- Research Article
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Stability of Approximate Quadratic Mappings
Journal of Inequalities and Applications volume 2010, Article number: 184542 (2010)
Abstract
We investigate the general solution of the quadratic functional equation , in the class of all functions between quasi-
-normed spaces, and then we prove the generalized Hyers-Ulam stability of the equation by using direct method and fixed point method.
1. Introduction
In 1940, Ulam [1] gave a talk before the Mathematics Club of the University of Wisconsin in which he discussed a number of unsolved problems. Among these was the following question concerning the stability of homomorphisms.
Let
be a group and let
be a metric group with metric
. Given
, does there exist a
such that if
satisfies
for all
, then a homomorphism
exists with
for all
?
In 1941, the first result concerning the stability of functional equations was presented by Hyers [2]. And then Aoki [3] and Bourgin [4] have investigated the stability theorems of functional equations with unbounded Cauchy differences. In 1978, Th. M. Rassias [5] provided a generalization of Hyers' Theorem which allows the Cauchy difference to be unbounded. It was shown by Gajda [6] as well as by Th. M. Rassias and Šemrl [7] that one cannot prove the Rassias' type theorem when . Găvruta [8] obtained generalized result of Th. M. Rassias' Theorem which allow the Cauchy difference to be controlled by a general unbounded function. J. M. Rassias [9, 10] established a similar stability theorem linear and nonlinear mappings with the unbounded Cauchy difference.
Let and
be real vector spaces. A function
is called a quadratic function if and only if
is a solution function of the quadratic functional equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ1_HTML.gif)
It is well known that a function between real vector spaces is quadratic if and only if there exists a unique symmetric biadditive function
such that
for all
, where the mapping
is given by
. See [11, 12] for the details. The Hyers-Ulam stability of the quadratic functional (1.1) was first proved by Skof [13] for functions
, where
is a normed space and
is a Banach space. Cholewa [14] demonstrated that Skof's theorem is also valid if
is replaced by an abelian group. Czerwik [15] proved the Hyers-Ulam stability of quadratic functional (1.1) by the similar way to Th. M. Rassias control function [5]. According to the theorem of Borelli and Forti [16], we obtain the following generalization of stability theorem for the quadratic functional (1.1): let
be an abelian group and
a Banach space; let
be a mapping with
satisfying the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ2_HTML.gif)
for all . Assume that one of the following conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ3_HTML.gif)
holds for all , then there exists a unique quadratic function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ4_HTML.gif)
for all . 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 [17–23].
In this paper, we consider a new quadratic functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ5_HTML.gif)
for all vectors in quasi--normed spaces. First, we note that a function
is a solution of the functional (1.5) in the class of all functions between vector spaces if and only if the function
is quadratic. Further, we investigate the generalized Hyers-Ulam stability of (1.5) by using direct method and fixed point method. As a result of the paper, we have a much better possible estimation of approximate quadratic mappings by quadratic mappings than that of Czerwik [15] and Skof [13].
2. Stability of (1.5)
Now, we consider some basic concepts concerning quasi--normed spaces and some preliminary results. We fix a real number
with
and let
denote either
or
. Let
be a linear space over
. A quasi-
-norm
is a real-valued function on
satisfying the following.
(1) for all
and
if and only if
.
(2) for all
and all
.
(3) There is a constant such that
for all
.
The pair is called a
space if
is a quasi-
-norm on
. The smallest possible
is called the modulus of concavity of
. A
space is a complete quasi-
-normed space. A quasi-
-norm
is called a
-norm
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ6_HTML.gif)
for all . In this case, a quasi-
-Banach space is called a
-Banach space. We can refer to [24, 25] for the concept of quasinormed spaces and
-Banach spaces. Given a
-norm, the formula
gives us a translation invariant metric on
. By the Aoki-Rolewicz theorem [25] (see also [24]), each quasinorm is equivalent to some
-norm. In [26], Tabor has investigated a version of the D. H. Hyers, Th. M. Rassias, and Z. Gajda theorem (see [5, 6]) in quasibanach spaces. Recently, J. M. Rassias and Kim [27] have obtained stability results of general additive equations in quasi-
-normed spaces.
