Open Access

Fuzzy Stability of an Additive-Quadratic-Quartic Functional Equation

Journal of Inequalities and Applications20102010:253040

https://doi.org/10.1155/2010/253040

Received: 27 August 2009

Accepted: 30 November 2009

Published: 12 January 2010

Abstract

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

1. Introduction and Preliminaries

Katsaras [1] defined a fuzzy norm on a vector space to construct a fuzzy vector topological structure on the space. Some mathematicians have defined fuzzy norms on a vector space from various points of view [24]. In particular, Bag and Samanta [5], following Cheng and Mordeson [6], gave an idea of fuzzy norm in such a manner that the corresponding fuzzy metric is of Kramosil and Michálek type [7]. They established a decomposition theorem of a fuzzy norm into a family of crisp norms and investigated some properties of fuzzy normed spaces [8].

We use the definition of fuzzy normed spaces given in [5, 9, 10] to investigate a fuzzy version of the generalized Hyers-Ulam stability forthe following functional equation

(1.1)

in the fuzzy normed vector space setting.

Definition 1.1 (see [5, 911]).

Let be a real vector space. A function is called a fuzzy norm on if for all and all ,

for ;

if and only if for all ;

if ;

;

is a nondecreasing function of and ;

for , is continuous on .

The pair is called a fuzzy normed vector space.

The properties of fuzzy normed vector spaces and examples of fuzzy norms are given in [9, 12].

Definition 1.2 (see [5, 911]).

Let be a fuzzy normed vector space. A sequence in is said to be convergent or converge if there exists an such that for all . In this case, is called thelimit of the sequence and we denote it by - .

Definition 1.3 (see [5, 9, 10]).

Let be a fuzzy normed vector space. A sequence in is called Cauchy if for each and each there exists an such that for all and all , we have .

It is wellknown that every convergent sequence in a fuzzy normed vector space is Cauchy. If each Cauchy sequence is convergent, then the fuzzy norm is said to be complete and the fuzzy normed vector space is called a fuzzy Banach space.

We say that a mapping between fuzzy normed vector spaces and is continuous at a point if for each sequence converging to in , the sequence converges to . If is continuous at each , then is said to be continuous on (see [8]).

The stability problem of functional equations originated from a question of Ulam [13] concerning the stability of group homomorphisms. Hyers [14] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [15] for additive mappings and by Th. M. Rassias [16] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [16] 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 G vruta [17] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th.M. Rassias' approach.

The functional equation

(1.2)

is called a quadratic functional equation. In particular, every solution of the quadratic functional equation is said to be aquadratic mapping. A generalized Hyers-Ulam stability problem for the quadratic functional equation was proved by Skof [18] for mappings , where is a normed space and is a Banach space. Cholewa [19] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [20] 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 [16, 2139]).

In [40], Lee et al. considered the following quartic functional equation:

(1.3)

It is easy to show that the function satisfies the 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.

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 1.4 (see [41, 42]).

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) ;

(2)the sequence converges to a fixed point of ;

(3) is the unique fixed point of in the set ;

(4) for all .

In 1996, G. Isac and Th. M. Rassias [43] 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 [12, 4448]).

This paper is organized as follows. In Section 2, we prove the generalized Hyers-Ulam stability of the additive-quadratic-quartic functional equation (1.1) in fuzzy Banach spaces for an odd case. In Section 3, we prove the generalized Hyers-Ulam stability of the additive-quadratic-quartic functional equation (1.1) in fuzzy Banach spaces for an even case.

Throughout this paper, assume that is a vector space and that is a fuzzy Banach space.

2. Generalized Hyers-Ulam Stability of the Functional Equation (1.1): An Odd Case

One can easily show that an odd mapping satisfies (1.1) if and only if the odd mapping mapping is an additive mapping, that is,

(2.1)

One can easily show that an even mapping satisfies (1.1) if and only if the even mapping is a quadratic-quartic mapping, that is,

(2.2)

It was shown in [49, Lemma  2.1] that and are quartic and quadratic, respectively, and that .

For a given mapping , we define

(2.3)

for all .

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in fuzzy Banach spaces: an odd case.

Theorem 2.1.

