- Research Article
- Open Access

# Solutions and Stability of Generalized Mixed Type QC Functional Equations in Random Normed Spaces

- YeolJe Cho
^{1}, - MadjidEshaghi Gordji
^{2}Email author and - Somaye Zolfaghari
^{3}

**2010**:403101

https://doi.org/10.1155/2010/403101

© Yeol Je Cho et al. 2010

**Received: **29 July 2010

**Accepted: **31 August 2010

**Published: **7 September 2010

## Abstract

## Keywords

- Banach Space
- Functional Equation
- Normed Space
- Cauchy Sequence
- Satisfy Inequality

## 1. Introduction

In 1940, the stability problem of functional equations originated from a question given by Ulam [1], that is, the stability of group homomorphisms.

Let be a group and be a metric group with the metric . For any , does there exist , such that, if a mapping satisfies inequality .

For all , then there exists a homomorphism with for all ?

In other words, under what condition, does there exists a homomorphism near an approximate homomorphism? The concept of stability for functional equation arises when we replace the functional equation by an inequality which acts as a perturbation of the equation.

In 1941, Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces.

Moreover, if is continuous in for any fixed then is linear.

In 1978, Rassias [3] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In 1991, Gajda [4] answered the question for the case
, which was raised by Rassias. This new concept is known as the *Hyers-Ulam-Rassias stability of functional equations* (see [5–17]).

*quadratic functional equation*. In particular, every solution of the quadratic equation (1.3) is said to be a

*quadratic mapping*. It is well known that a mapping between real vector spaces and is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 18]). The biadditive mapping is given by

Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof [19] for a mapping , where is a normed space and is a Banach space. Cholewa [20] noticed that the theorem of Skof is still true if the relevant domain is replaced by an abelian group. In [21], Czerwik proved the Hyers-Ulam-Rassias stability of (1.3) and Grabiec [22] generalized these results mentioned above.

where
is a mapping from a real vector space
into a real vector space
and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.5). The function
satisfies the functional equation (1.5)
which is thus called a *cubic functional equation*. Every solution of the cubic functional equation is said to be a *cubic function*. Also, Jun and Kim proved that a function
between real vector spaces
and
is a solution of (1.5) if and only if there exits a unique function
such that
for all
and
is symmetric for each fixed one variable and is additive for fixed two variables.

Definition 1.1 (see [27]).

A mapping
is called 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).

Definition 1.2 (see [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
satisfying the following conditions:

and
is the minimum
-norm. This space is called the *induced random normed space*.

Definition 1.3.

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

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

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

Theorem 1.4 (see [27]).

If is a RN-space and is a sequence in such that , then almost everywhere.

The stability of different functional equations in fuzzy normed spaces and random normed spaces has been studied in [13, 31–42].

where on random normed spaces. It is easy to see that the mapping is a solution of the functional equation (1.10). In Section 2, we investigate the general solution of functional equation (1.10) when is a mapping between vector spaces and, in Section 3, we establish the stability of the functional equation (1.10) in RN-spaces.

## 2. General Solutions

Before proceeding to the proof of Theorem 2.3 which is the main result in this section, we need the following two lemmas.

Lemma 2.1.

If an even function with satisfies (1.10), then is quadratic.

Proof.

which shows that is quadratic. This completes the proof.

Lemma 2.2.

If an odd function satisfies (1.10), then is cubic.

Proof.

Therefore, is a cubic function. This completes the proof.

Theorem 2.3.

A function with satisfies (1.10) for all if and only if there exist functions and such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is symmetric biadditive.

Proof.

It is clear that for all and it is easy to show that the functions and satisfy (1.10). Hence, by Lemmas 2.1 and 2.2, we know that the functions and are quadratic and cubic, respectively. Thus there exist a symmetric biadditive function such that for all and the function such that for all where the function is symmetric for each fixed one variable and is additive for fixed two variables. Therefore, we get for all

Conversely, let for all where the function is symmetric for each fixed one variable and is additive for fixed two variables and is biadditive. By a simple computation, one can show that the functions and satisfy the functional equation (1.10). Thus the function satisfies (1.10). This completes the proof.

## 3. Stability Problems

Theorem 3.1.

Proof.

Now, we show that is a quadratic mapping. In fact, replacing with and , respectively, in (3.3) and then taking the limit as , we find that satisfies (1.10) for all Therefore, the mapping is quadratic.

To prove (3.4) taking the limit as in (3.12), we get (3.4).

for all and . Therefore, by letting in inequality (3.15), we find that . This completes the proof.

Theorem 3.2.

Proof.

Now, we show that is a cubic mapping. In fact, replacing with and in (3.15), respectively, and then taking the limit , we find that satisfies (1.10) for all Therefore, the mapping is cubic.

Letting the limit in (3.32), we get inequality (3.18) by (3.26).

for all and . By letting in inequality (3.35), we know that . This completes the proof.

Theorem 3.3.

Proof.

Obviously, (3.38) follows from (3.40) and (3.42). This completes the proof.

## Declarations

### Acknowledgment

This paper was supported by the Korea Research Foundation Grant funded by the Korean Government (KRF-2008-313-C00050).

