Abstract

In this paper, the existence of a periodic solution for Abel’s differential equation is obtained first by using the fixed-point theorem. Then, by constructing the Lyapunov function, the uniqueness and stability of the periodic solution of the equation are obtained.

1. Introduction

The nonlinear Abel-type first-order differential equationplays an important role in many physical and technical applications [1, 2]. The mathematical properties of equation (1) have been intensively investigated in the mathematical and physical literature [3–15].

Recently, Ni [16] supposed that is a periodic particular solution of Abel’s differential equation under some conditions, then by means of the transformation method and the fixed-point theory, the author presented an alternative method of generating the other periodic solutions of Abel’s differential equation. Ni and Tian [17] got the sufficient conditions which guarantee the existence of three periodic solutions for Abel’s differential equation.

All the time, the existence, uniqueness, and stability of the periodic solution of differential equations have always been an important research hotspot in the field of differential equations. So, in this paper, equation (1) is considered and a new criterion is given to judge the existence, uniqueness, and stability of the periodic solution of equation (1) by using the fixed-point theorem and constructing the Lyapunov function, and some new results are obtained.

The rest of the paper is arranged as follows: in Section 2, some lemmas and abbreviations are introduced to be used later; in Section 3, the existence and stability of the unique periodic solution of equation (1) are obtained; in Section 4, two examples are given to verify the main results of the paper; the paper ends with a short conclusion.

2. Preliminaries

In this section, some definitions, lemmas, and abbreviations are given, which will be used to prove the main results.

Definition 1. (see [18]). Suppose is an -periodic continuous function on , thenmust exist, where is called the Fourier coefficient of and , such that , is called the Fourier index of ; there is a countable set , when and , and as long as , there must be , where is called the exponential set of .

Definition 2. (see [18]). A set of real numbers composed of linear combinations of integer coefficients of elements in is called a module or a frequency module of , which is denoted as , that is,

Lemma 1. (see [19]). Consider the following equation:where are the -periodic continuous functions, and if , then equation (4) has a unique -periodic continuous solution and can be written as follows:

Lemma 2. (see [20]). Suppose that an -periodic function sequence is convergent uniformly on any compact set of , is an -periodic function, and , then is convergent uniformly on .

Lemma 3. (see [21]). Suppose is a metric space, is a convex closed set of , and its boundary is , if is a continuous compact mapping, such that , then has at least a fixed point on .

For the sake of convenience, suppose that is an -periodic continuous function on , then it is denoted that

3. Main Results

In this section, the existence, uniqueness, and stability of the periodic solution of equation (1) are studied.

Theorem 1. Consider equation (1), and , and are all -periodic continuous functions on , suppose that the following conditions hold:then, equation (1) has a unique -periodic continuous solution which is uniformly asymptotic stable.

Proof. LetBy and (8), it follows thatBy zero point theorem, there is at least a zero point such thatBy and , it follows thatthus, has a unique zero point such that (10) holds. Since , and are all continuous functions on , and from (10) and (11), it follows that is a continuous function on .
By (10), it follows that is -periodic on , that is,If is not an -periodic function, thenthus, has two zero points, and , and it is a contradiction to the fact that has a unique zero point on . Therefore,that is to say, is an -periodic continuous function on .
In addition,Comparing the coefficients of the third power of of the left and right hands of (16), the following is obtained:By (16), let , and then satisfiesthus, equation (1) becomesBy simple computation and , it is obtained that the discriminant of the quadratic polynomial of one variable satisfies the following:Next, the existence of the unique periodic solution of equation (1) is proven by using the fixed-point theorem.
SupposeGiven any , the distance is defined as follows:thus, is a complete metric space. A convex closed set of S is taken as follows:Given any , consider the following equation:whereIt is easy to see that is an -periodic continuous function, and by and (20),and is bounded above and below by some negative constants for all and any , that is,Thus,By Lemma 1, equation (24) has a unique -periodic continuous solution as follows:By (18), it follows thatBy (25), it follows thatthus, it follows thatHence,By (23), (28), and (29), the following is obtained:thus
Define a mapping as follows:thus, if given any , then ; hence .
Now, it is proven that the mapping is a compact mapping.
Consider any sequence , then it follows thatOn the other hand, satisfieswhereIt is easy to see that is an -periodic continuous function, and by (23), (25), (27), (36), and (38), the following is obtained:thus,hence, is uniformly bounded; therefore, is uniformly bounded and equicontinuous on . By the theorem of Ascoli–Arzela, for any sequence , there exists a subsequence (also denoted by ) such that is convergent uniformly on any compact set of . By (40), combined with Lemma 2, is convergent uniformly on , that is to say, is relatively compact on .
Next, it is proven that is a continuous mapping.
Suppose andBy (35),where is between and , thus ; hence,By (41) and the above inequality, it follows thattherefore, is continuous. By (35), . According to Lemma 3, has at least a fixed point on and the fixed point is the periodic continuous solution of equation (1), andDefine a Lyapunov function as follows:where is the unique solution with an initial value of equation (1), and differentiating both sides of (46) along the solution of equation (1), the following is obtained:By , there is a positive number such that ; hence,therefore, the periodic solution of equation (1) is uniformly asymptotic stable; thus,so cannot be a periodic solution of equation (1), and equation (1) has a unique periodic solution which is uniformly asymptotic stable.
This is the end of the proof of Theorem 1.
Similarly, the following theorem is obtained.