Let be a function such that there exists an with
(2.4)
for all . Let be an odd mapping satisfying
(2.5)
for all and all . Then
(2.6)
exists for each and defines an additive mapping such that
(2.7)

for all and all .

Proof.

Letting in (2.5), we get
(2.8)

for all and 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  2.1 of [50].)

Now we consider the linear mapping such that

(2.11)

for all .

Let be given such that . Then

(2.12)
for all and all . Hence
(2.13)
for all and all . So implies that . This means that
(2.14)

for all .

It follows from (2.8) that

(2.15)

for all and all . So .

By Theorem 1.4, 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 and all .
  1. (2)
    as . This implies the equality
     
(2.19)
for all ;
  1. (3)
    , which implies the inequality
     
(2.20)

This implies that inequality (2.7) holds.

By (2.5),

(2.21)
for all , all and all . So
(2.22)
for all , all and all . Since for all and all ,
(2.23)

for all and all . Thus the mapping is additive, 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 and all . Then
(2.25)
exists for each and defines an additive mapping such that
(2.26)

for all and all .

Proof.

The proof follows from Theorem 2.1 by taking
(2.27)

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
(2.28)
for all . Let be an odd mapping satisfying (2.5). Then
(2.29)
exists for each and defines an additive mapping such that
(2.30)

for all and all .

Proof.

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

Consider the linear mapping such that

(2.31)

for all .

Let be given such that . Then

(2.32)
for all and all . Hence
(2.33)
for all and all . So implies that . This means that
(2.34)

for all .

It follows from (2.8) that

(2.35)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.
  1. (1)
    is a fixed point of , that is,
     
(2.36)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.37)
This implies that is a unique mapping satisfying (2.36) such that there exists a satisfying
(2.38)
for all and all .
  1. (2)
    as . This implies the equality
     
(2.39)
for all ;
  1. (3)
    , which implies the inequality
     
(2.40)

This implies that the inequality (2.30) holds.

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
(2.41)
exists for each and defines an additive mapping such that
(2.42)

for all and all .

Proof.

The proof follows from Theorem 2.3 by taking
(2.43)

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

3. Generalized Hyers-Ulam Stability of the Functional Equation (1.1):An Even Case

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the functional equation in fuzzy 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 (2.5). Then
(3.2)
exists for each and defines a quartic mapping such that
(3.3)

for all and all .

Proof.

Letting in (2.5), we get
(3.4)

for all and all .

Replacing by in (2.5), we get

(3.5)

for all and all .

By (3.4) and (3.5),

(3.6)
for all and all . Letting for all , we get
(3.7)

for all and all .

Consider the set

(3.8)
and introduce the generalized metric on
(3.9)

where, as usual, . It is easy to show that is complete. (see the proof of Lemma  2.1 of [50]).

Now we consider the linear mapping such that

(3.10)

for all .

Let be given such that . Then

(3.11)
for all and all . Hence
(3.12)
for all and all . So implies that . This means that
(3.13)

for all .

It follows from (3.7) that

(3.14)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.
  1. (1)
    is a fixed point of , that is,
     
(3.15)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.16)
This implies that is a unique mapping satisfying (3.15) such that there exists a satisfying
(3.17)
for all and all .
  1. (2)
    as . This implies the equality
     
(3.18)
for all .
  1. (3)
    , which implies the inequality
     
(3.19)

This implies that inequality (3.3) holds.

The rest of the proof is similar to that of 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
(3.20)
exists for each and defines a quartic mapping such that
(3.21)

for all and all .

Proof.

The proof follows from Theorem 3.1 by taking
(3.22)

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

Theorem 3.3.

Let be a function such that there exists an with
(3.23)
for all . Let be an even mapping satisfying and (2.5). Then
(3.24)
exists for each and defines a quartic mapping such that
(3.25)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

Consider the linear mapping such that

(3.26)

for all .

Let be given such that . Then

(3.27)
for all and all . Hence
(3.28)
for all and all . So implies that . This means that
(3.29)

for all .

