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Recent Advances in Fixed Point Theory in Abstract Spaces

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Volume 2021 |Article ID 5578527 | https://doi.org/10.1155/2021/5578527

Ni  Hua, "A Criterion for the Existence of the Unique Periodic Solution of One-Dimensional Periodic Differential Equation", Journal of Mathematics, vol. 2021, Article ID 5578527, 10 pages, 2021. https://doi.org/10.1155/2021/5578527

A Criterion for the Existence of the Unique Periodic Solution of One-Dimensional Periodic Differential Equation

Academic Editor: Huseyin Isik
Received10 Feb 2021
Revised03 Mar 2021
Accepted10 Mar 2021
Published22 Apr 2021

Abstract

In this paper, we discuss one-dimensional differential equation with -period. By using the fixed point theory, the existence of a periodic solution is obtained; by using the second Lyapunov method, the uniqueness and stability of the periodic solution are obtained.

1. Introduction

Consider the first-order nonautonomous differential equation:where and is continuous and is an -periodic function on .

As we all know, the first-order differential equation (1) is widely used to establish mathematical models in many fields, such as physics, biology, economy, and medicine. Because the nonautonomous differential equations have important practical applications, it has been a research subject of many scientists for a long time. The main problems studied are the existence, stability, and number of periodic solutions (see [18]).

In [9], the authors gave several methods to study periodic solutions of one-dimensional -periodic differential equations, including Poincare’s map, the stability of zero solution and its multiplicity, normal form, and averaging method. They summarized and improved some existing results and obtained some new conclusions as well.

As we all know, the existence, uniqueness, and stability of periodic solutions of differential equations have always been an important research hotspot in the field of differential equations (see [1020]). However, the above literatures are basically about the study of periodic solutions of specific equations, rather than the study of general differential equations.

In [21], the existence of a periodic solution for Abel’s differential equation is obtained first by using the fixed-point theorem. By constructing the Lyapunov function, the uniqueness, and stability of the periodic solution of the equation are obtained.

Stimulated by the works of [21], in this paper, we consider more general differential equation (1). By using the fixed point theorem and constructing Lyapunov function, we give a new criterion for the existence, uniqueness, and stability of the periodic solutions of the equation (1). These results generalize some related results in some literatures.

2. Some Lemmas and Abbreviations

Lemma 1 (see [22]). Consider the equation:where and are -periodic continuous functions on . If , then (2) has a unique -periodic continuous solution , and can be written as

Lemma 2 (see [23]). Assume that an -periodic sequence is convergent uniformly on any compact set of , is an -periodic function, and ; then, is convergent uniformly on .

Lemma 3 (see [24]). Assume is a metric space and is a convex closed set of ; its boundary is if is a continuous compact mapping, such that , then has at least a fixed point on .
For convenience, assume that is an -periodic continuous function on , we denote

3. Main Results

In this section, firstly, we use the mean value theorem to transform equation (1) into an equivalent equation. Then, we use the fixed point theorem to get the existence of periodic solution of equation (1). Finally, we use Lyapunov function method to obtain the uniqueness and stability of the periodic solution.

Theorem 1. Consider equation (1), is continuous, , and exists and is a continuous function, and is an -periodic continuous function on . Assume that the following conditions hold:then (1) equation (1) has a unique -periodic continuous solution and (2) is globally attractive.

