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Hyers-Ulam Stability of a Bi-Jensen Functional Equation on a Punctured Domain
Journal of Inequalities and Applications volume 2010, Article number: 476249 (2010)
Abstract
We obtain the Hyers-Ulam stability of a bi-Jensen functional equation: and simultaneously
. And we get its stability on the punctured domain.
1. Introduction
In 1940, Ulam [1] raised a question concerning the stability of homomorphisms: 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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ1_HTML.gif)
for all , then there is a homomorphism
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ2_HTML.gif)
for all ? The case of approximately additive mappings was solved by Hyers [2] under the assumption that
and
are Banach spaces. In 1949, 1950, and 1978, Bourgin [3], Aoki [4], and Rassias [5] gave a generalization of it under the conditions bounded by variables. Since then, the further generalization has been extensively investigated by a number of mathematicians, such as Găvruta, Rassias, and so forth, [6–25].
Throughout this paper, let be a normed space and
a Banach space. A mapping
is called a Jensen mapping if
satisfies the functional equation
For a given mapping
, we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ3_HTML.gif)
for all . A mapping
is called a bi-Jensen mapping if
satisfies the functional equations
and
.
In 2006, Bae and Park [26] obtained the generalized Hyers-Ulam stability of a bi-Jensen mapping. The following result is a special case of Theorem 6 in [26].
Theorem 1 A.
Let and let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ4_HTML.gif)
for all . Then there exist two bi-Jensen mappings
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ5_HTML.gif)
for all .
In Theorem A, they did not show that there exist a and a unique bi-Jensen mapping
such that
for all
. In 2008, Jun et al. [7, 8] improved Bae and Park's results.
In Section 2, we show that there exists a unique bi-Jensen mapping such that
for all
. In Section 3, we investigate the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain.
2. Stability of a Bi-Jensen Functional Equation
From Lemma 1 in [8], we get the following lemma.
Lemma.
Let be a bi-Jensen mapping. Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ6_HTML.gif)
for all and
.
Now we will give the Hyers-Ulam stability for a bi-Jensen mapping.
Theorem 2.2.
Let and let
be a mapping satisfying (1.4) for all
. Then there exists a unique bi-Jensen mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ7_HTML.gif)
for all with
. In particular, the mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ8_HTML.gif)
for all .
Proof.
Let be the map defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ9_HTML.gif)
for all and
. By (1.4), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ10_HTML.gif)
for all and
. For given integers
with
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ11_HTML.gif)
for all . By the above inequality, the sequence
is a Cauchy sequence for all
. Since
is complete, the sequence
converges for all
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ12_HTML.gif)
for all . Putting
and taking
in (2.6), we obtain the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ13_HTML.gif)
for all . By (1.4) and the definition of
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ14_HTML.gif)
for all . So
is a bi-Jensen mapping satisfying (2.2). Now, let
be another bi-Jensen mapping satisfying (2.2) with
. By Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ15_HTML.gif)
for all and
. As
, we may conclude that
for all
. Thus the bi-Jensen mapping
is unique.
Example 2.3.
Let be the bi-Jensen mappings defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ16_HTML.gif)
for all . Then
satisfy (1.4) for all
. In addition,
satisfy (2.2) for all
and
also satisfy (2.2) for all
. But we get
. Hence the condition
is necessary to show that the mapping
is unique.
3. Stability of a Bi-Jensen Functional Equation on the Punctured Domain
Let be a subset of
.
and
are punctured domain on the spaces
and
, respectively.
Throughout this paper, for a given mapping , let
be the mappings defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ17_HTML.gif)
for all .
Lemma.
Let be a subset of
satisfying the following condition: for every
, there exists a positive integer
such that
for all integer
with
, and such that
for all integer
with
. Let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ18_HTML.gif)
for all . Then there exists a unique bi-Jensen mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ19_HTML.gif)
for all . Moreover, the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ20_HTML.gif)
holds for all .
Proof.
