Open Access

Nonlinear -Random Stability of an ACQ Functional Equation

Journal of Inequalities and Applications20112011:194394

https://doi.org/10.1155/2011/194394

Received: 9 December 2010

Accepted: 6 February 2011

Published: 6 March 2011

Abstract

We prove the generalized Hyers-Ulam stability of the following additive-cubic-quartic functional equation: in complete latticetic random normed spaces.

1. Introduction

Random theory is a powerful hand set for modeling uncertainty and vagueness in various problems arising in the field of science and engineering. It has also very useful applications in various fields, for example, population dynamics, chaos control, computer programming, nonlinear dynamical systems, nonlinear operators, statistical convergence, and so forth. The random topology proves to be a very useful tool to deal with such situations where the use of classical theories breaks down. The usual uncertainty principle of Werner Heisenberg leads to a generalized uncertainty principle, which has been motivated by string theory and noncommutative geometry. In strong quantum gravity regime space-time points are determined in a random manner. Thus impossibility of determining the position of particles gives the space-time a random structure. Because of this random structure, position space representation of quantum mechanics breaks down, and therefore a generalized normed space of quasiposition eigenfunction is required. Hence, one needs to discuss on a new family of random norms. There are many situations where the norm of a vector is not possible to be found and the concept of random norm seems to be more suitable in such cases, that is, we can deal with such situations by modeling the inexactness through the random norm [1, 2].

The stability problem of functional equations originated from a question of Ulam [3] concerning the stability of group homomorphisms. Hyers [4] gave a first affirmative partial answer to the question of Ulam for Banach spaces. Hyers' theorem was generalized by Aoki [5] for additive mappings and by Th. M. Rassias [6] for linear mappings by considering an unbounded Cauchy difference. The paper of Th. M. Rassias [6] 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
(1.1)

A generalization of the Th. M. Rassias theorem was obtained by Găvruţa [7] by replacing the unbounded Cauchy difference by a general control function in the spirit of Th. M. Rassias approach.

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 [6, 824]).

In [25], Jun and Kim considered the following cubic functional equation:
(1.2)

It is easy to show that the function satisfies the 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 [26], 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.

The study of stability of functional equations is important problem in nonlinear sciences and application in solving integral equation via VIM [2729] PDE and ODE [3034]. 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 [35, 36]).

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 [37] 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 [3843]).

2. Preliminaries

The theory of random normed spaces (RN-spaces) is important as a generalization of deterministic result of linear normed spaces and also in the study of random operator equations. The RN-spaces may also provide us the appropriate tools to study the geometry of nuclear physics and have important application in quantum particle physics. The generalized Hyers-Ulam stability of different functional equations in random normed spaces, RN-spaces and fuzzy normed spaces has been recently studied by Alsina [44], Mirmostafaee and Moslehian [45] and Mirzavaziri and Moslehian [40], Miheţ and Radu [46], Miheţ et al. [47, 48], Baktash et al. [49], and Saadati et al. [50].

Let be a complete lattice, that is, a partially ordered set in which every nonempty subset admits supremum and infimum, and , . The space of latticetic random distribution functions, denoted by , is defined as the set of all mappings such that is left continuous and nondecreasing on , .

is defined as , where denotes the left limit of the function at the point . The space is partially ordered by the usual point-wise ordering of functions, that is, if and only if for all in . The maximal element for in this order is the distribution function given by
(2.1)

Definition 2.1 (see [51]).

A triangular norm ( -norm) on is a mapping satisfying the following conditions:

(a) (boundary condition);

(b) (commutativity);

(c) (associativity);

(d) and (monotonicity).

Let be a sequence in which converges to (equipped order topology). The -norm is said to be a continuous -norm if
(2.2)

for all .

A -norm can be extended (by associativity) in a unique way to an -array operation taking for the value defined by
(2.3)
can also be extended to a countable operation taking for any sequence in the value
(2.4)

The limit on the right side of (2.4) exists since the sequence is nonincreasing and bounded from below.

Note that we put whenever . If is a -norm then is defined for all and by 1, if and , if . A -norm is said to be of Hadžić-type (we denote by ) if the family is equicontinuous at (cf. [52]).

Definition 2.2 (see [51]).

A continuous -norm on is said to be continuous -representable if there exist a continuous -norm and a continuous -conorm on such that, for all , ,
(2.5)
For example,
(2.6)

for all , are continuous -representable.

Define the mapping from to by
(2.7)

Recall (see [52, 53]) that if is a given sequence in , is defined recurrently by and for .

