- Review Article
- Open Access

- Reza Saadati
^{1}Email author, - M M Zohdi
^{1}and - S M Vaezpour
^{1}

**2011**:194394

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

© Reza Saadati et al. 2011

**Received:**9 December 2010**Accepted:**6 February 2011**Published:**6 March 2011

## Abstract

## Keywords

- Functional Equation
- Unique Mapping
- Additive Mapping
- Unique Fixed Point
- Generalize Uncertainty Principle

## 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].

*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 [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, 8–24]).

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*.

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 [27–29] PDE and ODE [30–34]. Let
be a set A function
is called a *generalized metric* on
if
satisfies

We recall a fundamental result in fixed point theory.

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

(2)the sequence converges to a fixed point of ;

(3) is the unique fixed point of in the set ;

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 [38–43]).

## 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 , .

Definition 2.1 (see [51]).

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

*continuous*

*-norm*if

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]).

*continuous*

*-representable*if there exist a continuous -norm and a continuous -conorm on such that, for all , ,

for all , are continuous -representable.

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 ;

We note that from (LPN2) it follows that .

Example 2.4.

Then is a latticetic random normed space.

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.

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

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

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.

Proof.

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

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

This implies that inequality (3.7) holds.

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

Corollary 3.2.

Proof.

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

Theorem 3.3.

Proof.

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

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

This implies that inequality (3.37) holds.

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

Corollary 3.4.

Proof.

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

Theorem 3.5.

Proof.

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

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

This implies that inequality (3.54) holds.

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

Corollary 3.6.

Proof.

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

Theorem 3.7.

Proof.

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

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

This implies that inequality (3.71) holds.

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

Corollary 3.8.

Proof.

## 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.

Proof.

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

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

This implies that inequality (4.3) holds.

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

Corollary 4.2.

Proof.

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.

Corollary 4.4.

Proof.

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

## Authors’ Affiliations

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