Exponential Dichotomy of the Two-Dimensional Linear Differential System

Hua, Ni; Jing, Wang; Min, Bai

doi:https://doi.org/10.1155/2022/4411772

Journal of Mathematics

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Abstract Introduction Abbreviations Conclusions Data Availability Conflicts of Interest Authors’ Contributions Acknowledgments References Copyright Related Articles

Research Article | Open Access

Volume 2022 | Article ID 4411772 | https://doi.org/10.1155/2022/4411772

Exponential Dichotomy of the Two-Dimensional Linear Differential System

Ni Hua,¹Wang Jing,¹and Bai Min¹

Academic Editor: Chang Phang

Received06 Jun 2022

Accepted18 Aug 2022

Published13 Oct 2022

Abstract

This paper deals with a class of two-dimensional linear differential systems with periodic coefficients. A sufficient condition for the exponential dichotomy of the linear system is given by the fixed point theorem and variable substitution, and some new results are obtained.

1. Introduction

Consider the second-order linear differential equation

Equation (1) is often encountered in engineering, elasticity, electricity, and various oscillation problems.

Because of its importance, scholars have never stopped studying Equation (1) [1–4], and some scholars used transformation as follows:

Then Equation (1) becomes

Huang [5] used Lyapunov’s direct method to study the stability of the ordinary solution of Equation (3) and thereby obtained some sufficient conditions for the stability of the ordinary solution of Equation (1).

Exponential dichotomy is an important property of linear differential equation. The theory of exponential dichotomy is a generalization of the concept of hyperbolic rate of linear autonomous equation in linear nonautonomous equation and plays an important role in the analysis of nonautonomous equation. The theory of exponential dichotomy of linear differential equations can be traced back to Perron’s study of the stability of linear differential equations and the existence of bounded solutions of nonlinear differential equations. In 1934, Li established the theory of exponential dichotomy on the linear difference equation. The theory of exponential dichotomy has been playing an important role in the qualitative and stability research of differential and difference equations and has been applied to various research fields of mathematics with its rich theoretical ideas and complex mathematical skills [6].

Consider the homogeneous linear differential systemwhere is a square matrix of order , and is continuous on ; if there is a projection and positive constants , thenwhere is a fundamental solution matrix of (4), is the -order unit matrix, then (4) is said to have exponential dichotomy on . The exponential dichotomy of linear system on the whole axis is a powerful tool in the stability theory and also a very useful tool in the study of (almost) periodic differential systems [7].

Consider the following two-dimensional linear differential system

Remark 1. If , then Equation (6) becomes Equation (3). So, without losing generality, we just need to discuss Equation (6).
Shi [8] established some conclusions on the sign of characteristic exponents directly in terms of the coefficients. Theorem 1 in Ref. [8] is as follows:

Theorem 1. (see [8]). Consider Equation (6), , and are -periodic continuous functions on , and have continuous derivatives on , and assume that the following conditions hold true:

Then the characteristic exponents of system (6) must be positive and negative.

In this paper, we are devoted to obtaining the sufficient conditions for the exponential dichotomy of system (6) and generalized Theorem 1 in [8].

2. Some Lemmas and Abbreviations

Consider the following equation

Here, is an -periodic continuous -order square matrix function on , is an -periodic continuous -dimensional vector function on , and the corresponding homogeneous linear system of Equation (8) is as follows:

Lemma 1. (see [7]). If is an -periodic continuous -order square matrix function on , is an -periodic -dimensional continuous vector function on , and (9) has exponential dichotomy (5), then the nonhomogeneous linear periodic system (8) has a unique -periodic continuous solution, which can be expressed as follows:where , , and is a fundamental solution matrix of (9).

Lemma 2. (see [9]). Consider the following equationwhere are -periodic continuous functions on ; if , then Equation (11) has a unique -periodic continuous solution , then , and can be written as follows:

Lemma 3. (see [10]). Assume that -periodic sequence is convergent uniformly on any compact set of ,is an -periodic function, and , then is convergent uniformly on.

Lemma 4. (see [11]). Assume that is a metric space, is a convex closed set of , and its boundary is , and if is a continuous compact mapping, such that , then has at least a fixed point on .

Consider one-dimensional periodic differential equation as follows:

Here, is a continuous function, and .

Lemma 5. (see [12]).If has second-order continuous partial derivatives on , and , then Equation (13) has at most two periodic continuous solutions.

For the sake of convenience, assume that is an -periodic continuous function on , then we denote

The rest of the paper is arranged as follows. In section 3, we discuss the existence of two periodic solutions of Riccati’s equation. In section 4, we discuss the exponential dichotomy of two-dimensional linear differential system. In section 5, we consider the nonhomogeneous second-order linear differential equation and get some results about the existence of the unique periodic solution on the equation. Finally, we draw a conclusion of this paper.

