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Investigation of the Stability via Shadowing Property
Journal of Inequalities and Applications volume 2009, Article number: 156167 (2009)
Abstract
The shadowing property is to find an exact solution to an iterated map that remains close to an approximate solution. In this article, using shadowing property, we show the stability of the following equation in normed group: , where
,
and
is a mapping. And we prove that the even mapping which satisfies the above equation is quadratic and also the Hyers-Ulam stability of the functional equation in Banach spaces.
1. Introduction
The notion of pseudo-orbits very often appears in several areas of the dynamical systems. A pseudo-orbit is generally produced by numerical simulations of dynamical systems. One may consider a natural question which asks whether or not this predicted behavior is close to the real behavior of system. The above property is called the shadowing property (or pseudo-orbit tracing property). The shadowing property is an important feature of stable dynamical systems. Moreover, a dynamical system satisfying the shadowing property is in many respects close to a (topologically, structurally) stable system. It is well known that the shadowing property is a useful notion for the study about the stability theory, and the concept of the shadowing is close to this of the stability in dynamical systems.
In this paper, we are going to investigate the stability of functional equations using the shadowing property derived from dynamical systems.
The study of stability problems for functional equations is related to the following question raised by 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  a  
  such that if a mapping
satisfies the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ1_HTML.gif)
for all then a homomorphism
exists with
for all
-
D.
H. Hyers [2] provided the first partial solution of Ulam's question as follows. Let
and
be Banach spaces with norms
and
respectively. Hyers showed that if a function
satisfies the following inequality:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ2_HTML.gif)
for some and for all
then the limit
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ3_HTML.gif)
exists for each and
is the unique additive function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ4_HTML.gif)
for any Moreover, if
is continuous in
for each fixed
then
is linear.
Hyers' theorem was generalized in various directions. The very first author who generalized Hyers' theorem to the case of unbounded control functions was T. Aoki [3]. In 1978 Th. M. Rassias [4] by introducing the concept of the unbounded Cauchy difference generalized Hyers's Theorem for the stability of the linear mapping between Banach spaces. Afterward Th. M. Rassias's Theorem was generalized by many authors; see [5–7].
The quadratic function satisfies the functional equation
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ5_HTML.gif)
Hence this equation is called the quadratic functional equation, and every solution of the quadratic equation (1.5) is called a quadratic function.
A Hyers-Ulam stability theorem for the quadratic functional equation (1.5) was first proved by Skof [8] for functions where
is a normed space, and
is a Banach space. Cholewa [9] noticed that the theorem of Skof is still true if the relevant domain
is replaced by an abelian group. Several functional equations have been investigated in [10–12].
From now on, we let be an even integer, and
such that
We denote
In this paper, we investigate that a mapping
satisfies the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ6_HTML.gif)
for a mapping We will prove the stability in normed group by using shadowing property and also the Hyers-Ulam stability of each functional equation in Banach spaces.
2. A Generalized Quadratic Functional Equation in Several Variables
Lemma 2.1.
Let be an even integer number,
with
and
vector spaces. If an even mapping
which satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ7_HTML.gif)
then is quadratic, for all
Proof.
By letting in the equation (2.1), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ8_HTML.gif)
Since we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ9_HTML.gif)
that is, By the assumption
we have
Now, by letting
and
we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ10_HTML.gif)
for all Since
is even, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ11_HTML.gif)
for all From the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ12_HTML.gif)
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ13_HTML.gif)
Now letting and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ14_HTML.gif)
Hence
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ15_HTML.gif)
for all Then it is easily obtained that
is quadratic. This completes the proof.
We call this quadratic mapping a generalized quadratic mapping of r-type.
3. Stability Using Shadowing Property
In this section, we will take that is, we will investigate the generalized mappings of 1-type, and hence we will study the stability of the following functional equation by using shadowing property:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ16_HTML.gif)
for all where
is a commutative semigroup.
Before we proceed, we would like to introduce some basic definitions concerning shadowing and concepts to establish the stability; see [13]. After then we will investigate the stability of the given functional equation based on ideas from dynamical systems.
