- Research Article
- Open Access
Investigation of the Stability via Shadowing Property
© Sang-Hyuk Lee et al. 2009
- Received: 25 November 2008
- Accepted: 19 May 2009
- Published: 23 June 2009
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.
- Banach Space
- Functional Equation
- Unique Function
- Cauchy Sequence
- Group Homomorphism
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.
H. Hyers  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 . In 1978 Th. M. Rassias  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  for functions where is a normed space, and is a Banach space. Cholewa  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.
then is quadratic, for all
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.
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 . 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.
A 0-pseudo-orbit is called an orbit.
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 ).
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.
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.
for all then is a quadratic function.
for all Hence is a quadratic mapping which completes the proof.
Theorems 3.5 and 3.6 yield the following corollary.
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
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.
If is replaced by in the inequality (4.6), then the proof follows from the proof of Theorem 4.1.
The authors would like to thank the referee for his (or her) constructive comments and suggestions.
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