The Integrability and Existence of Periodic Solutions on a First-Order Nonlinear Differential Equation with a Polynomial Nonlinear Term

Abstract

This paper deals with a first-order differential equation with a polynomial nonlinear term. The integrability and existence of periodic solutions of the equation are obtained, and the stability of periodic solutions of the equation is derived.

1. Introduction

Consider the following first-order nonlinear differential equation:𝑑𝑥=𝑑𝑡^𝑛_𝑘=0𝑎_𝑘(𝑡)𝑥^𝑛−𝑘(𝑛∈𝑁,𝑛≥2), (1.1) when 𝑛=2, (1.1) becomes Ricatti’s equation, when 𝑛=3, (1.1) becomes the following nonlinear Abel type differential equation:𝑑𝑥(𝑡)𝑥𝑑𝑡=𝑎³+𝑏(𝑡)𝑥²+𝑐(𝑡)𝑥+𝑑(𝑡). (1.2) The nonlinear Abel type differential equation plays an important role in many physical and technical applications [1–9]. The mathematical properties of (1.2) have been intensively investigated in the mathematical and physical literature. Matsuno [10] analyzed a two-dimensional dynamical system associated with Abel nonlinear equation. Strobel and Reid [11], Reid and Strobel [12] have obtained superposition rules (prescriptions for combining a finite number of known particular solutions in such a way to obtain the general solution to a (system of) differential equation(s) without operation of integration) for the Abel type equation, involving four or two particular solutions. Mak et al. [13], Mak and Harko [14] have presented a solution-generating technique for Abel type ordinary differential equation, both suppose that 𝑦=𝑦₁(𝑥) is a particular solution of (1.2), by means of the transformations methods, and present an alternative method of generating the general solution of (1.2) from a particular one.

Zheltukhin and Trzetrzelewski [15] developed the geometric approach to study the dynamics of U(1)-invariant membranes. The approach reveals an important role of the Abel nonlinear differential equation of the first type with variable coefficients depending on time and one of the membrane extendedness parameters. The general solution of the Abel equation was constructed.

However, little work was done about the integrability and periodicity of (1.1). In this paper, we discuss the integrability and the periodic solutions of (1.1); the sufficient conditions which guarantee the integrability and the existence of the periodic solutions for (1.1) are obtained, and the stability of the periodic solutions of (1.1) is discussed. To the best of authors’ knowledge, this is the first paper considering the three periodic solutions of (1.1), some new results are obtained.

The present paper is organized as follows. In Section 2, we give three lemmas to be used later. In Section 3, the integrability of (1.1) is derived. In Section 4, the existence and stability of the periodic solutions of (1.1) are obtained. In Section 5, we conclude our results.

2. Preliminary Lemmas

For the sake of convenience, suppose that 𝑓 is a continuous 𝜔-periodic function defined on 𝑅, we denote 𝑓_𝑀=sup_{𝑡∈[0,𝜔]}𝑓(𝑡) and 𝑓_𝐿=inf_{𝑡∈[0,𝜔]}𝑓(𝑡); 𝑚(𝑓(𝑡))=(1/𝜔)∫^𝜔₀𝑓(𝑡)𝑑𝑡.

Consider the following:𝑑𝑥𝑎(𝑡)(𝑡)𝑥𝑑𝑡=𝑥+𝑏^𝑛−1, (2.1)𝑛∈𝑁, 𝑛≥2, 𝑎(𝑡), 𝑏(𝑡) are continuous functions defined on 𝑅.

Lemma 2.1. The domain 𝑅₊={𝑥∣𝑥>0} is positive invariant with respect to (2.1).

Proof. By (2.1), it follows that 𝑥(𝑡)=𝑥𝑡₀exp^𝑡_𝑡0𝑎(𝑠)+𝑏(𝑠)𝑥^𝑛−1(𝑠)𝑑𝑠, (2.2) the assertion is valid for all 𝑥(𝑡₀)>0,𝑡≥𝑡₀. The proof is completed.

Lemma 2.2. The domain 𝑅₋={𝑥∣𝑥<0} is negative invariant with respect to (2.1).

Lemma 2.3. Consider the following: 𝑑𝑥(𝑡)(𝑡)𝑑𝑡=𝑎𝑥+𝑏, (2.3)𝑎(𝑡), 𝑏(𝑡) are continuous 𝜔-periodic functions, if 𝑚(𝑎(𝑡))≠0, then (2.3) exists a unique 𝜔-periodic solution 𝜂(𝑡), and 𝜂(𝑡) can be written as follows: 𝜂(𝑡)=∫_𝑡^𝑡+𝜔𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏},𝑚(𝑎(𝑡))≠0 (2.4) or 𝜂(𝑡)=^𝑡−∞𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)(𝑎(𝑡−𝑑𝑠,𝑚))<0,_𝑡+∞𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑏(𝑠)𝑑𝑠,𝑚(𝑎(𝑡))>0. (2.5)

Proof. The proof of Lemma 2.3 is a few, such as that of the papers [16, 17] and others, But in [16], the author only proved the case of the almost periodic equation; in [17], the authors just proved that Lotka-volterra equation has a unique globally attractive periodic solution, but they did not give the very two expressions of the periodic solution as above. In order to make the following proof clear, here we give our proof.

