Open Access

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

Fixed Point Theory and Applications20092009:918785

DOI: 10.1155/2009/918785

Received: 23 August 2009

Accepted: 23 October 2009

Published: 19 November 2009

Abstract

Using the fixed point method, we prove the generalized Hyers-Ulam stability of the following additive-quadratic-cubic 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 for the 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 orconverge if there exists an such that for all . In this case, is called thelimit of the sequence and we denote it by - .

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 becomplete 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 , then the sequence converges to . If is continuous at each , then is said to be continuous on (see [8]).

In 1940, Ulam [13] 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.

We are given a group and a metric group with metric . Given , does there exist a such that if satisfies for all , then a homomorphism exists with for all ?

By now an affirmative answer has been given in several cases, and some interesting variations of the problem have also been investigated. We will call such an an approximate homomorphism.

In 1941, Hyers [14] considered the case of approximately additive mappings , where and are Banach spaces and satisfies the Hyers inequality

(1.2)

for all . It was shown that the limit

(1.3)

exists for all and that is the unique additive mapping satisfying

(1.4)

for all .

No continuity conditions are required for this result, but if is continuous in the real variable for each fixed , then is -linear, and if is continuous at a single point of , then is also continuous.

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 Găvruţa [17] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias' approach.

In 1982–1994, a generalization of the Hyers's result was proved by J. M. Rassias. He introduced the following weaker condition:

(1.5)

for all , controlled by a product of different powers of norms, where and real numbers , and retained the condition of continuity of in for each fixed . Besides he investigated that it is possible to replace in the above Hyers inequality by a nonnegative real-valued function such that the pertinent series converges and other conditions hold and still obtain stability results. In all the cases investigated in these results, the approach to the existence question was to prove asymptotic type formulas of the form

(1.6)

Theorem 1.3 (see [1823]).

Let be a real normed linear space and a real Banach space. Assume that is an approximately additive mapping for which there exist constants and such that and satisfies the Cauchy-Rassias inequality
(1.7)
for all . Then there exists a unique additive mapping satisfying
(1.8)

for all . If, in addition, is a mapping such that is continuous in for each fixed , then is an -linear mapping.

The functional equation

(1.9)

is called aquadratic 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 [24] for mappings , where is a normed space and is a Banach space. Cholewa [25] noticed that the theorem of Skof is still true if the relevant domain is replaced by an Abelian group. Czerwik [26] 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 [2769]).

In [70], Jun and Kim considered the following cubic functional equation:

(1.10)

It is easy to show that the function satisfies the functional (1.10), which is called a cubic functional equation and every solution of the cubic functional equation is said to be a cubic 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 [71, 72]).

Let be a complete generalized metric space and let be a strictly contractive mapping with Lipschitz constant . Then for each given element , either
(1.11)

for all nonnegative integers or there exists a positive integer such that

(1) ;
  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, Isac and Th. M. Rassias [73] 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 [7478]).

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

(2.1)

It was shown in [79, Lemma ] that and are cubic and additive, respectively, and that .

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

(2.2)

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 a cubic mapping such that
(2.7)

for all and all .

Proof.

Letting in (2.5), we get
(2.8)

for all and all .

Replacing by in (2.5), we get

(2.9)

for all and all .

By (2.8) and (2.9),

(2.10)
for all and all . Letting and for all , we get
(2.11)

for all and all .

Consider the set

(2.12)
and introduce the generalized metric on :
(2.13)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma of [80].)

Now we consider the linear mapping such that

(2.14)

for all .

Let be given such that . Then

(2.15)
for all and all . Hence
(2.16)
for all and all . So implies that . This means that
(2.17)

for all .

It follows from (2.11) that

(2.18)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(2.19)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.20)
This implies that is a unique mapping satisfying (2.19) such that there exists a satisfying
(2.21)

for all and all .

as . This implies the equality

(2.22)

for all .

, which implies the inequality

(2.23)

This implies that inequality (2.7) holds.

By (2.5),

(2.24)
for all , all and all . So
(2.25)
for all , all and all . Since for all and all ,
(2.26)

for all and 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
(2.27)
for all and all . Then
(2.28)
exists for each and defines a cubic mapping such that
(2.29)

for all and all .

Proof.

The proof follows from Theorem 2.1 by taking
(2.30)

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
(2.31)
for all . Let be an odd mapping satisfying (2.5). Then
(2.32)
exists for each and defines a cubic mapping such that
(2.33)

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

(2.34)

for all .

