The Scientific World Journal

Volume 2013 (2013), Article ID 372681, 6 pages

http://dx.doi.org/10.1155/2013/372681

Research Article

Locally Expansive Solutions for a Class of Iterative Equations

Wei Song and Sheng Chen

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

Received 25 August 2013; Accepted 30 September 2013

Academic Editors: F. J. Garcia-Pacheco, A. Ibeas, and F. Minhós

Copyright © 2013 Wei Song and Sheng Chen. 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.

Iterative equations which can be expressed by the following form , where , are investigated. Conditions for the existence of locally expansive solutions for such equations are given.

1. Introduction

Let be the set of all continuous self-mappings on a topological space . For any , let denote the th iterate of ; that is, . Equations having iteration as their main operation, that is, including iterates of the unknown mapping, are called iterative equations. It is one of the most interesting classes of functional equations [1–4], because it includes the problem of iterative roots [2, 5, 6], that is, finding some such that is identical to a given . The well-known Feigenbaum equation , arising in the discussion of period-doubling bifurcations [7, 8], is also an iterative equation.

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 [9, 10], polynomial-like iterative equations are important not only in the study of functional equations but also in the study of dynamical systems. For instance, such equations are encountered in the discussion on transversal homoclinic intersection for diffeomorphisms [11], normal form of dynamical systems [12], and dynamics of a quadratic mapping [13]. Some problems of invariant curves for dynamical systems also lead to such iterative equations [14].

For the case that is linear, where (1) can be written as

many results [15–17] have been given to present all of its continuous solutions. Conditions that ensure the uniqueness of such solutions are also given by [18, 19].

For the case that is nonlinear, the basic problems such as existence, uniqueness, and stability cannot be solved easily. In 1986, Zhang [20], under the restriction that , constructed an interesting operator called “structural operator” for (1) and used the fixed point theory in Banach space to get the solutions of (1). Hence, he overcame the difficulties encountered by the formers. By means of this method, Zhang and Si made a series of works concerning these qualitative problems, such as [21–24]. After that, (1) and other type equations were discussed extensively by employing this idea (see [25–31] and references therein).

On the other hand, great efforts have been made to solve the “leading coefficient problem” which was raised by [32, 33] as an open problem. The essence of solving this problem is to abolish the technical restriction and discuss (1) under the more natural assumption . As mentioned in [34, 35], a mapping is said to be locally expansive (resp., locally contractive) at its fixed point , if (resp., ). In 2004, Zhang [35] gave positive answers to this problem in local solutions in some cases of coefficients, but this paper only discussed the locally expansive case and the nonhyperbolic case. In 2009, Chen and Zhang [34] gave positive answers to this problem with more combinations between locally expansive mappings and locally contractive ones and combinations between increasing mappings and decreasing ones. The main tools used in the two papers above are Schröder transformation and Schauder fixed point theorem. In 2012, J. M. Chen and L. Chen [36] consider the locally contractive solutions of the iterative equation , and some results on locally contractive solutions of [34] were generalized. In 2007, Xu and Zhang [37] answered this problem by constructing solutions of (1). Their strategy is to construct the solutions piece by piece via a recursive formula obtained form (1). Following this idea, global increasing and decreasing solutions [38, 39] were also investigated.

Motivated by the above results, we will consider the existence of locally expansive solutions for the iterative equation of the following form:

where . Some results on locally expansive solutions in [34] are generalized.

1.1. Basic Assumptions, Definitions, and Notations

Firstly, we state some assumptions on the known function and the solution . Let be two intervals in , and let denote the set of all maps from to . It is well known that, for a compact interval , is a Banach space with the norm and is also a Banach space with the norm .

For convenience, let denote and , where . Let denote , where .

The assumption on is

(f1), where is an interval to be determined.

Assumptions on are

(H1); (H2)(H3) in a neighborhood of , where are nonnegative constants, .