From (2.3), it is easy for us to get the unique solution 𝑥(𝑡) with the initial value 𝑥(𝑡₀)=𝑥₀ which can be written as follows:𝑥(𝑡)=𝑒^{∫𝑡𝑡0𝑎(𝜏)𝑑𝜏}𝑥₀+^𝑡_𝑡0𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠, (2.6) thus we have𝑥(𝑡+𝜔)=𝑒^{∫𝑡0𝑡+𝜔𝑎(𝜏)𝑑𝜏}𝑥₀+_𝑡^𝑡+𝜔₀𝑏(𝑠)𝑒^{∫𝑠𝑡+𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠. (2.7) Let 𝑡=𝑡₀, by (2.6) and (2.7), if 𝑥(𝑡) is an 𝜔-periodic function, then it follows that 𝑥(𝑡₀)=𝑥(𝑡₀+𝜔), thus we can get𝑥₀=∫^𝑡0_𝑡^+𝜔₀𝑏(𝑠)𝑒^{∫𝑡0𝑠+𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}. (2.8) Substitute (2.8) into (2.6), it follows that𝑥(𝑡)=𝑒^{∫𝑡𝑡0𝑎(𝜏)𝑑𝜏}∫^𝑡0_𝑡^+𝜔₀𝑏(𝑠)𝑒^{∫𝑡0𝑠+𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}+^𝑡_𝑡0𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}=∫𝑑𝑠_{𝑡0𝑡+𝑤0}𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠+∫^𝑡_𝑡0𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}∫^𝑡_𝑡0𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}=∫^𝑡0_𝑡^+𝑤₀𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠+∫_𝑡^𝑡+𝜔_0+𝜔𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠−∫^𝑡_𝑡0𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}=∫_𝑡^𝑡+𝜔𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}. (2.9)

When 𝑚(𝑎(𝑡))<0, it follows that𝑥(𝑡)=∫_𝑡^𝑡+𝜔𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}=∫^𝑡+𝜔_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠−∫^𝑡_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}=∫^𝑡_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠−∫^𝑡_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}=∫^𝑡_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}∫^𝑡_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎(𝜏)𝑑𝜏}=^𝑡_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠, (2.10)

thus we have𝑥(𝑡+𝜔)=^𝑡+𝜔_−∞𝑏(𝑠)𝑒^{∫𝑠𝑡+𝜔𝑎(𝜏)𝑑𝜏}=𝑑𝑠_𝑡−∞𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}(𝑡).𝑑𝑠=𝑥 (2.11)

Hence, 𝑥(𝑡) is a unique 𝜔-periodic solution of (2.3); we rewrite it as follows:𝜂(𝑡)=^𝑡_−∞𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠. (2.12)

Similarly, we can prove that when 𝑚(𝑎(𝑡))>0, (2.3) has a unique 𝜔-periodic solution as follows:𝜂(𝑡)=−_𝑡^+∞𝑏(𝑠)𝑒^{∫𝑡𝑠𝑎(𝜏)𝑑𝜏}𝑑𝑠. (2.13)

Remark 2.4. Under the conditions of Lemma 2.3, all the infinite integrals above are convergent.

3. A Sufficient Condition of Integrability of the First-Order Nonlinear Differential Equation

In this section, we discuss the integrability of (1.1)

Theorem 3.1. Consider (1.1), where 𝑎₀(𝑡), 𝑎_𝑛(𝑡) are continuous functions defined on certain interval 𝐼, if there is a constant 𝛾 such that the following conditions hold: (𝐻₁)𝑎_𝑛−1(𝑡)=(−1)^𝑛−1(𝑛−1)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1, (𝐻₂)𝑎_𝑘(𝑡)=(−1)^𝑘𝐶^𝑘_𝑛𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑘, 𝑘=1,2,…,𝑛−2,
then
(i) the general solution of (1.1) can be written as follows:𝑥(𝑡)=±𝑒^{∫𝑎𝑛−1(𝑡)𝑑𝑡}𝐶−(𝑛−1)𝑎₀(𝑡)𝑒^{−∫𝑎𝑛−1(𝑡)𝑑𝑡}𝑑𝑡^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (3.1) where 𝐶 is an integration constant which makes the number of root greater than zero if 𝑛 is an odd number.(ii) Equation (1.1) has a particular solution𝜂(𝑡)=^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠. (3.2)

Proof. (i) Since the conditions (𝐻₁), (𝐻₂) hold, (1.1) can be written as follows: 𝑑𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠=𝑑𝑡𝑥−^𝑡_𝛾𝑎_𝑛(𝑠)×)𝑑𝑠(−1^𝑛𝑎₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1+𝑎₀(𝑡)𝑥−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1. (3.3)
Let 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (3.3) can be changed as follows: 𝑑𝑦)𝑑𝑡=𝑦(−1^𝑛𝑎₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1+𝑎₀(𝑡)𝑦^𝑛−1. (3.4) This is the Bernoulli-type equation, set 𝑢(𝑡)=𝑦^1−𝑛(𝑡), the Bernoulli-type equation (3.4) becomes 𝑑𝑢=()𝑑𝑡−1^𝑛−1(𝑛−1)𝑎₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1𝑢−(𝑛−1)𝑎₀(𝑡)=𝑎_𝑛−1(𝑡)𝑢−(𝑛−1)𝑎₀(𝑡), (3.5) according to the formula of general solution of linear differential equation, we can get that the general solution of (3.5) is given by 𝑢(𝑡)=𝑒^{∫𝑎𝑛−1(𝑡)𝑑𝑡}𝐶−(𝑛−1)𝑎₀(𝑡)𝑒^{−∫𝑎𝑛−1(𝑡)𝑑𝑡}𝑑𝑡. (3.6) By the variable transformation of 𝑢(𝑡)=𝑦^1−𝑛(𝑡), it follows that 𝑦(𝑡)=±𝑒^{∫𝑎𝑛−1(𝑡)𝑑𝑡}𝐶−(𝑛−1)𝑎₀(𝑡)𝑒^{−∫𝑎𝑛−1(𝑡)𝑑𝑡}𝑑𝑡^{−1/(𝑛−1)}, (3.7) since 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, we can get 𝑥(𝑡)=±𝑒^{∫𝑎𝑛−1(𝑡)𝑑𝑡}𝐶−(𝑛−1)𝑎₀(𝑡)𝑒^{−∫𝑎𝑛−1(𝑡)𝑑𝑡}𝑑𝑡^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠. (3.8)
Remark 3.2. If 𝑛 is an even number, take “+” in the front of above-corresponding formulas, if 𝑛 is an odd number, take “±”.
(ii) By (1.1), we set 𝑓(𝑡,𝑥)=∑^𝑛_𝑘=0𝑎_𝑘(𝑡)𝑥^𝑛−𝑘, thus 𝑓(𝑡,𝑥) is continuous on 𝐼×𝑅, 𝑓_𝑥(𝑡,𝑥) is also continuous on 𝐼×𝑅, hence (1.1) satisfies the principle of existence and uniqueness of differential equation, therefore, (1.1) has no singular solution.
From the variable transformation of 𝑢(𝑡)=𝑦^1−𝑛(𝑡), it requires 𝑦≠0, but from (3.4), we know 𝑦=0 is indeed a solution of (3.4). Since 𝑦=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, substitute 𝜂(𝑡)=∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 into (1.1), it just satisfies (1.1), so it is a particular solution of (1.1).
This is the end of the proof of Theorem 3.1.

Corollary 3.3. Consider Abel type differential equation (1.2), 𝑎(𝑡), 𝑑(𝑡) are continuous functions defined on certain interval 𝐼,  if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑏(𝑡)=−3𝑎(𝑡)∫^𝑡_𝛾𝑑(𝑠)𝑑𝑠, (𝐻₂)𝑐(𝑡)=2𝑎(𝑡)(∫^𝑡_𝛾𝑑(𝑠)𝑑𝑠)²,then the general solution of (1.2) can be written as follows: 𝑥(𝑡)=±𝑒^{∫𝑐(𝑡)𝑑𝑡}𝐶−2𝑎(𝑡)𝑒^{−∫𝑐(𝑡)𝑑𝑡}𝑑𝑡^−1/2+^𝑡_𝛾𝑑(𝑠)𝑑𝑠, (3.9) and ∫^𝑡_𝛾𝑑(𝑠)𝑑𝑠 is a particular solution of (1.2).

Corollary 3.4. Consider the following Ricatti’s equation: 𝑑𝑥(𝑡)𝑥𝑑𝑡=𝑎²+𝑏(𝑡)𝑥+𝑐(𝑡), (3.10)𝑎(𝑡), 𝑐(𝑡) are continuous functions defined on certain interval 𝐼, if there is a constant 𝛾 such that the following condition holds:(𝐻₁)𝑏(𝑡)=−𝑎(𝑡)∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠,then the general solution of (3.10) can be written as follows: 𝑥(𝑡)=𝑒^{∫𝑏(𝑡)𝑑𝑡}𝐶−𝑎(𝑡)𝑒^{−∫𝑏(𝑡)𝑑𝑡}𝑑𝑡⁻¹+^𝑡_𝛾𝑐(𝑠)𝑑𝑠, (3.11) and 𝜂(𝑡)=∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠 is a particular solution of (3.10).