Let us introduce some notations which will be used throughout this section. We denote the set of all nonnegative integers,
a complete normed space,
the closed ball centered at
with radius
and let
be given.
Definition 3.1.
Let    A sequence  
  in  
  is a  
- pseudo-orbit for  
if
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ17_HTML.gif)
A 0-pseudo-orbit is called an orbit.
Definition 3.2.
Let    be given. A function  
  is locally
-invertible at  
if for any point  
  in  
there exists a unique element  
in  
such that  
If  
is locally
-invertible at each  
then we say that  
  is locally
-invertible.
For a locally -invertible function
we define a function
in such a way that
denote the unique
from the above definition which satisfies
Moreover, we put
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ18_HTML.gif)
Theorem 3.3 (see [14]).
Let  be fixed, and let
be locally
-invertible. We assume additionally that  
Let  
and let
be an arbitrary
-pseudo-orbit. Then there exists a unique
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ19_HTML.gif)
Moreover,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ20_HTML.gif)
Let be a semigroup. Then the mapping
is called a (semigroup) norm if it satisfies the following properties:
(1)for all
(2)for all
(3)for all and also the equality holds when
where
is the binary operation on
Note that is called a group norm if
is a group with an identity
, and it additionally satisfies that
if and only if
From now on, we will simply denote the identity of
and the identity
of
by 0. We say that
is a normed (semi)group if
is a (semi)group with the (semi)group norm
Now, given an Abelian group
and
we define the mapping
by the formula
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ21_HTML.gif)
Since is a normed group, it is clear that
is locally
-invertible at 0, and
Also, we are going to need the following result. Tabor et al. proved the next lemma by using Theorem 3.3.
Lemma 3.4.
Let Let
be a commutative semigroup, and
a complete Abelian metric group. We assume that the mapping
is locally
-invertible and that
Let
satisfy the following two inequalities:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ22_HTML.gif)
where are endomorphisms in
, are endomorphisms in
We assume additionally that there exists
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ23_HTML.gif)
Then there exists a unique function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ24_HTML.gif)
Moreover, then satisfies
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ25_HTML.gif)
Proof.
Using the proof of [13, Theorem  2], one can easily show this lemma.
Let ,
an even integer,
an Abelian group, and
a complete normed Abelian group.
Theorem 3.5.
Let be arbitrary, and let
be a function such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ26_HTML.gif)
for all Then there exists a unique function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ27_HTML.gif)
for all
Proof.
By letting in (3.11), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ28_HTML.gif)
Thus Now, let
in (3.11). From the inequality
, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ29_HTML.gif)
Thus we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ30_HTML.gif)
for all To apply Lemma 3.4 for the function
we may let
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ31_HTML.gif)
Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ32_HTML.gif)
Thus we also obtain and so all conditions of Lemma 3.4 are satisfied. Hence we conclude that there exists a unique function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ33_HTML.gif)
and also we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ34_HTML.gif)
Theorem 3.6.
Suppose that is locally
-invertible,
is locally
-invertible, and
is locally
-invertible. If a function
satisfies the following equation:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ35_HTML.gif)
for all then
is a quadratic function.
Proof.
By letting in (3.20), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ36_HTML.gif)
By the uniqueness of the local division by we get
Also, setting
in (3.20),
implies that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ37_HTML.gif)
that is, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ38_HTML.gif)
for all By the uniqueness of the local division by
we get
for all
Now, by letting
and
in (3.20), we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ39_HTML.gif)
for all By the uniqueness of the local division by
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ40_HTML.gif)
for all Hence
is a quadratic mapping which completes the proof.
Theorems 3.5 and 3.6 yield the following corollary.
Corollary 3.7.
Let be a function satisfying (3.11), and let
be arbitrary. Suppose that
is locally
-invertible,
is locally
-invertible, and
is locally
-invertible. Then there exists a quadratic function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ41_HTML.gif)
for all
4. On Hyers-Ulam-Rassias Stabilities
In this section, let be a normed vector space with norm
a Banach space with norm
and
an even integer. For the given mapping
we define
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ42_HTML.gif)
for all
Theorem 4.1.
