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Stability of a 2-Dimensional Functional Equation in a Class of Vector Variable Functions

Journal of Inequalities and Applications20102010:167042

https://doi.org/10.1155/2010/167042

Received: 16 March 2010

Accepted: 24 June 2010

Published: 12 July 2010

Abstract

We prove the Hyers-Ulam stability of a 2-dimensional quadratic functional equation in a class of vector variable functions in Banach modules over a unital -algebra.

Keywords

Continuous FunctionReal NumberNormed SpaceQuadratic MappingStability Problem

1. Introduction

In 1940, Ulam proposed the 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. The authors investigated various functional equations and their Hyers-Ulam stability [38]. This Hyers-Ulam stability is a classical type of stability, but there is another kind of stability introduced by Risteski [9] for functional equations spanned over an -dimensional complex vector space too.

Let and be real or complex vector spaces. For a mapping , consider the quadratic functional equation
(11)
In 1989, Aczél and Dhombres [10] obtained the solution of (1.1) for the case that acts on . The result also holds when and are arbitrary real or complex vector spaces. For a mapping , consider the -dimensional quadratic functional equation:
(12)

The quadratic form given by is a solution of (1.2). In 2008, the authors of [8] acquired the general solution and proved the stability of the -dimensional quadratic functional equation (1.2) for the case that and are real vector spaces as follows.

The results of [8, Theor ms 3 and 4] also hold for complex vector spaces and . In this paper, we investigate the stability of (1.2) with two module actions in Banach modules over a unital -algebra.

2. Preliminaries

Let be a unital -algebra with a norm , and let and be left Banach -modules with norms and , respectively. Put , , , , and .

Definition 2.1.

A -dimensional vector variable quadratic mapping satisfying (1.2) is called -quadratic if for all and all .

Definition 2.2.

A unital -algebra is said to have real rank (see [11]) if the invertible self-adjoint elements are dense in .

For any element , , where and are self-adjoint elements; furthermore, , where and are positive elements (see [12, Lemma ]).

3. Results

Theorem 3.1.

Let be a function satisfying
(31)
for all . If the function is a Borel function, then it also satisfies
(32)

for all .

Proof.

By [8, Theo em 3], there exist two symmetric biadditive mappings such that for all . By the proof of The rem 3 in [8], we gain
(33)
for all and all . Letting in the equality (3.3), we get
(34)
for all and all . Putting in the equality (3.3) again, we have
(35)
for all . Since the function is measurable and satisfies (1.1), by [13], it is continuous. By the same reasoning, is also continuous. Let be fixed. Since is measurable, by [14, Theore 7.14.26], for every there is a closed set such that and is continuous. Since , one can choose satisfying . Take a sequence in converging to . By the equality (3.4), we get
(36)
for all . For each fixed , take a sequence in converging to . By (3.5) and the above equality, we have
(37)
Hence we see that
(38)

as desired.

Lemma 3.2.

Let and be normed spaces and a real number, and let be a mapping such that
(39)
for all . Suppose for . If is complete, then there exists a unique -variable quadratic mapping such that
(310)
for all . The mapping is given by
(311)

for all .

Proof.

Letting and in (3.9), we gain
(312)
for all . Putting in (3.12), we get
(313)
for all . Replacing by in the above inequality, we have
(314)
for all . By the above two inequalities, we see that
(315)
for all . Setting and in (3.9), we obtain that
(316)
for all . Replacing by in the above inequality, we see that
(317)
for all . By the last two inequalities, we know that
(318)
for all . By (3.12) and (3.18), we obtain that
(319)
for all . By (3.15) and the above inequality, we have
(320)
for all . Thus we obtain that
(321)
for all and all . Replacing by in the above inequality, we see that
(322)
for all and all . For given integers , we obtain that
(323)

for all .

Consider the case . By (3.23), the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by for all . By (3.9), we have
(324)
for all and all . Letting , we see that satisfies (1.2). Setting and taking in (3.23), one can obtain inequality (3.10). If is another 2-dimensional vector variable quadratic mapping satisfying (3.10), by [8, The rem 3], there are four symmetric biadditive mappings such that and for all . Thus we obtain that
(325)

for all . Hence the mapping is the unique 2-dimensional vector variable quadratic mapping, as desired.

