# On the Stability of a General Mixed Additive-Cubic Functional Equation in Random Normed Spaces

- TianZhou Xu
^{1}Email author, - JohnMichael Rassias
^{2}and - WanXin Xu
^{3}

**2010**:328473

https://doi.org/10.1155/2010/328473

© Tian Zhou Xu et al. 2010

**Received: **6 June 2010

**Accepted: **23 August 2010

**Published: **24 August 2010

## Abstract

## Keywords

## 1. Introduction

A basic question in the theory of functional equations is as follows: when is it true that a function, which approximately satisfies a functional equation, must be close to an exact solution of the equation?

If the problem accepts a unique solution, we say the equation is stable (see [1]). The first stability problem concerning group homomorphisms was raised by Ulam [2] in 1940 and affirmatively solved by Hyers [3]. The result of Hyers was generalized by Rassias [4] for approximate linear mappings by allowing the Cauchy difference operator to be controlled by . In 1994, a generalization of Rassias' theorem was obtained by G vru a [5], who replaced 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, e.g., [6–12] and references therein). In addition, J. M. Rassias et al. [13–16] generalized the Hyers stability result by introducing two weaker conditions controlled by the Ulam-Gavruta-Rassias (or UGR) product of different powers of norms and the JM Rassias (or JMR) mixed product-sum of powers of norms, respectively.

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 (see [17] and the references therein). The generalized Hyers-Ulam stability of different functional equations in random normed spaces, fuzzy normed spaces, and non-Archimedean fuzzy normed spaces has been recently studied in [14–28].

with in the quasi-Banach spaces. It is easy to see that the mapping is a solution of the functional equation (1.1), which is called a mixed additive-cubic functional equation, and every solution of the mixed additive-cubic functional equation is said to be a mixed additive-cubic mapping.

It is easy to show that the function satisfies the functional equation (1.2). We observe that in case (1.2) yields mixed additive-cubic equation (1.1). Therefore, (1.2) is a generalized form of the mixed additive-cubic equation.

In the present paper, we first prove a theorem on stability of equation in random normed spaces and derive from it results on stability of equation . Next, use those results to establish Ulam-Hyers stability for the general mixed additive-cubic functional equation (1.2) in the setting of random normed spaces. In this way some results will be obtained on stability of the linear functional equations also for the random normed spaces, which correspond, for example, to the papers [30–33].

## 2. Preliminaries

Definition 2.1 (see [17, 28]).

A function is a continuous triangular norm (briefly, a continuous -norm) if satisfies the following conditions:

(a) is commutative and associative;

Typical examples of continuous -norms are , and (the Lukasiewicz -norm).

Now, if is a -norm and is a given sequence of numbers in , we define a sequence recursively by and for all . is defined as .

Definition 2.2 (see [17, 28]).

A random normed space (briefly, RN-space) is a triple , where is a vector space, is a continuous -norm, and is a mapping from into such that the following conditions hold:

(RN1) for all if and only if ;

Example 2.3.

Let be a normed space. For all and , consider . Then is a random normed space, where is the minimum -norm. This space is called the induced random normed space.

Definition 2.4.

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

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

(3)An RN-space is said to be complete if and only if every Cauchy sequence in is convergent to a point in .

## 3. On the Stability of a General Mixed Additive-Cubic Equation in RN-Spaces

Theorem 3.1.

Proof.

for every . Taking the limit as in (3.8), by , we get (3.3).

Since , we get . Therefore, it follows from (3.9) that for all and so . This completes the proof.

Corollary 3.2.

Theorem 3.3.

Proof.

for all and . By Corollary 3.2, there exists a unique mapping such that and for all and .

for all and . Taking the limit as in (3.41), we conclude that fulfills (1.2), and so by [16, Lemma ], we see that the mapping is additive, which implies that the mapping is additive. This completes the proof.

Similar to Theorem 3.3, one can prove the following result.

Theorem 3.4.

for all and , where is defined as in Theorem 3.3.

Remark 3.5.

We can also prove Theorems 3.3 and 3.4 for and , respectively.

Theorem 3.6.

for all and , where is defined as in Theorem 3.3.

Proof.

for all and . Since the right hand side of the inequality tends to as tend to infinity, we find that . Therefore , and then . This completes the proof.

Remark 3.7.

We can formulate similar statements to Theorem 3.6 for .

Corollary 3.8.

Corollary 3.9.

Now, we give one example to illustrate the main results of Theorem 3.6. This example is a modification of the example of Zhang et al. [34].

Example 3.10.

Let be a Banach algebra, be a unit vector in and is defined as in Example 2.3. It is easy to see that is a complete -space.

## Declarations

### Acknowledgments

The authors would like to thank the area editor professor Radu Precup and two anonymous referees for their valuable comments and suggestions. T. Z. Xu was supported by the National Natural Science Foundation of China (10671013).

