Discrete Dynamics in Nature and Society

Volume 2014 (2014), Article ID 454569, 7 pages

http://dx.doi.org/10.1155/2014/454569

Research Article

Hyers-Ulam Stability of Iterative Equation in the Class of Lipschitz Functions

Chao Xia and Wei Song

Department of Mathematics, Harbin Institute of Technology, Harbin, Heilongjiang 150001, China

Received 14 March 2014; Revised 10 May 2014; Accepted 10 May 2014; Published 1 June 2014

Academic Editor: Krzysztof Ciepliński

Copyright © 2014 Chao Xia and Wei Song. 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.

Abstract.

Hyers-Ulam stability is a basic sense of stability for functional equations. In the present paper we discuss the Hyers-Ulam stability of a kind of iterative equations in the class of Lipschitz functions. By the construction of a uniformly convergent sequence of functions we prove that, for every approximate solution of such an equation, there exists an exact solution near it.

1. Introduction

It is pointed out that the theory of Ulam’s type stability (also quite often connected, e.g., with the names of Bourgin, Gvruţa, Aoki, Hyers, and Rassias) is a very popular subject of investigations at the moment in [1]. For more details of Ulam’s type stability, we refer the audience to [1–3]. In [1], the authors present a survey of some selected recent developments (results and methods) in the theory of Ulam’s type stability, such as the Forti method and the methods of fixed points and stability in non-Archimedean spaces. These results and methods have not been treated at all or have been treated only marginally. The very book of Jung [2] covers and offers almost all classic results on the Hyers-Ulam-Rassias stability in an integrated and self-contained fashion. And the authors of [3] discussed Hyers-Ulam stability for functional equations in single variable, including the forms of linear functional equations, nonlinear functional equations, and iterative equations. And they also clarified the relation between Hyers-Ulam stability and other senses of stability which are used for functional equations.

Let be the set of all continuous self-mappings on a topological space. For any , let denote theth iteration of ; that is, , , . Equations having iterations as their main operation, that is, including iterations of the unknown mapping, are called iterative equations. It is one of the most interesting classes of functional equations [4–7]. As a natural generalization of the problem of iterative roots, iterative equations of the following form:

are known as polynomial-like iterative equations. Here is an integer, , is a given mapping, and is unknown. As mentioned in [8, 9], polynomial-like iterative equations are important not only in the study of functional equations but also in the study of dynamical systems.

In 1986, Zhang [10] constructed an interesting operator called “structural operator” for (1) and used the fixed point theory in Banach space to get the solutions of (1). From then on (1) and other types of equations were discussed extensively by employing this idea (see [8, 9, 11–16] and references therein). In 2002, by means of a modification of Zhang’s method applied in [10], Kulczycki and Tabor [11] investigated the existence of Lipschitzian solutions of the iterative equation

where is a compact convex subset of and is a given Lipschitz function. They generalized the results of Zhang [10] to a more general case.

In 2002, Xu and Zhang [17] discussed the Hyers-Ulam stability for a nonlinear iterative equation which includes the polynomial-like iterative equation. In 2003, Agarwal et al. [3] investigated the Hyers-Ulam stability for a general form of iterative equations which include the polynomial-like iterative equation with variable coefficients. Motivated by the above results, in this paper, we will discuss the Hyers-Ulam stability of (2). As in [11], we give a result on the Hyers-Ulam stability of a functional equation with a more general form firstly. And, by means of this result, the Hyers-Ulam stability of (2) is investigated. In fact, we want to generalize the result of Xu and Zhang [17] to high-dimensional case.

2. Basic Assumptions, Definitions, and Lemmas

Let be a compact convex subset of , , with nonempty interior. Let

In , we use the supremum norm

where denotes the usual metric of . Obviously, is a complete metric space.

Definition 1 (see [11]). Let be a convex compact subset of with nonempty interior. For , and , one defines

where denotes the boundary of .

For a given function , we define its Lipschitz constant by

Lemma 2 (see [11]). Let , , and be arbitrary. One assumes additionally that is invertible. Then

Lemma 3 (see [11]). For every , the mapping

is well defined with Lipschitz constant.

Lemma 4 (see [11]). For , , , the mapping defined by

is Lipschitz with constant 1.

Lemma 5. Let , and be given. For any , is surjective.

Proof. Note that is homeomorphic to, where is the unit ball of . Without any loss of generality, let . Since , set ; we have , where denotes the degree of . Suppose that is not surjective. Let . Since , we know that ; then there exists a retraction mapping . Thus is a continuous mapping and . This means that , a contradiction. So is surjective.

Lemma 6. If , are homeomorphisms of with Lipschitz constant , then

Proof. Note that

So we have

Furthermore

This means that .

