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A Fixed Point Approach to the Stability of the Functional Equation ![](//media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_IEq1_HTML.gif)
Fixed Point Theory and Applications volume 2009, Article number: 912046 (2009)
Abstract
By applying the fixed point method, we will prove the Hyers-Ulam-Rassias stability of the functional equation under some additional assumptions on the function
and spaces involved.
1. Introduction
In 1940, Ulam [1] gave a wide ranging talk before the mathematics club of the University of Wisconsin in which he discussed a number of important unsolved problems. Among those was the question concerning the stability of group homomorphisms: "Let be a group and let
be a metric group with the metric
. Given
, does there exist a
such that if a function
satisfies the inequality
for all
, then there exists a homomorphism
with
for all
?''
The case of approximately additive functions was solved by Hyers [2] under the assumption that and
are Banach spaces. Indeed, he proved that each solution of the inequality
, for all
and
, can be approximated by an exact solution, say an additive function. Rassias [3] attempted to weaken the condition for the bound of the norm of the Cauchy difference as follows:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ1_HTML.gif)
and derived Hyers' theorem for the stability of the additive mapping as a special case. Thus in [3], a proof of the generalized Hyers-Ulam stability for the linear mapping between Banach spaces was obtained. A particular case of Rassias' theorem regarding the Hyers-Ulam stability of the additive mapping was proved by Aoki (see [4]).
The stability concept that was introduced by Rassias' theorem provided a large influence to a number of mathematicians to develop the notion of what is known today with the term Hyers-Ulam-Rassias stability of the linear mapping. Since then, the stability of several functional equations has been extensively investigated by several mathematicians. The terminology Hyers-Ulam-Rassias stability originates from these historical backgrounds. The terminology can also be applied to the case of other functional equations. For more detailed definitions of such terminologies, we can refer to [5–10].
Solutions of the functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ2_HTML.gif)
were investigated in [11, Section  2.2]. The stability problem for a general equation of the form was investigated by Cholewa [12] (see also [13]). Indeed, Cholewa proved the superstability of that equation under some additional assumptions on the functions and spaces involved.
In this paper, we will apply the fixed point method to prove the Hyers-Ulam-Rassias stability of the functional equation (1.2) for a class of functions of a vector space into a Banach space. To the best of authors' knowledge, no one has yet applied the fixed point method for studying the stability problems of (1.2). So, one of the aims of this paper is to apply the fixed point theory to this case.
Throughout this paper, let denote either
or
. Let
and
be a vector space over
and a Banach space over
, respectively.
2. Preliminaries
Let be a set. A function
is called a generalized metric on
if and only if
satisfies
() if and only if
;
() for all
;
() for all
.
Note that the only substantial difference of the generalized metric from the metric is that the range of generalized metric includes the infinity. We now introduce one of fundamental results of fixed point theory. For the proof, refer to [14]. For an extensive theory of fixed point theorems and other nonlinear methods the reader is referred to the book of Hyers et al. [15].
Theorem 2.1.
Let be a generalized complete metric space. Assume that
is a strictly contractive operator with the Lipschitz constant
. If there exists a nonnegative integer
such that
for some
, then the followings are true:
(a)the sequence converges to a fixed point
of
;
(b) is the unique fixed point of
in
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ3_HTML.gif)
(c)if , then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ4_HTML.gif)
Recently, Cădariu and Radu [16] applied the fixed point method to the investigation of the Cauchy additive functional equation [17, 18]. Using such a clever idea, they could present a short, simple proof for the Hyers-Ulam-Rassias stability of Cauchy and Jensen functional equations.
We remark that Isac and Rassias [19] were the first mathematicians who apply the Hyers-Ulam-Rassias stability approach for the proof of new fixed point theorems.
3. Main Results
In this section, by using an idea of Cădariu and Radu (see [16, 17]), we will prove the Hyers-Ulam-Rassias stability of the functional equation under the assumption that
is a bounded linear transformation.
