Open Access

Investigation of the Stability via Shadowing Property

Journal of Inequalities and Applications20092009:156167

https://doi.org/10.1155/2009/156167

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.

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
(1.1)
for all then a homomorphism exists with for all
  1. 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:

     
(1.2)
for some and for all then the limit
(1.3)
exists for each and is the unique additive function such that
(1.4)

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 [57].

The quadratic function satisfies the functional equation

(1.5)

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 [1012].

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:

(1.6)

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
(2.1)

then is quadratic, for all

Proof.

By letting in the equation (2.1), we have
(2.2)
Since we have
(2.3)
that is, By the assumption we have Now, by letting and we get
(2.4)
for all Since is even, we have
(2.5)
for all From the following equation:
(2.6)
we have
(2.7)
Now letting and we have
(2.8)
Hence
(2.9)

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:

(3.1)

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
(3.2)

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

(3.3)

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
(3.4)
Moreover,
(3.5)

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

(3.6)

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:
(3.7)
where are endomorphisms in , are endomorphisms in We assume additionally that there exists such that
(3.8)
Then there exists a unique function such that
(3.9)
Moreover, then satisfies
(3.10)

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
(3.11)
for all Then there exists a unique function such that
(3.12)

for all

Proof.

By letting in (3.11), we have
(3.13)
Thus Now, let in (3.11). From the inequality , we have
(3.14)
Thus we obtain
(3.15)
for all To apply Lemma 3.4 for the function we may let
(3.16)
Then we have
(3.17)
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
(3.18)
and also we have
(3.19)

Theorem 3.6.

Suppose that is locally -invertible, is locally -invertible, and is locally -invertible. If a function satisfies the following equation:
(3.20)

for all then is a quadratic function.

Proof.

By letting in (3.20), we have
(3.21)
By the uniqueness of the local division by we get Also, setting in (3.20), implies that
(3.22)
that is, we have
(3.23)
for all By the uniqueness of the local division by we get for all Now, by letting and in (3.20), we get
(3.24)
for all By the uniqueness of the local division by we have
(3.25)

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
(3.26)

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

(4.1)

for all

Theorem 4.1.

Let be an even mapping satisfying Assume that there exists a function such that
(4.2)
(4.3)
for all Then there exists a unique generalized quadratic mapping of -type such that
(4.4)

for all

Proof.

By letting and in (4.3), since is an even mapping and we have
(4.5)
for all Then we obtain that
(4.6)

for all

Using (4.6), we have

(4.7)
for all and all positive integer Hence we get
(4.8)
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
(4.9)
for all Since is even, so is By the definition of and (4.3), we have that
(4.10)

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

(4.11)

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
(4.12)
(4.13)
for all Then there exists a unique generalized quadratic mapping of -type such that
(4.14)

for all

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

(1)
School of Mecatronics Engineering, Changwon National University
(2)
Department of Mathematics Education, College of Education, Dankook University
(3)
Department of Mathematics, Chungnam National University

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Copyright

© Sang-Hyuk Lee et al. 2009

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