Define a set

Let , and be three constants, and define a set

The set is nonempty and is a convex compact subset of .

For , and , we define two functions as follows:

where . By the choices of and , we have and , where .

If the solution of (3) can be expressed as by the Schröder transformation, where is a constant to be determined, then (3) can be reduced to the following auxiliary equation:

If function is a solution of (3), then we can differentiate the equation. In fact, we can get that the derivative is a zero of the following polynomial:

We refer to the polynomial (8) as the characteristic polynomial of (3).

Finally, we give a basic lemma.

Lemma 1. Let be a convex open set, and let and belong to . If is continuous on and differentiable on , then there exists a such that

2. Main Results

Let .

Theorem 2. Suppose that . Suppose that there is a neighborhood of   that satisfies

for all , ; for all and all , .

Then, (3) has a locally expansive increasing solution near .

Theorem 3. Suppose that is odd and . Suppose that there is a neighborhood   of that satisfies

for all , ; for all , for all odd and for all even .

Then, (3) has a locally expansive decreasing solution near .

Theorem 4. Suppose that is even and . Suppose that there is a neighborhood   of that satisfies

for all , ; for all , for all odd and for all even .

Then, (3) has a locally expansive decreasing solution near .

3. Proof of the Main Results

Lemma 5. Under the conditions of Theorem 2 (Theorems 3 and 4, resp.), there is a constant (resp., in both cases) and such that for arbitrary given , (7) has a   solution on with and .

Proof. If is real and (7) has a local solution with and , then by differentiating the equation, we can see that is a root of characteristic polynomial (8).

If hypotheses of Theorem 2 hold, the hypothesis implies

But when , and this means that has a root . In the case of Theorems 3 and 4, has a root . Since for both of the cases and , and is a zero of (8), we have

The above inequality holds because of the choice of the sign of . This also means that . Now, we can choose a constant such that the following statements are true;

(1)) holds on , where ; (2)) holds on , where ; (3) holds on .

For a given , let

Furthermore, we can choose a such that for, for all , we have

Define a mapping as follows:

In order to show that is a self-mapping on , we calculate

Obviously, . Since , we have

Moreover, for all , by , we have

By and the choice of , we can get that

By the definition of , we get that

Summing up the above discussion, we get that .

Now, we will prove that is continuous. Considering , by Lemma 1 and , we have

Furthermore, by , we have

Finally, let

and we get that

Now, the continuity of is evident. By Schauder’s fixed point theorem, there exists a such that . This means that (7) with the chosen has a solution on with derivative at .

Proof of Theorems 2–4. Let be the solution of (7) obtained in Lemma 5. By the continuity of , we are able to choose a neighborhood of such that exists and is also on . Without any loss of generality, we can assume that . Hence, is a homeomorphism. Moreover, we can choose a neighborhood of which is so small that for all . Let for . Clearly is also and invertible on . Moreover, all iterates , are well defined on , and . Obviously, we have , and is locally expansive. Finally, for any , we have

Therefore, is a locally expansive solution of (3).

4. Examples

Example 1. Consider the following equation:

Obviously, . It is easy to verify that satisfy the assumptions of Theorem 2. This equation has at least one locally expansive increasing solution in a neighborhood of .

Example 2. Consider the following equation:

Obviously, . It is easy to verify that satisfy the assumptions of Theorem 3. This equation has at least one locally expansive decreasing solution in a neighborhood of .

Example 3. Consider the following equation:

Obviously, . It is easy to verify that satisfy the assumptions of Theorem 4. This equation has at least one locally expansive decreasing solution in a neighborhood of .

Conflict of Interests

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

Acknowledgments

The research of the authors was supported by National Natural Science Foundation of China (Grant nos. 11101105 and 11001064), by the Fundamental Research Funds for the Central Universities (Grant no. HIT. NSRIF. 2014085), and by the Scientific Research Foundation for the Returned Overseas Chinese Scholars.

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