Abstract and Applied Analysis

Volume 2009, Article ID 907167, 11 pages

http://dx.doi.org/10.1155/2009/907167

## The Stability of a Quadratic Functional Equation with the Fixed Point Alternative

^{1}Department of Mathematics, Research Institute for Natural Sciences, Hanyang University, Seoul 133-791, South Korea^{2}Department of Mathematics, Hanyang University, Seoul 133-791, South Korea

Received 15 September 2009; Accepted 1 December 2009

Academic Editor: W. A. Kirk

Copyright © 2009 Choonkil Park and Ji-Hye Kim. 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.

#### Abstract

Lee, An and Park introduced the quadratic functional equation and proved the stability of the quadratic functional equation in the spirit of Hyers, Ulam and Th. M. Rassias. Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation in Banach spaces.

#### 1. Introduction

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 Th. M. Rassias [4] for linear mappings by considering an unbounded Cauchy difference.

Theorem 1.1 (Th. M. Rassias). *Let be a mapping from a normed vector space into a Banach space subject to the inequality
**
for all , where and are constants with and . Then the limit
**
exists for all , and is the unique additive mapping which satisfies
**
for all . Also, if for each the function is continuous in , then is -linear.*

The above inequality (1.1) has provided a lot of influence in the development of what is now known as a * generalized Hyers-Ulam stability* of functional equations. Beginning around the year 1980, the topic of approximate homomorphisms, or the stability of the equation of homomorphism, was studied by a number of mathematicians. Găvruta [5] generalized the Rassias’ result.

Theorem 1.2 (see [6–8]). *Let be a real normed linear space and a real complete normed linear space. Assume that is an approximately additive mapping for which there exist constants and such that satisfies inequality
**
for all . Then there exists a unique additive mapping satisfying
**
for all . If, in addition, is a mapping such that the transformation is continuous in for each fixed , then is an -linear mapping.*

The functional equation

is called a * quadratic functional equation*. In particular, every solution of the quadratic functional equation is said to be a * quadratic function*. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [9] for mappings , where is a normed space and is a Banach space. Cholewa [10] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [11] proved the generalized Hyers-Ulam stability of the quadratic functional equation. Several functional equations have been investigated in [12–25].

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

We recall a fundamental result in fixed point theory.

Theorem 1.3 (see [26–28]). *Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
**
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 .*

Lee et al. [29] proved that a mapping satisfies

for all if and only if the mapping satisfies

for all .

Using the fixed point method, Park [14] proved the generalized Hyers-Ulam stability of the quadratic functional equation

in Banach spaces.

In this paper, using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation (1.8) in Banach spaces.

Throughout this paper, assume that is a normed vector space with norm and that is a Banach space with norm .

#### 2. Fixed Points and Generalized Hyers-Ulam Stability of a Quadratic Functional Equation

For a given mapping , we define

for all .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the quadratic functional equation .

Theorem 2.1. *Let be a mapping for which there exists a function with such that
**
for all . If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *Consider the set
and introduce the* generalized metric* on :
It is easy to show that is complete.

Now we consider the linear mapping such that
for all .

By [30, Theorem ],
for all .

Letting in (2.2), we get
for all . So
for all . Hence .

By Theorem 1.3, there exists a mapping such that

(1) is a fixed point of , that is,
for all . The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (2.10) such that there exists satisfying
for all .

(2) as . This implies the equality
for all .

(3) , which implies the inequality
This implies that the inequality (2.3) holds.

It follows from (2.2) and (2.13) that
for all . So for all .

By [29, Proposition ], the mapping is quadratic, as desired.

Corollary 2.2. *Let and be positive real numbers, and let be a mapping such that
**
for all . Then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *The proof follows from Theorem 2.1 by taking
for all . Then , and we get the desired result.

Theorem 2.3. *Let be a mapping for which there exists a function satisfying (2.2) and . If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *We consider the linear mapping such that
for all .

It follows from (2.8) that
for all . Hence .

By Theorem 1.3, there exists a mapping such that

(1) is a fixed point of , that is,
for all . The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (2.22) such that there exists satisfying
for all .

(2) as . This implies the equality
for all .

(3) , which implies the inequality
which implies that the inequality (2.19) holds.

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

Corollary 2.4. *Let and be positive real numbers, and let be a mapping satisfying (2.16). Then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *The proof follows from Theorem 2.3 by taking
for all . Then and, we get the desired result.

Theorem 2.5. *Let be a mapping for which there exists a function satisfying (2.2). If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *Consider the set
and introduce the * generalized metric * on :
It is easy to show that is complete.

Now we consider the linear mapping such that
for all .

By [30, Theorem ],
for all .

Letting in (2.2), we get
for all . So
for all . Hence .

By Theorem 1.3, there exists a mapping such that

(1) is a fixed point of , that is,
for all . The mapping is a unique fixed point of in the set
This implies that is a unique mapping satisfying (2.36) such that there exists satisfying
for all .

(2) as . This implies the equality
for all .

(3) , which implies the inequality
This implies that the inequality (2.29) holds.

It follows from (2.2) and (2.39) that
for all . So for all .

By [29, Proposition ], the mapping is quadratic, as desired.

Corollary 2.6. *Let and be positive real numbers, and let be a mapping satisfying (2.16). Then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *The proof follows from Theorem 2.5 by taking
for all . Then and, we get the desired result.

Corollary 2.7. *Let and be positive real numbers, and let be a mapping such that
**
for all . Then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *The proof follows from Theorem 2.5 by taking
for all . Then and, we get the desired result.

Theorem 2.8. *Let be a mapping for which there exists a function satisfying (2.2). If there exists an such that for all , then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *We consider the linear mapping such that
for all .

