Skip to main content
  • Research Article
  • Open access
  • Published:

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

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 .

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.27.4.222

    Article  MathSciNet  MATH  Google 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/00210064

    Article  MathSciNet  MATH  Google 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/S0002-9939-1978-0507327-1

    Article  MathSciNet  MATH  Google 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.1994.1211

    Article  MathSciNet  MATH  Google Scholar 

  6. Skof F: Proprietà locali e approssimazione di operatori. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890

    Article  MathSciNet  Google Scholar 

  7. Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.

    Article  MathSciNet  MATH  Google 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/BF02941618

    Article  MathSciNet  MATH  Google Scholar 

  9. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.

    Book  MATH  Google 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.2004.03.011

    Article  MathSciNet  MATH  Google 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.2006.04.079

    Article  MathSciNet  MATH  Google 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.

    Book  MATH  Google Scholar 

  13. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.

    MATH  Google 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.200510611

    Article  MathSciNet  MATH  Google 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.2005.02.001

    Article  MathSciNet  MATH  Google 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.2000.7046

    Article  MathSciNet  MATH  Google 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:1006499223572

    Article  MathSciNet  MATH  Google 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.1993.1070

    Article  MathSciNet  MATH  Google 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.

    Article  MathSciNet  Google 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.2004.12.062

    Article  MathSciNet  MATH  Google 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.

    Article  MathSciNet  MATH  Google Scholar 

  23. Rassias JM: Solution of the Ulam stability problem for quartic mappings. Glasnik Matematički 1999,34(54)(2):243–252.

    MathSciNet  MATH  Google Scholar 

  24. L Cădariu V Radu (2003) Fixed points and the stability of Jensen's functional equation Journal of Inequalities in Pure and Applied Mathematics NumberInSeries4

    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/S0002-9904-1968-11933-0

    Article  MathSciNet  MATH  Google 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/S0161171296000324

    Article  MathSciNet  MATH  Google 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-z

    Article  MathSciNet  MATH  Google 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.

    MathSciNet  MATH  Google 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.2008.01.100

    Article  MathSciNet  MATH  Google 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. preprint

Download references

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.

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to Choonkil Park.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License ( https://creativecommons.org/licenses/by/2.0 ), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and permissions

About this article

Cite this article

Lee, J., Kim, Jh. & Park, C. A Fixed Point Approach to the Stability of an Additive-Quadratic-Cubic-Quartic Functional Equation. Fixed Point Theory Appl 2010, 185780 (2010). https://doi.org/10.1155/2010/185780

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1155/2010/185780

Keywords