Note that ,
,
, and
for all
. Let
for any
. From (3.2), we get the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ21_HTML.gif)
for all , and we know that the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ22_HTML.gif)
holds for all . From (3.2), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ23_HTML.gif)
for all . From the above equalities, we obtain the equalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ24_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ25_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ26_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ27_HTML.gif)
for all and
.
Let be the set defined by
for each
. From the above equalities, we can define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ28_HTML.gif)
From the definition of , we get the equalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ29_HTML.gif)
for all . By (3.10), we get the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ30_HTML.gif)
for all and
, where
. And also we get the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ31_HTML.gif)
for all and
, where
. Hence the equality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ32_HTML.gif)
holds for all . From (3.8), (3.9), (3.10), and the definition of
, we easily get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ33_HTML.gif)
for all . And we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ34_HTML.gif)
for all with
, where
and
. From this, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ35_HTML.gif)
for all with
, where
. From the above equalities, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ36_HTML.gif)
for all . By the similar method, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ37_HTML.gif)
for all . Hence
is a bi-Jensen mapping. Let
be another bi-Jensen mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ38_HTML.gif)
for all . Using the above equality, we show that the equalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ39_HTML.gif)
hold for all and
as we desired, where
.
Corollary 3.2.
Let be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ40_HTML.gif)
for all . Then there exists a unique bi-Jensen mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ41_HTML.gif)
for all .
Example 3.3.
Let be the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ42_HTML.gif)
and let be the mapping defined by
for all
. Then the mappings
satisfy the conditions of Corollary 3.2 with
.
Now, we prove the Hyers-Ulam stability of a bi-Jensen functional equation on the punctured domain .
Theorem 3.4.
Let and
. Let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ43_HTML.gif)
for all . Then there exists a unique bi-Jensen mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ44_HTML.gif)
holds for all with
. The mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ45_HTML.gif)
for all .
Proof.
By (3.27), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ46_HTML.gif)
for all and
. For given integers
(
), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ47_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ48_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ49_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ50_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ51_HTML.gif)
for all . The sequences
,
, and
are Cauchy sequences for all
. Since
is complete, the above sequences converge for all
. From (3.34) and (3.35), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ52_HTML.gif)
for all . Using the inequalities (3.31)–(3.35) and the above equality, we can define the mappings
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ53_HTML.gif)
for all . By (3.27) and the definition of
, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ54_HTML.gif)
for all . Since
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ55_HTML.gif)
for all with
, where
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ56_HTML.gif)
for all . Similarly, the equalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ57_HTML.gif)
hold for all . By Lemma 3.1, There exist bi-Jensen mappings
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ58_HTML.gif)
for all . Since the equalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ59_HTML.gif)
hold, are bi-Jensen mappings. Putting
and taking
in (3.31), (3.32), and (3.33), one can obtain the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ60_HTML.gif)
for all . By (3.30) and the above equalities, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ61_HTML.gif)
for all , where
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ62_HTML.gif)
and . By (3.45), we get the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ63_HTML.gif)
for all and
, where
, and the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ64_HTML.gif)
for all and
. Hence
is a bi-Jensen mapping satisfying (3.28).
Now, let be another bi-Jensen mapping satisfying (3.28) with
. By Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ65_HTML.gif)
for all and
. As
, we may conclude that
for all
. By Lemma 3.1,
as we desired.
Example 3.5.
Let be the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ66_HTML.gif)
Let be the mapping defined by
for all
. Then
satisfies the conditions in Theorem 3.4, and
is a bi-Jensen mapping satisfying (3.28) but
.
Corollary 3.6.
Let be a mapping satisfying (3.13) and (3.27) for all
. Then there exists a bi-Jensen mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ67_HTML.gif)
for all .
Proof.
Let be as in the proof of Theorem 3.4. By (3.30), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ68_HTML.gif)
for . From the above inequalities and (3.45), we get the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F476249/MediaObjects/13660_2009_Article_2162_Equ69_HTML.gif)
for all .
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Kim, G., Lee, YH. Hyers-Ulam Stability of a Bi-Jensen Functional Equation on a Punctured Domain. J Inequal Appl 2010, 476249 (2010). https://doi.org/10.1155/2010/476249
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DOI: https://doi.org/10.1155/2010/476249