Abstract
This paper deals with a class of one element -degree polynomial differential equations. By the fixed point theory, we obtain periodic solutions of the equation. This paper generalizes some related conclusions of some papers.
1. Introduction
Consider the following one element -degree polynomial differential equation: where is the -periodic continuous functions on . When , equation (1) is a linear differential equation. With regard to the periodic solution of the equation, we propose the following:
Proposition 1 (see [1]). Consider the following: where and are -periodic continuous functions on ; if then equation (2) has a unique -periodic continuous solution , , and can be written as follows:
When , equation (1) is Riccati’s equation. Riccati’s equation plays an important role in fluid mechanics and in the theory of elastic vibration. There are many studies on this equation [2–9], and there is also a proposition about the periodic solutions of Riccati’s equation, as follows:
Proposition 2 (see [2]). Consider the following equation: where , , and are all -periodic continuous functions on . Suppose that the following conditions hold: then equation (4) has exactly two -periodic continuous solutions.
When , in [10], the author obtained the existence and more accurate range of two periodic solutions of equation (4) by means of the fixed point theorem.
It is easy for us to guess under what conditions is equation (1) satisfied, and are there existing periodic solutions of equation (1)?
In this paper, we consider the -degree polynomial differential equation for the special case of equation (1) as follows: and we give a new criterion to judge the existence of periodic solutions on equation (6); these conclusions generalize the relevant conclusions of References [1, 2, 10].
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 of periodic solutions on equation (6) is obtained. We end this paper with a short conclusion.
2. Some Lemmas and Abbreviations
Lemma 3 (see [11]). Suppose that an -periodic sequence is convergent uniformly on any compact set of , is an -periodic function, and , then is convergent uniformly on .
Lemma 4 (see [12]). 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 a fixed point on .
For the sake of convenience, suppose that is an -periodic continuous function on ; we denote
3. Periodic Solutions of the Polynomial Differential Equation
In this section, we discuss the existence of periodic solutions of equation (6).
Theorem 5. Consider equation (6), are all -periodic continuous functions on ; suppose that the following conditions hold: then equation (6) has exactly -periodic continuous solutions , and
Proof. By , it follows or . In order to avoid repetition, we only prove the case of . As the proof of the existence of every periodic solution is the same, for the sake of simplicity, we only prove the existence of the -th periodic solution of equation (6).
Here, we will divide the proof into two steps. (1)We prove the existence of periodic solutions of equation (6). Suppose given any , the distance is defined as follows: Thus, is a complete metric space. Take a convex closed set of as follows: Given any , consider the following: Here By , , and equation (12), we get that hence, we have and because are -periodic continuous functions on , it follows that are -periodic continuous functions on , by equation (16). According to Proposition 1, equation (13) has a unique -periodic continuous solution as follows: and By equations (12) and (14), it follows that hence we have By equations (12), (15), and (17), we get and hence, . Define a mapping as follows: Thus, if given any 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 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 equation (26), combined with Lemma 3, is convergent uniformly on , that is to say, is relatively compact on . Next, we prove that is a continuous mapping. Suppose , and Denote then we have and By equation (23), we have where is between and; thus, is between and hence we have By equation (29) and the above inequality, it follows that hence, is continuous; therefore, is a continuous compact mapping, and by equation (23), it is easy to see, ; according to Lemma 4, has a fixed point on , and the fixed point is the -periodic continuous solution of equation (6), and Similarly, we can prove the existence of the periodic solutions of equation (6), and we have (2)We prove that equation (6) has exactly periodic solutions. Let us discuss the possible range of of equation (6); we divide the initial value into the following parts: We will only prove the following cases. For the sake of convenience, suppose is an even number.
Remark 1. If is an odd number, the proof is similar, and we omit it here.
Let (i)If Consider equation (6). We have . Thus, may stay at or enter into at some time . If stays at , then . Thus, cannot be a periodic solution of equation (6). If enters into at some time , then there is not a such that ; thus can also not be a periodic solution of equation (6).(ii)If , then equation (6) has an -periodic continuous solution with initial value As by differential mean value theorem, it follows that By equation (39), we have Note that By equations (41) and (43), it follows that Now, suppose that there is another -periodic continuous solution of equation (6) which satisfies Because is a polynomial function with continuous partial derivatives to , equation (6) satisfies the existence and uniqueness of solutions to initial value problems of differential equations, thus By equations (41) and (45), it follows that Consider the following equation: Thus, we have By equations (43) and (45), it follows that By equations (41) and (50), it follows that By equations (49) and (51), it follows that By equations (46) and (52), this is a contradiction, thus cannot be a periodic solution of equation (6), that is to say, equation (6) has exactly a unique -periodic continuous solution which satisfies (iii)If Consider equation (6). We have . Thus, may stay at or enter into at some time . If stays at , we have , then cannot be a periodic solution of equation (6). If enters into at some time , then there is not a such that ; thus, can also not be a periodic solution of equation (6). Similarly, if , we can prove that equation (6) has exactly a unique -periodic continuous solution which satisfies . Similarly, if , then the solution of equation (6) with an initial value cannot be a periodic solution of equation (6).(iv)If Consider equation (6). We have . Thus, may stay at or . If stays at , we have , then cannot be a periodic solution of equation (6). If , then can also not be a periodic solution of equation (6). To sum up, equation (6) has exactly -periodic continuous solutions which satify
This is the end of the proof of Theorem 5.
Remark 2. If , Theorem 5 is exactly Proposition 1.
Remark 3. If , Theorem 5 is exactly Proposition 2.
4. Conclusion
Consider the following Riccati’s differential equation: about the periodic solutions on equation (54), there is a conclusion as follows:
Theorem 6 (see [13]). Consider equation (54); , , and are all -periodic continuous functions on . Suppose that the following conditions hold: then equation (54) has at most two -periodic continuous solutions.
What conditions are the coefficient functions of the equation satisfied? The equation has exactly two periodic solutions. Proposition 2 gives the answer.
Consider the following Abel’s differential equation: about the periodic solutions on equation (56), we have the following result.
Theorem 7 (see [13]). Consider equation (56), , , , and are all -periodic continuous functions on . Suppose that the following conditions hold: then equation (56) has at most three -periodic continuous solutions.
We cannot help but ask: What conditions are the coefficient functions of the equation satisfied? The equation has exactly three periodic solutions. We show the following answer.
Corollary 8. Consider the following Abel’s differential equation: Here, are all -periodic continuous functions on . Suppose that the following conditions hold: then equation (58) has exactly three -periodic continuous solutions , , and , and
For , equation (1) has not always the most periodic solutions (see [13]). But in this paper, we obtain a new criterion for the existence of periodic solutions of periodic equation (6); the size range of periodic solutions is also given. It can be said that this paper is a generalization of the conclusions of the related articles on periodic solutions.
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 authors declare that they have no competing interest.
Authors’ Contributions
The authors contributed to each part of this paper equally. The authors read and approved the final manuscript.
Acknowledgments
This research was supported by a special project supported by the senior personnel of Jiangsu University (14JDG176).