A negation on is any decreasing mapping satisfying and . If , for all , then is called an involutive negation. In the following, is endowed with a (fixed) negation .

Definition 2.3.

A latticetic random normed space is a triple , where is a vector space and is a mapping from into such that the following conditions hold:

(LRN1) for all if and only if ;

(LRN2) for all in , and ;

(LRN3) for all and .

We note that from (LPN2) it follows that .

Example 2.4.

Let and operation be defined by
(2.8)
Then is a complete lattice (see [51]). In this complete lattice, we denote its units by and . Let be a normed space. Let for all , and be a mapping defined by
(2.9)

Then is a latticetic random normed space.

If is a latticetic random normed space, then
(2.10)

is a complete system of neighborhoods of null vector for a linear topology on generated by the norm .

Definition 2.5.

Let be a latticetic random normed space.

(1)A sequence in is said to be convergent to in if, for every and , there exists a positive integer such that whenever .

(2)A sequence in is called Cauchy sequence if, for every and , there exists a positive integer such that whenever .

(3)A latticetic random normed spaces is said to be complete if and only if every Cauchy sequence in is convergent to a point in .

Theorem 2.6.

If is a latticetic random normed space and is a sequence such that , then .

Proof.

The proof is the same as classical random normed spaces, see [54].

Lemma 2.7.

Let be a latticetic random normed space and . If
(2.11)

then and .

Proof.

Let for all . Since , we have , and by (LRN1) we conclude that .

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

One can easily show that an even mapping satisfies (1.1) if and only if the even mapping is a quartic mapping, that is,
(3.1)
and that an odd mapping satisfies (1.1) if and only if the odd mapping is an additive-cubic mapping, that is,
(3.2)

It was shown in Lemma 2.2 of [55] that and are cubic and additive, respectively, and that .

For a given mapping , we define
(3.3)

for all .

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

Theorem 3.1.

Let be a linear space, a complete LRN-space and a mapping from to is denoted by such that, for some ,
(3.4)
Let be an odd mapping satisfying
(3.5)
for all and all . Then
(3.6)
exists for each and defines a cubic mapping such that
(3.7)

for all and all .

Proof.

Letting in (3.5), we get
(3.8)

for all and all .

Replacing by in (3.5), we get
(3.9)

for all and all .

By (3.8) and (3.9),
(3.10)
for all and all . Letting and for all , we get
(3.11)

for all and all .

Consider the set
(3.12)
and introduce the generalized metric on :
(3.13)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma 2.1 of [46].)

Now we consider the linear mapping such that
(3.14)

for all .

Let be given such that . Then
(3.15)
for all and all . Hence
(3.16)
for all and all . So implies that
(3.17)
This means that
(3.18)

for all .

It follows from (3.11) that
(3.19)
for all and all . So
(3.20)

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,
(3.21)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(3.22)
This implies that is a unique mapping satisfying (3.21) such that there exists a satisfying
(3.23)

for all and all ;

(2) as . This implies the equality
(3.24)

for all ;

(3) , which implies the inequality
(3.25)

This implies that inequality (3.7) holds.

From , by (3.5), we deduce that
(3.26)
and so, by (LRN3) and (3.4), we obtain
(3.27)
It follows that
(3.28)
for all , all and all . Since ,
(3.29)
for all and all . Then
(3.30)

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

Corollary 3.2.

Let and let be a real number with . Let be a normed vector space with norm and let be an LRN-space in which and . Let be an odd mapping satisfying
(3.31)
for all and all . Then
(3.32)
exists for each and defines a cubic mapping such that
(3.33)

for all and all .

Proof.

The proof follows from Theorem 3.1 by taking
(3.34)

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

Theorem 3.3.

Let be a linear space, a complete LRN-space and a mapping from to is denoted by such that, for some ,
(3.35)
Let be an odd mapping satisfying (1.1). Then
(3.36)
exists for each and defines a cubic mapping such that
(3.37)

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
(3.38)

for all .

Let be given such that . Then
(3.39)
for all and all . Hence
(3.40)
for all and all . So implies that
(3.41)
This means that
(3.42)

for all .

It follows from (3.11) that
(3.43)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,
(3.44)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(3.45)
This implies that is a unique mapping satisfying (3.44) such that there exists a satisfying
(3.46)

for all and all ;

(2) as . This implies the equality
(3.47)

for all ;

(3) , which implies the inequality
(3.48)

This implies that inequality (3.37) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.4.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an odd mapping satisfying (3.31). Then
(3.49)
exists for each and defines a cubic mapping such that
(3.50)

for all and all .

Proof.

