Open Access

Stability Problems of Quintic Mappings in Quasi- -Normed Spaces

Journal of Inequalities and Applications20102010:368981

https://doi.org/10.1155/2010/368981

Received: 12 March 2010

Accepted: 23 April 2010

Published: 26 May 2010

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 quintic function such that , for all .

1. Introduction

In 1940 Ulam [1] proposed the problem concerning the stability of group homomorphisms as follows.Let be a group and let be a metric group with the metric . Given that , does there exist a such that if a function satisfies the inequality for all then there is a homomorphism with for all ? In other words, when is it true that a mapping satisfying a functional equation approximately must be close to the solution of the given functional equation ? In 1941, Hyers [2] considered the case of approximately additive mappings under the assumption that and are Banach spaces.

The famous Hyers stability result that appeared in [2] was generalized by Aoki [3] for the stability of the additive mapping involving a sum of powers of -norms. In 1978, Th. M. Rassias [4] provided a generalization of Hyers' Theorem for the stability of the linear mapping, which allows the Cauchy difference to be unbounded. This result of Th. M. Rassias lead mathematicians working in stability of functional equations to establish what is known today as Hyers-Ulam-Rassias stability or Cauchy-Rassias stability as well as to introduce new definitions of stability concepts. During the last three decades, several stability problems of a large variety of functional equations have been extensively studied and generalized by a number of authors [514]. In particular, J. M. Rassias [15] 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 [16] 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.

Similar to the quartic functional equation, we may define quintic functional equation as follows.

Definition 1.1.

Let be a linear space and let be a real complete linear space. Then a mapping is called quintic if the quintic functional equation
(1.3)

holds for all

Note that the mapping is called quintic because the following algebraic identity
(1.4)

holds for all .

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 1.2.

Let be a linear space over a field A - -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 aquasi- -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 [1719].

In this paper, we consider the following quintic functional equation:
(1.5)

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 quintic function satisfying (1.5).

2. Quintic Functional Equations

Lemma 2.1.

Let be a quintic mapping satisfying (1.3). Then

(1) , for all and ,

(2)

(3) is an odd mapping,

Proof.
  1. (1)
    Letting in (1.3), we have
    (2.1)
     
that is, , for all . Now inductively replacing by , we have the desired result. (2) Putting in (1.3),
(2.2)
Hence (3) Letting in (1.3), we get
(2.3)
for all By (1) and (2), we have
(2.4)

for all Thus it is an odd mapping.

Note that for all and .

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 (1.5). After that we will study the stability by using a subadditive function. For a given mapping , let
(3.1)

for all .

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 quintic mapping which satisfies (1.3) and the inequality
(3.3)

for all

Proof.

By letting in the inequality (3.2), we have
(3.4)
that is,
(3.5)
for all Now, replacing by and multiplying in the inequality (3.5), we get
(3.6)
for all Combining the two equations (3.5) and (3.6), we have
(3.7)
for all Inductively, since we have
(3.8)
for all For all and with and inductively switching and and multiplying in the inequality (3.5), we have
(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 (1.3) for all that is, is the quintic mapping. Also, the inequality (3.8) implies the inequality (3.3). Now, it remains to show the uniqueness. Assume that there exists satisfying (1.3) and (3.3). Lemma 2.1 implies that and for all . 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 quintic mapping which satisfies (1.3) 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 [19].

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 quintic mapping which satisfies (1.3) 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 quintic mapping. Then
(3.20)
for all Hence the mapping is a quintic mapping. Note that the inequality (3.18) implies the inequality (3.16) by letting and taking Assume that there exists satisfying (1.5) 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 quintic mapping which satisfies (1.3) 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.

Authors’ Affiliations

(1)
Faculty of Liberal Education, University of Incheon
(2)
Department of Mathematical Education, Dankook University

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© In Goo Cho et al. 2010

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