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Stability of a Bi-Jensen Functional Equation II
Journal of Inequalities and Applications volume 2009, Article number: 976284 (2009)
Abstract
We investigate the stability of the bi-Jensen functional equation II
in the spirit of Găvruta.
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%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ1_HTML.gif)
for all , then there is a homomorphism
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_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 1978, Rassias [3] gave a generalization of Hyers' theorem by allowing the Cauchy difference to be controlled by a sum of powers like
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ3_HTML.gif)
Găvruta [4] provided a further generalization of Rassias' theorem in which he replaced the bound by a general function.
Throughout this paper, let and
be a normed space and a Banach space, respectively. A mapping
is called a Cauchy mapping (resp., a Jensen mapping) if
satisfies the functional equation
(resp.,
For a given mapping , we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ4_HTML.gif)
for all . A mapping
is called a bi-Jensen mapping if
satisfies the functional equations
and
.
Bae and Park [5] obtained the generalized Hyers-Ulam stability of a bi-Jensen mapping. Jun et al. [6] improved the results of Bae and Park in the spirit of Rassias.
In this paper, we investigate the stability of a bi-Jensen functional equation ,
in the spirit of Găvruta.
2. Stability of a Bi-Jensen Functional Equation
Jun et al. [7] established the basic properties of a bi-Jensen mapping in the following lemma.
Lemma 2.1.
Let be a bi-Jensen mapping. Then, the following equalities hold:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ5_HTML.gif)
for all and
.
Now we have the stability of a bi-Jensen mapping.
Theorem 2.2.
Let be two functions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ6_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ7_HTML.gif)
for all . Let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ8_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%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ9_HTML.gif)
for all with
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ10_HTML.gif)
The mapping is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ11_HTML.gif)
for all .
Proof.
Let be the maps defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ12_HTML.gif)
for all . By (2.4), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ13_HTML.gif)
for all and
. For given integers
(
),
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ14_HTML.gif)
for all . By (2.2) and (2.3), the sequences
,
, and
are Cauchy sequences for all
. Since
is complete, the sequences
,
, and
converge for all
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ15_HTML.gif)
for all . Putting
and taking
in (2.10), one can obtain the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ16_HTML.gif)
for all . By (2.4) and the definitions of
and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ17_HTML.gif)
for all . So
is a bi-Jensen mapping satisfying (2.5), where
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ18_HTML.gif)
Now, let be another bi-Jensen mapping satisfying (2.5) with
. By Lemma 2.1, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ19_HTML.gif)
for all and
. As
, we may conclude that
for all
. Thus such a bi-Jensen mapping
is unique.
Remark 2.3.
Let be the functions defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ20_HTML.gif)
for all . Let
be the bi-Jensen mappings defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ21_HTML.gif)
for all . Then,
, and
satisfy (2.2), (2.3), (2.4) for all
. In addition,
satisfy (2.5) for all
and
also satisfy (2.5) for all
. But we get
. Hence, the condition
is necessary to show that the mapping F is unique.
We have another stability result applying for several cases.
Theorem 2.4.
Let be two functions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ22_HTML.gif)
for all . Let
be a mapping satisfying (2.4) for all
. Then, there exists a unique bi-Jensen mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ23_HTML.gif)
for all . The mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ24_HTML.gif)
for all .
Proof.
By (2.4) and the similar method in Theorem 2.2, we define the maps by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ25_HTML.gif)
for all . By (2.4) and the definitions of
,
, and
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ26_HTML.gif)
for all . By the similar method in Theorem 2.2,
is a bi-Jensen mapping satisfying (2.19), where
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ27_HTML.gif)
Now, let be another bi-Jensen mapping satisfying (2.19). Using Lemma 2.1,
, and
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ28_HTML.gif)
for all and
. As
, we may conclude that
for all
. Thus such a bi-Jensen mapping
is unique.
Theorem 2.5.
Let be two functions satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ29_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ30_HTML.gif)
for all . Let
be a mapping satisfying (2.4) for all
. Then, there exists a bi-Jensen mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ31_HTML.gif)
for all , where the mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ32_HTML.gif)
for all .
Proof.
We can obtain as in Theorem 2.4 and
as in Theorem 2.2. Hence,
is a bi-Jensen mapping satisfying (2.27), where
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ33_HTML.gif)
Theorem 2.6.
Let be two functions satisfying (2.2) and (2.26) for all
. Let
be a mapping satisfying (2.4) for all
. Then, there exists a bi-Jensen mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ34_HTML.gif)
for all , where the mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ35_HTML.gif)
for all .
Theorem 2.7.
Let be two functions satisfying (2.3) and (2.25) for all
. Let
be a mapping satisfying (2.4) for all
. Then, there exists a bi-Jensen mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ36_HTML.gif)
for all , where the mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ37_HTML.gif)
for all .
Applying Theorems 2.2–2.7, we easily get the following corollaries.
Corollary 2.8.
Let and
. Let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ38_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%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ39_HTML.gif)
for all .
Proof.
Applying Theorem 2.2 (Theorems 2.4 and 2.5, resp.) for the case (
and
, resp.), we obtain the desired result.
Corollary 2.9.
Let (
), and
. Let
be a mapping such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_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%2F2009%2F976284/MediaObjects/13660_2008_Article_2046_Equ41_HTML.gif)
for all .
Proof.
Applying Theorems 2.6 and 2.7, we obtain the desired result.
References
Ulam SM: A Collection of Mathematical Problems. Interscience, New York, NY, USA; 1968.
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(4):222–224. 10.1073/pnas.27.4.222
Rassias ThM: On the stability of the linear mapping in Banach spaces. Proceedings of the American Mathematical Society 1978,72(2):297–300. 10.1090/S0002-9939-1978-0507327-1
Găvruta P: A generalization of the Hyers-Ulam-Rassias stability of approximately additive mappings. Journal of Mathematical Analysis and Applications 1994,184(3):431–436. 10.1006/jmaa.1994.1211
Bae J-H, Park W-G: On the solution of a bi-Jensen functional equation and its stability. Bulletin of the Korean Mathematical Society 2006,43(3):499–507.
Jun K-W, Jung I-S, Lee Y-H: Stability of a bi-Jensen functional equation. preprint preprint
Jun K-W, Lee Y-H, Oh J-H: On the Rassias stability of a bi-Jensen functional equation. Journal of Mathematical Inequalities 2008,2(3):363–375.
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Jun, KW., Jung, IS. & Lee, YH. Stability of a Bi-Jensen Functional Equation II. J Inequal Appl 2009, 976284 (2009). https://doi.org/10.1155/2009/976284
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DOI: https://doi.org/10.1155/2009/976284