Proof. (1) By and differential mean value theorem, equation (1) is transformed into the equation:Noting that is continuous, , exists and is a continuous function, and from (6), it follows that is continuous andDefining, the distance is defined as follows:and thus, is a complete metric space.Take a convex closed set of as follows:, and consider the equation:whereIt follows from (7), (10), and (12) that is an -periodic continuous function on ; thus, there is a positive number such thatIt follows from thatSince and are -periodic continuous functions, it followsis an -periodic continuous function. It follows from (14) and Lemma 1 that equation (11) has a unique -periodic continuous solution as follows:It follows from (6), (10), and (11) thatand hence, we haveIt follows from (10), (14), and (16) thatand therefore, .Define a mapping as follows:and thus, if given any , then ; hence, .Now, we prove that the mapping is a compact mapping.Consider any sequence ; then, it followsOn the other hand, satisfieswhereSince is continuous, , exists and is a continuous function, and from (23) and , it follows that is an -periodic continuous function on .It follows from (10), (13), (22), and (24) thatThus, we haveHence, 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 (27), combined with Lemma 2, is convergent uniformly on , that is to say, is relatively compact on .Next, we prove that is a continuous mapping.Assume , andSince has first-order continuous partial derivative on , it follows from (28) thatthat is,It follows from (21) thatwhere is between and .Since are -periodic continuous functions on , it followsThus, there is a (also, assume it is in the above) when , such thatthat is,SetHence, we haveIt follows from (30) and above inequality thatTherefore, is continuous. By (21), easy to see, . According to Lemma 3, has at least a fixed point on , the fixed point is the -periodic continuous solution of equation (1), and(2)Construct a Lyapunov function as follows:where is the unique solution with initial value of equation (1) and is the periodic solution of equation (1).
Differentiating both sides of (39) along the solution of equation (1), we havewhere is between and .
By , it followsThus, cannot be a periodic solution of equation (1), and it is proved that equation (1) has a unique -periodic continuous solution which is globally attractive.
This is the end of the proof of Theorem 1.

Theorem 2. Consider equation (1), is continuous and , and exists and is a continuous function; is an -periodic continuous function on . Assume that the following conditions hold:Then, (1) equation (1) has a unique -periodic continuous solution and (2) is unstable.

Proof. (1)  By and differential mean value theorem, equation (1) is transformed into the equationNoting that is continuous, , exists and is a continuous function, and from (43), it follows that is continuous andDefining, the distance is defined as follows:Thus, is a complete metric space.Take a convex closed set of as follows:, and consider the equationwhereIt follows from (44), (47), and (49) that is an -periodic continuous function on ; thus, there is a positive number such thatIt follows from thatSince and are -periodic continuous functions, it followsis an -periodic continuous function. It follows from (51) and Lemma 1 that equation (48) has a unique -periodic continuous solution as follows:It follows from (43), (47), and (48) thatHence, we haveIt follows from (47), (51), and (53) thatTherefore, .Define a mapping as follows:Thus, if given any , then , and hence, .Now, we prove that the mapping is a compact mapping.Consider any sequence , and then, it followsOn the other hand, satisfieswhereSince is continuous, , exists and is a continuous function, and from (60) and , it follows that is an -periodic continuous function on .It follows from (47), (50), (59), and (61) thatThus, we haveHence, is uniformly bounded, and 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 (64), combined with Lemma 2, is convergent uniformly on , that is to say, is relatively compact on .Next, we prove that is a continuous mapping.Assume ,Since has first-order continuous partial derivative on , we havethat is,It follows from (58) thatwhere is between and .Since are -periodic continuous functions on , it followsand thus, there is a (also, assume it is in the above), when , such thatthat is,Setand hence, we haveIt follows from (67) and the above inequality thatTherefore, is continuous. By (58), easy to see, . According to Lemma 3, has at least a fixed point on , the fixed point is the -periodic continuous solution of equation (1), and(2)Construct a Lyapunov function as follows:where is the unique solution with initial value of equation (1) and is the periodic solution of equation (1).
Differentiating both sides of (76) along the solution of equation (1), we havewhere is between and .
By , it followsand thus, cannot be a periodic solution of equation (1), and it is proved that equation (1) has a unique -periodic continuous solution which is unstable.
This is the end of the proof of Theorem 2.

4. Examples

The following examples show the feasibility of our main results.

Example 1. Consider the equationwhereand hence, equation (79) satisfies all the conditions of Theorem 1. It follows from Theorem 1 that equation (79) has a unique -periodic solution which is globally attractive.
Clearly, according to the solution curve of equation (79), if given any initial value (e.g., ), then the solution curve of equation (79) does tend to the curve of the periodic solution (see Figure 1).

Example 2. Consider the equationwhereHence, equation (81) satisfies all the conditions of Theorem 2. It follows from Theorem 2 that equation (81) has a unique -periodic solution