The proof follows from Theorem 3.3 by taking
(3.51)

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

Theorem 3.5.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,
(3.52)
Let be an odd mapping satisfying (3.5). Then
(3.53)
exists for each and defines an additive mapping such that
(3.54)

for all and all .

Proof.

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

Letting and for all in (3.10), we get
(3.55)

for all and all .

Now we consider the linear mapping such that
(3.56)

for all .

Let be given such that . Then
(3.57)
for all and all . Hence
(3.58)
for all and all . So implies that . This means that
(3.59)

for all .

It follows from (3.55) that
(3.60)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,
(3.61)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(3.62)
This implies that is a unique mapping satisfying (3.61) such that there exists a satisfying
(3.63)

for all and all ;

(2) as . This implies the equality
(3.64)

for all ;

(3) , which implies the inequality
(3.65)

This implies that inequality (3.54) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.6.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an odd mapping satisfying (3.31). Then
(3.66)
exists for each and defines an additive mapping such that
(3.67)

for all and all .

Proof.

The proof follows from Theorem 3.5 by taking
(3.68)

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

Theorem 3.7.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,
(3.69)
Let be an odd mapping satisfying (3.5). Then
(3.70)
exists for each and defines an additive mapping such that
(3.71)

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
(3.72)

for all .

Let be given such that . Then
(3.73)
for all and all . Hence
(3.74)
for all and all . So implies that
(3.75)
This means that
(3.76)

for all .

It follows from (3.55) that
(3.77)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,
(3.78)
for all . Since is odd, is an odd mapping. The mapping is a unique fixed point of in the set
(3.79)
This implies that is a unique mapping satisfying (3.78) such that there exists a satisfying
(3.80)

for all and all ;

(2) as . This implies the equality
(3.81)

for all ;

(3) , which implies the inequality
(3.82)

This implies that inequality (3.71) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 3.8.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an odd mapping satisfying (3.31). Then
(3.83)
exists for each and defines an additive mapping such that
(3.84)

for all and all .

Proof.

The proof follows from Theorem 3.7 by taking
(3.85)

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

4. 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 complete RN-spaces: an even case.

Theorem 4.1.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,
(4.1)
Let be an even mapping satisfying and (3.5). Then
(4.2)
exists for each and defines a quartic mapping such that
(4.3)

for all and all .

Proof.

Letting in (3.5), we get
(4.4)

for all and all .

Letting in (3.5), we get
(4.5)

for all and all .

By (4.4) and (4.5),
(4.6)

for all and all .

Consider the set
(4.7)
and introduce the generalized metric on
(4.8)

where, as usual, . It is easy to show that is complete. (See the proof of Lemma 2.1 of [46].)

Now we consider the linear mapping such that
(4.9)

for all .

Let be given such that . Then
(4.10)
for all and all . Hence
(4.11)
for all and all . So implies that
(4.12)
This means that
(4.13)

for all .

It follows from (4.6) that
(4.14)

for all and all . So .

By Theorem 1.1, there exists a mapping satisfying the following:

(1) is a fixed point of , that is,
(4.15)
for all . Since is even, is an even mapping. The mapping is a unique fixed point of in the set
(4.16)
This implies that is a unique mapping satisfying (4.15) such that there exists a satisfying
(4.17)

for all and all ;

(2) as . This implies the equality
(4.18)

for all ;

(3) , which implies the inequality
(4.19)

This implies that inequality (4.3) holds.

The rest of the proof is similar to the proof of Theorem 3.1.

Corollary 4.2.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an even mapping satisfying and (3.31). Then
(4.20)
exists for each and defines a quartic mapping such that
(4.21)

for all and all .

Proof.

The proof follows from Theorem 4.1 by taking
(4.22)

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

Similarly, we can obtain the following. We will omit the proof.

Theorem 4.3.

Let be a linear space, an LRN-space and let be a mapping from to is denoted by such that, for some ,
(4.23)
Let be an even mapping satisfying and (3.5). Then
(4.24)
exists for each and defines a quartic mapping such that
(4.25)

for all and all .

Corollary 4.4.

Let , and let be a real number with . Let be a normed vector space with norm , and let be an LRN-space in which and . Let be an even mapping satisfying and (3.31). Then
(4.26)
exists for each and defines a quartic mapping such that
(4.27)

for all and all .

Proof.