3. Two Periodic Solutions of Riccati’s Equation

Topological degree theory and fixed point theorem are often used by scholars to find (almost) periodic solutions of differential equations. Stimulated by the works of [13–15], in this section, we consider the existence of two periodic solutions of the following Riccati’s equation:

The existence of two periodic solutions of Riccati equation is obtained by using the fixed point theorem.

Theorem 2. Consider Equation (15), , and are all -periodic continuous functions on , and assume that the following condition holds true:

Remark 2. Without loss of generality, assume .
Then Equation (15) has exactly one positive, one negative, and two -periodic continuous solutions as follows:(1)One negative -periodic continuous solution is , and then(2)Another positive -periodic continuous solution is , and then

Remark 3. In Ref. [16], the author got a proposition as follows:

Proposition 1. (see [16]). Consider Equation (15), , and are all -periodic continuous functions on , and assume that the following conditions hold true:then Equation (15) has exactly one positive, one negative, and two -periodic continuous solutions , and then

The proof of Theorem 1 in this paper is different from that of Theorem 1 in [16], and from Equation (17), (18), and (20), the ranges of periodic solutions obtained by us is more accurate. It can be seen that Theorem 1 in this paper is an extension of Theorem 1 in paper [16].

Proof. By , Equation (15) can be turned intoDenoteIt follows from thatIt follows from (22) and Equation (21) thatNow, we divide the proof into three steps.(1)We prove the existence of the periodic solution of Equation (15). Assume that , then the distance is defined as follows: Thus, is a complete metric space. Take a convex closed set of as follows: , then consider the following equation: It follows from (23) and (27) that Hence, we have Since , and are -periodic continuous functions, it follows that are -periodic continuous functions, by (30), according to Lemma 2, (24) has a unique -periodic continuous solution as follows: and It follows from (22) and (27) that Hence, we have It follows from (27), (29), and (32) that Hence, . We define a mapping as follows: Thus, if , then ; hence, . Now, we prove that the mapping is a compact mapping. Consider any sequence , then it is given as On the other hand, satisfies Thus, we have 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 which is convergent uniformly on any compact set of . From (40), combined with Lemma 3, is convergent uniformly on ; that is, is relatively compact on . Next, we prove that is a continuous mapping. Assume , and then It follows from (37) that Here, is between and ; thus, is between and ; hence, we have By (41) and the above inequality, we get Therefore, is continuous. From equation (37), . According to Lemma 4, has at least a fixed point on , and the fixed point is the -periodic continuous solution of Equation (15), and then we get(2)We prove the existence of the periodic solution of Equation (15). Assume that When , the distance is defined as follows: Thus, is a complete metric space. Take a convex closed set of as follows: , then consider the following equation: It follows from and (23) and (51) that Hence, we have Since , and are -periodic continuous functions, we get From (54), according to Lemma 2, we get unique -periodic continuous solution as follows: and It follows from (22) and (51) that Hence, we have It follows from (51), (53), and (56) that Hence, . We define a mapping as follows: Thus, if , then ; hence, . Now, we prove that the mapping is a compact mapping. Consider any sequence , then it follows that On the other hand, satisfies Thus, we have Hence, e is uniformly bounded; therefore, is uniformly bounded and equicontinuous on . By the theorem of Ascoli–Arzela, for any sequence, , then there exists a subsequence (also denoted by ) such that is convergent uniformly on any compact set . From (40), combined with Lemma 3, is convergent uniformly on ; that is, is relatively compact on . Next, we prove that is a continuous mapping. Assume that , and then It follows from (61) that Here, is between and ; thus, is between and ; hence, we have From (65) and the above inequality, we get Therefore, is continuous. From Equation (61), . According to Lemma 4, has at least a fixed point on , then the fixed point is the -periodic continuous solution of Equation (15), and then(3)We prove Equation (15) has exactly two periodic solutions and . Let then From Equation (68), according to Lemma 5, Equation (15) has at most two periodic continuous solutions, and we have known that Equation (15) has two periodic continuous solutions: ; thus, it follows that Equation (15) has exactly two periodic solutions , and thenThis is the end of the proof of Theorem 2.

4. Exponential Dichotomy

In this section, we consider Equation (6) and get the sufficient condition for the exponential dichotomy of Equation (6).

Theorem 3. Consider Equation (6),, and are -periodic continuous functions on , and have continuous derivatives on , then assume that the following conditions hold true:Then, Equation (6) has exponential dichotomy.