Next, consider the case . Since , by inequality (3.20), we gain
(326)
for all . Thus we get
(327)
for all and all . Replacing by in the above inequality, we have
(328)
for all and all . For given integers , we obtain that
(329)
for all . By (3.29), the sequence is a Cauchy sequence for all . Since is complete, the sequence converges for all . Define by for all . By (3.9), we have
(330)
for all and all . Letting , we see that satisfies (1.2). Setting and taking in (3.29), one can obtain inequality (3.10). If is another 2-dimensional vector variable quadratic mapping satisfying (3.10), by in [8, Th orem 3], there are four symmetric biadditive mappings such that and for all . Thus we obtain that
(331)

for all . Hence the mapping is the unique 2-dimensional vector variable quadratic mapping, as desired.

Theorem 3.3.

Let be a real number, and let be a mapping such that
(332)

for all and all . If is continuous in for each fixed , then there exists a unique -dimensional vector variable -quadratic mapping satisfying (1.2) and (3.10) for all .

Proof.

Suppose . By Lemma 3.2, it follows from the inequality of the statement for that there exists a unique -dimensional vector variable quadratic mapping satisfying (1.2) and inequality (3.10) for all .

Let be fixed. And let be any continuous linear functional, that is, is an arbitrary element of the dual space of . For , consider two functions and defined by and for all . By the assumption that is continuous in for each fixed , the functions and are continuous for all . Note that and for all and all . By [8], the sequences and are Cauchy sequences for all . Define two functions and by and for all . Note that and for all . Since satisfies (1.2), we get
(333)
for all . Since and are the pointwise limits of continuous functions, they are Borel functions. Thus the functions and as measurable quadratic functions are continuous (see [13]), so have the forms and for all . Since satisfies (1.2), by [8, Th orem 3], there exist two symmetric biadditive mappings such as for all . Hence we have
(334)
for all . Since is any continuous linear functional, the -dimensional quadratic mapping satisfies for all . Therefore we obtain
(335)
for all and all . Let be an arbitrary positive integer. Replacing and by and , respectively, and letting in inequality (3.32), we gain
(336)
for all and all . Note that there is a constant such that the condition
(337)
for each and each (see [12, Definiti n 12]). For all and all , we get
(338)
as . Hence we have
(339)
for all and all . Since for each , by (3.35), we obtain
(340)

for all nonzero and all . By (3.35), we get for all . Therefore the mapping is the unique -dimensional vector variable -quadratic mapping satisfying (1.2) and (3.10).

The proof of the case is similar to that of the case .

Theorem 3.4.

Let be a real number and of real rank , and let be a mapping such that
(341)

for all and all . For each fixed , let the sequence converge uniformly on . If is continuous in for each fixed , then there exists a unique -dimensional vector variable mapping satisfying (1.2) and (3.10) such that for all and all .

Proof.

Suppose . By [8, T eorem 4], there exists a unique -dimensional quadratic mapping satisfying (1.2) and inequality (3.10) on . Let be fixed. And let be an arbitrary element of the dual space of . For , consider the functions defined by for all . By the assumption that is continuous in for each fixed , the function is continuous for all . Note that for all and all . By [8], the sequence is a Cauchy sequence for all . Define a function by for all . Note that for all . Thus we have
(342)
for all . Since is the pointwise limit of continuous functions, it is a Borel function. By Theorem 3.1, we gain for all . Hence we get
(343)
for all . Since is any continuous linear functional, the -dimensional quadratic mapping satisfies for all . Therefore we obtain
(344)
for all and all . Let be an arbitrary positive integer. Replacing and by and , respectively, and letting in the inequality (3.41), we get
(345)
for all and all . By condition (3.37), for all and all , we have
(346)
Hence we obtain that
(347)

for all and all .

Let . Since is dense in , there exist two sequences and in such that and as . Put and . Then and as . Set and . Then and as and . Since is uniformly converges on and is continuous in , we see that is also continuous in for each . In fact, we gain
(348)
for all . Thus we get
(349)
for all . By equality (3.47), we have
(350)
as for all By equality (3.49) and the above convergence, we see that
(351)

for all . By equality (3.47) and the above convergence, we obtain for all and all .

The proof of the case is similar to that of the case .

Authors’ Affiliations

(1)
Department of Mathematics Education, College of Education, Mokwon University, Daejeon, Republic of Korea
(2)
College of Liberal Arts, Kyung Hee University, Yongin, Republic of Korea

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Copyright

© Won-Gil Park and Jae-Hyeong Bae. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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