Abstract

The existence of two periodic solutions of the Riccati’s equation when the coefficients are -periodic and have different signs is obtained. One of these solutions is unstable on and the other one is attractive on some region. Finally, an example is presented.

1. Introduction

The nonlinear Riccati type first-order differential equation plays an important role in fluid mechanics and the theory of elastic vibration. There are a lot of research about this equation [16]: in [1, 2], the sufficient conditions of the existence of periodic solutions of the system were given; also in [1], the stability of periodic solutions of (1) was obtained, and there is no globally asymptotic stable periodic solution; [3] studied some special types of Riccati equations and got the general solution and the existence of periodic solutions of (1); [4] studied (1) with characteristic multiplier; [5] studied the high dimensional Riccati equation and obtained some sufficient conditions of the existence of periodic solutions of the equation; [6] obtained some criteria for the existence of periodic solutions of (1).

An extensive study of the set of periodic solutions of (1) was initiated in [7] and continued in [812]. In those papers the coefficients are real. The complex ones were considered in [13, 14]. The problem of nonexistence of periodic solutions was investigated in [9, 12, 1519]. There are some papers (e.g., [2024]) where stability and asymptotic behaviour of solutions were considered.

Recently, M. R. Mokhtarzadeh, M. R. Pournaki, and A. Razani [25] dealt with scalar Riccati differential equations and assumed that a, b, and c are -periodic continuous real functions on R and give certain conditions to guarantee the existence of at least one periodic solution for (1); Pawel Wilczynski [26] gave a few sufficient conditions for the existence of two periodic solutions of the Riccati ordinary differential equation in the plane and gave also examples of the equation without periodic solutions for the Riccati ordinary differential equation.

Consider a class of Riccati equation as follows: where are periodic continuous functions, about the existence of periodic solutions of (2); there are two results.

Proposition 1 (see [1]). Consider (2), suppose and then (2) has two periodic continuous solutions , and

Proposition 2 (see [6]). Consider the following equation: where is an periodic continuous function, if and then (5) has two periodic continuous solutions , and where , , ,

There are also articles on the periodic solutions of Riccati equation (2)(see [27, 28]).

It is well known that scholars often use the fixed point theory to study the existence of periodic solutions on differential equation (see [2931]).

Stimulated by the works of [2931], in this paper, we consider (2), and by using the fixed point theory, we obtain the existence of two periodic continuous solutions of Riccati type equation: one is attractive on some region and unstable on another region, and another is unstable. We give the ranges of the size of the two periodic continuous solutions: one is positive, another is negative; they are symmetrical about , and following are our main results.

Conclusion. Consider (2), are periodic continuous functions, suppose that the following conditions hold: and then (2) has two periodic continuous solutions,, and meanwhile, we get the stability of two periodic solutions of (2).

Then, we consider (1) and give two results about the existence of two periodic solutions on (1). These conclusions generalize the relevant conclusions of related papers.

2. Some Lemmas, Definitions, and Abbreviations

Lemma 3 (see [32]). Consider the following (10) where are periodic continuous functions. If ,then (10) has a unique periodic continuous solution , , and can be written as follows.

Lemma 4 (see [33]). 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 5 (see [34]). Suppose is a metric space, is a convex closed set of , its boundary is ; if is a continuous compact mapping, such that , then has a fixed point on .

Definition 6 (see [33, page 43]). Suppose is an periodic continuous function on ; then must exists, is called the Fourier coefficient of , the such that is called the Fourier index of . There is a countable set ,when , , as long as , there must be , and is called the exponential set of .

Definition 7 (see [33, page 47]). 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, For the sake of convenience, suppose that is an -periodic continuous function on ; we denote

3. Periodic Solutions of Riccati’s Type Equation

Theorem 8. Consider (2), are periodic continuous functions, and suppose that the following conditions hold: and then (2) has two periodic continuous solutions.
(1) One periodic continuous solution is , and is attractive if given initial value on , and unstable if given initial value on , where is any given initial value of (2) and (2) Another periodic continuous solution is , and is unstable on .

