Quadratic-Quartic Functional Equations in RN-Spaces

Abstract

We obtain the general solution and 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 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. 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 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 1978, 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]). The functional equation

(1.3)

is related to a symmetric biadditive mapping. It is natural that this equation is called a quadratic functional equation. In particular, every solution of the quadratic functional equation (1.3) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exits a unique symmetric biadditive mapping such that for all (see [5, 13]). The biadditive mapping is given by

(1.4)

The Hyers-Ulam-Rassias stability problem for the quadratic functional equation (1.3) was proved by Skof for mappings , where is a normed space and is a Banach space (see [14]). Cholewa [15] noticed that the theorem of Skof is still true if relevant domain is replaced an abelian group. In [16], Czerwik proved the Hyers-Ulam-Rassias stability of the functional equation (1.3). Grabiec [17] has generalized the results mentioned above.

In [18], Park and Bae considered the following quartic functional equation

(1.5)

In fact, they proved that a mapping between two real vector spaces and is a solution of (1.5) if and only if there exists a unique symmetric multiadditive mapping such that for all . It is easy to show that the function satisfies the functional equation (1.5), which is called a quartic functional equation (see also [19]). In addition, Kim [20] has obtained the Hyers-Ulam-Rassias stability for a mixed type of quartic and quadratic functional equation.

The Hyers-Ulam-Rassias stability of different functional equations in random normed and fuzzy normed spaces has been recently studied in [21–26]. It should be noticed that in all these papers the triangle inequality is expressed by using the strongest triangular norm .

The aim of this paper is to investigate the stability of the additive-quadratic functional equation in random normed spaces (in the sense of Sherstnev) under arbitrary continuous -norms.

In this sequel, we adopt the usual terminology, notations, and conventions of the theory of random normed spaces, as in [22, 23, 27–29]. Throughout this paper, is the space of distribution functions, that is, the space of all mappings such that is left-continuous and nondecreasing on and . Also, 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 point-wise 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

(1.6)

Definition 1.1 (see [28]).

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 [30, 31]) that if is a -norm and is a given sequence of numbers in , then is defined recurrently by and for . is defined as It is known [31] that for the Lukasiewicz -norm, the following implication holds:

(1.7)

Definition 1.2 (see [29]).

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 ;

(RN2) for all , ;

(RN3) for all and

Every normed space defines a random normed space , where

(1.8)

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

Definition 1.3.

Let be an RN-space.

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

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

An 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 [28]).

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

Recently, Gordji et al. establish the stability of cubic, quadratic and additive-quadratic functional equations in RN-spaces (see [32, 33]).

In this paper, we deal with the following functional equation:

(1.9)

on RN-spaces. It is easy to see that the function is a solution of (1.9).

In Section 2, we investigate the general solution of the functional equation (1.9) when is a mapping between vector spaces and in Section 3, we establish the stability of the functional equation (1.9) in RN-spaces.

2. General Solution

We need the following lemma for solution of (1.9). Throughout this section, and are vector spaces.

Lemma 2.1.

If a mapping satisfies (1.9) for all then is quadratic-quartic.

Proof.

We show that the mappings defined by and defined by are quadratic and quartic, respectively.

Letting in (1.9), we have . Putting in (1.9), we get . Thus the mapping is even. Replacing by in (1.9), we get

(2.1)

for all . Interchanging with in (1.9), we obtain

(2.2)

for all . Since is even, by (2.2), one gets

(2.3)

for all It follows from (2.1) and (2.3) that

(2.4)

for all . This means that

(2.5)

for all . Therefore, the mapping is quadratic.

To prove that is quartic, we have to show that

(2.6)

for all Since is even, the mapping is even. Now if we interchange with in the last equation, we get

(2.7)

for all . Thus, it is enough to prove that satisfies (2.7). Replacing and by and in (1.9), respectively, we obtain

(2.8)

for all . Since for all ,

(2.9)

for all . By (2.8) and (2.9), we get

(2.10)

for all . By multiplying both sides of (1.9) by , we get

(2.11)

for all . If we subtract the last equation from (2.10), we obtain

(2.12)

for all .

Therefore, the mapping is quartic. This completes the proof of the lemma.

Theorem 2.2.