The proof follows from Theorem 4.3 by taking
(4.28)

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

Authors’ Affiliations

(1)
Department of Mathematics, Science and Research Branch, Islamic Azad University

References

  1. El Naschie MS: On a fuzzy Kähler-like manifold which is consistent with the two slit experiment. International Journal of Nonlinear Sciences and Numerical Simulation 2005,6(2):95–98. 10.1515/IJNSNS.2005.6.2.95View ArticleGoogle Scholar
  2. Sigalotti LDG, Mejias A: On El Naschie's conjugate complex time, fractal space-time and faster-than-light particles. International Journal of Nonlinear Sciences and Numerical Simulation 2006,7(4):467–472. 10.1515/IJNSNS.2006.7.4.467View ArticleGoogle Scholar
  3. Ulam SM: A Collection of Mathematical Problems. Interscience Publishers, New York, NY, USA; 1960:xiii+150.MATHGoogle Scholar
  4. 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 ArticleGoogle Scholar
  5. 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/00210064MATHMathSciNetView ArticleGoogle Scholar
  6. 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/S0002-9939-1978-0507327-1MATHMathSciNetView ArticleGoogle Scholar
  7. 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.1994.1211MATHMathSciNetView ArticleGoogle Scholar
  8. Hyers DH, Isac G, Rassias ThM: Stability of Functional Equations in Several Variables. Birkhäuser, Boston, Mass, USA; 1998:vi+313.MATHView ArticleGoogle Scholar
  9. Jung S-M: Hyers-Ulam-Rassias Stability of Functional Equations in Mathematical Analysis. Hadronic Press, Palm Harbor, Fla, USA; 2001:ix+256.MATHGoogle Scholar
  10. Park C: Hyers-Ulam-Rassias stability of homomorphisms in quasi-Banach algebras. Bulletin des Sciences Mathématiques 2008,132(2):87–96.MATHView ArticleGoogle Scholar
  11. 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.200510611MATHMathSciNetView ArticleGoogle Scholar
  12. Park C, Cui J: Generalized stability of -ternary quadratic mappings. Abstract and Applied Analysis 2007, 2007:-6.Google Scholar
  13. Park C, Najati A: Homomorphisms and derivations in -algebras. Abstract and Applied Analysis 2007, 2007:-12.Google Scholar
  14. Rassias JM: On approximation of approximately linear mappings by linear mappings. Bulletin des Sciences Mathématiques 1984,108(4):445–446.MATHGoogle Scholar
  15. Rassias JM: Refined Hyers-Ulam approximation of approximately Jensen type mappings. Bulletin des Sciences Mathématiques 2007,131(1):89–98.MATHView ArticleGoogle 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.MATHView ArticleGoogle Scholar
  17. Rassias ThM: Problem 16; 2, Report of the 27th International Symposium on Functional Equations. Aequationes Mathematicae 1990, 39: 292–293.Google Scholar
  18. Rassias ThM: On the stability of the quadratic functional equation and its applications. Universitatis Babeş-Bolyai. Studia. Mathematica 1998,43(3):89–124.MATHGoogle Scholar
  19. 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.2000.6788MATHMathSciNetView ArticleGoogle Scholar
  20. 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.2000.7046MATHMathSciNetView ArticleGoogle Scholar
  21. Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000,62(1):23–130. 10.1023/A:1006499223572MATHMathSciNetView ArticleGoogle Scholar
  22. 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/S0002-9939-1992-1059634-1MATHMathSciNetView ArticleGoogle Scholar
  23. 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.1993.1070MATHMathSciNetView ArticleGoogle Scholar
  24. 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.1998.6129MATHMathSciNetView ArticleGoogle Scholar
  25. 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
  26. Lee SH, Im SM, Hwang IS: Quartic functional equations. Journal of Mathematical Analysis and Applications 2005,307(2):387–394. 10.1016/j.jmaa.2004.12.062MATHMathSciNetView ArticleGoogle Scholar
  27. Saadati R, Vaezpour SM, Rhoades BE: -stability approach to variational iteration method for solving integral equations. Fixed Point Theory and Applications 2009, 2009:-9.Google Scholar
  28. He JH: Variational iteration method—a kind of nonlinear analytical technique: some examples. International Journal of Non-Linear Mechanics 1999, 34: 699–708. 10.1016/S0020-7462(98)00048-1MATHView ArticleGoogle Scholar
  29. He J-H: A review on some new recently developed nonlinear analytical techniques. International Journal of Nonlinear Sciences and Numerical Simulation 2000,1(1):51–70. 10.1515/IJNSNS.2000.1.1.51MATHMathSciNetView ArticleGoogle Scholar