Proof. By , (2) can be turned into (1) Suppose Given any , the distance is defined as follows: and thus is a complete metric space. Take a convex closed set of S as follows: Given any and considering the following equation by and (22), we get that and hence we have Since are periodic continuous functions, it follows that are periodic continuous functions; by (25), according to Lemma 3, (23) has a unique positive periodic continuous solution as follows andBy (22), it follows thatand hence we have By (22), (24), and (27), we get and and hence,
Define a mapping as follows and thus, given any , ; hence .
Now, we prove that the mapping is a compact operator.
Consider any sequence ; then we have the following. On the other hand, satisfies thus we have and 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 R. Also combined with Lemma 4, is convergent uniformly on R; that is to say, is relatively compact on .
Next, we prove that is a continuous operator.
Suppose , and and by (33),we have where is between and ; thus is between and , and hence we have By (37), it follows that and, therefore, is continuous; by (33), it is easy to see that , and according to Lemma 5, has at least a fixed point on ; the fixed point is the periodic continuous solution of (2), and Let where is the unique solution of (2) with initial value , and is the periodic solution of (2); differentiating both sides of (42) along the solution of (2), we get This is Bernoulli’s equation; let , and it can be turned into the following equation Note that according to Lemma 3, (44) has a unique periodic continuous solution as follows it is easy to know and thus the infinite integral is convergent; thereby, the infinite integral is convergent.
In addition, and and thus we have By (46), we know (44) has a unique periodic continuous solution , and by the transformations ,we know (2) has another periodic continuous solution as follows Since are periodic solutions of (2), we have Since is a periodic solution of (2), we only consider its maximum and minimum values in a cycle; suppose is the minimum value of , is the maximum value of , is the minimum value point of , and is the maximum value point of , where ; then we have thus it follows thatand it is easy to see that From (46), let then we get that the possible extremum of satisfying so thus we have and it is easy to see that are also extreme points of , is the maximum value of , is the minimum value of , and thus is the minimum value of and is the maximum value of . By (51), it follows that are two possible extremums of ; moreover, let and then we get the following equation that all possible extreme points of function satisfy Take the negative sign of (64); since they are the possible extremums of , by (61), (62), and (64), we get (2) We prove the stability of two periodic solutions and of (2).
First, we prove the stability of the periodic solution of (2).
It is easy to know that the unique solution of (44) with initial value is By (42) and , the unique solution of (43) with initial value is By (42), we have By (45), we have Following we will discuss the sign of in three cases:
(i) If ,that is by (50), (71), and (72), it follows thatand, therefore, the periodic solution of (2) is attractive if given the initial value
(ii) If from (71), (68), (72), and (69), we haveNow, we discuss in two cases.
(I) If , then ; thus we have from (66), (79), when , it follows thatby (76), (77), we have and thus the periodic solution of (2) is attractive if given the initial value If , then , (81) also holds; by (i) and (I) of (ii), the unique periodic solution of (2) is attractive if given the initial value (II) If , then thus , by (77), and according to zero point theorem, there exists a , such that therefore, when , we have thus by (75) and (84), it follows thatand thus the periodic solution of (2) is unstable if initial value .
(iii) If , that is, , at this time, the unique solution of (2) with initial value is just the periodic solution , and is also unstable.
By (II) of (ii) and (iii), we get that if given the initial value is unstable.
Next, we prove the stability of the periodic solutions of (2).
By (51), it follows thatwhere is the unique solution of (2) with initial value . From the above proof, we know that when ,, that is to say, given any , there is a , such that as , so, when , we have and, therefore, it follows thatNote that is bounded and positive on , and thus is unstable if .
When , there are two cases.
(I) If , by (87), there exists a , such that since is bounded and positive on , we have and thus is unstable.
(II) If , by (89), is also unstable.
Thus is unstable if .
Therefore, the periodic solution of (2) is unstable on .
This is the end of the proof of Theorem 8.

Theorem 9. Under the conditions of Theorem 8, (2) has exactly two periodic continuous solutions: and .

Proof. The proof of the existence of and is seen in Theorem 8; now, we prove that (2) has exactly two periodic continuous solutions: and .
We know that if , the unique solution of (2) is , and if ,the unique solution of (2) is
(i) If , by (87), the unique solution of (2) satisfies and thus cannot be a periodic solution.
(ii) If , we know that is attractive; thus the unique solution of (2) is satisfied and hence cannot be a periodic solution; otherwise, there is a certain such that for any .
Therefore, (2) has exactly two periodic continuous solutions, and
This is the end of the proof of Theorem 9.

Theorem 10. Consider (2), are periodic continuous functions, suppose that the following conditions hold: and then (2) has two periodic continuous solutions.
(1) One periodic continuous solution is , and is unstable on .
(2) Another -periodic continuous solution is , and is attractive if given initial value on , and it is unstable if given initial value on , where is any given initial value of (2), and

