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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 HyersUlam stability of the functional equation in Banach spaces.
1. Introduction
The notion of pseudoorbits very often appears in several areas of the dynamical systems. A pseudoorbit 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 pseudoorbit 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
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:
for some and for all then the limit
exists for each and is the unique additive function such that
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 HyersUlam 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 HyersUlam 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
then is quadratic, for all
Proof.
By letting in the equation (2.1), we have
Since we have
that is, By the assumption we have Now, by letting and we get
for all Since is even, we have
for all From the following equation:
we have
Now letting and we have
Hence
for all Then it is easily obtained that is quadratic. This completes the proof.
We call this quadratic mapping a generalized quadratic mapping of rtype.
3. Stability Using Shadowing Property
In this section, we will take that is, we will investigate the generalized mappings of 1type, 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.
Let A sequence in is a  pseudoorbit for if
A 0pseudoorbit 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 locallyinvertible.
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 fixed, and let be locally invertible. We assume additionally that Let and let be an arbitrary pseudoorbit. Then there exists a unique such that
Moreover,
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
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:
where are endomorphisms in , are endomorphisms in We assume additionally that there exists such that
Then there exists a unique function such that
Moreover, then satisfies
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
for all Then there exists a unique function such that
for all
Proof.
By letting in (3.11), we have
Thus Now, let in (3.11). From the inequality , we have
Thus we obtain
for all To apply Lemma 3.4 for the function we may let
Then we have
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
and also we have
Theorem 3.6.
Suppose that is locally invertible, is locally invertible, and is locally invertible. If a function satisfies the following equation:
for all then is a quadratic function.
Proof.
By letting in (3.20), we have
By the uniqueness of the local division by we get Also, setting in (3.20), implies that
that is, we have
for all By the uniqueness of the local division by we get for all Now, by letting and in (3.20), we get
for all By the uniqueness of the local division by we have
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
for all
4. On HyersUlamRassias 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
for all
Theorem 4.1.
Let be an even mapping satisfying Assume that there exists a function such that
for all Then there exists a unique generalized quadratic mapping of type such that
for all
Proof.
By letting and in (4.3), since is an even mapping and we have
for all Then we obtain that
for all
Using (4.6), we have
for all and all positive integer Hence we get
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
for all Since is even, so is By the definition of and (4.3), we have that
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.
Let be an even mapping satisfying Assume that there exists a function such that
for all Then there exists a unique generalized quadratic mapping of type such that
for all
Proof.
If is replaced by in the inequality (4.6), then the proof follows from the proof of Theorem 4.1.
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The authors would like to thank the referee for his (or her) constructive comments and suggestions.
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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
Keywords
 Banach Space
 Functional Equation
 Unique Function
 Cauchy Sequence
 Group Homomorphism