From now on, let be a quasi-
-normed space with norm
and let
be a
-Banach space with norm
unless we give any specific reference. Now, we are ready to investigate the generalized Hyers-Ulam stability problem for the functional (1.5) using direct method.
Theorem 2.1.
Assume that a function satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ7_HTML.gif)
for all and that
satisfies the following control conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ8_HTML.gif)
for all . Then there exists a unique quadratic function
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ9_HTML.gif)
for all , where
. The function
is defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ10_HTML.gif)
for all
Proof.
Putting in (2.2), we get
. Replacing
by
in (2.2), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ11_HTML.gif)
for all . Dividing (2.6) by
, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ12_HTML.gif)
for all where
. Now letting
and dividing
in (2.7), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ13_HTML.gif)
for all . Therefore we prove from the inequality (2.8) that for any integers
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ14_HTML.gif)
Since the right-hand side of (2.9) tends to zero as , the sequence
is Cauchy for all
and thus converges by the completeness of
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ15_HTML.gif)
Letting in (2.2), respectively, and dividing both sides by
and after then taking the limit in the resulting inequality, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ16_HTML.gif)
and so the function is quadratic.
Taking the limit in (2.9) with as
, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ17_HTML.gif)
which yields the estimation (2.4).
To prove the uniqueness of the quadratic function subject to (2.4), let us assume that there exists a quadratic function
which satisfies (1.5) and the inequality (2.4). Obviously, we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ18_HTML.gif)
for all . Hence it follows from (2.4) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ19_HTML.gif)
for all . Therefore letting
, one has
for all
, completing the proof of uniqueness.
Theorem 2.2.
Assume that a function satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ20_HTML.gif)
for all and that
satisfies conditions
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ21_HTML.gif)
for all . Then there exists a unique quadratic function
satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ22_HTML.gif)
for all . The function
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ23_HTML.gif)
for all
Proof.
In this case, since
and so
by assumption.
Replacing by
in (2.6), we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ24_HTML.gif)
for . Therefore we prove from inequality (2.19) that for any integers
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ25_HTML.gif)
for all . Since the right-hand side of (2.20) tends to zero as
, the sequence
is Cauchy for all
and thus converges by the completeness of
. Define
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ26_HTML.gif)
for all .
Thereafter, applying the same argument as in the proof of Theorem 2.1, we obtain the desired result.
We now introduce a fundamental result of fixed point theory. We refer to [28] for the proof of it, and the reader is referred to papers [29–31].
Theorem 2.3.
Let be a generalized complete metric space (i.e.,
may assume infinite values). Assume that
is a strictly contractive operator with the Lipschitz constant
Then for a given element
one of the following assertions is true:
(A1) for all
;
(A2) there exists a nonnegative integer such that
(A2.1) for all
;
(A2.2) the sequence converges to a fixed point
of
;
(A2.3) is the unique fixed point of
in the set
(A2.4) for all
For an extensive theory of fixed point theorems and other nonlinear methods, the reader is referred to the book of Hyers et al. [32]. In 1996, Isac and Th. M. Rassias [33] applied the stability theory of functional equations to prove fixed point theorems and study some new applications in nonlinear analysis. Cădariu and Radu [29, 31] and Radu [34] applied the fixed point theorem of alternative to the investigation of Cauchy and Jensen functional equations. Recently, Jung et al. [35–40] and Jung and Rassias [41] have obtained the generalized Hyers-Ulam stability of functional equations via the fixed point method.
Now we are ready to investigate the generalized Hyers-Ulam stability problem for the functional (1.5) using the fixed point method.
Theorem 2.4.
Let be a function with
for which there exists a function
such that there exists a constant
satisfying the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ27_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ28_HTML.gif)
for all . Then there exists a unique quadratic function
defined by
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ29_HTML.gif)
for all
Proof.