Proof. By , (2) can be turned into Suppose Given any , the distance is defined as follows: and thus is a complete metric space. Take a convex closed set of S as follows Given any , consider the following equation By and (107), we get that and hence we have Since are periodic continuous functions, it follows that are periodic continuous functions; by (110), according to Lemma 3, (108) has a unique positive periodic continuous solution as follows and By (107), it follows thatand hence we have By (109), (107), and (112), we get and and, hence,
Define a map as follows and thus if given any , then , and hence .
Now, we prove that the mapping T is a compact operator.
Consider any sequence , then it follows thatOn the other hand, satisfies thus we have and 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 R. Also combined with Lemma 4, is convergent uniformly on R; that is to say, is relatively compact on .
Next, we prove that T is a continuous operator.
Suppose , and by (118), we have where is between and , thus is between and , hence we have by (122), it follows thatand, therefore, is continuous. By (118), it is easy to see that , and according to Lemma 5, has at least a fixed point on ; the fixed point is the periodic continuous solution of (2), and Let where is the unique solution of (2) with initial value , and is the periodic solution of (2); differentiating both sides of (127) along the solution of (2), we get This is Bernoulli’s equation. Let , and it can be turned into the following equation Note that according to Lemma 3, (129) has a unique periodic continuous solution as follows it is easy to know thatand thus the infinite integral is convergent; thereby, the infinite integral is convergent.
In addition, and thus we have by (131), we know that (129) has a unique periodic continuous solution , and by the transformations ,we know that (2) has another periodic continuous solution as follows Since are periodic solution of (2), we have Since is a periodic solution of (2), we only consider its maximum and minimum values in a cycle. Suppose is the minimum value of , is the maximum value of , is the minimum value point of , and is the maximum value point of , where ; then we have thus it followsand it is easy to see thatFrom (131), let Then we get that the possible extremums of satisfy so thus we have and it is easy to see that are also extreme points of , is the maximum value of , and is the minimum value of ; thus is the minimum value of and is the maximum value of . By (136), it follows that are two possible extremums of . Moreover, let and then we get from the following equation that all possible extreme points of function satisfy Take the negative sign of (149). Since they are the possible extremums of , by (146), (147), and (149), we get We prove the stability of two periodic solutions and of (2).
First, we prove the stability of the periodic solution of (2).
It is easy to know that the unique solution of (115) with initial value is By (127) and , the unique solution of (128) with initial value is By (127), (136), and (151), we have By (130), we have Following we discuss the sign of in following cases:
(i) If , that is, by (135), (156), and (157), it follows thatand, therefore, the periodic solution of (2) is attractive if given the initial value
(ii) If ,that is, , at this time, the unique solution of (2) with initial value is just the periodic solution , and .
(iii) If from (156), (153), (157), and (154), we knowNow, we discuss in two cases.
(I) If , then . Thus we have from (151), when , it follows thattherefore, we have and thus the periodic solution of (2) is attractive if given the initial value By (i), (ii), and (I) of (iii), the periodic solution of (2) is attractive if given the initial value (II) If , then thus by (165), there is a , such that according to zero point theorem, there exists a , such that therefore, when , we have and thus By (160) and (169), it follows that and thus the periodic solution of (2) is unstable if initial value .
(iiii)In addition, if , then , and we have is unstable.
By (II) of (iii) and (iiii), we get that if given the initial value is unstable.
Next, we prove the stability of the periodic solutions of (2). when ,; that is to say, given any , there is a , such that as , so, when , we have therefore, it follows thatand note that is bounded and positive on , so is unstable if .
When , there are two cases:
(I) If , by (174), there exists a , such that since is bounded and positive on , by (136), we have and thus is unstable.
(II) If , by (176), is also unstable.
Thus is unstable if .
Therefore, the periodic solution of (2) is unstable on .
This is the end of the proof of Theorem 10.

Theorem 11. Under the conditions of Theorem 10, (2) has exactly two periodic continuous solutions, and .

Proof. The proof of the existence of and is seen in Theorem 10. Now, we prove that (2) has exactly two periodic continuous solutions, and
We know that if , the unique solution of (2) is , and if , the unique solution of (2) is
(I) If , by (174), the unique solution of (2) satisfies and thus cannot be periodic solution.
(II) If , we know that is attractive; thus the unique solution of (2) is satisfied as , and hence cannot be periodic solution; otherwise, there is a certain such that for any .
Therefore, (2) has exactly two periodic continuous solutions, and
This is the end of the proof of Theorem 11.

4. Periodic Solutions on Riccati’s Equation

From the proofs of Theorems 811, we can get two results about the existence of periodic solutions on (1).

Theorem 12. Consider (1); are periodic continuous functions, and are derivable on ; suppose that the following conditions hold: and then (1) has exactly two periodic continuous solutions.

Proof. (1) can be turned into (186) can be also turned into Let then (187) is turned into By , (189) satisfies all the conditions of Theorems 8 and 9; according to Theorems 8 and 9, (189) has exactly two periodic solutions ; and by (188), (186) has exactly two periodic solutions Similarly, we can get the following.

Theorem 13. Consider (1); are periodic continuous functions, and are derivable on ; suppose that the following conditions hold: and then (1) has exactly two periodic continuous solutions.

Proof. The proof is similar to that of Theorem 12, so we omit it here.

5. Example

The following example shows the feasibility of our main results.

Example 1. Consider the following equation: Here, , and it is easy to calculate that

Clearly, conditions - of Theorems 8 and 9 are satisfied. It follows from Theorems 8 and 9 that (192) has exactly two periodic continuous solutions and , and is attractive on , and unstable on , where is any given initial value of (192), and

Another periodic continuous solution is , and is unstable on .

From this example, using Matlab, we can deduce the value ; when initial value , the solution curve of (192) tends to the curve of the periodic solution as is achieved at a certain value (see Figure 1); when initial value , the solution curve of (192) arrives at at some time (see Figure 2).

6. Concluding Remarks

In this paper, when the coefficient functions of Riccati’s type equation satisfy we obtain the existence and more accurate range of two periodic solutions of the equation by means of the fixed point theorem. This is a great improvement on the paper [1, 6] and provides a criterion for judging the existence and size range of periodic solutions of the equation, which has great application value in engineering technological and physical fields.

Data Availability

The data used to support the findings of this study are included within the article.

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 authors thank Jiangsu University, the Senior Talent Foundation of Jiangsu University (14JDG176), for sponsoring this research work.