A mapping satisfies (1.9) for all if and only if there exist a unique symmetric multiadditive mapping and a unique symmetric bi-additive mapping such that

(2.13)

for all .

Proof.

Let satisfy (1.9) and assume that are mappings defined by

(2.14)

for all By Lemma 2.1, we obtain that the mappings and are quadratic and quartic, respectively, and

(2.15)

for all

Therefore, there exist a unique symmetric multiadditive mapping and a unique symmetric bi-additive mapping such that and for all [5, 18]. So

(2.16)

for all The proof of the converse is obvious.

3. Stability

Throughout this section, assume that is a real linear space and is a complete RN-space.

Theorem 3.1.

Let be a mapping with for which there is ( is denoted by ) with the property:

(3.1)

for all and all If

(3.2)

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

(3.3)

for all and all

Proof.

Putting in (3.1), we obtain

(3.4)

for all and all Letting in (3.1), we get

(3.5)

for all and all Putting in (3.1), we obtain

(3.6)

for all and all Replacing by in (3.6), we see that

(3.7)

for all and all It follows from (3.5) and (3.7) that

(3.8)

for all and all If we add (3.4) to (3.8), then we have

(3.9)

Let

(3.10)

for all and all . Then we get

(3.11)

for all and all Let be a mapping defined by . Then we conclude that

(3.12)

for all and all . Thus we have

(3.13)

for all and all Hence

(3.14)

for all , all and all This means that

(3.15)

for all all and all By the triangle inequality, from it follows that

(3.16)

for all and all In order to prove the convergence of the sequence , we replace with in (3.16) to obtain that

(3.17)

Since the right-hand side of the inequality (3.17) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all .

Now we show that is a quadratic mapping. Replacing with and in (3.1), respectively, we get

(3.18)

Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1, the mapping is quadratic.

Letting the limit as in (3.16), we get (3.3) by (3.10).

Finally, to prove the uniqueness of the quadratic mapping subject to (3.3), let us assume that there exists another quadratic mapping which satisfies (3.3). Since for all and all from (3.3), it follows that

(3.19)

for all and all . Letting in (3.19), we conclude that , as desired.

Theorem 3.2.

Let be a mapping with for which there is ( is denoted by ) with the property:

(3.20)

for all and all If

(3.21)

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

(3.22)

for all and all

Proof.

Putting in (3.20), we obtain

(3.23)

for all and all . Letting in (3.20), we get

(3.24)

for all and all . Putting in (3.20), we obtain

(3.25)

for all and all . Replacing by in (3.25), we get

(3.26)

for all and all . It follows from (3.5) and (3.26) that

(3.27)

for all and all . If we add (3.23) to (3.27), then we have

(3.28)

Let

(3.29)

for all and all . Then we get

(3.30)

for all and all Let be a mapping defined by . Then we conclude that

(3.31)

for all and all . Thus we have

(3.32)

for all and all Hence

(3.33)

for all , all and all This means that

(3.34)

for all all and all By the triangle inequality, from it follows that

(3.35)

for all and all In order to prove the convergence of the sequence , we replace with in (3.35) to obtain that

(3.36)

Since the right-hand side of (3.36) tends to as and tend to infinity, the sequence is a Cauchy sequence. Thus we may define for all .

Now we show that is a quartic mapping. Replacing with and in (3.20), respectively, we get

(3.37)

Taking the limit as , we find that satisfies (1.9) for all . By Lemma 2.1 we get that the mapping is quartic.

Letting the limit as in (3.35), we get (3.22) by (3.29).

Finally, to prove the uniqueness of the quartic mapping subject to let us assume that there exists a quartic mapping which satisfies (3.22). Since and for all and all from (3.22), it follows that

(3.38)

for all and all . Letting in (3.38), we get that , as desired.

Theorem 3.3.

Let be a mapping with for which there is ( is denoted by ) with the property:

(3.39)

for all and all If

(3.40)

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

(3.41)

for all and all

Proof.

By Theorems 3.1 and 3.2, there exist a quadratic mapping and a quartic mapping such that

(3.42)

for all and all . It follows from the last inequalities that

(3.43)

for all and all . Hence we obtain (3.41) by letting and for all The uniqueness property of and is trivial.