# Stability of a Mixed Type Functional Equation on Multi-Banach Spaces: A Fixed Point Approach

- Liguang Wang
^{1}Email author, - Bo Liu
^{1}and - Ran Bai
^{1}

**2010**:283827

**DOI: **10.1155/2010/283827

© LiguangWang et al. 2010

**Received: **11 December 2009

**Accepted: **29 March 2010

**Published: **6 April 2010

## Abstract

Using fixed point methods, we prove the Hyers-Ulam-Rassias stability of a mixed type functional equation on multi-Banach spaces.

## 1. Introduction and Preliminaries

The stability problem of functional equations 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's theorem was generalized by Aoki [3] for additive mappings and by Rassias [4] for linear mappings by considering an unbounded Cauchy difference. The paper of Rassias has provided a lot of influence in the development of what we call generalized Hyers-Ulam-Rassias stability of functional equations. In 1990, Rassias [5] asked whether such a theorem can also be proved for . In 1991, Gajda [6] gave an affirmative solution to this question when , but it was proved by Gajda [6] and Rassias and Šemrl [7] that one cannot prove an analogous theorem when . In 1994, a generalization was obtained by Gavruta [8], who replaced the bound by a general control function . 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. Some of the open problems in this field were solved in the papers mentioned [9–15].

The notion of multi-normed space was introduced by Dales and Polyakov (see in [16–19]). This concept is somewhat similar to operator sequence space and has some connections with operator spaces and Banach lattices. Motivations for the study of multi-normed spaces and many examples were given in [16]. Let be a complex linear space, and let , we denote by the linear space consisting of -tuples , where . The linear operations on are defined coordinate-wise. When we write for an element in , we understand that appears in the th coordinate. The zero elements of either E or are both denoted by when there is no confusion. We denote by the set and by the group of permutations on .

Definition 1.1.

such that is a norm on for each , such that for each , and such that for each , the following axioms are satisfied:

In this case, we say that is a multi-normed space.

Suppose that is a multi-normed space and . It is easy to show that

It follows from (b) that if is a Banach space, then is a Banach space for each ; in this case is said to be a multi-Banach space.

In the following, we first recall some fundamental result in fixed-point theory.

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

We recall the following theorem of Diaz and Margolis [20].

Theorem 1.2 (see [20]).

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

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

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

Baker [21] was the first author who applied the fixed-point method in the study of Hyers-Ulam stability (see also [22]). In 2003, Cadariu and Radu applied the fixed-point method to the investigation of the Jensen functional equation (see [23, 24]). By using fixed point methods, the stability problems of several functional equations have been extensively investigated by a number of authors (see [25–27]).

In this paper, we will show the Hyers-Ulam-Rassias stability of a mixed type functional equation on multi-Banach spaces using fixed-point methods.

## 2. A Mixed Type Functional Equation

First we give some lemma needed later.

Lemma 2.1 (see [28] Lemma 6.1).

If an even function satisfies(2.1), then is quartic-quadratic function.

Lemma 2.2 (see [28] Lemma 6.2).

If an odd function satisfies (2.1), then is cubic-additive function.

Theorem 2.3.

Proof.

and satisfies (2.1). Since is also even and , we have that is quadratic by Lemma 2.1. Then is quadratic.

Theorem 2.4.

Proof.

The proof is similar to that of Theorem 2.3.

Theorem 2.5.

Proof.

Let and for all . Then we have (2.17). The uniqueness of and is easy to show.

Theorem 2.6.

Proof.

The proof is similar to that of Theorems 2.3 and 2.4.

Theorem 2.7.

Proof.

for all . Let and . The rest is similar to that of the proof of Theorem 2.5.

Theorem 2.8.

Proof.

By (2.26) and (2.28), we have (2.24).This completes the proof.

## Declarations

### Acknowledgments

This work was supported in part by the Scientific Research Project of the Department of Education of Shandong Province (no. J08LI15). The authors are grateful to the referees for their valuable suggestions.