Let be given such that . Then

(2.35)
for all and all . Hence
(2.36)
for all and all . So implies that . This means that
(2.37)

for all .

It follows from (2.11) that

(2.38)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(2.39)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.40)
This implies that is a unique mapping satisfying (2.39) such that there exists a satisfying
(2.41)

for all and all .

as . This implies the equality

(2.42)

for all .

, which implies the inequality

(2.43)

This implies that the inequality (2.33) holds.

The rest of the proof is similar to that of 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 (2.27). Then
(2.44)
exists for each and defines a cubic mapping such that
(2.45)

for all and all .

Proof.

The proof follows from Theorem 2.3 by taking
(2.46)

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
(2.47)
for all . Let be an odd mapping satisfying (2.5). Then
(2.48)
exists for each and defines an additive mapping such that
(2.49)

for all and all .

Proof.

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

Letting and for all in (2.10), we get

(2.50)

for all and all .

Now we consider the linear mapping such that

(2.51)

for all .

Let be given such that . Then

(2.52)
for all and all . Hence
(2.53)
for all and all . So implies that . This means that
(2.54)

for all .

It follows from (2.50) that

(2.55)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following

is a fixed point of , that is,

(2.56)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.57)
This implies that is a unique mapping satisfying (2.56) such that there exists a satisfying
(2.58)

for all and all .

as . This implies the equality

(2.59)

for all ;

, which implies the inequality

(2.60)

This implies that inequality (2.49) holds.

The rest of the proof is similar to that of 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 (2.27). Then
(2.61)
exists for each and defines an additive mapping such that
(2.62)

for all and all .

Proof.

The proof follows from Theorem 2.5 by taking
(2.63)

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

Theorem 2.7.

Let be a function such that there exists an with
(2.64)
for all . Let be an odd mapping satisfying (2.5). Then
(2.65)
exists for each and defines an additive mapping such that
(2.66)

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

(2.67)

for all .

Let be given such that . Then

(2.68)
for all and all . Hence
(2.69)
for all and all . So implies that . This means that
(2.70)

for all .

It follows from (2.50) that

(2.71)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(2.72)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(2.73)
This implies that is a unique mapping satisfying (2.72) such that there exists a satisfying
(2.74)

for all and all .

as . This implies the equality

(2.75)

for all .

, which implies the inequality

(2.76)

This implies that inequality (2.66) holds.

The rest of the proof is similar to that of 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 (2.27). Then
(2.77)
exists for each and defines an additive mapping such that
(2.78)

for all and all .

Proof.

The proof follows from Theorem 2.7 by taking
(2.79)

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 quadratic mapping such that
(3.3)

for all and all .

Proof.

Replacing by in (2.5), we get
(3.4)

for all and all .

It follows from (3.4) that

(3.5)

for all and all .

Consider the set

(3.6)
and introduce the generalized metric on :
(3.7)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma of [80].)

Now we consider the linear mapping such that

(3.8)

for all .

Let be given such that . Then

(3.9)
for all and all . Hence
(3.10)
for all and all . So implies that . This means that
(3.11)

for all .

It follows from (3.5) that .

By Theorem 1.4, there exists a mapping satisfying the following:

is a fixed point of , that is,

(3.12)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.13)
This implies that is a unique mapping satisfying (3.12) such that there exists a satisfying
(3.14)

for all and all .

as . This implies the equality

(3.15)

for all .

, which implies the inequality

(3.16)

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 (2.27). Then
(3.17)
exists for each and defines a quadratic mapping such that
(3.18)

for all and all .

Proof.

The proof follows from Theorem 3.1 by taking
(3.19)

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
(3.20)
for all . Let be an even mapping satisfying and (2.5). Then
(3.21)
exists for each and defines a quadratic mapping such that
(3.22)

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

(3.23)

for all .

Let be given such that . Then

(3.24)
for all and all . Hence
(3.25)
for all and all . So implies that . This means that
(3.26)

for all .

It follows from (3.4) that

(3.27)

for all and all . So .

By Theorem 1.4, there exists a mapping satisfying the following.

is a fixed point of , that is,

(3.28)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(3.29)
This implies that is a unique mapping satisfying (3.28) such that there exists a satisfying
(3.30)

for all and all .

as . This implies the equality

(3.31)

for all .

, which implies the inequality

(3.32)

This implies that inequality (3.22) 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 (2.27). Then
(3.33)
exists for each and defines a quadratic mapping such that
(3.34)

for all and all .

