- Research Article
- Open access
- Published:
Stability of a 2-Dimensional Functional Equation in a Class of Vector Variable Functions
Journal of Inequalities and Applications volume 2010, Article number: 167042 (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.
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 [3–8]. 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ1_HTML.gif)
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:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ2_HTML.gif)
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, Theorms 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ3_HTML.gif)
for all . If the function
is a Borel function, then it also satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ4_HTML.gif)
for all .
Proof.
By [8, Theoem 3], there exist two symmetric biadditive mappings
such that
for all
. By the proof of The
rem 3 in [8], we gain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ5_HTML.gif)
for all and all
. Letting
in the equality (3.3), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ6_HTML.gif)
for all and all
. Putting
in the equality (3.3) again, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ8_HTML.gif)
for all . For each fixed
, take a sequence
in
converging to
. By (3.5) and the above equality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ9_HTML.gif)
Hence we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ10_HTML.gif)
as desired.
Lemma 3.2.
Let and
be normed spaces and
a real number, and let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ11_HTML.gif)
for all . Suppose
for
. If
is complete, then there exists a unique
-variable quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ12_HTML.gif)
for all . The mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ13_HTML.gif)
for all .
Proof.
Letting and
in (3.9), we gain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ14_HTML.gif)
for all . Putting
in (3.12), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ15_HTML.gif)
for all . Replacing
by
in the above inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ16_HTML.gif)
for all . By the above two inequalities, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ17_HTML.gif)
for all . Setting
and
in (3.9), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ18_HTML.gif)
for all . Replacing
by
in the above inequality, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ19_HTML.gif)
for all . By the last two inequalities, we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ20_HTML.gif)
for all . By (3.12) and (3.18), we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ21_HTML.gif)
for all . By (3.15) and the above inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ22_HTML.gif)
for all . Thus we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ23_HTML.gif)
for all and all
. Replacing
by
in the above inequality, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ24_HTML.gif)
for all and all
. For given integers
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ25_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ26_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ27_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ28_HTML.gif)
for all . Thus we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ29_HTML.gif)
for all and all
. Replacing
by
in the above inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ30_HTML.gif)
for all and all
. For given integers
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ32_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ33_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ34_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ35_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ36_HTML.gif)
for all . Since
is any continuous linear functional, the
-dimensional quadratic mapping
satisfies
for all
. Therefore we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ37_HTML.gif)
for all and all
. Let
be an arbitrary positive integer. Replacing
and
by
and
, respectively, and letting
in inequality (3.32), we gain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ38_HTML.gif)
for all and all
. Note that there is a constant
such that the condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ39_HTML.gif)
for each and each
(see [12, Definiti
n 12]). For all
and all
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ40_HTML.gif)
as . Hence we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ41_HTML.gif)
for all and all
. Since
for each
, by (3.35), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ42_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ44_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ45_HTML.gif)
for all . Since
is any continuous linear functional, the
-dimensional quadratic mapping
satisfies
for all
. Therefore we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ46_HTML.gif)
for all and all
. Let
be an arbitrary positive integer. Replacing
and
by
and
, respectively, and letting
in the inequality (3.41), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ47_HTML.gif)
for all and all
. By condition (3.37), for all
and all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ48_HTML.gif)
Hence we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ49_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ50_HTML.gif)
for all . Thus we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ51_HTML.gif)
for all . By equality (3.47), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ52_HTML.gif)
as for all
By equality (3.49) and the above convergence, we see that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F167042/MediaObjects/13660_2010_Article_2068_Equ53_HTML.gif)
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
.
References
Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience Publishers, New York, NY, USA; 1960:xiii+150.
Hyers DH: On the stability of the linear functional equation. Proceedings of the National Academy of Sciences of the United States of America 1941, 27: 222–224. 10.1073/pnas.27.4.222
Bae J-H, Park W-G: On stability of a functional equation with variables. Nonlinear Analysis. Theory, Methods & Applications 2006,64(4):856–868. 10.1016/j.na.2005.06.028
Bae J-H, Park W-G: On a cubic equation and a Jensen-quadratic equation. Abstract and Applied Analysis 2007, 2007:-10.
Park W-G, Bae J-H: On a Cauchy-Jensen functional equation and its stability. Journal of Mathematical Analysis and Applications 2006,323(1):634–643. 10.1016/j.jmaa.2005.09.028
Park W-G, Bae J-H: A multidimensional functional equation having quadratic forms as solutions. Journal of Inequalities and Applications 2007, 2007:-8.
Park W-G, Bae J-H: A functional equation originating from elliptic curves. Abstract and Applied Analysis 2008, 2008:-10.
Park W-G, Bae J-H: A functional equation related to quadratic forms without the cross product terms. Honam Mathematical Journal 2008,30(2):219–225.
Risteski IB: A new class of quasicyclic complex vector functional equations. Mathematical Journal of Okayama University 2008, 50: 1–61.
Aczél J, Dhombres J: Functional Equations in Several Variables, Encyclopedia of Mathematics and Its Applications. Volume 31. Cambridge University Press, Cambridge, UK; 1989:xiv+462.
Davidson KR: C⋆-Algebras by Example, Fields Institute Monographs. Volume 6. American Mathematical Society, Providence, RI, USA; 1996:xiv+309.
Bonsall F, Duncan J: Complete Normed Algebras. Springer, New York, NY, USA; 1973:x+301.
Kurepa S: On the quadratic functional. Publications de l'Institut Mathématique 1961, 13: 57–72.
Bogachev VI: Measure Theory. Vol. II. Springer, Berlin, Germany; 2007.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
About this article
Cite this article
Park, WG., Bae, JH. Stability of a 2-Dimensional Functional Equation in a Class of Vector Variable Functions. J Inequal Appl 2010, 167042 (2010). https://doi.org/10.1155/2010/167042
Received:
Accepted:
Published:
DOI: https://doi.org/10.1155/2010/167042