- Research Article
- Open Access

# Fixed Points, Inner Product Spaces, and Functional Equations

- Choonkil Park
^{1}Email author

**2010**:713675

https://doi.org/10.1155/2010/713675

© Choonkil Park. 2010

**Received:**1 February 2010**Accepted:**5 July 2010**Published:**20 July 2010

## Abstract

Rassias introduced the following equality , , for a fixed integer . Let be real vector spaces. It is shown that, if a mapping satisfies the following functional equation for all with , which is defined by the above equality, then the mapping is realized as the sum of an additive mapping and a quadratic mapping. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the above functional equation in real Banach spaces.

## Keywords

- Banach Space
- Functional Equation
- Stability Problem
- Additive Mapping
- Positive Real Number

## 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' 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 [4] has provided a lot of influence on the development of what we call the *generalized Hyers-Ulam stability* or *Hyers-Ulam-Rassias stability* of functional equations. A generalization of the Rassias theorem was obtained by G
vruţa [5] by replacing the unbounded Cauchy difference by a general control function in the spirit of Rassias' approach.

is called a quadratic functional equation. 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 generalized Hyers-Ulam stability of the above quadratic functional equation and of two functional equations of quadratic type was obtained by C dariu and Radu [9].

for all with , where is a real vector space.

for all with (see [10]).

which is called a *functional equation of quadratic type*. In fact,
in
satisfies the functional equation of quadratic type. In particular, every solution of the functional equation of quadratic type is said to be a *quadratic-type mapping*. One can easily show that if
satisfies the quadratic functional equation then
satisfies the functional equation of quadratic type. 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 [11–24]).

Let
be a set. Then, a function
is called a *generalized metric* on
if
satisfies the following:

(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 Rassias [27] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications. By using the fixed point method, the stability problems of several functional equations have been extensively investigated by a number of authors (see [17, 28–31]).

Throughout this paper, assume that is a fixed integer greater than . Let be a real normed vector space with norm , and let be a real Banach space with norm .

In this paper, we investigate the functional equation (1.4). Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation (1.4) in real Banach spaces.

## 2. Fixed Points and Functional Equations Associated with Inner Product Spaces

We investigate the functional equation (1.4).

Lemma 2.1.

for all with , then the mapping is realized as the sum of an additive mapping and a quadratic-type mapping.

Proof.

Let and for all . Then, is an odd mapping and is an even mapping satisfying and (2.1).

for all . So, is an additive mapping.

for all . So, is a quadratic-type mapping.

for all .

Let and for all . Then, is an odd mapping and is an even mapping satisfying . If , then and .

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

Theorem 2.2.

for all .

Proof.

By the same method given in [17, 28, 32], one can easily show that is complete.

for all .

for all .

for all . Hence, .

- (1)

for all . Since is an even mapping, is an even mapping.

This implies that inequality (2.7) holds.

for all with . So, for all with . By Lemma 2.1, the mapping is a quadratic-type mapping.

Therefore, there exists a unique quadratic-type mapping satisfying (2.7).

Corollary 2.3.

for all .

Proof.

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

Remark 2.4.

for all .

for all .

Theorem 2.5.

for all .

Proof.

By the same method given in [17, 28, 32], one can easily show that is complete.

for all .

for all .

for all . Hence, .

- (1)

- (3)

This implies that inequality (2.28) holds.

for all with . So, for all with . By Lemma 2.1, the mapping is an additive mapping.

Therefore, there exists a unique additive mapping satisfying (2.28), as desired.

Corollary 2.6.

for all .

Proof.

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

Remark 2.7.

for all .

for all .

Combining Theorems 2.2 and 2.5, we obtain the following result.

Theorem 2.8.

for all .

Corollary 2.9.

for all .

Proof.

Define , and apply Theorem 2.8 to get the desired result.

Combining Remarks 2.4 and 2.7, we obtain the following result.

Remark 2.10.

for all .

for all .

## Declarations

### Acknowledgment

This paper was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

## Authors’ Affiliations

## References

- Ulam SM:
*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1960.MATHGoogle 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**(1):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 - Cǎdariu L, Radu V:
**Fixed points and the stability of quadratic functional equations.***Analele Universităţii de Vest din Timişoara*2003,**41**(1):25–48.MathSciNetMATHGoogle Scholar - Rassias ThM:
**On characterizations of inner product spaces and generalizations of the H. Bohr inequality.**In*Topics in Mathematical Analysis*.*Volume 11*. Edited by: Rassias ThM. World Scientific, Teaneck, NJ, USA; 1989:803–819.View ArticleGoogle 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 - Moslehian MS:
**On the orthogonal stability of the Pexiderized quadratic equation.***Journal of Difference Equations and Applications*2005,**11**(11):999–1004. 10.1080/10236190500273226MathSciNetView ArticleMATHGoogle Scholar - Park C-G:
**Homomorphisms between Poisson -algebras.***Bulletin of the Brazilian Mathematical Society*2005,**36**(1):79–97. 10.1007/s00574-005-0029-zMathSciNetView ArticleMATHGoogle Scholar - Park C, Cho YS, Han M-H:
**Functional inequalities associated with Jordan-von Neumann-type additive functional equations.***Journal of Inequalities and Applications*2007,**2007:**-13.Google Scholar - Park C, Cui J:
**Generalized stability of -ternary quadratic mappings.***Abstract and Applied Analysis*2007,**2007:**-6.Google Scholar - Park C, Najati A:
**Homomorphisms and derivations in -algebras.***Abstract and Applied Analysis*2007,**2007:**-12.Google Scholar - Radu V:
**The fixed point alternative and the stability of functional equations.***Fixed Point Theory*2003,**4**(1):91–96.MathSciNetMATHGoogle Scholar - Rassias ThM:
**Problem 16; 2, Report of the 27th International Symp. on Functional Equations.***Aequationes Mathematicae*1990,**39:**292–293; 309.Google Scholar - Rassias ThM:
**On the stability of the quadratic functional equation and its applications.***Studia Universitatis Babeş-Bolyai. Mathematica*1998,**43**(3):89–124.MathSciNetMATHGoogle Scholar - Rassias ThM:
**The problem of S. M. Ulam for approximately multiplicative mappings.***Journal of Mathematical Analysis and Applications*2000,**246**(2):352–378. 10.1006/jmaa.2000.6788MathSciNetView ArticleMATHGoogle 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 - Rassias ThM, Shibata K:
**Variational problem of some quadratic functionals in complex analysis.***Journal of Mathematical Analysis and Applications*1998,**228**(1):234–253. 10.1006/jmaa.1998.6129MathSciNetView ArticleMATHGoogle Scholar - Cădariu 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):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:
**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 - Cădariu L, Radu V:
**On the stability of the Cauchy functional equation: a fixed point approach.**In*Iteration Theory (ECIT '02), Grazer Mathematische Berichte*.*Volume 346*. Karl-Franzens-Universitaet Graz, Graz, Austria; 2004:43–52.Google 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.