## Authors’ Affiliations

## References

- Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8*. Interscience, 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 - Rassias ThM (Ed):
*Functional Equations, Inequalities and Applications*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2003:x+224.MATHGoogle 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 - Rassias ThM, Šemrl P:
**On the behavior of mappings which do not satisfy Hyers-Ulam stability.***Proceedings of the American Mathematical Society*1992,**114**(4):989–993. 10.1090/S0002-9939-1992-1059634-1MathSciNetView ArticleMATHGoogle Scholar - Gavruta 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 - Gavruta P:
**An answer to a question of Th. M. Rassias and J. Tabor on mixed stability of mappings.***Buletinul ŞtiinŞific al Universităţii "Politehnica" din Timişoara. Seria Matematică-Fizică*1997,**42(56)**(1):1–6.MathSciNetMATHGoogle Scholar - Gavruta P:
**On the Hyers-Ulam-Rassias stability of mappings.**In*Recent Progress in Inequalities, Mathematics and Its Applications*.*Volume 430*. Kluwer Academic Publishers, Dordrecht, The Netherlands; 1998:465–469.View ArticleGoogle Scholar - Gavruta P:
**An answer to a question of John M. Rassias concerning the stability of Cauchy equation.***Advances in Equations and Inequalities, Hadronic Mathematics Series*1999, 67–71.Google Scholar - Gavruta P:
**On a problem of G. Isac and Th. M. Rassias concerning the stability of mappings.***Journal of Mathematical Analysis and Applications*2001,**261**(2):543–553. 10.1006/jmaa.2001.7539MathSciNetView ArticleMATHGoogle Scholar - Gavruta P:
**On the Hyers-Ulam-Rassias stability of the quadratic mappings.***Nonlinear Functional Analysis and Applications*2004,**9**(3):415–428.MathSciNetMATHGoogle Scholar - Gavruta P, Hossu M, Popescu D, Căprău C:
**On the stability of mappings and an answer to a problem of Th. M. Rassias.***Annales Mathématiques Blaise Pascal*1995,**2**(2):55–60. 10.5802/ambp.47View ArticleMathSciNetMATHGoogle Scholar - Gavruta L, Gavruta P:
**On a problem of John M. Rassias concerning the stability in Ulam sense of Euler-Lagrange equation.**In*Functional Equations, Difference Inequalities and Ulam Stability Notions*. Nova Sciences; 2010:47–53.Google Scholar - Dales HG, Polyakov ME: Multi-normed spaces and multi-Banach algebras. preprintGoogle Scholar
- Dales HG, Moslehian MS:
**Stability of mappings on multi-normed spaces.***Glasgow Mathematical Journal*2007,**49**(2):321–332. 10.1017/S0017089507003552MathSciNetView ArticleMATHGoogle Scholar - Moslehian MS, Nikodem K, Popa D:
**Asymptotic aspect of the quadratic functional equation in multi-normed spaces.***Journal of Mathematical Analysis and Applications*2009,**355**(2):717–724. 10.1016/j.jmaa.2009.02.017MathSciNetView ArticleMATHGoogle Scholar - Moslehian MS:
**Superstability of higher derivations in multi-Banach algebras.***Tamsui Oxford Journal of Mathematical Sciences*2008,**24**(4):417–427.MathSciNetMATHGoogle 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 - Baker JA:
**The stability of certain functional equations.***Proceedings of the American Mathematical Society*1991,**112**(3):729–732. 10.1090/S0002-9939-1991-1052568-7MathSciNetView ArticleMATHGoogle Scholar - Agarwal RP, Xu B, Zhang W:
**Stability of functional equations in single variable.***Journal of Mathematical Analysis and Applications*2003,**288**(2):852–869. 10.1016/j.jmaa.2003.09.032MathSciNetView ArticleMATHGoogle Scholar - Cadariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 4):-7.Google Scholar
- Cadariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.**In*Iteration Theory (ECIT '02), Die Grazer Mathematischen Berichte*.*Volume 346*. Karl-Franzens-Universitaet Graz, Graz, Austria; 2004:43–52.Google Scholar - Jung S-M, Rassias JM:
**A fixed point approach to the stability of a functional equation of the spiral of Theodorus.***Fixed Point Theory and Applications*2008,**2008:**-7.Google Scholar - Park C, Rassias JM:
**Stability of the Jensen-type functional equation in -algebras: a fixed point approach.***Abstract and Applied Analysis*2009,**2009:**-17.Google Scholar - Park C, An JS:
**Stability of the Cauchy-Jensen functional equation in -algebras: a fixed point approach.***Fixed Point Theory and Applications*2008,**2008:**-11.Google Scholar - Eshaghi-Gordji M, Kaboli-Gharetapeh S, Moslehian MS, Zolfaghari S: Stability of a mixed type additive, quadratic, cubic and quartic functional equation. In Nonlinear Analysis and Variational Problems, Springer Optimization and Its Applications, 35. Edited by: Pardalos PM, Rassias ThM, Khan AA. Springer, Berlin, Germany; 2009:65–80.Google Scholar

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