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Journal of Applied Mathematics
Volume 2018, Article ID 6725989, 11 pages
https://doi.org/10.1155/2018/6725989
Research Article

The Fixed Point Theory and the Existence of the Periodic Solution on a Nonlinear Differential Equation

Faculty of Science, Jiangsu University, 301 Xuefu Road, Zhenjiang, Jiangsu 212013, China

Correspondence should be addressed to Ni Hua; moc.621@979auhin

Received 3 June 2018; Revised 12 July 2018; Accepted 17 July 2018; Published 1 August 2018

Academic Editor: Mehmet Sezer

Copyright © 2018 Ni Hua. 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

This paper deals with a nonlinear differential equation, by using the fixed point theory. The existence of the periodic solution of the nonlinear differential equation is obtained; these results are new.

1. Introduction

The nonlinear Abel-type first-order differential equation plays an important role in many physical and technical applications; because of its importance, many scholars have studied it [17].

Recently, A. Cima, A. Gasull, and F. Manosas [8] gave the maximum number of polynomial solutions of some integrable polynomial Abel differential equations; Jaume Giné Claudia and Valls [9] studied the center problem for Abel polynomial differential equations of second kind; Jianfeng Huang and Haihua Liang [10] were devoted to the investigation of Abel equation by means of Lagrange interpolation formula; they gave a criterion to estimate the number of limit cycles of the Abel’s equations; Berna Bülbül and Mehmet Sezer [11] introduced a numerical power series algorithm which is based on the improved Taylor matrix method for the approximate solution of Abel-type differential equations; Ni et al. [12] discussed the existence and stability of the periodic solutions of (1) and obtained the sufficient conditions which guaranteed the existence and stability of the periodic solutions for (1) from a particular one.

In this paper, we consider the following more general nonlinear differential equation: firstly, we give two results about the existence and uniqueness of the periodic solution of (2) by using the fixed point theory; then, we use Lyapunov function method and discuss the stability of the periodic solution; further, we discuss (1) and get the existence of the periodic solution of (1); some new results are obtained.

2. Preliminaries

Lemma 1 (see [12]). Consider the following linear differential equation: where are periodic continuous functions; if , then for (3) there exists a unique periodic continuous solution , and can be written as follows:For the sake of convenience, suppose that is an -periodic continuous function on ; we denote

The rest of the paper is arranged as follows. We will study the existence and stability property of the periodic solution of system (2) in the next section and discuss the existence of the periodic solution of system (1) on Section 4. We end this paper with two examples.

3. Periodic Solution on a Nonlinear Differential Equation

Theorem 2. Consider (2); are both periodic continuous functions on and uniformly with respect to ; suppose that the following conditions hold: then for (2) there exists a unique -periodic continuous solution , and is uniformly asymptotic stable.

Proof. LetGiven any , the norm is defined as follows:thus is a Banach space. Consider the following equation: by , according to Lemma 1, (10) has a unique periodic continuous solution as follows: and Define a map as follows: thus if given any , then ; hence ; given any , we have here, is between and ; thus , so we have By , it follows that by (16), is a compression mapping; according to the compression mapping principle, has a fixed point on , the fixed point is the unique periodic continuous solution of (2), and
Define a Lyapunov function as follows: where is the unique solution with the initial value of (2); differentiating both sides of (17) along the solution of (2), we get By and , we have ; there is a positive number such that , and hence we have Therefore, the periodic solution is uniformly asymptotic stable.

Theorem 3. Consider (2); are both periodic continuous functions on and uniformly, with respect to , suppose that the following conditions hold: then for (2) there exists a unique -periodic continuous solution , and is unstable.

Proof. Let Given any , the norm is defined as follows:thus is a Banach space. Consider the following equation: by , according to Lemma 1, (24) has a unique periodic continuous solution as follows: and Define a map as follows: thus if given any , then ; hence ; given any , we have here, is between and ; thus , so we have By , it follows thatby (30), is a compression mapping; according to the compression mapping principle, has a fixed point on , the fixed point is the unique periodic continuous solution of (2), and
Define a Lyapunov function as follows: where is the unique solution with the initial value of (2); differentiating both sides of (31) along the solution of (2), we get By and , we have ; there is a positive number such that , and hence we have Therefore, the periodic solution is unstable.
Consider the following nonlinear differential equation: it is easy for us to draw the two corollaries about the existence and stability of the periodic solution on (34).

Corollary 4. Consider (34); is periodic continuous functions on and uniformly with respect to , is an periodic continuous function on ; suppose that the following conditions hold: then for (34) there exists a unique -periodic continuous solution , and is uniformly asymptotic stable.

Corollary 5. Consider (34); is periodic continuous functions on and uniformly with respect to , is an periodic continuous function on , suppose that the following conditions hold: then for (34) there exists a unique -periodic continuous solution , and is unstable.

4. Periodic Solution on Abel’s Differential Equation

Theorem 6. Consider (1); are all periodic continuous functions; suppose that the following conditions hold: here, ; then for (1) there exists a unique -periodic continuous solution , and is uniformly asymptotic stable.

Proof. Define a set given any , the norm is defined as follows: thus is a Banach space. Consider the following equation: By and , we get that and by Lemma 1, (43) has a unique continuous periodic solution as follows: by (46), we get Define a map as follows: thus if given any , then ; hence ; given any , we have here, is between and ; thus , so we have therefore, By , it follows thatby (51) and (52), is a compression mapping; according to the compression mapping principle, has a fixed point on , the fixed point is the unique periodic continuous solution of (1), and .
Define a Lyapunov function as follows: where is the unique solution with the initial value of (1); differentiating both sides of (53) along the solution of (1), we get By , there is a positive number such that , and hence we have Therefore, the periodic solution of (1) is uniformly asymptotic stable.

Theorem 7. Consider (1); are all periodic continuous functions; suppose that the following conditions hold: where ; then for (1) there exists a unique -periodic continuous solution , and is unstable.

Proof. Define a set given any , the norm is defined as follows:thus is a Banach space. Consider the following equation: By and , we get that and by Lemma 1, (60) has a unique periodic continuous solution as follows: and Define a map as follows: thus if given any , then ; hence ; given any , we have where is between and ; thus
, so we have therefore,