Open Access

A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Functional Equation

Fixed Point Theory and Applications20102010:185780

DOI: 10.1155/2010/185780

Received: 24 August 2009

Accepted: 10 January 2010

Published: 21 January 2010

Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic-quartic functional equation in Banach spaces.

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.

The functional equation
(1.1)

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 [919]).

In [20], Jun and Kim considered the following cubic functional equation
(1.2)

which is called a cubic functional equation, and every solution of the cubic functional equation is said to be a cubic mapping.

In [21], Lee et al. considered the following quartic functional equation
(1.3)

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

(1) if and only if ;

(2) for all ;

(3) for all .

We recall a fundamental result in fixed point theory.

Theorem 1.1 (see [24, 25]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
(1.4)

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 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 [2732]).

This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-cubic-quartic functional equation
(1.5)

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

For a given mapping , we define
(2.1)

for all .

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.

Let be a function such that there exists an with
(2.2)
for all . Let be an odd mapping satisfying
(2.3)
for all . Then there is a unique cubic mapping such that
(2.4)

for all .

Proof.

Letting in (2.3), we get
(2.5)

for all .

Replacing by in (2.3), we get
(2.6)

for all .

By (2.5) and (2.6),
(2.7)
for all . Letting and for all , we get
(2.8)

for all .

Consider the set
(2.9)
and introduce the generalized metric on :
(2.10)

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

Now we consider the linear mapping such that
(2.11)

for all .

Let be given such that . Then
(2.12)
for all . Hence
(2.13)
for all . So implies that . This means that
(2.14)

for all .

It follows from (2.8) that
(2.15)

for all . So .

By Theorem 1.1, there exists a mapping satisfying the following.
  1. (1)
    is a fixed point of , that is,
    (2.16)
     
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.17)
This implies that is a unique mapping satisfying (2.16) such that there exists a satisfying
(2.18)
for all .
  1. (2)
    as . This implies the equality
    (2.19)
     
for all .
  1. (3)
    , which implies the inequality
    (2.20)
     

This implies that the inequality (2.4) holds.

By (2.3),
(2.21)
for all and all . So
(2.22)
for all and all . So
(2.23)

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

Corollary 2.2.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying
(2.24)
for all . Then there is a unique cubic mapping such that
(2.25)

for all .

Proof.

The proof follows from Theorem 2.1 by taking
(2.26)

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

Theorem 2.3.

Let be a function such that there exists an with
(2.27)
for all . Let be an odd mapping satisfying (2.3). Then there is a unique cubic mapping such that
(2.28)

for all .

Proof.

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

Consider the linear mapping such that
(2.29)

for all .

It follows from (2.8) that
(2.30)

for all . So .

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

Corollary 2.4.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.24). Then there is a unique cubic mapping such that
(2.31)

for all .

Proof.

The proof follows from Theorem 2.3 by taking
(2.32)

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

Theorem 2.5.

Let be a function such that there exists an with
(2.33)
for all . Let be an odd mapping satisfying (2.3). Then there is a unique additive mapping such that
(2.34)

for all .

Proof.

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

Letting and for all in (2.7), we get
(2.35)

for all .

Now we consider the linear mapping such that
(2.36)

for all .

It follows from (2.35) that
(2.37)

for all . So .

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

Corollary 2.6.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.24). Then there is a unique additive mapping such that
(2.38)

for all .

Theorem 2.7.

Let be a function such that there exists an with
(2.39)
for all . Let be an odd mapping satisfying (2.3). Then there is a unique additive mapping such that
(2.40)

for all .

Proof.

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

Consider the linear mapping such that
(2.41)

for all .

It follows from (2.35) that
(2.42)

for all . So .

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

Corollary 2.8.

Let and let be a real number with . Let be a normed vector space with norm . Let be an odd mapping satisfying (2.24). Then there is a unique additive mapping such that
(2.43)

for all .

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.

Let be a function such that there exists an with
(3.1)
for all . Let be an even mapping satisfying and (2.3). Then there is a unique quartic mapping such that
(3.2)

for all .

Proof.

Letting in (2.3), we get
(3.3)

for all .

Replacing by in (2.3), we get
(3.4)

for all .

By (3.4) and (3.5),
(3.5)
for all . Letting for all , we get
(3.6)

for all .