Lemma 7. Suppose that , are continuous mappings and , whereis a positive constant. Then

Proof. We use induction to prove this lemma. For , . Suppose that this is true, for ; then

Let be a subset of and satisfy

(A1):for all , , ;(A2): if is invertible, then ;(A3): is closed and convex.

And let

It is easy to see that .

Let and let be a map. For any we will consider a functional equation of the following form:

where is unknown.

Definition 8. If, for every such that , there exists a solution of (17) such that , where is a constant and does not depend on the choice of , then (17) is said to have Hyers-Ulam stability.

3. Main Results

Lemma 9. Let be a convex compact subset of with nonempty interior. Let and with . Let be given. If there exists a decreasing function and a constant such that

(P1):for any , (P2): .

Then the following two facts hold:

(c1):for any , is a homeomorphism with ;(c2): for any given , the sequence

 is well defined and each is surjective.

Proof. (c1): since , we get that . By inequality (P2) and the range of the function , we know that both and hold. Therefore , which implies that is nonempty. For any and any , since and is decreasing, we have

This means that is injective. Since , by Lemma 5, we obtain that is surjective. Thus is a homeomorphism. And follows from Lemma 2.

(c2): by Lemma 3, the mapping

is continuous. By Lemma 4 and assumptions of , the mapping

is well defined and continuous.

Now, by the assumption (P1) and the fact , it is easy to see that, for every ,

Since (A2) holds and by Lemma 2, we know that all elements of are invertible. Moreover we can get that

and by means of (A1) we have

Finally, by inequality (P2), we obtain that the mapping

is well defined.

Take any . By means of the mapping (25), it is easy to see that the sequence

is well defined and each . By virtue of Lemma 5, it is easy to see that all are surjective.

Theorem 10. Let be a convex compact subset of with nonempty interior. Let and with . Let be given and let there exist a decreasing function and a constant such that (P1) and (P2) of Lemma 9 hold. Moreover, if there is a constant such that

then, for any with

where is a constant, there exists a unique continuous solution of (17) such that

where

Proof. Since all the assumptions of Lemma 9 hold, so we can construct a sequence of functions as follows. We take

first and then by means of equality (18) repeatedly, define

By the conclusions (c1) and (c2) of Lemma 9, both and are well defined for all . And each is a homeomorphism of onto itself with .

Now we claim that both

hold for . We prove these two inequalities inductively.

For , by virtue of the definition of , we have

And noting that and is surjective for , we have

Assume that they are true for integer . Then, by the same arguments, we can get that

Thus (33) is proved by induction.

For any positive integers and with , we have

Since , it follows from inequality (37) that

As a Cauchy sequence, converges uniformly in the Banach space . Let

Clearly, . From

we know that is a solution of (17). Furthermore,

This proves (29).

Concerning uniqueness, we assume that there is another continuous solution for (17), such that

where only depends on . By Lemma 5, we know that is also surjective. By means of Lemma 6 and the assumption that is Lipschitzian, it follows that

This implies that

However, ; this implies that ; that is, .

4. Iterative Equations in

Theorem 11. Let be a convex compact subset of with nonempty interior. Let be arbitrary with . Assume that , and there exists a constant such that

(a1):(a2): (a3):

hold. Then the equation

has Hyers-Ulam stability.

Proof. Let . For , define

We claim that, for , . For , by the compactness and convexity of and the assumption (a1), we have

where is the convex hull of the set Moreover, by means of the assumption (a1) and the definition of the mapping and the set , it is easy to see that . Now we prove the continuity of . By the compactness of , we first denote

By the assumption (a1), for any , there exists a positive integer such that

Note that each , , , is continuous. Then, for any , there is a such that when the fact

holds. Thus we can get that

This means that the fact holds.

By virtue of the mapping , (45) can be rewritten as

Since , this implies that And by the assumption (a2), we see that . For , define

Since we obtain that and it is easy to see that is decreasing.

For any and any with , we claim that . It is enough to show that, for ,

But

It follows from the assumption (a2) that

Thus (P1) and (P2) of Lemma 9 also hold. Moreover, we can get that

hold. Let . By the assumption (a2), we know that there must be a positive integer such that . So we know that . Furthermore, by the assumption (a3), we have . Then, by Theorem 10, we get the conclusion.

Example 12. For + = , where and denotes the distance of the point from , obviously, . By simple calculation, we get that, for any , , the following two inequalities:

hold. This implies that . Then we consider the equation

Denote , , . Define − . Let and . Then we have

By Theorem 11, (59) has Hyers-Ulam stability.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgment

The research of the authors was supported by National Natural Science Foundation of China (Grant nos. 11101105 and 11001064).

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