It follows from (3.7) that

(3.30)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.
  1. (1)
    is a fixed point of , that is,
     
(3.31)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.32)
This implies that is a unique mapping satisfying (3.31) such that there exists a satisfying
(3.33)
for all and all .
  1. (2)
    as . This implies the equality
     
(3.34)
for all ;
  1. (3)
    , which implies the inequality
     
(3.35)

This implies that inequality (3.25) holds.

The rest of the proof is similar to that of 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
(3.36)
exists for each and defines a quartic mapping such that
(3.37)

for all and all .

Proof.

The proof follows from Theorem 3.3 by taking
(3.38)

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

Theorem 3.5.

Let be a function such that there exists an with
(3.39)
for all . Let be an even mapping satisfying and (2.5). Then
(3.40)
exists for each and defines a quadratic mapping such that
(3.41)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

Letting for all in (3.6), we get

(3.42)

for all and all .

Now we consider the linear mapping such that

(3.43)

for all .

Let be given such that . Then

(3.44)
for all and all . Hence
(3.45)
for all and all . So implies that . This means that
(3.46)

for all .

It follows from (3.42) that

(3.47)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.
  1. (1)
    is a fixed point of , that is,
     
(3.48)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.49)
This implies that is a unique mapping satisfying (3.48) such that there exists a satisfying
(3.50)
for all and all .
  1. (2)
    as . This implies the equality
     
(3.51)
for all .
  1. (3)
    , which implies the inequality
     
(3.52)

This implies that inequality (3.41) holds.

The rest of the proof is similar to that of 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
(3.53)
exists for each and defines a quadratic mapping such that
(3.54)

for all and all .

Proof.

The proof follows from Theorem 3.5 by taking
(3.55)

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

Theorem 3.7.

Let be a function such that there exists an with
(3.56)
for all . Let be an even mapping satisfying and (2.5). Then
(3.57)
exists for each and defines a quadratic mapping such that
(3.58)

for all and all .

Proof.

Let be the generalized metric space defined in the proof of Theorem 3.1.

Consider the linear mapping such that

(3.59)

for all .

Let be given such that . Then

(3.60)
for all and all . Hence
(3.61)
for all and all . So implies that . This means that
(3.62)

for all .

It follows from (3.42) that

(3.63)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.
  1. (1)
    is a fixed point of , that is,
     
(3.64)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.65)
This implies that is a unique mapping satisfying (3.64) such that there exists a satisfying
(3.66)
for all and all .
  1. (2)
    as . This implies the equality
     
(3.67)
for all .
  1. (3)
    , which implies the inequality
     
(3.68)

This implies that inequality (3.58) holds.

The rest of the proof is similar to that of 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
(3.69)
exists for each and defines a quadratic mapping such that
(3.70)

for all and all .

Proof.

The proof follows from Theorem 3.7 by taking
(3.71)

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

Declarations

Acknowledgment

This work was supported by the Hanyang University in 2009.