Proof.

The proof follows from Theorem 3.3 by taking
(3.35)

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

Declarations

Acknowledgment

This work was supported by Basic Science Research Program through the National Research Foundation of Korea funded by the Ministry of Education, Science and Technology (NRF-2009-0070788).

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. Rassias JM: On approximation of approximately linear mappings by linear mappings. Journal of Functional Analysis 1982,46(1):126–130. 10.1016/0022-1236(82)90048-9MathSciNetView ArticleMATHGoogle Scholar
  19. Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984,108(4):445–446.MathSciNetMATHGoogle Scholar
  20. Rassias JM: Solution of a problem of Ulam. Journal of Approximation Theory 1989,57(3):268–273. 10.1016/0021-9045(89)90041-5MathSciNetView ArticleMATHGoogle Scholar
  21. Rassias JM: On the stability of the Euler-Lagrange functional equation. Chinese Journal of Mathematics 1992,20(2):185–190.MathSciNetMATHGoogle Scholar
  22. Rassias JM: Solution of a stability problem of Ulam. Discussiones Mathematicae 1992, 12: 95–103.MathSciNetMATHGoogle Scholar
  23. Rassias JM: Complete solution of the multi-dimensional problem of Ulam. Discussiones Mathematicae 1994, 14: 101–107.MathSciNetMATHGoogle Scholar
  24. Skof F: Proprietà locali e approssimazione di operator. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890MathSciNetView ArticleGoogle Scholar
  25. Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar
  26. 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
  27. 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
  28. Eshaghi-Gordji M, Abbaszadeh S, Park C: On the stability of a generalized quadratic and quartic type functional equation in quasi-Banach spaces. preprint
  29. Hyers DH, Isac G, Rassias ThM: 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
  30. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
  31. 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
  32. Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Bulletin des Sciences Mathématiques 2008,132(2):87–96. 10.1016/j.bulsci.2006.07.004View ArticleMathSciNetMATHGoogle Scholar
  33. Park C, Cui J: Generalized stability of -ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:-6.Google Scholar
  34. Park C, Najati A: Homomorphisms and derivations in -algebras. Abstract and Applied Analysis 2007, 2007:-12.Google Scholar
  35. Czerwik S: Functional Equations and Inequalities in Several Variables. World Scientific, River Edge, NJ, USA; 2002:x+410.View ArticleMATHGoogle Scholar
  36. Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995,50(1–2):143–190. 10.1007/BF01831117MathSciNetView ArticleMATHGoogle Scholar
  37. Gajda Z: On stability of additive mappings. International Journal of Mathematics and Mathematical Sciences 1991,14(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar
  38. Găvruţa P: An answer to a question of John M. Rassias concerning the stability of Cauchy equation. In Advances in Equations and Inequalities, Hadronic Mathematics Series. Hadronic Press, Palm Harbor, Fla, USA; 1999:67–71.Google Scholar
  39. Gilányi A: On the stability of monomial functional equations. Publicationes Mathematicae Debrecen 2000,56(1–2):201–212.MathSciNetMATHGoogle Scholar
  40. Gruber PM: Stability of isometries. Transactions of the American Mathematical Society 1978, 245: 263–277.MathSciNetView ArticleMATHGoogle Scholar
  41. Jun K-W, Kim H-M: On the stability of Euler-Lagrange type cubic mappings in quasi-Banach spaces. Journal of Mathematical Analysis and Applications 2007,332(2):1335–1350. 10.1016/j.jmaa.2006.11.024MathSciNetView ArticleMATHGoogle Scholar
  42. Jun K-W, Kim H-M, Rassias JM: Extended Hyers-Ulam stability for Cauchy-Jensen mappings. Journal of Difference Equations and Applications 2007,13(12):1139–1153. 10.1080/10236190701464590MathSciNetView ArticleMATHGoogle Scholar
  43. Jung S-M: On the Hyers-Ulam stability of the functional equations that have the quadratic property. Journal of Mathematical Analysis and Applications 1998,222(1):126–137. 10.1006/jmaa.1998.5916MathSciNetView ArticleMATHGoogle Scholar
  44. Jung S-M: On the Hyers-Ulam-Rassias stability of a quadratic functional equation. Journal of Mathematical Analysis and Applications 1999,232(2):384–393. 10.1006/jmaa.1999.6282MathSciNetView ArticleMATHGoogle Scholar
  45. 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
  46. Lee Y-S, Chung S-Y: Stability of an Euler-Lagrange-Rassias equation in the spaces of generalized functions. Applied Mathematics Letters 2008,21(7):694–700. 10.1016/j.aml.2007.07.022MathSciNetView ArticleMATHGoogle Scholar
