- Research Article
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Fixed Points and Stability of a Generalized Quadratic Functional Equation
Journal of Inequalities and Applications volume 2009, Article number: 193035 (2009)
Abstract
Using the fixed point method, we prove the generalized Hyers-Ulam stability of the generalized quadratic functional equation in Banach modules, where
are nonzero rational numbers with
.
1. Introduction
The stability problem of functional equations originated from a question of Ulam [1] concerning the stability of group homomorphisms: let be a group and let
be a metric group with the metric
. Given
, does there exist
such that if a mapping
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ1_HTML.gif)
for all , then there is a homomorphism
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ2_HTML.gif)
for all
Hyers [2] gave a first affirmative answer to the question of Ulam for Banach spaces. Let and
be Banach spaces. Assume that
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ3_HTML.gif)
for some and all
. Then there exists a unique additive mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ4_HTML.gif)
for all .
Aoki [3] and Th. M. Rassias [4] provided a generalization of the Hyers' theorem for additive and linear mappings, respectively, by allowing the Cauchy difference to be unbounded.
Theorem 1.1 (Th. M. Rassias [4]).
Let be a mapping from a normed vector space
into a Banach space
subject to the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ5_HTML.gif)
for all , where
and
are constants with
and
. Then the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ6_HTML.gif)
exists for all and
is the unique additive mapping which satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ7_HTML.gif)
for all . If
, then the inequality (1.5) holds for
and (1.7) for
. Also, if for each
the mapping
is continuous in
, then
is
- linear.
Theorem 1.2 (J. M. Rassias [5–7]).
Let be a real normed linear space and let
be a real Banach space. Assume that
is a mapping for which there exist constants
and
such that
and
satisfies the functional inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ8_HTML.gif)
for all . Then there exists a unique additive mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ9_HTML.gif)
for all . If, in addition,
is a mapping such that the transformation
is continuous in
for each fixed
then
is linear.
In 1994, a generalization of Theorems 1.1 and 1.2 was obtained by Găvruţa [8], who replaced the bounds and
by a general control function
.
The functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ10_HTML.gif)
is called a quadratic functional equation. Quadratic functional equations were used to characterize inner product spaces [9–11]. In particular, every solution of the quadratic equation (1.10) is said to be a quadratic mapping. It is well known that a mapping between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive mapping
such that
for all
(see [9, 12]). The biadditive mapping
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ11_HTML.gif)
The generalized Hyers-Ulam stability problem for the quadratic functional equation (1.10) was proved by Skof for mappings where
is a normed space and
is a Banach space (see [13]). Cholewa [14] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an Abelian group. J. M. Rassias [15] and Czerwik [16], proved the stability of the quadratic functional equation (1.10). Grabiec [17] has generalized these results mentioned above. J. M. Rassias [18] introduced and investigated the stability problem of Ulam for the Euler-Lagrange quadratic mappings:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ12_HTML.gif)
In addition, J. M. Rassias [19] generalized the Euler-Lagrange quadratic mapping (1.12) and investigated its stability problem. The Euler-Lagrange quadratic mapping (1.12) has provided a lot of influence in the development of general Euler-Lagrange quadratic equations (mappings) which is now known as Euler-Lagrange-Rassias quadratic functional equations (mappings).
Jun and Lee [20] proved the generalized Hyers-Ulam stability of a pexiderized quadratic equation. The stability problems of several functional equations have been extensively investigated by a number of authors and there are many interesting results concerning this problem (see [8, 20–47]). We also refer the readers to the books [48–51].
Let be a set. A function
is called a generalized metric on
if
satisfies
(i) if and only if
,
(ii) for all
,
(iii) for all
We recall the following theorem by Margolis and Diaz.
Theorem 1.3 (see [52]).
Let be a complete generalized metric space and let
be a strictly contractive mapping with Lipschitz constant
. Then for each given element
, either
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ13_HTML.gif)
for all nonnegative integers or there exists a nonnegative integer
such that
(1) for all
,
(2)the sequence converges to a fixed point
of
,
(3) is the unique fixed point of
in the set
,
(4) for all
.
Throughout this paper, we assume that are nonzero rational numbers with
and that
is a unital Banach algebra with unit
, norm
, and
. Assume that
is a normed left
-module and
is a (unit linked) Banach left
-module. A quadratic mapping
is called
-quadratic if
for all
and all
.
In this paper, we investigate an -quadratic mapping associated with the generalized quadratic functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ14_HTML.gif)
and using the fixed point method (see [24, 25, 38, 53–55]), we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach
-modules associated with the functional equation (1.14). In 1996, Isac and Th. M. Rassias [56] were the first to provide applications of stability theory of functional equations for the proof of new fixed point theorems with applications.
For convenience, we use the following abbreviation for a given and a mapping
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ15_HTML.gif)
for all .
2. Fixed Points and Stability of the Generalized Quadratic Functional Equation (1.14)
Proposition 2.1.