Authors’ Affiliations

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

References

  1. Katsaras AK: Fuzzy topological vector spaces. II. Fuzzy Sets and Systems 1984, 12(2):143–154. 10.1016/0165-0114(84)90034-4MathSciNetView ArticleMATHGoogle Scholar
  2. Felbin C: Finite-dimensional fuzzy normed linear space. Fuzzy Sets and Systems 1992, 48(2):239–248. 10.1016/0165-0114(92)90338-5MathSciNetView ArticleMATHGoogle Scholar
  3. Krishna SV, Sarma KKM: Separation of fuzzy normed linear spaces. Fuzzy Sets and Systems 1994, 63(2):207–217. 10.1016/0165-0114(94)90351-4MathSciNetView ArticleMATHGoogle Scholar
  4. Xiao J-Z, Zhu X-H: Fuzzy normed space of operators and its completeness. Fuzzy Sets and Systems 2003, 133(3):389–399. 10.1016/S0165-0114(02)00274-9MathSciNetView ArticleMATHGoogle Scholar
  5. Bag T, Samanta SK: Finite dimensional fuzzy normed linear spaces. Journal of Fuzzy Mathematics 2003, 11(3):687–705.MathSciNetMATHGoogle Scholar
  6. Cheng SC, Mordeson JN: Fuzzy linear operators and fuzzy normed linear spaces. Bulletin of the Calcutta Mathematical Society 1994, 86(5):429–436.MathSciNetMATHGoogle Scholar
  7. Kramosil I, Michálek J: Fuzzy metrics and statistical metric spaces. Kybernetika 1975, 11(5):336–344.MathSciNetMATHGoogle Scholar
  8. Bag T, Samanta SK: Fuzzy bounded linear operators. Fuzzy Sets and Systems 2005, 151(3):513–547. 10.1016/j.fss.2004.05.004MathSciNetView ArticleMATHGoogle Scholar
  9. Mirmostafaee AK, Mirzavaziri M, Moslehian MS: Fuzzy stability of the Jensen functional equation. Fuzzy Sets and Systems 2008, 159(6):730–738. 10.1016/j.fss.2007.07.011MathSciNetView ArticleMATHGoogle Scholar
  10. Mirmostafaee AK, Moslehian MS: Fuzzy versions of Hyers-Ulam-Rassias theorem. Fuzzy Sets and Systems 2008, 159(6):720–729. 10.1016/j.fss.2007.09.016MathSciNetView ArticleMATHGoogle Scholar
  11. Mirmostafaee AK, Moslehian MS: Fuzzy approximately cubic mappings. Information Sciences 2008, 178(19):3791–3798. 10.1016/j.ins.2008.05.032MathSciNetView ArticleMATHGoogle Scholar
  12. 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
  13. 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
  14. 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
  15. 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
  16. 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
  17. 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
  18. 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
  19. Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984, 27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar
  20. Czerwik St: 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
  21. Baktash E, Cho YJ, Jalili M, Saadati R, Vaezpour SM: On the stability of cubic mappings and quadratic mappings in random normed spaces. Journal of Inequalities and Applications 2008, 2008:-11.Google Scholar
  22. Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. Advances in Difference Equations to appear to appearGoogle Scholar
  23. Hyers DH, Isac G, Rassias TM: Stability of Functional Equations in Several Variables, Progress in Nonlinear Differential Equations and Their Applications. Volume 34. Birkhäuser, Boston, Mass, USA; 1998:vi+313.MATHGoogle Scholar
  24. 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
  25. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
  26. Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Bulletin des Sciences Mathématiques 2008, 132(2):87–96.View ArticleMATHMathSciNetGoogle Scholar
  27. Park C, Cui J: Generalized stability of -ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:-6.Google Scholar
  28. Park C, Najati A: Homomorphisms and derivations in -algebras. Abstract and Applied Analysis 2007, 2007:-12.Google Scholar
  29. Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984, 108(4):445–446.MATHMathSciNetGoogle Scholar
  30. Rassias JM: Refined Hyers-Ulam approximation of approximately Jensen type mappings. Bulletin des Sciences Mathématiques 2007, 131(1):89–98.View ArticleMATHMathSciNetGoogle Scholar
  31. Rassias JM, Rassias MJ: Asymptotic behavior of alternative Jensen and Jensen type functional equations. Bulletin des Sciences Mathématiques 2005, 129(7):545–558.View ArticleMATHMathSciNetGoogle Scholar
  32. Rassias ThM: Problem 16; 2, report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 1990, 39(2–3):292–293.Google Scholar
  33. Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Universitatis Babes-Bolyai 1998, 43(3):89–124.MATHMathSciNetGoogle Scholar
  34. 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
  35. 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
  36. 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
  37. Rassias ThM, Šemrl P: On the behavior of mappings which do not satisfy Hyers-Ulam stability. Proceedings of the American Mathematical Society 1992, 114(4):989–993. 10.1090/S0002-9939-1992-1059634-1MathSciNetView ArticleMATHGoogle Scholar
  38. 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
  39. 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
  40. 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
  41. 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):Google Scholar
  42. 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
  43. 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
  44. Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory, Grazer Mathematische Berichte. Volume 346. Karl-Franzens-Universitaet, Graz, Austria; 2004:43–52.Google Scholar
  45. 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
  46. 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
  47. Park C: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
  48. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003, 4(1):91–96.MathSciNetMATHGoogle Scholar
  49. 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
  50. 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.MathSciNetView ArticleMATHGoogle Scholar

Copyright

© Choonkil Park. 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.