**Received: **19 September 2010

**Accepted: **31 December 2010

**Published: **5 January 2011

## Abstract

## Keywords

## 1. Introduction and Preliminaries

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation? If the problem accepts a unique solution, we say the equation is stable (see [1]). The first stability problem concerning group homomorphisms was raised by Ulam [2] in 1940 and affirmatively solved by Hyers [3]. The result of Hyers was generalized by Rassias [4] for approximate linear mappings by allowing the Cauchy difference operator to be controlled by . In 1994, a generalization of Rassias' theorem was obtained by Găvruţa [5], who replaced by a general control function in the spirit of Rassias' approach. 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, e.g., [6–30] and references therein).

In 1996, Isac and Rassias [30] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. The stability problems of several various functional equations have been extensively investigated by a number of authors using fixed point methods (see [8, 10, 17, 22]).

is said to be a quadratic functional equation because the quadratic function is a solution of the functional equation (1.1). Every solution of the quadratic functional equation is said to be a quadratic mapping. A quadratic functional equation was used to characterize inner product spaces.

was introduced by Rassias [26]. It is easy to show that the function is a solution of (1.3). Every solution of the quartic functional equation is said to be a quartic mapping.

Moreover, we prove the stability of the quintic and sextic functional equations in quasi- -normed spaces via fixed point method, and also using Gajda's example to give two counterexamples for a singular case. Since is a solution of (1.4), we say that it is a quintic functional equation. Similarly, is a solution of (1.5), and we say that it is a sextic functional equation. Every solution of the quintic or sextic functional equation is said to be a quintic or sextic mapping, respectively.

For the sake of convenience, we recall some basic concepts concerning quasi- -normed spaces (see [24]).

Definition 1.1.

Let be a fix real number with , and let denote either or . Let be a linear space over . A quasi- -norm is a real-valued function on satisfying the following:

(1) for all and if and only if ;

(3)there is a constant such that for all .

A quasi- -normed space is a pair , where is a quasi- -norm on . The smallest possible is called the modulus of concavity of . A quasi- -Banach space is a complete quasi- -normed space.

for all . In this case, a quasi- -Banach space is called a -Banach space. We can refer to [7, 17] for the concept of quasinormed spaces and -Banach spaces.

Given a -norm, the formula gives us a translation invariant metric on . By the Aoki-Rolewicz theorem, each quasi-norm is equivalent to some -norm. Since it is much easier to work with -norms than quasi-norms, henceforth we restrict our attention mainly to -norms.

## 2. General Solutions to Quintic and Sextic Functional Equations

In this section, let and be vector spaces. In the following theorem, we investigate the general solutions of the functional equation (1.4) and (1.5). Some basic facts on -additive symmetric mappings can be found in [29].

Theorem 2.1.

A function is a solution of the functional equation (1.4) if and only if is of the form for all , where is the diagonal of the 5-additive symmetric map .

Proof.

where is an arbitrary element of and is the diagonal of the -additive symmetric map for . By and for all , we get , and the function is odd. Thus we have . It follows that . By (2.7) and whenever and , we obtain . Hence , and so for all . Therefore, .

Conversely, assume that for all , where is the diagonal of the 5-additive symmetric map . From , , , , , and , we see that satisfies (1.4), which completes the proof of Theorem 2.1.

Theorem 2.2.

A function is a solution of the functional equation (1.5) if and only if is of the form for all , where is the diagonal of the 6-additive symmetric map .

Proof.

where is an arbitrary element of and is the diagonal of the -additive symmetric map for . By and for all , we get and the function is even. Thus we have , , and . It follows that . By (2.19) and whenever and , we obtain . Hence , and so for all . Therefore, .

Conversely, assume that for all , where is the diagonal of the 6-additive symmetric map . From , , , , , , and , we see that satisfies (1.5), which completes the proof of Theorem 2.2.

## 3. Stability of the Quintic Functional Equation

Lemma 3.1.

Proof.

It is easy to show that is a complete generalized metric space (see [8–10]).

This means that is a strictly contractive self-mapping of with Lipschitz constant .

This implies that the inequality (3.3) holds.

Since, for every , , we get . This completes the proof.

Theorem 3.2.

Proof.

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

Corollary 3.3.

Corollary 3.4.

Corollary 3.5.

for all , where and are defined as in Corollaries 3.3 and 3.4.

The following example shows that the assumption cannot be omitted in Corollary 3.4. This example is a modification of the example of Gajda [6] for the additive functional inequality (see also [7]).

Example 3.6.

for all , but there do not exist a quintic mapping and a constant such that for all .

Proof.

which contradicts (3.53).

## 4. Stability of the Sextic Functional Equation

Theorem 4.1.

Proof.

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

Corollary 4.2.

Corollary 4.3.

Corollary 4.4.

for all , where and are defined as in Corollaries 4.2 and 4.3.

For the case , similar to Example 3.6, we have the following counterexample.

Example 4.5.

for all , but there do not exist a sextic mapping and a constant such that for all .

Proof.

which contradicts (4.41).

## Declarations

### Acknowledgment

The first author was supported by the National Natural Science Foundation of China (10671013, 60972089).

## Authors’ Affiliations

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