## Authors’ Affiliations

## References

- Moszner Z: On the stability of functional equations.
*Aequationes Mathematicae*2009, 77(1–2):33–88. 10.1007/s00010-008-2945-7MathSciNetView ArticleMATHGoogle Scholar - Ulam SM:
*A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics*. Interscience Publishers, New York, NY, USA; 1960:xiii+150.Google Scholar - 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 - 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 - Găvruta 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 - Fechner W: On the Hyers-Ulam stability of functional equations connected with additive and quadratic mappings.
*Journal of Mathematical Analysis and Applications*2006, 322(2):774–786. 10.1016/j.jmaa.2005.09.054MathSciNetView ArticleMATHGoogle Scholar - Jung SM: 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 - Sikorska J: On a direct method for proving the Hyers-Ulam stability of functional equations. Journal of Mathematical Analysis and Applications 372(1):99–109.Google Scholar
- Brzdęk J: On a method of proving the Hyers-Ulam stability of functional equations on restricted domains.
*The Australian Journal of Mathematical Analysis and Applications*2009, 6(1, article no. 4):10.MathSciNetMATHGoogle Scholar - Brzdęk J, Pietrzyk A: A note on stability of the general linear equation.
*Aequationes Mathematicae*2008, 75(3):267–270. 10.1007/s00010-007-2894-6MathSciNetView ArticleMATHGoogle Scholar - Forti G-L: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations.
*Journal of Mathematical Analysis and Applications*2004, 295(1):127–133. 10.1016/j.jmaa.2004.03.011MathSciNetView ArticleMATHGoogle Scholar - Forti G-L: Elementary remarks on Ulam-Hyers stability of linear functional equations.
*Journal of Mathematical Analysis and Applications*2007, 328(1):109–118. 10.1016/j.jmaa.2006.04.079MathSciNetView ArticleMATHGoogle Scholar - Ravi K, Arunkumar M, Rassias JM: Ulam stability for the orthogonally general Euler-Lagrange type functional equation.
*International Journal of Mathematics and Statistics*2008, 3(A06):36–46.MathSciNetMATHGoogle Scholar - Xu TZ, Rassias JM, Xu WX: Stability of a general mixed additive-cubic functional equation in non-Archimedean fuzzy normed spaces.
*Journal of Mathematical Physics*2010, 51:-19.Google Scholar - Xu TZ, Rassias JM, Xu WX: A fixed point approach to the stability of a general mixed additive-cubic functional equation in quasi fuzzy normed spaces. to appear in International Journal of Physical SciencesGoogle Scholar
- Xu TZ, Rassias JM, Xu WX: Intuitionistic fuzzy stability of a general mixed additive-cubic equation.
*Journal of Mathematical Physics*2010, 51(6):21.MathSciNetView ArticleMATHGoogle Scholar - Mohamadi M, Cho YJ, Park C, Vetro P, Saadati R: Random stability of an additive-quadratic-quartic functional equation.
*Journal of Inequalities and Applications*2010, 2010:-18.Google Scholar - 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 - Cădariu L, Radu V: Fixed points and stability for functional equations in probabilistic metric and random normed spaces.
*Fixed Point Theory and Applications*2009, 2009:-18.Google Scholar - Eshaghi Gordji M, Savadkouhi MB: Stability of mixed type cubic and quartic functional equations in random normed spaces.
*Journal of Inequalities and Applications*2009, 2009:-9.Google Scholar - Gordji Eshaghi M, Rassias JM, Savadkouhi MB: Approximation of the quadratic and cubic functional equations in RN-spaces.
*European Journal of Pure and Applied Mathematics*2009, 2(4):494–507.MathSciNetMATHGoogle Scholar - 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 - Miheţ D: The stability of the additive Cauchy functional equation in non-Archimedean fuzzy normed spaces.
*Fuzzy Sets and Systems*2010, 161: 2206–2212. 10.1016/j.fss.2010.02.010MathSciNetView ArticleMATHGoogle Scholar - Mirmostafaee AK, Moslehian MS: Stability of additive mappings in non-Archimedean fuzzy normed spaces.
*Fuzzy Sets and Systems*2009, 160(11):1643–1652. 10.1016/j.fss.2008.10.011MathSciNetView ArticleMATHGoogle Scholar - Moslehian MS, Rassias ThM: Stability of functional equations in non-Archimedean spaces.
*Applicable Analysis and Discrete Mathematics*2007, 1(2):325–334. 10.2298/AADM0702325MMathSciNetView ArticleMATHGoogle Scholar - Moslehian MS, Nikodem K, Popa D: Asymptotic aspect of the quadratic functional equation in multi-normed spaces.
*Journal of Mathematical Analysis and Applications*2009, 355(2):717–724. 10.1016/j.jmaa.2009.02.017MathSciNetView ArticleMATHGoogle Scholar - Park C: Fixed points and the stability of an AQCQ-functional equation in non-Archimedean normed spaces.
*Abstract and Applied Analysis*2010, 2010:-15.Google Scholar - Saadati R, Vaezpour SM, Cho YJ: 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 - Najati A, Eskandani GZ: Stability of a mixed additive and cubic functional equation in quasi-Banach spaces.
*Journal of Mathematical Analysis and Applications*2008, 342(2):1318–1331. 10.1016/j.jmaa.2007.12.039MathSciNetView ArticleMATHGoogle Scholar - Brzdęk J, Jung S-M: A note on stability of a linear functional equation of second order connected with the Fibonacci numbers and Lucas sequences. Journal of Inequalities and ApplicationsGoogle Scholar
- Brzdęk J, Popa D, Xu B: Hyers-Ulam stability for linear equations of higher orders.
*Acta Mathematica Hungarica*2008, 120(1–2):1–8. 10.1007/s10474-007-7069-3MathSciNetView ArticleMATHGoogle Scholar - Jung SM: Functional equation and its Hyers-Ulam stability.
*Journal of Inequalities and Applications*2009, 2009:-10.Google Scholar - Jung S-M: Hyers-Ulam stability of Fibonacci functional equation.
*Bulletin of the Iranian Mathematical Society*2009, 35(2):217–227.MathSciNetMATHGoogle Scholar - Zhang S-S, Rassias JM, Saadati R: Stability of a cubic functional equation in intuitionistic random normed spaces.
*Journal of Applied Mathematics and Mechanics*2010, 31(1):21–26. 10.1007/s10483-010-0103-6MathSciNetView ArticleMATHGoogle Scholar

## Copyright

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.