## Authors’ Affiliations

## References

- Ulam SM:
*Problems in Modern Mathematics*. John Wiley & Sons, New York, NY, USA; 1964:xvii+150.MATHGoogle 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 TM: 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 - Gajda Z: On stability of additive mappings.
*International Journal of Mathematics and Mathematical Sciences*1991, 14(3):431–434. 10.1155/S016117129100056XMathSciNetView ArticleMATHGoogle Scholar - Aczél J, Dhombres J:
*Functional Equations in Several Variables, Encyclopedia of Mathematics and its Applications*.*Volume 31*. Cambridge University Press, Cambridge, UK; 1989:xiv+462.View ArticleMATHGoogle Scholar - 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 - 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 - Gordji ME, Kaboli Gharetapeh S, Rashidi E, Karimi T, Aghaei M: Ternary Jordan derivations in -ternary algebras.
*Journal of Computational Analysis and Applications*2010, 12(2):463–470.MathSciNetMATHGoogle Scholar - 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 - Găvruta P, Găvruta L: A new method for the generalized Hyers-Ulam-Rassias stability.
*International Journal of Nonlinear Analysis and Applications*2010, 1(2):11–18.MATHGoogle Scholar - 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, Boston, Mass, USA; 1998:vi+313.MATHGoogle Scholar - Isac G, Rassias TM: On the Hyers-Ulam stability of -additive mappings.
*Journal of Approximation Theory*1993, 72(2):131–137. 10.1006/jath.1993.1010MathSciNetView ArticleMATHGoogle Scholar - Khodaei H, Rassias ThM: Approximately generalized additive functions in several variables.
*International Journal of Nonlinear Analysis and Applications*2010, 1: 22–41.MATHGoogle Scholar - Park C, Najati A: Generalized additive functional inequalities in Banach algebras.
*International Journal of Nonlinear Analysis and Applications*2010, 1: 54–62.MATHGoogle Scholar - Park C, Gordji ME: Comment on approximate ternary Jordan derivations on Banach ternary algebras [ Bavand Savadkouhi et al. J. Math. Phys. 50, 042303 (2009)].
*Journal of Mathematical Physics*2010, 51:-7.Google Scholar - Rassias TM: 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 - Rassias TM: 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 - Kannappan P: Quadratic functional equation and inner product spaces.
*Results in Mathematics. Resultate der Mathematik*1995, 27(3–4):368–372.MathSciNetView ArticleMATHGoogle Scholar - 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 - Cholewa PW: Remarks on the stability of functional equations.
*Aequationes Mathematicae*1984, 27(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar - 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 - Grabiec A: The generalized Hyers-Ulam stability of a class of functional equations.
*Publicationes Mathematicae Debrecen*1996, 48(3–4):217–235.MathSciNetMATHGoogle Scholar - 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 - Chang S-S, Cho YJ, Kang SM:
*Nonlinear Operator Theory in Probabilistic Metric Spaces*. Nova Science Publishers, Huntington, NY, USA; 2001:x+338.MATHGoogle Scholar - Gordji ME, Ebadian A, Zolfaghari S: Stability of a functional equation deriving from cubic and quartic functions.
*Abstract and Applied Analysis*2008, 2008:-17.Google Scholar - Gordji ME, Khodaei H: Solution and stability of generalized mixed type cubic, quadratic and additive functional equation in quasi-Banach spaces.
*Nonlinear Analysis*2009, 71(11):5629–5643. 10.1016/j.na.2009.04.052MathSciNetView ArticleMATHGoogle Scholar - 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 - Sherstnev AN: On the notion of a random normed space.
*Doklady Akademii Nauk SSSR*1963, 149: 280–283.MathSciNetMATHGoogle Scholar - Hadžić O, Pap E:
*Fixed Point Theory in Probabilistic Metric Spaces, Mathematics and Its Applications*.*Volume 536*. Kluwer Academic, Dordrecht, The Netherlands; 2001:x+273.Google Scholar - 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.MathSciNetMATHGoogle 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 - Gordji ME, Ghaemi MB, Majani H: Generalized Hyers-Ulam-Rassias theorem in Menger probabilistic normed spaces.
*Discrete Dynamics in Nature and Society*2010, 2010:-11.Google Scholar - Khodaei H, Kamyar M: Fuzzy approximately additive mappings.
*International Journal of Nonlinear Analysis and Applications*2010, 1: 44–53.MATHGoogle Scholar - Mohamadi M, Cho YJ, Park C, Vetro F, Saadati R: Random stability on an additive-quadratic-quartic functional equation.
*Journal of Inequalities and Applications*2010, 2010:-18.Google Scholar - Miheţ D, Saadati R, Vaezpour SM: The stability of the quartic functional equation in random normed spaces.
*Acta Applicandae Mathematicae*2010, 110(2):797–803. 10.1007/s10440-009-9476-7MathSciNetView ArticleMATHGoogle Scholar - Miheţ D, Saadati R, Vaezpour SM: The stability of an additive functional equation in Menger probabilistic -normed spaces. Mathematics, Slovak. In pressGoogle Scholar
- Park C: Fuzzy stability of an additive-quadratic-quartic functional equation.
*Journal of Inequalities and Applications*2010, 2010:-22.Google Scholar - Park C: Fuzzy stability of additive functional inequalities with the fixed point alternative.
*Journal of Inequalities and Applications*2009, 2009:-17.Google Scholar - Park C: A fixed point approach to the fuzzy stability of an additive-quadratic-cubic functional equation.
*Fixed Point Theory and Applications*2009, 2009:-24.Google Scholar - Saadati R, Vaezpour SM, Cho YJ: Erratum: 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 - Shakeri S, Saadati R, Park C: Stability of the quadratic functional equation in non-Archimedean −fuzzy normed spaces.
*International Journal of Nonlinear Analysis and Applications*2010, 1: 72–83.MATHGoogle Scholar - Zhang S-S, Rassias JM, Saadati R: Stability of a cubic functional equation in intuitionistic random normed spaces.
*Applied Mathematics and Mechanics. English Edition*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.