# 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

## 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

We recall a fundamental result in fixed point theory.

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

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

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

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

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.

Proof.

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

This implies that the inequality (2.4) holds.

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

Corollary 2.2.

Proof.

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

Theorem 2.3.

Proof.

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

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

Corollary 2.4.

Proof.

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

Theorem 2.5.

Proof.

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

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

Corollary 2.6.

Theorem 2.7.

Proof.

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

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

Corollary 2.8.

## 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.

Proof.

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

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

Corollary 3.2.

Theorem 3.3.

Proof.

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

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

Corollary 3.4.

Theorem 3.5.

Proof.

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

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

Corollary 3.6.

Theorem 3.7.

Proof.

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

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

Corollary 3.8.

## 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.

Proof.

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

Corollary 4.2.

Theorem 4.3.

Proof.

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

Corollary 4.4.

## 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

## References

- Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. 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 - Aoki T:
**On the stability of the linear transformation in Banach spaces.***Journal of the Mathematical Society of Japan*1950,**2:**64–66. 10.2969/jmsj/00210064MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**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 - Skof F:
**Proprietà locali e approssimazione di operatori.***Rendiconti del Seminario Matematico e Fisico di Milano*1983,**53:**113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar - Cholewa PW:
**Remarks on the stability of functional equations.***Aequationes Mathematicae*1984,**27**(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar - Czerwik S:
**On the stability of the quadratic mapping in normed spaces.***Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*1992,**62:**59–64. 10.1007/BF02941618MathSciNetView ArticleMATHGoogle Scholar - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle 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 - Hyers DH, Isac G, Rassias ThM:
*Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications, 34*. Birkhäuser, Boston, Mass, USA; 1998:vi+313.View ArticleMATHGoogle Scholar - Jung S-M:
*Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis*. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar - Park C:
**Hyers-Ulam-Rassias stability of a generalized Apollonius-Jensen type additive mapping and isomorphisms between -algebras.***Mathematische Nachrichten*2008,**281**(3):402–411. 10.1002/mana.200510611MathSciNetView ArticleMATHGoogle Scholar - Park C, Najati A:
**Homomorphisms and derivations in -algebras.***Abstract and Applied Analysis*2007,**2007:**-12.Google Scholar - Rassias JM, Rassias MJ:
**Asymptotic behavior of alternative Jensen and Jensen type functional equations.***Bulletin des Sciences Mathématiques*2005,**129**(7):545–558. 10.1016/j.bulsci.2005.02.001View ArticleMathSciNetMATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations in Banach spaces.***Journal of Mathematical Analysis and Applications*2000,**251**(1):264–284. 10.1006/jmaa.2000.7046MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62**(1):23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM, Šemrl P:
**On the Hyers-Ulam stability of linear mappings.***Journal of Mathematical Analysis and Applications*1993,**173**(2):325–338. 10.1006/jmaa.1993.1070MathSciNetView ArticleMATHGoogle Scholar - Jun K-W, Kim H-M:
**The generalized Hyers-Ulam-Rassias stability of a cubic functional equation.***Journal of Mathematical Analysis and Applications*2002,**274**(2):267–278.MathSciNetView ArticleGoogle Scholar - Lee SH, Im SM, Hwang IS:
**Quartic functional equations.***Journal of Mathematical Analysis and Applications*2005,**307**(2):387–394. 10.1016/j.jmaa.2004.12.062MathSciNetView ArticleMATHGoogle Scholar - Chung JK, Sahoo PK:
**On the general solution of a quartic functional equation.***Bulletin of the Korean Mathematical Society*2003,**40**(4):565–576.MathSciNetView ArticleMATHGoogle Scholar - Rassias JM:
**Solution of the Ulam stability problem for quartic mappings.***Glasnik Matematički*1999,**34(54)**(2):243–252.MathSciNetMATHGoogle Scholar - Cădariu, L, Radu, V 2003Fixed points and the stability of Jensen's functional equationJournal of Inequalities in Pure and Applied Mathematics(1, article 4)Google Scholar
- Diaz JB, Margolis B:
**A fixed point theorem of the alternative, for contractions on a generalized complete metric space.***Bulletin of the American Mathematical Society*1968,**74:**305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar - Isac G, Rassias ThM:
**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 - Cădariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.**In*Iteration Theory (ECIT '02), Grazer Math. Ber.*.*Volume 346*. Karl-Franzens-Univ. Graz, Graz, Austria; 2004:43–52.Google Scholar - Cădariu L, Radu V:
**Fixed point methods for the generalized stability of functional equations in a single variable.***Fixed Point Theory and Applications*2008,**2008:**-15.Google Scholar - Mirzavaziri M, Moslehian MS:
**A fixed point approach to stability of a quadratic equation.***Bulletin of the Brazilian Mathematical Society*2006,**37**(3):361–376. 10.1007/s00574-006-0016-zMathSciNetView ArticleMATHGoogle Scholar - Park C:
**Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras.***Fixed Point Theory and Applications*2007,**2007:**-15.Google Scholar - Park C:
**Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach.***Fixed Point Theory and Applications*2008,**2008:**-9.Google Scholar - Radu V:
**The fixed point alternative and the stability of functional equations.***Fixed Point Theory*2003,**4**(1):91–96.MathSciNetMATHGoogle 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. 10.1016/j.jmaa.2008.01.100MathSciNetView ArticleMATHGoogle Scholar - Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S:
**Stability of an additive-cubic-quartic functional equation.***Advances in Difference Euqations*2009,**2009:**-20.Google Scholar - Eshaghi-Gordji M, Abbaszadeh S, Park C:
**On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces.***Journal of Inequalities and Applications*2009,**2009:**-26.Google Scholar - Eshaghi-Gordji M, Kaboli Gharetapeh S, Rassias JM, Zolfaghari S:
**Solution and stability of a mixed type additive, quadratic, and cubic functional equation.***Advances in Difference Equations*2009,**2009:**-17.Google Scholar - Eshaghi-Gordji M: Stability of a functional equation deriving from quartic and additive functions. preprintGoogle Scholar

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