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 [1–7].
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, By , it follows that by (68) and (69), 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 .
(2) Define a Lyapunov function as follows: where is the unique solution with the initial value of (1); differentiating both sides of (70) 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 unstable.
5. Examples
The following examples show the feasibility of our main results.
Example 1. Consider the following equation:It is easy to calculate that , and Clearly, conditions of Theorem 2 are satisfied. It follows from Theorem 2 that system (73) has a unique periodic solution , which is uniformly asymptotic stable. Clearly, by the graph of solution curve, given any initial value (e.g., ), the solution curve of (73) tends to the curve of the periodic solution (see Figure 1).
Example 2. Consider the following equation: It is easy to calculate that , and Clearly, conditions of Theorem 6 are satisfied. It follows from Theorem 6 that (75) has a unique periodic solution , which is uniformly asymptotic stable.
Clearly, by the graph of solution curve, given any initial value ( e.g., ), the solution curve of (75) tends to the curve of the periodic solution (see Figure 2).
6. Conclusion
Abel’s differential equation plays an important role in many physical and technical applications; because of this, the paper also deals with Abel’s differential equation; we first give two results about the existence and uniqueness of the periodic solution on a nonlinear differential equation by using the fixed point theory; then, we get the existence of the periodic solution of Abel’s differential equation, the results having great application value.
Data Availability
Data sharing is not applicable to this article as no datasets were generated or analysed during the current study.
Conflicts of Interest
The author declares that there are no conflicts of interest.
Acknowledgments
This work is supported by special project supported by senior personnel of Jiangsu University (14JDG176).