The Existence of n Periodic Solutions on One Element n-Degree Polynomial Differential Equation

This paper deals with a class of one element n-degree polynomial differential equations. By the fixed point theory, we obtain n periodic solutions of the equation. This paper generalizes some related conclusions of some papers.


Introduction
Consider the following one element n-degree polynomial differential equation: where a i ðtÞ ði = 0, 1, 2,⋯, nÞ is the ω-periodic continuous functions on R. When n = 1, 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 a 1 ðtÞ and a 0 ðtÞ are ω-periodic continuous functions on R; if Ð ω 0 a 1 ðtÞdt ≠ 0, then equation (2) has a unique ω-periodic continuous solution ηðtÞ, mod ðηÞ ⊆ mod ða 1 ðtÞ, a 0 ðtÞÞ, and ηðtÞ can be written as follows: When n = 2, 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][3][4][5][6][7][8][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 a 2 ðtÞ, a 1 ðtÞ, and a 0 ðtÞ are all ω-periodic continuous functions on R. Suppose that the following conditions hold: then equation (4) has exactly two ω-periodic continuous solutions.
When a 1 ðtÞ ≡ 0, 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 n periodic solutions of equation (1)?
In this paper, we consider the n-degree polynomial differential equation for the special case of equation (1) as follows: and we give a new criterion to judge the existence of n 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 n periodic solutions on equation (6) is obtained. We end this paper with a short conclusion.

Some Lemmas and Abbreviations
Lemma 3 (see [11]). Suppose that an ω-periodic sequence f f n ðtÞg is convergent uniformly on any compact set of R, f ðtÞ is an ω-periodic function, and mod ð f n Þ ⊆ mod ð f Þ ðn = 1, 2,⋯Þ, then ff n ðtÞg is convergent uniformly on R.
Lemma 4 (see [12]). Suppose V is a metric space, C is a convex closed set of V, and its boundary is ∂C; if T : V ⟶ V is a continuous compact mapping, such that Tð∂CÞ ⊆ C, then T has a fixed point on C.
For the sake of convenience, suppose that f ðtÞ is an ω-periodic continuous function on R; we denote

Periodic Solutions of the Polynomial Differential Equation
In this section, we discuss the existence of n periodic solutions of equation (6).

Theorem 5.
Consider equation (6), aðtÞ, γ i ðtÞ ði = 1, 2,⋯,nÞ are all ω-periodic continuous functions on R; suppose that the following conditions hold: then equation (6) has exactly n ω-periodic continuous solutions Φ i ðtÞ ði = 1, 2,⋯,nÞ , and Proof. By ðH 1 Þ, it follows aðtÞ > 0 or aðtÞ < 0. In order to avoid repetition, we only prove the case of aðtÞ > 0. 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 n-th periodic solution Φ n ðtÞ of equation (6).
Here, we will divide the proof into two steps.
(1) We prove the existence of n periodic solutions of equation (6). Suppose given any φðtÞ, ψðtÞ ∈ S, the distance is defined as follows: Thus, ðS, ρÞ is a complete metric space. Take a convex closed set B n of S as follows: Given any φðtÞ ∈ B n , consider the following: Here By ðH 1 Þ, ðH 2 Þ, and equation (12), we get that hence, we have ð ω By equations (12) and (14), it follows that hence we have By equations (12), (15), and (17), we get and hence, ηðtÞ ∈ B n .
Define a mapping as follows: Thus, if given any φðtÞ ∈ B n , then ðTφÞðtÞ ∈ B n , hence T : B n ⟶ B n . Now, we prove that the mapping T is a compact mapping.
Consider any sequence fφ k ðtÞg ⊆ B n ðk = 1, 2,⋯Þ, then it follows that on the other hand, ðTφ k ÞðtÞ = x φ k ðtÞ satisfies Thus, we have hence fðdx φ k ðtÞÞ/dtg is uniformly bounded; therefore, fx φ k ðtÞg is uniformly bounded and equicontinuous on R, by the theorem of Ascoli-Arzela, for any sequence fx φ k ðtÞg ⊆ B n , there exists a subsequence (also denoted by fx φ k ðtÞg) such that fx φ k ðtÞg is convergent uniformly on any compact set of R, by equation (26), combined with Lemma 3, fx φ k ðtÞg is convergent uniformly on R, that is to say, T is relatively compact on B n .
Next, we prove that T is a continuous mapping.
Suppose fφ k ðtÞg ⊆ B n , φðtÞ ∈ B n , and Denote then we have and

Advances in Mathematical Physics
By equation (23), we have where ξ is between Ð t s f k ðτÞdτ and Ð t s f ðτÞdτ; thus, ξ is between and hence we have By equation (29) and the above inequality, it follows that hence, T is continuous; therefore, T : B n ⟶ B n is a continuous compact mapping, and by equation (23), it is easy to see, Tð∂B n Þ ⊆ B n ; according to Lemma 4, T has a fixed point on B n , and the fixed point is the ω-periodic continuous solution Φ n ðtÞ of equation (6), and Similarly, we can prove the existence of the periodic solutions Φ i ðtÞ ði = 1, 2,⋯, n − 1Þ of equation (6), and we have (2) We prove that equation (6) has exactly n periodic solutions.
Let us discuss the possible range of xðtÞ of equation (6); we divide the initial value xðt 0 Þ = x 0 into the following parts: We will only prove the following cases. For the sake of convenience, suppose n is an even number.