- Research Article
- Open Access

# Stability of Quadratic Functional Equations via the Fixed Point and Direct Method

- Eunyoung Son
^{1}, - Juri Lee
^{1}and - Hark-Mahn Kim
^{1}Email author

**2010**:635720

https://doi.org/10.1155/2010/635720

© Eunyoung Son et al. 2010

**Received:**13 October 2009**Accepted:**19 January 2010**Published:**1 March 2010

## Abstract

Cădariu and Radu applied the fixed point theorem to prove the stability theorem of Cauchy and Jensen functional equations. In this paper, we prove the generalized Hyers-Ulam stability via the fixed point method and investigate new theorems via direct method concerning the stability of a general quadratic functional equation.

## Keywords

- Banach Space
- Functional Equation
- Fixed Point Theorem
- Stability Theorem
- Unbounded Function

## 1. Introduction

In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.

*Let*
*be a group and let*
*be a metric group with metric*
*. Given*
*, does there exist a*
*such that if*
*satisfies*
*for all*
*, then a homomorphism*
*exists with*
*for all*
*?*

The concept of stability for functional equations arises when we replace the functional equation by an inequality which acts as a perturbation of the equation. Thus we say that a functional equation is stable if any mapping approximately satisfying the equation is near to a true solution such that and for some function depending on the given function In 1941, the first result concerning the stability of functional equations for the case where and are Banach spaces was presented by Hyers [2]. In fact, he proved that each solution of the inequality for all can be approximated by a unique additive function defined by such that for every . Moreover, if is continuous in for each fixed , then the function is linear. And then Aoki [3], Bourgin [4], and Forti [5] have investigated the stability theorems of functional equations which generalize the Hyers' result. In 1978, Rassias [6] attempted to weaken the condition for the bound of Cauchy difference controlled by a sum of unbounded function and provided a generalization of Hyers' theorem. In 1991, Gajda [7] gave an affirmative solution to this question for by following the same approach as in [6]. Rassias [8] established a similar stability theorem for the unbounded Cauchy difference controlled by a product of unbounded function . G vruţa [9] provided a further generalization of Rassias' theorem by replacing the bound of Cauchy difference by a general control function. During the last two decades a number of papers and research monographs have been published on various generalizations and applications of the generalized Hyers-Ulam stability to a number of functional equations and mappings (see [10–15]).

Let and be real vector spaces. A function is called a quadratic function if and only if is a solution function of the quadratic functional equation

for all It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function such that for all , where the mapping is given by . See [16, 17] for the details.

The Hyers-Ulam stability of the quadratic functional equation (1.1) was first proved by Skof [18] for functions , where is a normed space and is a Banach space. Cholewa noticed that Skof's theorem is also valid if is replaced by an Abelian group. Czerwik [19] proved the generalized Hyers-Ulam stability of quadratic functional equation (1.1) in the spirit of Rassias approach. On the other hand, according to the theorem of Borelli and Forti [20], we know the following generalization of stability theorem for quadratic functional equation. Let be a 2-divisible Abelian group and a Banach space, and let be a mapping with satisfying the inequality

for all . Assume that one of the series

holds for all , then there exists a unique quadratic function such that

for all . 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 [21–27].

In 1996, Isac and Rassias [28] applied the stability theory of functional equations to prove fixed point theorems and study some new applications in nonlinear analysis.Radu [29], Cãdariu and Radu [30, 31] applied the fixed point theorem of alternative to the investigation of Cauchy and Jensen functional equations. Recently, Jung et al. [32],Jung [33, 34],Jung and Lee [35],Jung and Min [36],Jung and Rassias [37] have obtained the generalized Hyers-Ulam stability of functional equations via the fixed point method.

Now, we see that the norm defined by a real inner product space satisfies the following equality:

for all vectors Thus employing the last equality, we introduce to consider the following functional equation

with several variables for any fixed with . It is obvious that if in (1.6), then the solution function is even and thus it reduces to (1.1). Conversely, we observe that the general solution of (1.6) in the class of all functions between vector spaces is exactly a quadratic function. In this paper, we are going to investigate the general solution of (1.6) and then we are to prove the generalized Hyers-Ulam stability of (1.6) for a large class of functions from vector spaces into complete -normed spaces by using fixed point method, and direct method.

