# 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

## References

- Moszner Z: On the stability of functional equations.
*Aequationes Mathematicae*2009, 77(1–2):33–88. 10.1007/s00010-008-2945-7MathSciNetView ArticleMATHGoogle Scholar - Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics*. Interscience Publishers, New York, NY, USA; 1960:xiii+150.Google Scholar - Hyers DH: On the stability of the linear functional equation.
*Proceedings of the National Academy of Sciences of the United States of America*1941, 27: 222–224. 10.1073/pnas.27.4.222MathSciNetView ArticleMATHGoogle Scholar - Rassias TM: On the stability of the linear mapping in Banach spaces.
*Proceedings of the American Mathematical Society*1978, 72(2):297–300. 10.1090/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - Găvruţa P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings.
*Journal of Mathematical Analysis and Applications*1994, 184(3):431–436. 10.1006/jmaa.1994.1211MathSciNetView ArticleMATHGoogle Scholar - Gajda Z: On stability of additive mappings.
*International Journal of Mathematics and Mathematical Sciences*1991, 14(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar - Jun K-W, Kim H-M: On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces.
*Journal of Mathematical Analysis and Applications*2007, 332(2):1335–1350. 10.1016/j.jmaa.2006.11.024MathSciNetView ArticleMATHGoogle Scholar - Xu TZ, Rassias JM, Xu WX: A fixed point approach to the stability of a general mixed AQCQ-functional equation in non-archimedean normed spaces.
*Discrete Dynamics in Nature and Society*2010, 2010:-24.Google Scholar - Miheţ D, Radu V: On the stability of the additive Cauchy functional equation in random normed spaces.
*Journal of Mathematical Analysis and Applications*2008, 343(1):567–572.MathSciNetView ArticleMATHGoogle Scholar - Park C: Fixed points and the stability of an AQCQ-functional equation in non-Archimedean normed spaces.
*Abstract and Applied Analysis*2010, 2010:-15.Google Scholar - Sikorska J: On a direct method for proving the Hyers-Ulam stability of functional equations.
*Journal of Mathematical Analysis and Applications*2010, 372(1):99–109. 10.1016/j.jmaa.2010.06.056MathSciNetView ArticleMATHGoogle Scholar - Brzdęk J: On a method of proving the Hyers-Ulam stability of functional equations on restricted domains.
*The Australian Journal of Mathematical Analysis and Applications*2009, 6(1, article 4):10.MathSciNetMATHGoogle Scholar - Forti G-L: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations.
*Journal of Mathematical Analysis and Applications*2004, 295(1):127–133. 10.1016/j.jmaa.2004.03.011MathSciNetView ArticleMATHGoogle Scholar - Forti G-L: Elementary remarks on Ulam-Hyers stability of linear functional equations.
*Journal of Mathematical Analysis and Applications*2007, 328(1):109–118. 10.1016/j.jmaa.2006.04.079MathSciNetView ArticleMATHGoogle Scholar - Ravi K, Arunkumar M, Rassias JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation.
*International Journal of Mathematics and Statistics*2008, 3(A08):36–46.MathSciNetMATHGoogle Scholar - Xu TZ, Rassias JM, Xu WX: Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces. Journal of Mathematical Physics 2010, 51(6):-19.Google Scholar
- Xu TZ, Rassias JM, Xu WX: A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces. International Journal of Physical Sciences 2011., 6(2, 12 pages):Google Scholar
- Xu TZ, Rassias JM, Xu WX: Intuitionistic fuzzy stability of a general mixed additive-cubic equation. Journal of Mathematical Physics 2010, 51(6):-21.Google Scholar
- Xu TZ, Rassias JM, Xu WX: On the stability of a general mixed additive-cubic functional equation in random normed spaces.
*Journal of Inequalities and Applications*2010, 2010:-16.Google Scholar - Mohamadi M, Cho YJ, Park C, Vetro P, Saadati R: Random stability of an additive-quadratic quartic functional equation.
*Journal of Inequalities and Applications*2010, 2010:-18.Google Scholar - Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces.
*Journal of Inequalities and Applications*2008, 2008:-11.Google Scholar - Cădariu L, Radu V: Fixed points and stability for functional equations in probabilistic metric and random normed spaces.
*Fixed Point Theory and Applications*2009, 2009:-18.Google Scholar - Eshaghi Gordji M, Savadkouhi MB: Stability of mixed type cubic and quartic functional equations in random normed spaces.
*Journal of Inequalities and Applications*2009, 2009:-9.Google Scholar - Rassias JM, Kim H-M: Generalized Hyers-Ulam stability for general additive functional equations in quasi--normed spaces.
*Journal of Mathematical Analysis and Applications*2009, 356(1):302–309. 10.1016/j.jmaa.2009.03.005MathSciNetView ArticleMATHGoogle Scholar - Rassias JM: Solution of the Ulam stability problem for cubic mappings.
*Glasnik Matematički. Serija III*2001, 36(1):63–72.MathSciNetMATHGoogle Scholar - Rassias JM: Solution of the Ulam stability problem for quartic mappings.
*Glasnik Matematički. Serija III*1999, 34(2):243–252.MathSciNetMATHGoogle Scholar - Saadati R, Vaezpour SM, Cho YJ: On the stability of cubic mappings and quartic mappings in random normed spaces.
*Journal of Inequalities and Applications*2009, 2009:-6.Google Scholar - Jung S-M, Brzdęk J: A note on stability of a linear functional equation of second order connected with the Fibonacci numbers and Lucas sequences.
*Journal of Inequalities and Applications*2010, 2010:-10.Google Scholar - Xu TZ, Rassias JM, Xu WX: A generalized mixed quadratic-quartic functional equation. to appear in Bulletin of the Malaysian Mathematical Sciences SocietyGoogle Scholar
- Isac G, Rassias TM: Stability of -additive mappings: applications to nonlinear analysis.
*International Journal of Mathematics and Mathematical Sciences*1996, 19(2):219–228. 10.1155/S0161171296000324MathSciNetView ArticleMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. 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.