# Investigation of the Stability via Shadowing Property

- Sang-Hyuk Lee
^{1}, - Heejeong Koh
^{2}and - Se-Hyun Ku
^{3}Email author

**2009**:156167

**DOI: **10.1155/2009/156167

© Sang-Hyuk Lee et al. 2009

**Received: **25 November 2008

**Accepted: **19 May 2009

**Published: **23 June 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.

*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*

- 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:

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

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:

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

## 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:

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.

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

Theorem 3.3 (see [14]).

Let
be a semigroup. Then the mapping
is called a (*semigroup*) *norm* if it satisfies the following properties:

(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

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.

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.

Proof.

Theorem 3.6.

for all then is a quadratic function.

Proof.

for all Hence is a quadratic mapping which completes the proof.

Theorems 3.5 and 3.6 yield the following corollary.

Corollary 3.7.

## 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

Theorem 4.1.

Proof.

Using (4.6), we have

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

for all as Thus the generalized quadratic mapping is unique.

Theorem 4.2.

Proof.

If is replaced by in the inequality (4.6), then the proof follows from the proof of Theorem 4.1.

## Declarations

### Acknowledgment

The authors would like to thank the referee for his (or her) constructive comments and suggestions.

## Authors’ Affiliations

## References

- Ulam SM:
*Problems in Morden Mathematics*. John Wiley & Sons, New York, NY, USA; 1960.Google Scholar - 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.27.4.222MathSciNetView ArticleMATHGoogle Scholar - 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/00210064MathSciNetView ArticleMATHGoogle Scholar - 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/S0002-9939-1978-0507327-1MathSciNetView ArticleMATHGoogle Scholar - 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.2000.7046MathSciNetView ArticleMATHGoogle Scholar - Rassias ThM:
**On the stability of functional equations and a problem of Ulam.***Acta Applicandae Mathematicae*2000,**62**(1):23–130. 10.1023/A:1006499223572MathSciNetView ArticleMATHGoogle Scholar - 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.2000.6788MathSciNetView ArticleMATHGoogle Scholar - Skof F:
**Local properties and approximation of operators.***Rendiconti del Seminario Matematico e Fisico di Milano*1983,**53:**113–129. 10.1007/BF02924890MathSciNetView ArticleMATHGoogle Scholar - Cholewa PW:
**Remarks on the stability of functional equations.***Aequationes Mathematicae*1984,**27**(1–2):76–86.MathSciNetView ArticleMATHGoogle Scholar - 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.2006.02.003MathSciNetView ArticleMATHGoogle Scholar - Forti GL:
**Hyers-Ulam stability of functional equations in several variables.***Aequationes Mathematicae*1995,**50**(1–2):143–190. 10.1007/BF01831117MathSciNetView ArticleMATHGoogle Scholar - 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.2004.03.011MathSciNetView ArticleMATHGoogle Scholar - 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.2006.05.022MathSciNetView ArticleMATHGoogle Scholar - Tabor J:
**Locally expanding mappings and hyperbolicity.***Topological Methods in Nonlinear Analysis*2007,**30**(2):335–343.MathSciNetMATHGoogle Scholar

## Copyright

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.