## 2. Stability of (1.6) by Fixed Point Method

For notational convenience, given a mapping , we define the difference operator of (1.6) by

for all , which is called the approximate remainder of the functional equation (1.6) and acts as a perturbation of the equation.

We now introduce a fundamental result of fixed point theory. We refer to [38] for the proof of it. For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [39].

Theorem.

Let be a generalized complete metric space (i.e., may assume infinite values). Assume that is a strictly contractive operator, that is, there exists a Lipschitz constant with such that for all Then for a given element one of the following assertions is true:

there exists a nonnegative integer such that

the sequence converges to a fixed point of ;

is the unique fixed point of in

Throughout this paper, we consider a -Banach space. Let be a real number with and let denote either real field or complex field . Suppose is a vector space over . A function is called a -norm if and only if it satisfies

A -Banach space is a -normed space which is complete with respect to the -norm. Now we are ready to investigate the generalized Hyers-Ulam stability problem for the functional equation (1.6) using the fixed point method. From now on, let X be a linear space and let Y be a -Banach space over unless we give any specific reference where is a fixed real number with

Theorem 2.2.

Proof.

for all that is, Thus we see that for any and so is strictly contractive with constant on .

which yields inequality (2.4).

for all , which implies that is a solution of (1.6) and so the mapping is quadratic.

To prove the uniqueness of , assume now that is another quadratic mapping satisfying inequality (2.4). Then is a fixed point of and Since the mapping is a unique fixed point of in the set we see that by Theorem 2.1 The proof is complete.

The following theorem is an alternative result of Theorem 2.2.

Theorem.

Proof.

for all that is, Thus we see that for any and so is strictly contractive with constant on .

which yields the inequality (2.16).

Replacing instead of in the last part of Theorem 2.2, we can prove that is a unique quadratic function satisfying (2.16) for all

As applications, one has the following corollaries concerning the stability of (1.6).

Corollary.

Proof.

Letting and then applying Theorem 2.2 with contractive constant , we obtain easily the result.

Corollary.

Proof.

Letting for all and then applying Theorem 2.2 with contractive constant and Theorem 2.3 with contractive constant , we obtain easily the results.

## 3. Stability of (1.6) by Direct Method

In the next two theorems, let be a mapping satisfying one of the conditions

Theorem.

Proof.

for all . Therefore letting , one has for all , completing the proof of uniqueness.

Theorem.

Proof.

for all . Therefore the function is quadratic.

for all and all . Therefore letting , one has for all . This completes the proof.

In the following corollary, we have a stability result of (1.6) with difference operator bounded by the sum of powers of -norms.

Corollary.

Proof.

Letting for all and then applying Theorems 3.1 and 3.2, we obtain easily the results.

We observe that if and in Corollary 3.3, then the stability result obtained by the fixed point method in Corollary 2.5 is somewhat different from the stability result obtained by direct method in Corollary 3.3. The stability result in Corollary 3.3 is sharper than that of Corollary 2.5.

In the next corollary, we get a stability result of (1.6) with difference operator bounded by the product of powers of -norms.

Corollary.

for all and for all if , where if .

Proof.

We remark that satisfies condition (3.1) for the case or condition (3.2) for the case . By Theorems 3.1 and 3.2, we get the results.

We observe that if in Corollary 3.4, then the stability result obtained by the fixed point method with contractive constants respectively, coincides with the stability result (3.28) obtained by direct method.

## Declarations

### Acknowledgment

This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0070940).