Theorem 3.1.
Let and
be a vector space over
and a Banach space over
, respectively, and let
be a Banach space over
. Assume that
is a bounded linear transformation, whose norm is denoted by
, satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ5_HTML.gif)
for all and that there exists a real number
with
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ6_HTML.gif)
for all . Moreover, assume that
is a given function satisfying
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ7_HTML.gif)
for all . If
and a function
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ8_HTML.gif)
for any , then there exists a unique solution
of (1.2) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ9_HTML.gif)
for all .
Proof.
First, we denote by the set of all functions
and by
the generalized metric on
defined as
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ10_HTML.gif)
Then, as in the proof of [20, Theorem  3.1], we can show that is a generalized complete metric space. Now, let us define an operator
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ11_HTML.gif)
for every .
We assert that is strictly contractive on
. Given
, let
be an arbitrary constant with
, that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ12_HTML.gif)
for each . By (3.2), (3.3), (3.7), and (3.8), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ13_HTML.gif)
for all , that is, in view of (3.6),
for any
, where
is the Lipschitz constant with
. Thus,
is strictly contractive.
We now verify that . If we substitute
for
and
in (3.4), then it follows from (3.3) and (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ14_HTML.gif)
for every , that is,
.
Taking in Theorem 2.1, (a) implies that there exists a function
, which is a fixed point of
, such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ15_HTML.gif)
Due to Theorem 2.1(c), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ16_HTML.gif)
which implies the validity of (3.5). According to Theorem 2.1(b), is the unique fixed point of
with
.
We now assert that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ17_HTML.gif)
for all and
. Indeed, it follows from (3.1), (3.2), (3.3), (3.4), and (3.7) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ18_HTML.gif)
for any . We assume that (3.13) is true for some
. Then, it follows from (3.1), (3.2), (3.3), (3.7), and (3.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ19_HTML.gif)
which proves the validity of (3.13) for all .
Finally, we prove that for any
. Since
is continuous as a bounded linear transformation, it follows from (3.11) and (3.13) that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ20_HTML.gif)
which ends our proof.
Obviously, for nonnegative constants and
,
satisfies the conditon (3.3).
Corollary 3.2.
Let and
be a vector space over
and a Banach space over
, respectively, and let
be a Banach space over
. Assume that
is a bounded linear transformation, whose norm is denoted by
, satisfying the condition (3.1) and that there exists a real number
satisfying the condition (3.2). If
and a function
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ21_HTML.gif)
for all and for some nonnegative real constants
and
, then there exists a unique solution
of (1.2) such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ22_HTML.gif)
for all .
4. An Example
Assume that and consider the Banach spaces
and
, where we define
for all
. Let
and
be fixed complex numbers with
and let
be a linear transformation defined by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ23_HTML.gif)
Then it is easy to show that satisfies the condition (3.1).
If and
are complex numbers satisfying
, then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ24_HTML.gif)
Thus, we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ25_HTML.gif)
which implies the boundedness of the linear transformation .
On the other hand, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ26_HTML.gif)
for any ; that is, we can choose
for the value of
and then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ27_HTML.gif)
If a function satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ28_HTML.gif)
for all and for some
, then our Corollary 3.2 (with
and
) implies that there exists a unique function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ29_HTML.gif)
for all and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F912046/MediaObjects/13663_2009_Article_1190_Equ30_HTML.gif)
for any .
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Acknowledgments
The authors would like to express their cordial thanks to the referees for their useful comments which have improved the first version of this paper. This work was supported by the National Research Foundation of Korea Grant funded by the Korean Government (no. 2009-0071206).
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Jung, SM., Min, S. A Fixed Point Approach to the Stability of the Functional Equation .
Fixed Point Theory Appl 2009, 912046 (2009). https://doi.org/10.1155/2009/912046
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DOI: https://doi.org/10.1155/2009/912046