  30. He J-H, Wu X-H: Variational iteration method: new development and applications. Computers & Mathematics with Applications 2007,54(7–8):881–894. 10.1016/j.camwa.2006.12.083MATHMathSciNetView ArticleGoogle Scholar
  31. He J-H: Variational iteration method—some recent results and new interpretations. Journal of Computational and Applied Mathematics 2007,207(1):3–17. 10.1016/j.cam.2006.07.009MATHMathSciNetView ArticleGoogle Scholar
  32. Ozer H: Application of the variational iteration method to the boundary value problems with jump discontinuities arising in solid mechanics. International Journal of Nonlinear Sciences and Numerical Simulation 2007,8(4):513–518. 10.1515/IJNSNS.2007.8.4.513View ArticleGoogle Scholar
  33. Biazar J, Ghazvini H: He's variational iteration method for solving hyperbolic differential equations. International Journal of Nonlinear Sciences and Numerical Simulation 2007, 8: 311–314. 10.1515/IJNSNS.2007.8.3.311Google Scholar
  34. Odibat ZM, Momani S: Application of variational iteration method to nonlinear differential equations of fractional order. International Journal of Nonlinear Sciences and Numerical Simulation 2006,7(1):27–34. 10.1515/IJNSNS.2006.7.1.27MathSciNetView ArticleGoogle Scholar
  35. Cădariu L, Radu V: Fixed points and the stability of Jensen's functional equation. Journal of Inequalities in Pure and Applied Mathematics 2003,4(1, article 4):7.Google Scholar
  36. 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/S0002-9904-1968-11933-0MATHMathSciNetView ArticleGoogle Scholar
  37. 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/S0161171296000324MATHMathSciNetView ArticleGoogle Scholar
  38. Cădariu L, Radu V: On the stability of the Cauchy functional equation: a fixed point approach. In Iteration Theory. Volume 346. Karl-Franzens-Universität Graz, Graz, Austria; 2004:43–52.Google Scholar
  39. 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
  40. 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-zMATHMathSciNetView ArticleGoogle Scholar
  41. 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
  42. Park C: Generalized Hyers-Ulam stability of quadratic functional equations: a fixed point approach. Fixed Point Theory and Applications 2008, 2008:-9.Google Scholar
  43. Radu V: The fixed point alternative and the stability of functional equations. Fixed Point Theory 2003,4(1):91–96.MATHMathSciNetGoogle Scholar
  44. Alsina C: On the stability of a functional equation arising in probabilistic normed spaces. In General Inequalities. Volume 80. Birkhäuser, Basel, Switzerland; 1987:263–271.Google Scholar
  45. Mirmostafaee AK, Moslehian MS: Fuzzy approximately cubic mappings. Information Sciences 2008,178(19):3791–3798. 10.1016/j.ins.2008.05.032MATHMathSciNetView ArticleGoogle Scholar
  46. 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.MATHMathSciNetView ArticleGoogle Scholar
  47. Miheţ D, Saadati R, Vaezpour SM: The stability of the quartic functional equation in random normed spaces. Acta Applicandae Mathematicae 2011,110(2):797–803.Google Scholar
  48. Miheţ D, Saadati R, Vaezpour SM: The stability of anadditive functional equation in Menger probabilistic -normed spaces. Mathematica Slovaca. In pressGoogle Scholar
  49. 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
  50. Saadati R, Vaezpour SM, Cho YJ: Erratumml: A note to paper "On the stability of cubic mappings and quartic mappings in random normed spaces". Journal of Inequalities and Applications 2009, 2009:-6.Google Scholar
  51. Deschrijver G, Kerre EE: On the relationship between some extensions of fuzzy set theory. Fuzzy Sets and Systems 2003,133(2):227–235. 10.1016/S0165-0114(02)00127-6MATHMathSciNetView ArticleGoogle Scholar
  52. Hadžić O, Pap E: Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications. Volume 536. Kluwer Academic Publishers, Dordrecht, The Netherlands; 2001:x+273.Google Scholar
  53. Hadžić O, Pap E, Budinčević M: Countable extension of triangular norms and their applications to the fixed point theory in probabilistic metric spaces. Kybernetika 2002,38(3):363–382.MATHMathSciNetGoogle Scholar
  54. Schweizer B, Sklar A: Probabilistic Metric Spaces, North-Holland Series in Probability and Applied Mathematics. North-Holland Publishing, New York, NY, USA; 1983:xvi+275.Google Scholar
  55. Eshaghi-Gordji M, Kaboli-Gharetapeh S, Park C, Zolfaghri S: Stability of an additive-cubic-quartic functional equation. Advances in Difference Equations 2009, 2009: 20.Google Scholar

Copyright

© Reza Saadati et al. 2011

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.