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

Corollary 2.9. *Let and be positive real numbers, and let be a mapping satisfying (2.16). Then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *The proof follows from Theorem 2.8 by taking
for all . Then , and we get the desired result.

Corollary 2.10. *Let and be positive real numbers, and let be a mapping satisfying (2.44). Then there exists a unique quadratic mapping satisfying (1.8) and
**
for all .*

*Proof. *The proof follows from Theorem 2.8 by taking
for all . Then , and we get the desired result.

#### Acknowledgment

The first author was supported by Hanyang University in 2009.

#### References

- S. M. Ulam,
*Problems in Modern Mathematics*, John Wiley & Sons, New York, NY, USA, 1960. - D. H. Hyers, “On the stability of the linear functional equation,”
*Proceedings of the National Academy of Sciences of the United States of America*, vol. 27, pp. 222–224, 1941. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - T. Aoki, “On the stability of the linear transformation in Banach spaces,”
*Journal of the Mathematical Society of Japan*, vol. 2, pp. 64–66, 1950. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “On the stability of the linear mapping in Banach spaces,”
*Proceedings of the American Mathematical Society*, vol. 72, no. 2, pp. 297–300, 1978. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. Găvruta, “A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 184, no. 3, pp. 431–436, 1994. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Journal of Functional Analysis*, vol. 46, no. 1, pp. 126–130, 1982. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “On approximation of approximately linear mappings by linear mappings,”
*Bulletin des Sciences Mathématiques*, vol. 108, no. 4, pp. 445–446, 1984. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. M. Rassias, “Solution of a problem of Ulam,”
*Journal of Approximation Theory*, vol. 57, no. 3, pp. 268–273, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - F. Skof, “Local properties and approximation of operators,”
*Rendiconti del Seminario Matematico e Fisico di Milano*, vol. 53, pp. 113–129, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - P. W. Cholewa, “Remarks on the stability of functional equations,”
*Aequationes Mathematicae*, vol. 27, no. 1-2, pp. 76–86, 1984. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - S. Czerwik, “On the stability of the quadratic mapping in normed spaces,”
*Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg*, vol. 62, pp. 59–64, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - D. Miheţ and V. Radu, “On the stability of the additive Cauchy functional equation in random normed spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 343, no. 1, pp. 567–572, 2008. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Park, “Fixed points and Hyers-Ulam-Rassias stability of Cauchy-Jensen functional equations in Banach algebras,”
*Fixed Point Theory and Applications*, vol. 2007, Article ID 50175, 15 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Park, “Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach,”
*Fixed Point Theory and Applications*, vol. 2008, Article ID 493751, 9 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Park, Y. S. Cho, and M.-H. Han, “Functional inequalities associated with Jordan-von Neumann-type additive functional equations,”
*Journal of Inequalities and Applications*, vol. 2007, Article ID 41820, 13 pages, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - C. Park and J. Cui, “Generalized stability of ${C}^{*}$-ternary quadratic mappings,”
*Abstract and Applied Analysis*, vol. 2007, Article ID 23282, 6 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - C. Park and J. Hou, “Homomorphisms between ${C}^{*}$-algebras associated with the Trif functional equation and linear derivations on ${C}^{*}$-algebras,”
*Journal of the Korean Mathematical Society*, vol. 41, no. 3, pp. 461–477, 2004. View at Google Scholar · View at MathSciNet - C. Park and A. Najati, “Homomorphisms and derivations in ${C}^{*}$-algebras,”
*Abstract and Applied Analysis*, vol. 2007, Article ID 80630, 12 pages, 2007. View at Publisher · View at Google Scholar · View at MathSciNet - C. Park and J. Park, “Generalized Hyers-Ulam stability of an Euler-Lagrange type additive mapping,”
*Journal of Difference Equations and Applications*, vol. 12, no. 12, pp. 1277–1288, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “On the stability of the quadratic functional equation and its applications,”
*Studia Universitatis Babeş-Bolyai Mathematica*, vol. 43, no. 3, pp. 89–124, 1998. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “The problem of S. M. Ulam for approximately multiplicative mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 246, no. 2, pp. 352–378, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “On the stability of functional equations in Banach spaces,”
*Journal of Mathematical Analysis and Applications*, vol. 251, no. 1, pp. 264–284, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias, “On the stability of functional equations and a problem of Ulam,”
*Acta Applicandae Mathematicae*, vol. 62, no. 1, pp. 23–130, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias and P. Šemrl, “On the Hyers-Ulam stability of linear mappings,”
*Journal of Mathematical Analysis and Applications*, vol. 173, no. 2, pp. 325–338, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - Th. M. Rassias and K. Shibata, “Variational problem of some quadratic functionals in complex analysis,”
*Journal of Mathematical Analysis and Applications*, vol. 228, no. 1, pp. 234–253, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Cǎdariu and V. Radu, “On the stability of the Cauchy functional equation: a fixed point approach,” in
*Iteration Theory*, vol. 346 of*Grazer Mathematische Berichte*, pp. 43–52, Karl-Franzens-Universitaet Graz, Graz, Austria, 2004. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. B. Diaz and B. Margolis, “A fixed point theorem of the alternative for contractions on a generalized complete metric space,”
*Bulletin of the American Mathematical Society*, vol. 74, pp. 305–309, 1968. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - V. Radu, “The fixed point alternative and the stability of functional equations,”
*Fixed Point Theory*, vol. 4, no. 1, pp. 91–96, 2003. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - J. Lee, J. S. An, and C. Park, “On the stability of quadratic functional equations,”
*Abstract and Applied Analysis*, vol. 2008, Article ID 628178, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet - L. Cǎdariu and V. Radu, “Fixed points and the stability of Jensen's functional equation,”
*Journal of Inequalities in Pure and Applied Mathematics*, vol. 4, no. 1, article 4, 2003. View at Google Scholar