A mapping satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ16_HTML.gif)
for all if and only if
is quadratic.
Proof.
Let satisfy (2.1). Since
letting
in (2.1), we get
. Letting
in (2.1), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ17_HTML.gif)
for all . It follows from (2.1) that
for all
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ18_HTML.gif)
for all We decompose
into the even part and the odd part by putting
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ19_HTML.gif)
for all It is clear that
for all
It is easy to show that the mappings
and
satisfy (2.2) and (2.3). Thus we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ20_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ21_HTML.gif)
for all Letting
in (2.5), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ22_HTML.gif)
for all . It follows from (2.2), (2.5), and (2.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ23_HTML.gif)
for all Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ24_HTML.gif)
for all . So
is quadratic. We claim that
For this, it follows from (2.2) and (2.6) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ25_HTML.gif)
for all . So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ26_HTML.gif)
for all . Letting
in (2.11), we get
for all
. So it follows from (2.11) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ27_HTML.gif)
for all . Replacing
by
and
by
in (2.12), we infer that
is additive. To complete the proof we have two cases.
Case 1 ().
Since is additive and satisfies (2.1), letting
and replacing
by
in (2.1), we get
for all
. Since
, we get
Case 2 ().
Since is additive and satisfies (2.2), we have
for all
. Since
, we get
Hence and this proves that
is quadratic.
Conversely, let be quadratic. Then there exists a unique symmetric biadditive mapping
such that
for all
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ28_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ29_HTML.gif)
for all . Hence
satisfies (2.1).
Corollary 2.2.
Let be a mapping satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ30_HTML.gif)
for all and all
If for each
the mapping
is continuous in
, then
is
-quadratic.
Proof.
Let By Proposition 2.1,
is quadratic. Thus
is
-quadratic. Let
and let
be a sequence of rational numbers such that
Since
is
-quadratic and the mapping
is continuous in
for each
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ31_HTML.gif)
for all . So
is
-quadratic. Letting
in (2.15), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ32_HTML.gif)
for all and all
It is clear that (2.17) is also true for
For each element
Since
is
-quadratic and
for all
and all
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ33_HTML.gif)
for all and all
So the
-quadratic mapping
is also
-quadratic. This completes the proof.
Now we prove the generalized Hyers-Ulam stability of -quadratic mappings in Banach
-modules.
Theorem 2.3.
Let be a mapping with
for which there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ34_HTML.gif)
for all and all
. Let
be a constant such that
for all
If for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ35_HTML.gif)
for all .
Proof.
It follows from that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ36_HTML.gif)
for all .
Letting in (2.19), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ37_HTML.gif)
for all and all
. Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ38_HTML.gif)
for all and all
. Let
We introduce a generalized metric on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ39_HTML.gif)
It is easy to show that is a generalized complete metric space [24].
Now we consider the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ40_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%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ41_HTML.gif)
for all . By the assumption and the last inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ42_HTML.gif)
for all . So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ43_HTML.gif)
for any . It follows from (2.23) (by letting
) that
. According to Theorem 1.3, the sequence
converges to a fixed point
of
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ44_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%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ45_HTML.gif)
that is, the inequality (2.20) holds true for all . It follows from the definition of
, (2.19), and (2.21) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ46_HTML.gif)
for all and all
. By Proposition 2.1 (by letting
), the mapping
is quadratic. Let
be a continuous linear functional. For any
, we consider the mapping
defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ47_HTML.gif)
Since is quadratic and
is linear,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ48_HTML.gif)
for all So
is quadratic. Also
is measurable since it is the pointwise limit of the sequence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ49_HTML.gif)
It follows from [48, Corollary 10.2] that for all
Then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ50_HTML.gif)
for all Hence
for all
and all
By Corollary 2.2, the mapping
is
-quadratic.
Corollary 2.4.
Let and
be nonnegative real numbers such that
and let
be a mapping satisfying the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ51_HTML.gif)
for all and all
. If for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ52_HTML.gif)
for all .
Proof.
Letting and
in (2.36), we get
Now, the proof follows from Theorem 2.3 by taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ53_HTML.gif)
for all . Then we can choose
and we get the desired result.
Remark 2.5.
Let be a mapping with
for which there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ54_HTML.gif)
for all and all
. Let
be a constant such that
for all
. By a similar method to the proof of Theorem 2.3, one can show that if for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ55_HTML.gif)
for all .
For the case (where
are nonnegative real numbers and
with
, there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ56_HTML.gif)
for all .
Corollary 2.6.
Let and let
be nonnegative real numbers such that
and let
be a mapping satisfying the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ57_HTML.gif)
for all and all
. If for each
the mapping
is continuous in
, then
is
-quadratic.
Theorem 2.7.
Let be an even mapping for which there exists a function
satisfying (2.19) and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ58_HTML.gif)
for all and all
. Let
be a constant such that the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ59_HTML.gif)
satisfying for all
If for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ60_HTML.gif)
for all .
Proof.
