- Research Article
- Open access
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On the Asymptoticity Aspect of Hyers-Ulam Stability of Quadratic Mappings
Journal of Inequalities and Applications volume 2010, Article number: 454875 (2011)
Abstract
We investigate the Hyers-Ulam stability of the quadratic functional equation on restricted domains. Applying these results, we study of an asymptotic behavior of these quadratic mappings.
1. Introduction
The question concerning the stability of group homomorphisms was posed by Ulam [1]. Hyers [2] solved the case of approximately additive mappings on Banach spaces. Aoki [3] provided a generalization of the Hyers' theorem for additive mappings. In [4], Rassias generalized the result of Hyers for linear mappings by allowing the Cauchy difference to be unbounded (see also [5]). The result of Rassias has been generalized by Gvruţa [6] who permitted the norm of the Cauchy difference
to be bounded by a general control function under some conditions. This stability concept is also applied to the case of various functional equations by a number of authors. For more results on the stability of functional equations, see [7–32]. We also refer the readers to the books [33–37].
It is easy to see that the function defined by
with
an arbitrary constant is a solution of the functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ1_HTML.gif)
So, it is natural that each equation is called a quadratic functional equation. In particular, every solution of the quadratic equation (1.1) is said to be a quadratic function. It is well known that a function between real vector spaces
and
is quadratic if and only if there exists a unique symmetric biadditive function
such that
for all
(see [21, 33, 35]).
A stability theorem for the quadratic functional equation (1.1) was proved by Skof [38] for functions , where
is a normed space and
is a Banach space. Cholewa [11] noticed that the result of Skof holds (with the same proof) if
is replaced by an abelian group
. In [12], Czerwik generalized the result of Skof by allowing growth of the form
for the norm of
, where
and
. In 1998, Jung [39] investigated the Hyers-Ulam stability for additive and quadratic mappings on restricted domains (see also [40–42]). Rassias [43] investigated the Hyers-Ulam stability of mixed type mappings on restricted domains. In [44], the authors considered the asymptoticity of Hyers-Ulam stability close to the asymptotic derivability.
2. Stability of (1.1) on Restricted Domains
In this section, we investigate the Hyers-Ulam stability of the functional equation (1.1) on a restricted domain. As an application, we use the result to the study of an asymptotic behavior of that equation.
Theorem 2.1.
Given a real normed vector space and a real Banach space
, let
and
with
be fixed. If a mapping
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ2_HTML.gif)
for all such that
, where
, then there exists a unique quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ3_HTML.gif)
for all with
and
. Moreover, if
is measurable or if
is continuous in
for each fixed
, then
for all
and
.
Proof.
Letting in (2.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ4_HTML.gif)
for all with
. If we put
with
and
in (2.1), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ5_HTML.gif)
It follows from (2.3) and (2.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ6_HTML.gif)
for all with
. Replacing
by
in (2.5), we infer the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ7_HTML.gif)
for all with
and all integers
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ8_HTML.gif)
for all with
and all integers
. It follows from (2.7) that the sequence
converges for all
with
. Let us denote
for all
with
. It is clear that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ9_HTML.gif)
for all with
. Letting
and
in (2.7), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ10_HTML.gif)
for all with
.
Now, suppose that such that
, then by (2.1) and the definition of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ11_HTML.gif)
We have to extend the mapping to the whole space
. Given any
with
, let
denote the largest integer such that
. Consider the mapping
defined by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ12_HTML.gif)
Let with
and let
. We have two cases.
Case 1.
If , we have from (2.8) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ13_HTML.gif)
Case 2.
If , then
is the largest integer satisfying
, and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ14_HTML.gif)
Therefore, for all
with
. From the definition of
and (2.8), it follows that
for all
. Now, suppose that
with
and choose a positive integer
such that
. By the definition of
and its property, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ15_HTML.gif)
So by the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ16_HTML.gif)
for all with
. Since
, (2.15) holds true for
. Let
with
. It follows from (2.1) and (2.15) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ17_HTML.gif)
Letting in (2.16), we get
for all
with
. Since
, the same is true for
. So,
is even and this implies that (2.16) is true for all
. Therefore,
is quadratic. By the definition
when
, thus (2.2) follows from (2.9). To prove the uniqueness of
, let
be another quadratic mapping satisfying (2.2) for all
. Let
with
and choose a positive integer
such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ18_HTML.gif)
for all . Since
and
are quadratic, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ19_HTML.gif)
for all . Therefore,
. Since
, we have
for all
. The proof of our last assertion follows from the proof of Theorem  1 in [12].
