On the Stability of Generalized Quartic Mappings in Quasi--Normed Spaces

Abstract

We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi--normed spaces and then the stability by using a subadditive function for the generalized quartic function such that , where , , , for all .

1. Introduction

One of the interesting questions concerning the stability problems of functional equations is as follows: when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation? Such an idea was suggested in 1940 by Ulam [1] as follows. Letbe a group and letbe a metric group with the metric. Given, does there exist asuch that if a functionsatisfies the inequalityfor all, then there is a homomorphismwithfor all? In other words, we are looking for situations when the homomorphisms are stable; that is, if a mapping is almost a homomorphism, then there exists a true homomorphism near it. In 1941, Hyers [2] considered the case of approximately additive mappings in Banach spaces and satisfying the well-known weak Hyers inequality controlled by a positive constant. The famous Hyers stability result that appeared in [2] was generalized in the stability involving a sum of powers of norms by Aoki [3]. In 1978, Rassias [4] provided a generalization of Hyers Theorem which allows the Cauchy difference to be unbounded. During the last decades, stability problems of various functional equations have been extensively studied and generalized by a number of authors [5–10]. In particular, Rassias [11] introduced the quartic functional equation

(1.1)

It is easy to see that is a solution of (1.1) by virtue of the identity

(1.2)

For this reason, (1.1) is called a quartic functional equation. Also Chung and Sahoo [12] determined the general solution of (1.1) without assuming any regularity conditions on the unknown function. In fact, they proved that the function is a solution of (1.1) if and only if where the function is symmetric and additive in each variable. Lee and Chung [13] introduced a quartic functional equation as follows:

(1.3)

for fixed integer with

Let be a real number with and let be either or We will consider the definition and some preliminary results of a quasi--norm on a linear space.

Definition 1.1.

Let be a linear space over a field A quasi--norm is a real-valued function on satisfying the followings.

(1) for all and if and only if

(2) for all and all

(3) There is a constant such that for all

The pair is called a quasi--normed space if is a quasi--norm on The smallest possible is called the modulus of concavity of A quasi-Banach space is a complete quasi--normed space.

A quasi--norm is called a -norm ( if for all In this case, a quasi--Banach space is called a -Banach space; see [14–16].

In this paper, we consider the following the generalized quartic functional equation:

(1.4)

for fixed integers and such that for all We investigate the generalized Hyers-Ulam-Rassias stability problem in quasi--normed spaces and then the stability by using a subadditive function for the generalized quartic function satisfying (1.4).

For the same reason as (1.1) and (1.2), we call (1.4) generalized quartic functional equation.

2. Quartic Functional Equations

Let be real vector spaces. In this section, we will investigate that the functional equation (1.1) is equivalent to the presented functional equation (1.4).

Lemma 2.1.

A mapping satisfies the functional equation (1.1) if and only if satisfies

(2.1)

where for all

Proof.

We will show it by induction on Assume that it holds for all less than equal Now, letting be in (2.1),

(2.2)

and also replacing by in (2.1),

(2.3)

for all Adding (2.2) and (2.3), we have

(2.4)

for all By induction steps, we have

(2.5)

Hence we have

(2.6)

for all Thus they are equivalent.

Theorem 2.2.

If a mapping satisfies the functional equation (1.4), then satisfies the functional equation (2.1).

Proof.

By letting in (2.1), we have Since and Putting in (2.1),

(2.7)

Now, replacing by in (2.7),

(2.8)

By (2.7) and (2.8), we have that is, Hence is even. This implies that that is, for all Now, we will show that (2.1) implies (1.4). By letting in (2.1), we have

(2.9)

Switching and in the previous equation,

(2.10)

By (2.1) with , the previous equation implies that

(2.11)

Hence we have

(2.12)

for all

Corollary 2.3.

If a mapping satisfies the functional equation (1.1), then satisfies the functional equation (1.4).

3. Stabilities

Throughout this section, let be a quasi--normed space and let be a quasi--Banach space with a quasi--norm . Let be the modulus of concavity of . We will investigate the generalized Hyers-Ulam-Rassias stability problem for the functional equation (1.4). After then we will study the stability by using a subadditive function. For a given mapping and all fixed integers and with let

(3.1)

Theorem 3.1.

Suppose that there exists a mapping for which a mapping satisfies

(3.2)

and the series converges for all Then there exists a unique generalized quartic mapping which satisfies (1.4) and the inequality

(3.3)

for all

Proof.

By letting in the inequality (3.2), since we have

(3.4)

that is,

(3.5)

for all Now, putting and multiplying in the inequality (3.5), we get

(3.6)

for all Combining (3.5) and (3.6), we have

(3.7)

for all Inductively, since we have

(3.8)

for all For all and with and switching and and multiplying in the inequality (3.5), inductively,

(3.9)

for all Since the right-hand side of the previous inequality tends to 0 as hence is a Cauchy sequence in the quasi--Banach space Thus we may define

(3.10)

for all Since replacing and by and , respectively, and dividing by in the inequality (3.2), we have

(3.11)

for all By taking the definition of implies that satisfies (1.4) for all that is, is the generalized quartic mapping. Also, the inequality (3.8) implies the inequality (3.3). Now, it remains to show the uniqueness. Assume that there exists satisfying (1.4) and (3.3). It is easy to show that for all and as in the proof of Theorem 2.2. Then

(3.12)

for all By letting we immediately have the uniqueness of .

Theorem 3.2.

Suppose that there exists a mapping for which a mapping satisfies

(3.13)

and the series converges for all Then there exists a unique generalized quartic mapping which satisfies (2.1) and the inequality

(3.14)

for all

Proof.

If is replaced by in the inequality (3.5), then the proof follows from the proof of Theorem 3.1.

Now we will recall a subadditive function and then investigate the stability under the condition that the space is a -Banach space. The basic definitions of subadditive functions follow from [16].

A function having a domain and a codomain that are both closed under addition is called

(1)a subadditive function if

(2)a contractively subadditive function if there exists a constant with such that

(3)an expansively superadditive function if there exists a constant with such that

for all

Theorem 3.3.

Suppose that there exists a mapping for which a mapping satisfies

(3.15)

for all and the map is contractively subadditive with a constant such that Then there exists a unique generalized quartic mapping which satisfies (1.4) and the inequality

(3.16)

for all

Proof.

By the inequalities (3.5) and (3.9) of the proof of Theorem 3.1, we have

(3.17)

that is,

(3.18)

for all and for all and with Hence is a Cauchy sequence in the space Thus we may define

(3.19)

for all Now, we will show that the map is a generalized quartic mapping. Then

(3.20)

for all Hence the mapping is a generalized quartic mapping. Note that the inequality (3.18) implies the inequality (3.16) by letting and taking Assume that there exists satisfying (1.4) and (3.16). We know that for all Then

(3.21)

that is,

(3.22)

for all By letting we immediately have the uniqueness of .

Theorem 3.4.

Suppose that there exists a mapping for which a mapping satisfies

(3.23)

for all and the map is expansively superadditive with a constant such that Then there exists a unique generalized quartic mapping which satisfies (1.4) and the inequality

(3.24)

for all

Proof.

By letting in (3.23), we have

(3.25)

and then replacing by ,

(3.26)

for all For all and with inductively we have

(3.27)

for all The remains follow from the proof of Theorem 3.3.