Let be an even mapping satisfying
Assume that there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ43_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ44_HTML.gif)
for all Then there exists a unique generalized quadratic mapping of
-type
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ45_HTML.gif)
for all
Proof.
By letting and
in (4.3), since
is an even mapping and
we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ46_HTML.gif)
for all Then we obtain that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ47_HTML.gif)
for all
Using (4.6), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ48_HTML.gif)
for all and all positive integer
Hence we get
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ49_HTML.gif)
for all and all positive integers
and
with
Hence the sequence
is a Cauchy sequence. From the completeness of
we may define a mapping
by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ50_HTML.gif)
for all Since
is even, so is
By the definition of
and (4.3), we have that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ51_HTML.gif)
for all Since
the mapping
is a generalized quadratic mapping of
-type by Lemma 2.1. Also, letting
and passing the limit
in (4.8), we get (4.4).
To prove the uniqueness, suppose that is another generalized quadratic mapping of
-type satisfying (4.4). Then we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ52_HTML.gif)
for all as
Thus the generalized quadratic mapping
is unique.
Theorem 4.2.
Let be an even mapping satisfying
Assume that there exists a function
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ53_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ54_HTML.gif)
for all Then there exists a unique generalized quadratic mapping of
-type
such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F156167/MediaObjects/13660_2008_Article_1899_Equ55_HTML.gif)
for all
Proof.
If is replaced by
in the inequality (4.6), then the proof follows from the proof of Theorem 4.1.
References
Ulam SM: Problems in Morden Mathematics. John Wiley & Sons, New York, NY, USA; 1960.
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: 222–224. 10.1073/pnas.27.4.222
Aoki T: On the stability of the linear transformation in Banach spaces. Journal of the Mathematical Society of Japan 1950, 2: 64–66. 10.2969/jmsj/00210064
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
Rassias ThM: On the stability of functional equations in Banach spaces. Journal of Mathematical Analysis and Applications 2000,251(1):264–284. 10.1006/jmaa.2000.7046
Rassias ThM: On the stability of functional equations and a problem of Ulam. Acta Applicandae Mathematicae 2000,62(1):23–130. 10.1023/A:1006499223572
Rassias ThM: The problem of S. M. Ulam for approximately multiplicative mappings. Journal of Mathematical Analysis and Applications 2000,246(2):352–378. 10.1006/jmaa.2000.6788
Skof F: Local properties and approximation of operators. Rendiconti del Seminario Matematico e Fisico di Milano 1983, 53: 113–129. 10.1007/BF02924890
Cholewa PW: Remarks on the stability of functional equations. Aequationes Mathematicae 1984,27(1–2):76–86.
Chu H-Y, Kang DS: On the stability of an
-dimensional cubic functional equation. Journal of Mathematical Analysis and Applications 2007,325(1):595–607. 10.1016/j.jmaa.2006.02.003
Forti GL: Hyers-Ulam stability of functional equations in several variables. Aequationes Mathematicae 1995,50(1–2):143–190. 10.1007/BF01831117
Forti G-L: Comments on the core of the direct method for proving Hyers-Ulam stability of functional equations. Journal of Mathematical Analysis and Applications 2004,295(1):127–133. 10.1016/j.jmaa.2004.03.011
Tabor J, Tabor J: General stability of functional equations of linear type. Journal of Mathematical Analysis and Applications 2007,328(1):192–200. 10.1016/j.jmaa.2006.05.022
Tabor J: Locally expanding mappings and hyperbolicity. Topological Methods in Nonlinear Analysis 2007,30(2):335–343.
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Lee, SH., Koh, H. & Ku, SH. Investigation of the Stability via Shadowing Property. J Inequal Appl 2009, 156167 (2009). https://doi.org/10.1155/2009/156167
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DOI: https://doi.org/10.1155/2009/156167