- Research Article
- Open Access

# Stability of Mixed Type Cubic and Quartic Functional Equations in Random Normed Spaces

- M. Eshaghi Gordji
^{1}Email author and - M. B. Savadkouhi
^{1}

**2009**:527462

https://doi.org/10.1155/2009/527462

© M. Eshaghi Gordji and M. B. Savadkouhi 2009

**Received:**22 June 2009**Accepted:**5 August 2009**Published:**31 August 2009

## Abstract

We obtain the stability result for the following functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary -norms .

## Keywords

- Banach Space
- Functional Equation
- Additive Mapping
- Triangle Inequality
- Mixed Type

## 1. Introduction

for all Moreover if is continuous in for each fixed then is linear. In Rassias [3] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. In Gajda [4] answered the question for the case , which was raised by Rassias. This new concept is known as Hyers-Ulam-Rassias stability of functional equations (see [5–12]).

and they established the general solution and the generalized Hyers-Ulam-Rassias stability for the functional equation (1.3) The function satisfies the functional equation (1.3) which is thus called a cubic functional equation. Every solution of the cubic functional equation is said to be a cubic function. Jun and Kim proved that a function between real vector spaces X and Y is a solution of (1.3) 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.

In fact they proved that a function between real vector spaces and is a solution of (1.4) if and only if there exists a unique symmetric multiadditive function such that for all (see also [15–18]). It is easy to show that the function satisfies the functional equation (1.4) which is called a quartic functional equation and every solution of the quartic functional equation is said to be a quartic function.

In the sequel we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [19–21]. Throughout this paper, is the space of distribution functions that is, the space of all mappings , such that is leftcontinuous and nondecreasing on and is a subset of consisting of all functions for which , where denotes the left limit of the function at the point , that is, . The space is partially ordered by the usual pointwise 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

Definition 1.1 (see [20]).

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

(a) is commutative and associative;

(b) is continuous;

(c) for all ;

(d) whenever and for all .

Definition 1.2 (see [21]).

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:

for all if and only if ;

for all , ;

for all and

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

Definition 1.3.

Let be a RN-space.

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

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

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

Theorem 1.4 (see [20]).

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

on random normed spaces. It is easy to see that the function is a solution of the functional equation (1.8) In the present paper we establish the stability of the functional equation (1.8) in random normed spaces.

## 2. Main Results

for all and all

Theorem 2.1.

for all and all

Proof.

Taking the limit as , we find that satisfies (1.8) for all Therefore the mapping is cubic.

To prove (2.3) take the limit as in (2.9) Finally, to prove the uniqueness of the cubic function subject to (2.3) let us assume that there exists a cubic function which satisfies (2.3) Since and for all and from (2.3) it follows that

for all and all . By letting in above inequality, we find that .

Theorem 2.2.

for all and all

Proof.

Taking the limit as , we find that satisfies (1.8) for all Hence, the mapping is quartic.

To prove (2.14) take the limit as in (2.20) Finally, to prove the uniqueness property of subject to (2.14) let us assume that there exists a quartic function which satisfies (2.14) Since and for all and from (2.14) it follows that

for all and all . Taking the limit as , we find that .

Theorem 2.3.

for all and all

Proof.

Obviously, (2.25) follows from (2.28) and (2.31).

## Declarations

### Acknowledgment

The second author would like to thank the Office of Gifted Students at Semnan University for its financial support.

## Authors’ Affiliations

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