Let us define to be the set of all functions
and introduce a generalized metric
on
as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ30_HTML.gif)
Then it is easy to show that is complete (see [37, Proof of Theorem
]). Now we define an operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ31_HTML.gif)
for all First, we assert that
is strictly contractive with constant
on
. Given
, let
be an arbitrary constant with
that is,
Then it follows from (2.23) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ32_HTML.gif)
for all that is,
for any
with
Thus we see that
for any
and so
is strictly contractive with constant
on
.
Next, if we put in (2.22) and we divide both sides by
, then we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ33_HTML.gif)
for all which implies
Thus applying Theorem 2.3 to the complete generalized metric space with contractive constant
, we see from
of Theorem 2.3 that there exists a function
which is a fixed point of
, that is,
such that
as
By mathematical induction we know that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ34_HTML.gif)
for all
Since as
by
of Theorem 2.3, there exists a sequence
such that
as
and
for every
Hence, it follows from the definition of
that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ35_HTML.gif)
for all This implies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ36_HTML.gif)
for all
In turn, it follows from (2.22) and (2.23) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ37_HTML.gif)
for all , which implies that
is a solution of (1.5) and so the mapping
is quadratic.
By of Theorem 2.3, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ38_HTML.gif)
which yields the inequality (2.24).
To prove the uniqueness of , assume now that
is another quadratic mapping satisfying the inequality (2.24). Then
is a fixed point of
with
in view of the inequality (2.24). This implies that
and so
by
of Theorem 2.3. The proof is complete.
By a similar way, one can prove the following theorem using the fixed point method.
Theorem 2.5.
Let be a function with
for which there exists a function
such that there exists a constant
satisfying the inequalities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ39_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ40_HTML.gif)
for all . Then there exists a unique quadratic function
defined by
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ41_HTML.gif)
for all
Proof.
We use the same notations for and
as in the proof of Theorem 2.4. Thus
is a complete generalized metric space. Let us define an operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ42_HTML.gif)
for all Then it follows from (2.35) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ43_HTML.gif)
for all that is,
Thus we see that
for any
and so
is strictly contractive with constant
on
.
Next, if we put in (2.34) and we divide both sides by
, then we get by virtue of (2.35)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ44_HTML.gif)
for all which implies
Thereafter, applying the same argument as in the proof of Theorem 2.4, we obtain the desired results.
3. Applications of Main Results
In the following corollary, we have a stability result of (1.5) in the sense of Th. M. Rassias.
Corollary 3.1.
Let and
be real numbers such that
and
for
. Assume that a function
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ45_HTML.gif)
for all and for all
if
. Then there exists a unique quadratic function
which satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ46_HTML.gif)
for all and for all
if
. The function
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ47_HTML.gif)
for all , where
if
.
Proof.
If , then we get
by putting
in (3.1). Letting
for all
and then applying Theorem 2.1 we obtain easily the desired results.
Corollary 3.2.
Let and
be real numbers such that
and
for
. Assume that a function
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ48_HTML.gif)
for all . Then there exists a unique quadratic function
which satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ49_HTML.gif)
for all . The function
is given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ50_HTML.gif)
for all .
In the following corollary, we have a stability result of (1.5) in the sense of Hyers.
Corollary 3.3.
Let be a nonnegative real number. Assume that a function
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ51_HTML.gif)
for all . Then there exists a unique quadratic function
, defined by
which satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ52_HTML.gif)
for all .
In the next corollary, we get a stability result of (1.5) in the sense of J. M. Rassias.
Corollary 3.4.
Let be real numbers such that
and
, where
. Suppose that a function
satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ53_HTML.gif)
for all , and for all
if
. Then there exists a unique quadratic function
which satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2010%2F184542/MediaObjects/13660_2009_Article_2076_Equ54_HTML.gif)
for all and all
if
, where
if
.
Proof.
Letting and applying Theorems 2.1 and 2.2, we get the results.
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Acknowledgment
This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0070940).
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Kim, HM., Kim, M. & Lee, J. Stability of Approximate Quadratic Mappings. J Inequal Appl 2010, 184542 (2010). https://doi.org/10.1155/2010/184542
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DOI: https://doi.org/10.1155/2010/184542