Since it follows from (2.19) that
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ61_HTML.gif)
for all and all
. Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ62_HTML.gif)
for all and all
. Letting
and replacing
by
and
by
in (2.47), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ63_HTML.gif)
for all , where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ64_HTML.gif)
Letting in (2.48), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ65_HTML.gif)
for all Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ66_HTML.gif)
for all Let
We introduce a generalized metric on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ67_HTML.gif)
Now we consider the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ68_HTML.gif)
Similar to the proof of Theorem 2.3, we deduce that the sequence converges to a fixed point
of
which is
-quadratic. Also
is the unique fixed point of
in the set
and satisfies (2.45).
Corollary 2.8.
Let and let
be nonnegative real numbers and let
be an even mapping satisfying the inequality (2.36) for all
and all
. If for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ69_HTML.gif)
for all .
Proof.
Letting and
in (2.36), we get
Now the proof follows from Theorem 2.7 by taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ70_HTML.gif)
for all . Then we can choose
and we get the desired result.
Remark 2.9.
Let be an even mapping with
for which there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ71_HTML.gif)
for all and all
. Let
be a constant such that the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ72_HTML.gif)
satisfying for all
By a similar method to the proof of Theorem 2.7, one can show that if for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ73_HTML.gif)
for all .
For the case (where
are nonnegative real numbers and
, there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ74_HTML.gif)
for all .
Corollary 2.10.
Let and let
be nonnegative real numbers such that
and let
be an even mapping satisfying the inequality (2.42) for all
and all
. If for each
the mapping
is continuous in
, then
is
-quadratic.
We may omit the evenness of the mapping in Theorem 2.7.
Theorem 2.11.
Let be a mapping for which there exists a function
satisfying (2.19) and (2.43) for all
and all
. Let
be a constant such that the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ75_HTML.gif)
satisfying for all
If for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ76_HTML.gif)
for all .
Proof.
Since it follows from (2.19) that
We decompose
into the even part
and the odd part
It follows from (2.19) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ77_HTML.gif)
for all and all
. By Theorem 2.7, there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ78_HTML.gif)
for all . We get from (2.62) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ79_HTML.gif)
for all and all
, where
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ80_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ81_HTML.gif)
for all . Letting
in (2.66), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ82_HTML.gif)
for all . Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ83_HTML.gif)
for all . Let
We introduce a generalized metric on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ84_HTML.gif)
Now we consider the mapping defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ85_HTML.gif)
Similar to the proof of Theorem 2.3, we deduce that the sequence converges to a fixed point
of
which is quadratic and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ86_HTML.gif)
Also is odd since
is odd. Therefore,
since
is quadratic too. Now (2.61) follows from (2.63) and (2.71).
Corollary 2.12.
Let and let
be nonnegative real numbers and let
be a mapping satisfying the inequality (2.36) for all
and all
. If for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ87_HTML.gif)
for all .
Proof.
Letting and
in (2.36), we get
Now the proof follows from Theorem 2.11 by taking
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ88_HTML.gif)
for all . Then we can choose
and we get the desired result.
Remark 2.13.
Let be a mapping with
for which there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ89_HTML.gif)
for all and all
. Let
be a constant such that the mapping
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ90_HTML.gif)
satisfying for all
By a similar method to the proof of Theorem 2.11, one can show that if for each
the mapping
is continuous in
, then there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ91_HTML.gif)
for all . Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ92_HTML.gif)
for all .
For the case (where
are nonnegative real numbers and
, there exists a unique
-quadratic mapping
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ93_HTML.gif)
for all .
For the case , we have the following counterexample which is a modification of the example of Czerwik [16].
Example 2.14.
Let be defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ94_HTML.gif)
where is a positive real number. Consider the function
by the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ95_HTML.gif)
where It is clear that
is continuous and bounded by
on
. We prove that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ96_HTML.gif)
for all To see this, if
or
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ97_HTML.gif)
Now suppose that Then there exists a nonnegative integer
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ98_HTML.gif)
Therefore,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ99_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ100_HTML.gif)
for all From the definition of
and (2.83), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ101_HTML.gif)
Therefore, satisfies (2.81). Let
be a quadratic function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ102_HTML.gif)
for all Then there exists a constant
such that
for all
(see [57]). So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ103_HTML.gif)
for all Let
with
If
, then
for all
So
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F193035/MediaObjects/13660_2008_Article_1914_Equ104_HTML.gif)
which contradicts (2.88).
Corollary 2.15.
Let and let
be nonnegative real numbers such that
and let
be a mapping satisfying the inequality (2.42) for all
and all
. If for each
the mapping
is continuous in
, then
is
-quadratic.
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Acknowledgments
The authors would like to thank the referees for bringing some useful references to their attention. The second author was supported by Korea Research Foundation Grant KRF-2008-313-C00041.
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Najati, A., Park, C. Fixed Points and Stability of a Generalized Quadratic Functional Equation. J Inequal Appl 2009, 193035 (2009). https://doi.org/10.1155/2009/193035
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DOI: https://doi.org/10.1155/2009/193035