  47. Nakmahachalasint P: On the generalized Ulam-Gavruta-Rassias stability of mixed-type linear and Euler-Lagrange-Rassias functional equations. International Journal of Mathematics and Mathematical Sciences 2007, 2007:-10.Google Scholar
  48. Park C-G: Hyers-Ulam-Rassias stability of a generalized Euler-Lagrange type additive mapping and isomorphisms between -algebras. Bulletin of the Belgian Mathematical Society. Simon Stevin 2006,13(4):619–632.MathSciNetMATHGoogle Scholar
  49. Pietrzyk A: Stability of the Euler-Lagrange-Rassias functional equation. Demonstratio Mathematica 2006,39(3):523–530.MathSciNetMATHGoogle Scholar
  50. Rassias JM: On the stability of a multi-dimensional Cauchy type functional equation. In Geometry, Analysis and Mechanics. World Scientific, River Edge, NJ, USA; 1994:365–376.Google Scholar
  51. Rassias JM: On the stability of the general Euler-Lagrange functional equation. Demonstratio Mathematica 1996,29(4):755–766.MathSciNetMATHGoogle Scholar
  52. Rassias JM: Solution of a Cauchy-Jensen stability Ulam type problem. Archivum Mathematicum 2001,37(3):161–177.MathSciNetMATHGoogle Scholar
  53. Rassias JM: Alternative contraction principle and Ulam stability problem. Mathematical Sciences Research Journal 2005,9(7):190–199.MathSciNetGoogle Scholar
  54. Rassias JM: Refined Hyers-Ulam approximation of approximately Jensen type mappings. Bulletin des Sciences Mathématiques 2007,131(1):89–98. 10.1016/j.bulsci.2006.03.011View ArticleMathSciNetMATHGoogle Scholar
  55. Rassias JM, Rassias MJ: On some approximately quadratic mappings being exactly quadratic. The Journal of the Indian Mathematical Society 2002,69(1–4):155–160.MathSciNetMATHGoogle Scholar
  56. Rassias JM, Rassias MJ: On the Ulam stability of Jensen and Jensen type mappings on restricted domains. Journal of Mathematical Analysis and Applications 2003,281(2):516–524. 10.1016/S0022-247X(03)00136-7MathSciNetView ArticleMATHGoogle Scholar
  57. Rassias JM, Rassias MJ: Asymptotic behavior of Jensen and Jensen type functional equations. Panamerican Mathematical Journal 2005,15(4):21–35.MathSciNetMATHGoogle Scholar
  58. 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
  59. Rassias MJ, Rassias JM: On the Ulam stability for Euler-Lagrange type quadratic functional equations. The Australian Journal of Mathematical Analysis and Applications 2005,2(1):1–10.MathSciNetMATHGoogle Scholar
  60. Rassias ThM: Problem 16; 2, Report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 1990,39(2–3):292–293.Google Scholar
  61. Rassias ThM: On the stability of the quadratic functional equation and its applications. Studia Universitatis Babeş-Bolyai 1998,43(3):89–124.MathSciNetMATHGoogle Scholar
  62. 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
  63. 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
  64. 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
  65. 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
  66. 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
  67. 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
  68. Ravi K, Arunkumar M: On the Ulam-Gavruta-Rassias stability of the orthogonally Euler-Lagrange type functional equation. International Journal of Applied Mathematics & Statistics 2007, 7: 143–156.MathSciNetGoogle Scholar
  69. Roh J, Shin HJ: Approximation of Cauchy additive mappings. Bulletin of the Korean Mathematical Society 2007,44(4):851–860. 10.4134/BKMS.2007.44.4.851MathSciNetView ArticleMATHGoogle Scholar
  70. 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):867–878. 10.1016/S0022-247X(02)00415-8MathSciNetView ArticleMATHGoogle Scholar
  71. Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Fixed Point Theory 2003,4(1, article 4):-7.
  72. 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
  73. 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
  74. Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. Grazer Mathematische Berichte 2004, 346: 43–52.MathSciNetMATHGoogle Scholar
  75. 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
  76. 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
  77. Park C: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
  78. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003,4(1):91–96.MathSciNetMATHGoogle Scholar
  79. Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. preprint
  80. 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

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© Choonkil Park. 2009

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