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

It follows from (3.16) that
(3.7)

for all . So .

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

Corollary 3.2.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is unique quartic mapping such that
(3.8)

for all .

Theorem 3.3.

Let be a function such that there exists an with
(3.9)
for all . Let be an even mapping satisfying and (2.3). Then there is a unique quartic mapping such that
(3.10)

for all .

Proof.

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

Consider the linear mapping such that
(3.11)

for all .

It follows from (3.16) that
(3.12)

for all . So .

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

Corollary 3.4.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is a unique quartic mapping such that
(3.13)

for all .

Theorem 3.5.

Let be a function such that there exists an with
(3.14)
for all . Let be an even mapping satisfying and (2.3). Then there is a unique quadratic mapping such that
(3.15)

for all .

Proof.

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

Letting for all in (3.6), we get
(3.16)

for all .

Now we consider the linear mapping such that
(3.17)

for all .

It follows from (3.16)that
(3.18)

for all . So .

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

Corollary 3.6.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is a unique quadratic mapping such that
(3.19)

for all .

Theorem 3.7.

Let be a function such that there exists an with
(3.20)
for all . Let be an even mapping satisfying and (2.3). Then there is a unique quadratic mapping such that
(3.21)

for all .

Proof.

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

Consider the linear mapping such that
(3.22)

for all .

It follows from (3.16) that
(3.23)

for all . So .

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

Corollary 3.8.

Let and let be a real number with . Let be a normed vector space with norm . Let be an even mapping satisfying and (2.24). Then there is a unique quadratic mapping such that
(3.24)

for all .

4. Generalized Hyers-Ulam Stability of the Functional Equation (1.5)

One can easily show that an odd mapping satisfies (1.5) if and only if the odd mapping is an additive-cubic mapping, that is,
(4.1)

It was shown in of [34, Lemma ] that and are cubic and additive, respectively, and that .

One can easily show that an even mapping satisfies (1.5) if and only if the even mapping is a quadratic-quartic mapping, that is,
(4.2)

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].

Let and . Then is odd and is even. and satisfy the functional equation (1.5). Let and . Then . Let and . Then . Thus
(4.3)

So we obtain the following results.

Theorem 4.1.

Let be a function such that there exists an with
(4.4)
for all . Let be a mapping satisfying and (2.3). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that
(4.5)

for all .

Proof.

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

Corollary 4.2.

Let and let be a real number with . Let be a mapping satisfying and (2.24). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that
(4.6)

for all .

Theorem 4.3.

Let be a function such that there exists an with
(4.7)
for all . Let be a mapping satisfying and (2.3). Then there exist an additive mapping , a quadratic mapping , a cubic mapping , and a quartic mapping such that
(4.8)

for all .

Proof.

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

Corollary 4.4.

Let and let be a real number with . Let be a mapping satisfying and (2.24). Then there exist an additive mapping , a quadratic mapping , a cubic mapping and a quartic mapping such that
(4.9)

for all .

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

(1)
Department of Mathematics, Daejin University
(2)
Department of Mathematics, Research Institute for Natural Sciences, Hanyang University

References

  1. 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
  2. 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
  3. 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
  4. 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
  5. 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
  6. 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
  7. Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar
  8. 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
  9. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar
  10. 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
  11. 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
  12. 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
  13. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
  14. 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
  15. Park C, Najati A: Homomorphisms and derivations in -algebras. Abstract and Applied Analysis 2007, 2007:-12.Google Scholar
  16. 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
  17. 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
  18. 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
  19. 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
  20. 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
  21. 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
  22. 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
  23. Rassias JM: Solution of the Ulam stability problem for quartic mappings. Glasnik Matematički 1999,34(54)(2):243–252.MathSciNetMATHGoogle Scholar
  24. 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
  25. 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
  26. 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
  27. 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
  28. 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
  29. 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
  30. 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
  31. Park C: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
  32. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003,4(1):91–96.MathSciNetMATHGoogle Scholar
  33. 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
  34. 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
  35. 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
  36. 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
  37. Eshaghi-Gordji M: Stability of a functional equation deriving from quartic and additive functions. preprintGoogle Scholar

Copyright

© Jung Rye Lee et al. 2010

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.