Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces

Abstract

We obtain the generalized Hyers-Ulam stability of the Cauchy-Jensen functional equation .

1. Introduction

In 1940, Ulam proposed the general Ulam stability problem (see [1]).

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 is a homomorphism with for all ?

In 1941, this problem was solved by Hyers [2] in the case of Banach space. Thereafter, we call that type the Hyers-Ulam stability.

Throughout this paper, let and be vector spaces. A mapping is called an additive mapping (respectively, an affine mapping) if satisfies the Cauchy functional equation (respectively, the Jensen functional equation ). Aoki [3] and Rassias [4, 5] extended the Hyers-Ulam stability by considering variables for Cauchy equation. Using the method introduced in [3], Jung [6] obtained a result for Jensen equation. It also has been generalized to the function case by Găvruta [7] and Jung [8] for Cauchy equation, and by Lee and Jun [9] for Jensen equation.

Definition 1.1.

A mapping is called a Cauchy-Jensen mapping if satisfies the system of equations

(11)

When , the function given by is a solution of (1.1). In particular, letting , we get a function given by .

For a mapping , consider the functional equation

(12)

Definition 1.2 (see [10, 11]).

Let be a real linear space. A quasi-norm is real-valued function on X satisfying the following.

(i) for all and if and only if .

(ii) for all and all .

(iii)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

(13)

for all . In this case, a quasi-Banach space is called a -Banach space.

The authors [12] obtained the solutions of (1.1) and (1.2) as follows.

Theorem 1 A.

A mapping satisfies (1.1) if and only if there exist a biadditive mapping and an additive mapping such that for all .

Theorem 1 B.

A mapping satisfies (1.1) if and only if it satisfies (1.2).

In this paper, we investigate the generalized Hyers-Ulam stability of (1.1) and (1.2).

2. Stability of (1.1) and (1.2)

Throughout this section, assume that is a quasi-normed space with quasi-norm and that is a -Banach space with -norm . Let be the modulus of concavity of .

Let and be two functions such that

(21)

(22)

for all , and

(23)

(24)

for all .

Theorem 2.1.

Suppose that a mapping satisfies the inequalities

(25)

(26)

for all . Then the limits

(27)

exist for all and the mappings and are Cauchy-Jensen mappings satisfying

(28)

(29)

for all .

Proof.

Letting and replacing by in (2.5) then,

(210)

for all . Replacing by in the above inequality and dividing by , we get

(211)

for all and all nonnegative integers . Since is a -Banach space, we have

(212)

for all and all nonnegative integers and with . Therefore we conclude from (2.3) and (2.12) that the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by

(213)

for all . Letting and passing the limit in (2.12), we get (2.8). Now, we show that is a Cauchy-Jensen mapping. It follows from (2.1), (2.11), and (2.13) that

(214)

for all . So for all .

On the other hand it follows from (2.1), (2.5), (2.6), and (2.13) that

(215)

for all . Thus is a Cauchy-Jensen mapping. Next, setting in (2.6) and replacing by and by in (2.6), one can obtain that

(216)

respectively, for all . By two above inequalities,

(217)

for all . By the same method as above, one can find a Cauchy-Jensen mapping which satisfies (2.9). In fact, for all .

From now on, let be a function such that

(218)

(219)

for all .

We will use the following lemma in order to prove Theorem 2.3.

Lemma 2.2 (see [13]).

Let and let be nonnegative real numbers. Then

(220)

Theorem 2.3.

Suppose that a mapping satisfies and the inequality

(221)

for all . Then the limit exists for all and the mapping is the unique Cauchy-Jensen mapping satisfying

(222)

where

(223)

for all .

Proof.

Letting in (2.21), we get

(224)

for all . Putting and in (2.24), we get

(225)

for all . Replacing by and by in (2.24), we get

(226)

for all . By (2.25) and (2.26), we have

(227)

for all . Setting and in (2.24), we get

(228)

for all . By (2.27) and the above inequality, we get

(229)

for all . Replacing by in (2.26), we get

(230)

for all . By (2.29) and the above inequality, we have

(231)

for all . Replacing by in the above inequality, we get

(232)

for all . By (2.25) and the above inequality, we get

(233)

where

(234)

for all . Replacing by and by in the above inequality and dividing ,we get

(235)

for all and all nonnegative integers . Since is a -norm, we have

(236)

for all and all nonnegative integers and with . Therefore we conclude from (2.18) and (2.36) that the sequence is a Cauchy sequence in for all . Since is complete, the sequence converges in for all . So one can define the mapping by

(237)

for all . Letting , passing the limit in (2.36), and applying lemma, we get (2.22). Now, we show that is a Cauchy-Jensen mapping. By lemma, we infer that for all . It follows from (2.18), (2.35), and the above equality that

(238)

for all . So for all .

On the other hand it follows from (2.18), (2.21), and (2.37) that

(239)

for all . Hence the mapping satisfies (1.2). To prove the uniqueness of , let be another Cauchy-Jensen mapping satisfying (2.22). It follows from (2.19) that

(240)

for all . Hence for all . So it follows from (2.22) and (2.37) that

(241)

for all . So .