We now introduce one of the fundamental results of fixed point theory by Margolis and Diaz.
Theorem 2.2 (see [22]).
Let be a complete generalized metric space and let
be a strictly contractive mapping with Lipschitz constant
. If there exists a nonnegative integer
such that
for some
, then the following are true:
(1)the sequence converges to a fixed point
of
,
(2) is the unique fixed point of
in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ20_HTML.gif)
(3) for all
.
By using the idea of Cădariu and Radu [45], we applied a fixed point method to the investigation of the generalized Hyers-Ulam stability of the functional equation (1.1) on a restricted domain.
Theorem 2.3.
Given a real normed vector space and a real Banach space
, let
be fixed and let
be a mapping which satisfies the inequality (2.1) for all
, where
is a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ21_HTML.gif)
for all , where
is a constant number, then there exists a unique quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ22_HTML.gif)
for all with
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ23_HTML.gif)
and for all
. Moreover, if
is measurable or if
is continuous in
for each fixed
, then
for all
and
.
Proof.
It follows from (2.20) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ24_HTML.gif)
for all . Let
. Letting
for
in (2.1), we get the following inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ28_HTML.gif)
It follows from (2.24) and (2.25) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ29_HTML.gif)
By (2.26) and (2.27), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ30_HTML.gif)
It follows from (2.25) and (2.29) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ31_HTML.gif)
Using (2.28) and (2.30), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ32_HTML.gif)
By (2.24), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ33_HTML.gif)
Hence, we obtain from (2.31) and (2.32) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ34_HTML.gif)
So, it follows from (2.28) and (2.33) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ35_HTML.gif)
for all . Let
. We introduce a generalized metric on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ36_HTML.gif)
We assert that is a generalized complete metric space. Let
be a Cauchy sequence in
and
be given, then there exists an integer
such that
for all
. This implies that
for all
and all
. Therefore,
is a Cauchy sequence in
for all
. Since
is a Banach space,
converges for all
. Thus, we can define a function
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ37_HTML.gif)
Since
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ38_HTML.gif)
for all and all
, we get
for all
. That is, the Cauchy sequence
converges to
in
. Hence,
is complete. We now consider the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ39_HTML.gif)
Let and let
be an arbitrary constant with
. From the definition of
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ40_HTML.gif)
for all . By the assumption (2.20) and the last inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ41_HTML.gif)
for all . So
. That is,
is a strictly contractive on
. It follows from (2.34) that
. Therefore, according to Theorem 2.2, there exists a function
such that the sequence
converges to
and
. Indeed,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ42_HTML.gif)
and , for all
. Also,
is the unique fixed point of
in the set
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ43_HTML.gif)
By (2.1), (2.23) and using the definition of , we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ44_HTML.gif)
for all . We will define a mapping
such that
. Similar to the proof of Theorem 2.1 for a given
with
, let
denote the largest integer such that
. Consider the mapping
defined by
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ45_HTML.gif)
Let with
and let
. We have two cases.
Case 1.
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_IEq229_HTML.gif)
. Since for all
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ46_HTML.gif)
Case 2.
If , then
is the largest integer satisfying
, and we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ47_HTML.gif)
Therefore, for all
with
. Using
for all
and the definition of
, we get that
for all
. Now, suppose that
with
and choose a positive integer
such that
. By the definition of
and its property, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ48_HTML.gif)
So by the definition of , we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ49_HTML.gif)
for all with
. Since
, (2.48) holds true for
. Let
with
,
,
. It follows from (2.1), (2.23), and (2.48) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ50_HTML.gif)
Since and
for all
, we conclude that (2.49) is true for all
. Let
with
. Putting
in (2.49), we get
. Therefore, by letting
in (2.49), we get
for all
with
. Since
, the same is true for
. So,
is even and this implies that (2.49) is true for all
. Therefore,
is quadratic. To prove the uniqueness of
, let
be another quadratic mapping satisfying (2.21), for all
. Let
with
and choose a positive integer
such that
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ51_HTML.gif)
for all . Since
and
are quadratic, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ52_HTML.gif)
for all . Therefore, (2.23) implies that
. Since
, we have
for all
. Our last assertion is trivial in view of Theorem 2.1.