## 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 - Bourgin DG: Classes of transformations and bordering transformations.
*Bulletin of the American Mathematical Society*1951, 57: 223–237. 10.1090/S0002-9904-1951-09511-7MathSciNetView ArticleMATHGoogle Scholar - Forti GL: An existence and stability theorem for a class of functional equations.
*Stochastica*1980, 4(1):23–30. 10.1080/17442508008833155MathSciNetView 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 - 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 JM: On approximation of approximately linear mappings by linear mappings.
*Bulletin des Sciences Mathématiques*1984, 108(4):445–446.MathSciNetMATHGoogle 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 - Czerwik S:
*Functional Equations and Inequalities in Several Variables*. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar - Eshaghi Gordji M, Karimi T, Kaboli Gharetapeh S: Approximately -Jordan homomorphisms on Banach algebras.
*Journal of Inequalities and Applications*2009, 2009:-8.Google 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 - Najati A, Park C: Fixed points and stability of a generalized quadratic functional equation.
*Journal of Inequalities and Applications*2009, 2009:-19.Google Scholar - Park C, An JS, Moradlou F: Additive functional inequalities in Banach modules.
*Journal of Inequalities and Applications*2008, 2008:-10.Google Scholar - Aczél J, Dhombres J:
*Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications*.*Volume 31*. Cambridge University Press, Cambridge, UK; 1989:xiv+462.View ArticleMATHGoogle Scholar - Hyers DH, Rassias ThM: Approximate homomorphisms.
*Aequationes Mathematicae*1992, 44(2–3):125–153. 10.1007/BF01830975MathSciNetView ArticleMATHGoogle Scholar - Skof F: Local properties and approximation of operators.
*Rendiconti del Seminario Matematico e Fisico di Milano*1983, 53: 113–129. 10.1007/BF02924890MathSciNetView 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 - Borelli C, Forti GL: On a general Hyers-Ulam stability result.
*International Journal of Mathematics and Mathematical Sciences*1995, 18(2):229–236. 10.1155/S0161171295000287MathSciNetView ArticleMATHGoogle Scholar - Bouikhalene B, Elqorachi E, Rassias JM: The superstability of d'Alembert's functional equation on the Heisenberg group.
*Applied Mathematics Letters*2010, 23(1):105–109. 10.1016/j.aml.2009.08.013MathSciNetView ArticleMATHGoogle Scholar - Cao H-X, Lv J-R, Rassias JM: Superstability for generalized module left derivations and generalized module derivations on a Banach module. I.
*Journal of Inequalities and Applications*2009, 2009:-10.Google Scholar - Gilányi A: On the stability of monomial functional equations.
*Publicationes Mathematicae Debrecen*2000, 56(1–2):201–212.MathSciNetMATHGoogle 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 - Kim H-M, Rassias JM, Cho Y-S: Stability problem of Ulam for Euler-Lagrange quadratic mappings.
*Journal of Inequalities and Applications*2007, 2007:-15.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 - Savadkouhi MB, Eshaghi Gordji M, Rassias JM, Ghobadipour N: Approximate ternary Jordan derivations on Banach ternary algebras.
*Journal of Mathematical Physics*2009, 50(4):-9.MathSciNetView 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 - Radu V: The fixed point alternative and the stability of functional equations.
*Fixed Point Theory*2003, 4(1):91–96.MathSciNetMATHGoogle 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):-7.MathSciNetMATHGoogle 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 - Jung S-M, Kim T-S, Lee K-S: A fixed point approach to the stability of quadratic functional equation.
*Bulletin of the Korean Mathematical Society*2006, 43(3):531–541.MathSciNetView ArticleMATHGoogle Scholar - Jung S-M: A fixed point approach to the stability of isometries.
*Journal of Mathematical Analysis and Applications*2007, 329(2):879–890. 10.1016/j.jmaa.2006.06.098MathSciNetView ArticleMATHGoogle Scholar - Jung S-M: A fixed point approach to the stability of a Volterra integral equation.
*Fixed Point Theory and Applications*2007, 2007:-9.Google Scholar - Jung S-M, Lee Z-H: A fixed point approach to the stability of quadratic functional equation with involution.
*Fixed Point Theory and Applications*2008, 2008:-11.Google Scholar - Jung S-M, Min S: A fixed point approach to the stability of the functional equation .
*Fixed Point Theory and Applications*2009, 2009:-8.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 - 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(2):305–309. 10.1090/S0002-9904-1968-11933-0MathSciNetView ArticleMATHGoogle Scholar - Hyers DH, Isac G, Rassias ThM:
*Topics in Nonlinear Analysis and Applications*. World Scientific, River Edge, NJ, USA; 1997:xiv+699.View 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.