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Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces
Journal of Inequalities and Applications volume 2010, Article number: 151547 (2010)
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
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
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
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
for all , and
for all .
Theorem 2.1.
Suppose that a mapping satisfies the inequalities
for all . Then the limits
exist for all and the mappings and are Cauchy-Jensen mappings satisfying
for all .
Proof.
Letting and replacing by in (2.5) then,
for all . Replacing by in the above inequality and dividing by , we get
for all and all nonnegative integers . Since is a -Banach space, we have
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
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
for all . So for all .
On the other hand it follows from (2.1), (2.5), (2.6), and (2.13) that
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
respectively, for all . By two above inequalities,
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
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
Theorem 2.3.
Suppose that a mapping satisfies and the inequality
for all . Then the limit exists for all and the mapping is the unique Cauchy-Jensen mapping satisfying
where
for all .
Proof.
Letting in (2.21), we get
for all . Putting and in (2.24), we get
for all . Replacing by and by in (2.24), we get
for all . By (2.25) and (2.26), we have
for all . Setting and in (2.24), we get
for all . By (2.27) and the above inequality, we get
for all . Replacing by in (2.26), we get
for all . By (2.29) and the above inequality, we have
for all . Replacing by in the above inequality, we get
for all . By (2.25) and the above inequality, we get
where
for all . Replacing by and by in the above inequality and dividing ,we get
for all and all nonnegative integers . Since is a -norm, we have
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
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
for all . So for all .
On the other hand it follows from (2.18), (2.21), and (2.37) that
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
for all . Hence for all . So it follows from (2.22) and (2.37) that
for all . So .
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Bae, JH., Park, WG. Stability of a Cauchy-Jensen Functional Equation in Quasi-Banach Spaces. J Inequal Appl 2010, 151547 (2010). https://doi.org/10.1155/2010/151547
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DOI: https://doi.org/10.1155/2010/151547