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

Abstract

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

1. Introduction

The stability problem of functional equations originated from a question of Ulam [1] in concerning the stability of group homomorphisms. Let be a group and let be a metric group with the metric Given , does there exist a , such that if a mapping satisfies the inequality for all , then there exists a homomorphism with for all In other words, under what condition does there exist 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 Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let be a mapping between Banach spaces such that

(1.1)

for all and for some Then there exists a unique additive mapping such that

(1.2)

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]).

Jun and Kim [13] introduced the following cubic functional equation:

(1.3)

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.

Park and Bea [14] introduced the following quartic functional equation:

(1.4)

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

(1.5)

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 .

Typical examples of continuous -norms are , and (the Lukasiewicz -norm). Recall (see [22, 23]) that if is a -norm and is a given sequence of numbers in , is defined recurrently by and for is defined as It is known [23] that for the Lukasiewicz -norm the following implication holds:

(1.6)

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

Every normed spaces defines a random normed space where

(1.7)

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.

The generalized Hyers-Ulam-Rassias stability of different functional equations in random normed spaces has been recently studied in [24–29]. Recently, Eshaghi Gordji et al. [30] established the stability of mixed type cubic and quartic functional equations (see also [31]). In this paper we deal with the following functional equation:

(1.8)

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

From now on, we suppose that is a real linear space, is a complete RN-space, and is a function with for which there is ( denoted by ) with the property

(2.1)

for all and all

Theorem 2.1.

Let be odd and let

(2.2)

for all and all , then there exists a unique cubic mapping such that

(2.3)

for all and all

Proof.

Setting in (2.1), we get

(2.4)

for all If we replace in (2.4) by and divide both sides of (2.4) by 3, we get

(2.5)

for all and all Thus we have

(2.6)

for all and all Therefore,

(2.7)

for all and all Therefore we have

(2.8)

for all and all As by the triangle inequality it follows

(2.9)

for all and In order to prove the convergence of the sequence , we replace with in (2.9) to find that

(2.10)

Since the right-hand side of the inequality tends to as and tend to infinity, the sequence is a Cauchy sequence. Therefore, we may define for all . Now, we show that is a cubic map. Replacing with and respectively in (2.1) it follows that

(2.11)

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

(2.12)

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

Theorem 2.2.

Let be even and let

(2.13)

for all and all , then there exists a unique quartic mapping such that

(2.14)

for all and all

Proof.

By putting in (2.1) we obtain

(2.15)

for all Replacing in (2.15) by to get

(2.16)

for all and all Hence,

(2.17)

for all and all Therefore,

(2.18)

for all and all So we have

(2.19)

for all and all As by the triangle inequality it follows that

(2.20)

for all and We replace with in (2.20) to obtain

(2.21)

Since the right-hand side of the inequality tends to as and tend to infinity, the sequence is a Cauchy sequence. Therefore, we may define for all . Now, we show that is a quartic map. Replacing with and respectively, in (2.1) it follows that

(2.22)

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

(2.23)

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

Theorem 2.3.

Let

(2.24)

for all and all , then there exist a unique cubic mapping and a unique quartic mapping such that

(2.25)

for all and all

Proof.

Let

(2.26)

for all Then and

(2.27)

for all Hence, in view of Theorem 2.1, there exists a unique quartic function such that

(2.28)

Let

(2.29)

for all Then and

(2.30)

for all From Theorem 2.2, it follows that there exists a unique cubic mapping such that

(2.31)

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