Proof. From Equation (6), the following equation can be obtained:where .
Letthen Equation (72) becomesLetthen Equation (73) becomesThis is Riccati’s equation. From , Equation (75) satisfies all the conditions of Theorem 2. According to Theorem 2, Equation (75) has exactly one negative, one positive, and two -periodic continuous solutions and .
It follows from Equation (74) that Equation (73) has two continuous solutions:Here, .
It follows from Equation (72) that Equation (71) has two continuous solutions:Here, .
Morveover, we getThus, are linearly independent.
Sincethen Equation (6) has two sets of linearly independent solutions as follows:Taking the fundamental solution matrix of Equation (6), we get as follows:Thus, we haveTake projection , we get as follows:and then it followsSince are -periodic continuous functions on , it follows that there exist such that the following inequalities hold true:Thus, Equation (6) has exponential dichotomy, and the dichotomy constants are .
This is the end of the proof of Theorem 3.
If we turn Equation (6) into a second-order linear differential equation about , we can get the following:

Theorem 4. Consider Equation (6), and , and are -periodic continuous functions on , and and have continuous derivatives on , then assume that the following conditions hold true:then Equation (6) has exponential dichotomy.

Proof. From Equation (6), the following equation can be obtained:whereLetthen Equation (89) becomesLetthen Equation (87) becomesand this is Riccati’s equation. From , Equation (92) satisfies all the conditions of Theorem 2. According to Theorem 2, Equation (92) has exactly one negative, one positive, and two -periodic continuous solutions and .
It follows from Equation (92) two continuous solutions as follows:andHere,
It follows from Equation (89) that Equation (87) has two continuous solutions as follows:andHere, .
Moreover,Thus, are linearly independent.
SinceThen Equation (6) has two sets of linearly independent solutions given as follows:Take the fundamental solution matrix of Equation (6) as follows:Thus, we haveTake projection asThen it followsSince are -periodic continuous functions on , it follows that there exist such that the following inequalities hold true:Thus, Equation (6) has exponential dichotomy, and the dichotomy constants are .
This is the end of the proof of Theorem 4.
Next, we will calculate the characteristic exponents of system (6). We first give the definition of the characteristic exponent of the linear system.

Definition 1. (see[17]). Consider the following linear differential system:where is a square matrix of order , and continuous on is a nonzero solution of (105). Definewhere is called the characteristic exponent of .
According the proof of Theorem 4, we get as follows.

Theorem 5. Consider Equation (6), and and are -periodic continuous functions on , and and have continuous derivatives on , assume that the following conditions hold true:

Then the characteristic exponents of system Equation (6) must be positive and negative.

Proof. According to the proof of Theorem 4, by (95)–(97), and (99), Equation (6) has two linearly independent solutions as follows:where . Thus, we haveSimilarly, we haveThus, the characteristic exponents of system (6) must be positive and negative.
This is the end of the proof of Theorem 5.
Similar as Theorem 5, we get as follows.

Theorem 6. Consider Equation (6), and , and are -periodic continuous functions on , and and have continuous derivatives on , assume that the following conditions hold true:then the characteristic exponents of system (1.5) must be positive and negative.

Remark 4. Comparing Theorem 5 of this paper with Theorem 1 of (88), it can be found that the conditions of Theorem 5 of this paper is weaker than those of Theorem 1 of (88). It can be seen that Theorem 5 of this paper is the generalization of Theorem 1 of (88).

5. Existence of a Unique Periodic Solution

Consider the following equation

By Theorem 3 and Theorem 4, according to Lemma 1, we can easily get as follows.

Theorem 7. Consider Equation (114), and and are -periodic continuous functions on , and and have continuous derivatives on , assume that the following conditions hold true:then Equation (114) has a unique -periodic continuous solution.

Theorem 8. Consider Equation (114), and and are -periodic continuous functions on , and and have continuous derivatives on , assume that the following conditions hold true:Then Equation (114) has a unique -periodic continuous solution.

In Equation (114), if , then (112) becomes the following equation:

Set then (115) is equivalent to the following equation:

So we can easily get what as follows.

Theorem 9. Consider Equation (116), and , and are -periodic continuous functions on , and assume that the following condition holds true:Then Equation (116) has a unique -periodic continuous solution.

Further, if , then we can obtain what as follows.

Theorem 10. Consider the following Hill’s equation

are both -periodic continuous functions on , and assume that the following condition holds true:

Then equation (117) has a unique -periodic continuous solution.

6. Conclusions

In this paper, the exponential dichotomy of two-dimensional linear system is studied. To verify whether the system has exponential dichotomy, the key is to find out the fundamental solution matrix of the system, but it is not easy to find the fundamental solution matrix of the system. In this paper, a new method for judging the exponential dichotomy of two-dimensional linear system is provided; that is, the two-dimensional linear system is transformed into a second-order linear system by using variable substitution; Then, the second-order linear system is transformed into Riccati’s equation, and the existence of two linearly independent periodic solutions of Riccati’s equation is obtained by using the fixed point theorem; thus, the fundamental solution matrix of the two-dimensional linear system is obtained. Under certain conditions, the two-dimensional linear system has exponential dichotomy.

This method to verify the exponential dichotomy of two-dimensional linear system in this paper is new and feasible. It will have certain application value.

Data Availability

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Authors’ Contributions

The authors contributed to each part of this paper equally. The authors read and approved the final manuscript.

Acknowledgments

The research was supported by the 20th Batch of Scientific Research Projects of Jiangsu University (Y20A044).

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