Corollary 2.4.
Given a real normed vector space and a real Banach space
, let
and
with
be fixed. Suppose that a mapping
satisfies the inequality (2.1) for all
, then there exists a unique quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ53_HTML.gif)
for all with
and
. Moreover, if
is measurable or if
is continuous in
for each fixed
, then
for all
and
.
Remark 2.5.
We may replace the condition (2.20) by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ54_HTML.gif)
for all and
. Using the direct method, there exists a unique quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ55_HTML.gif)
for all with
. For the case
, where
and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ56_HTML.gif)
Using ideas from the papers [39, 43], we prove the generalized Hyers-Ulam stability of (1.1) on restricted domains. We first prove the following lemma.
Lemma 2.6.
Given a real normed vector space and a real Banach space
, let
and
be fixed. If a mapping
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ57_HTML.gif)
for all with
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ58_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ59_HTML.gif)
Proof.
Assume that . If
, then we choose a
with
. Otherwise, let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ60_HTML.gif)
It is clear that and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ61_HTML.gif)
Also
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ62_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ63_HTML.gif)
So, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ64_HTML.gif)
So, satisfies (2.57) for all
.
Theorem 2.7.
Given a real normed vector space and a real Banach space
, let
and
with
be given. Assume that a mapping
satisfies the inequality (2.56) for all
with
, then there exists a unique quadratic mapping
such that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ65_HTML.gif)
for all .
Proof.
By Lemma 2.6, satisfies (2.57) for all
. Letting
in (2.57), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ66_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ67_HTML.gif)
We can use the argument given in the proof of Theorem 2.1 to arrive the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ68_HTML.gif)
for all and all integers
. It follows from (2.67) that the sequence
converges for all
. So, we can define the mapping
by
for all
. Letting
and
in (2.67), we get (2.64).
For the case and
in Theorem 2.7, it is obvious that our inequality (2.64) is sharper than the corresponding inequalities of Jung [39] and Rassias [43].
Skof [38] has proved an asymptotic property of the additive mappings, and Jung [39] has proved an asymptotic property of the quadratic mappings (see also [41]). Using the method in [39], the proof of the following corollary follows from Theorem 2.7 by letting and
.
Corollary 2.8 (see [39]).
Given a real normed vector space and a real Banach space
, a mapping
satisfies (1.1) if and only if the asymptotic condition
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ69_HTML.gif)
holds true.
3.
-Asymptotically Quadratic Mappings
We apply our results to the study of -asymptotical derivatives. Let
be a real normed vector space and let
be a real Banach space
. Let
be arbitrary.
Definition 3.1.
A mapping is called
-asymptotically close to a mapping
if and only if
.
Definition 3.2.
A mapping is called
-asymptotically derivable if the mapping
is
-asymptotically close to a quadratic mapping
. In this case, we say that
is a
-asymptotical derivative of
.
Definition 3.3.
A mapping is called
-asymptotically quadratic if and only if, for every
, there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F454875/MediaObjects/13660_2010_Article_2153_Equ70_HTML.gif)
for all with
.
Definition 3.4.
A mapping is called quadratic outside a ball if there exists
such that
for all
with
.
We have the following result.
Theorem 3.5.
If is quadratic outside a ball and
is
-asymptotically close to
, then
is
-asymptotically quadratic.
The following result follows from Corollary 2.4.
Corollary 3.6.
If is quadratic outside a ball and
is
-asymptotically close to
, then
has a
-asymptotical derivative.
References
Ulam SM: A Collection of Mathematical Problems, Interscience Tracts in Pure and Applied Mathematics, no. 8. Interscience, New York, NY, USA; 1960:xiii+150.
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Rahimi, A., Najati, A. & Bae, JH. On the Asymptoticity Aspect of Hyers-Ulam Stability of Quadratic Mappings. J Inequal Appl 2010, 454875 (2011). https://doi.org/10.1155/2010/454875
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DOI: https://doi.org/10.1155/2010/454875