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Abstract and Applied Analysis
Volume 2012 (2012), Article ID 954352, 13 pages
http://dx.doi.org/10.1155/2012/954352
Research Article

Smooth Solutions of a Class of Iterative Functional Differential Equations

School of Mathematics, Chongqing Normal University, Chongqing 401331, China

Received 14 December 2011; Accepted 5 March 2012

Academic Editor: Miroslava Růžičková

Copyright © 2012 Houyu Zhao. 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

By Faà di Bruno’s formula, using the fixed-point theorems of Schauder and Banach, we study the existence and uniqueness of smooth solutions of an iterative functional differential equation .

1. Introduction

There has been a lot of monographs and research articles to discuss the kinds of solutions of functional differential equations since the publication of Jack Hale's paper [1]. Several papers discussed the iterative functional differential equations of the form where . More specifically, Eder [2] considered the functional differential equation and proved that every solution either vanishes identically or is strictly monotonic. Furthermore, Fečkan [3] and Wang [4] studied the equation with different conditions. Staněk [5] considered the equation and obtained every solution either vanishes identically or is strictly monotonic. Si and his coauthors [6, 7] studied the following equations: and established sufficient conditions for the existence of analytic solutions. Especially in [8, 9], the smooth solutions of the following equations: have been studied by the fixed-point theorems of Schauder and Banach.

A smooth function is taken to mean one that has a number of continuous derivatives and for which the highest continuous derivative is also Lipschitz. Let if are continuous, is the set in which and maps a closed interval I into I. As in [9], we, using the same symbols, denote the norm then with is a Banach space, and is a subset of . For given , let

For convenience, we will make use of the notation where , and are nonnegative integers. Let be a closed interval in . By induction, we may prove that where is a uniquely defined multivariate polynomial with nonnegative coefficients. The proof can be found in [8].

In order to seek a solution of (1.6), in such that is a fixed point of the function , that is, , it is natural to seek an interval of the form with .

Let us define

2. Smooth Solutions of (1.6)

In this section, we will prove the existence theorem of smooth solutions for (1.6). First of all, we have the inequalities in the following for all : and the proof can be found in [9].

Theorem 2.1. Let , where and satisfy where , then (1.6) has a solution in provided the following conditions hold:
(i) where , and the sum is over all nonnegative integer solutions of the Diophantine equation and ,
(ii) where and ,
(iii) where and ,

Proof. Define an operator from into by We will prove that for any , where the second inequality is from (2.4) and . Thus, .
It is easy to see that and by Faà di Bruno's formula, for , we have where the sum is over all nonnegative integer solutions of the Diophantine equation and .
Furthermore, note , by (2.6) and (2.7), where and . Thus, for , By (2.8), we have where and .
Finally, where and . By (2.9), we see that Now, we can say that is an operator from into itself.
Next, we will show that is continuous. Let , then where and .
Moreover, we can find some constants such that where are the positive constants depend on , and . Then Here, . So we can say that is continuous.
It is easy to see that is closed and convex. We now show that is a relatively compact subset of . For any , Next, for any in , we have Hence, is bounded in and equicontinuous on , and by the Arzela-Ascoli theorem, we know is relatively compact in , since is the subset of , and we can say that is relatively compact in .
From Schauder's fixed-point theorem, we conclude that for some in . By differentiating both sides of the above equality, we see that is the desired solution of (1.6). This completes the proof.

Theorem 2.2. Let , where and satisfy (2.4), then (1.6) has a unique solution in provided the conditions (2.6)–(2.9) hold and in (2.23).

Proof. Since , we see that defined by (2.10) is contraction mapping on the close subset of . Thus, the fixed point in the proof of Theorem 2.1 must be unique. This completes the proof.

Remark 2.3. By Theorem 2.1 or Theorem 2.2, the existence and uniqueness of smooth solutions of an iterative functional differential equation of the form (1.6) can be obtained. If , we can also find that the solution is -smooth.
Now, we will show that the conditions in Theorem 2.1 do not self-contradict. Consider the following equation: where , and . Moreover, we take . Then, (2.4) is satisfied, and define the interval . Now, take , then (2.6) and (2.7) are satisfied.
Finally, if we take as positive, and then (2.8) and (2.9) are satisfied.
Thus, we have shown that when and are defined as above, then there will be a solution for (2.28) in .

Acknowledgment

This work was partially supported by the Natural Science Foundation of Chongqing Normal University (Grant no. 12XLB003).

References

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