4. The Existence and Stability of Periodic Solutions of the First-Order Nonlinear Differential Equation

DefineΩ₁=𝑥∣𝑥>^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠,Ω₂=𝑥∣𝑥>inf_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑎_𝑛(𝑠),Ω𝑑𝑠₃=𝑥∣𝑥<^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠,Ω₄=𝑥∣𝑥<sup_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.1)

where 𝑎_𝑛(𝑡), 𝛾, 𝜔 are the same function and numbers as the following Theorem 4.1, and 𝑡₀ is any given initial time of (1.1).

Theorem 4.1. Consider (1.1), 𝑛 is an odd number, and 𝑛>2, 𝑎₀(𝑡), 𝑎_𝑛(𝑡) are continuous 𝜔-periodic functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑎_𝑛−1(𝑡)=(𝑛−1)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1, (𝐻₂)𝑎_𝑘(𝑡)=(−1)^𝑘𝐶^𝑘_𝑛𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑘, 𝑘=1,2,…,𝑛−2,(𝐻₃)𝑚(𝑎_𝑛(𝑡))=0, 𝑎_𝑛(𝑡)≢0,(𝐻₄)𝑎₀(𝑡)<0,
then
(i) equation (1.1) has three 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), 𝜒(𝑡), and they can be written as follows:𝜂(𝑡)=^𝑡_𝛾𝑎_𝑛(𝑠)𝜙(𝑡)=−()𝑑𝑠,𝑛−1^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝜒(𝑡)−()𝑑𝑠,=−𝑛−1^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠; (4.2) (ii) if given any initial value 𝑥(𝑡₀)∈Ω₁, then the periodic solution 𝜙(𝑡) of (1.1) is globally attractive in Ω₂; (iii) if given any initial value 𝑥(𝑡₀)∈Ω₃, then the periodic solution 𝜒(𝑡) of (1.1) is globally attractive in Ω₄.

Proof. (i) Since the conditions (𝐻₁), (𝐻₂) hold, (1.1) can be written as follows: 𝑑𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠=𝑑𝑡𝑥−^𝑡_𝛾𝑎_𝑛(𝑠)×𝑑𝑠−𝑎₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1+𝑎₀(𝑡)𝑥−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1. (4.3) Let 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.3) can be changed as follows: 𝑑𝑦𝑑𝑡=𝑦−𝑎₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1+𝑎₀(𝑡)𝑦^𝑛−1. (4.4) This is the Bernoulli-type equation, set 𝑢(𝑡)=𝑦^1−𝑛(𝑡), the Bernoulli-type equation (4.4) becomes 𝑑𝑢=()𝑎𝑑𝑡𝑛−1₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1𝑢−(𝑛−1)𝑎₀(𝑡),𝑑𝑢𝑑𝑡=𝑎_𝑛−1(𝑡)𝑢−(𝑛−1)𝑎₀(𝑡). (4.5) By (𝐻₃), (𝐻₄), it follows that _𝛾^𝑡+𝜔𝑎_𝑛(𝑠)𝑑𝑠=^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠+_𝑡^𝑡+𝜔𝑎_𝑛(𝑠)=𝑑𝑠_𝑡𝛾𝑎_𝑛(𝑠)𝑑𝑠+^𝜔₀𝑎_𝑛(𝑠)=𝑑𝑠_𝑡𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.6) thus ∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 is an 𝜔-periodic function, therefore, (𝑛−1)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1 is also an 𝜔-periodic function and 𝑎_𝑛−1(𝑡)=(𝑛−1)𝑎₀(𝑡)^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1≤0, (4.7) since 𝑎₀(𝑡)<0, 𝑎_𝑛(𝑡)≠0 (not identically equal to zero), thus we have 𝑚𝑎_𝑛−1(𝑡)<0, (4.8) according to Lemma 2.3, it follows that (4.5) has a unique 𝜔-periodic solution as follows 𝜑(𝑡)=−(𝑛−1)^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠, (4.9) from (4.9), it is easy to know that 𝜑(𝑡)>0, since 𝜑(𝑡) is periodic on 𝑅, it is bounded on 𝑅, so it is positive and bounded, since 𝑢(𝑡)=𝑦^1−𝑛(𝑡), and by Lemmas 2.1 and 2.2, it follows that (4.4) has two 𝜔-periodic solutions as follows: 𝜓(𝑡)=−(𝑛−1)^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}𝜓𝑡₀,𝜓>0_∗(𝑡)=−−(𝑛−1)^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}𝜓^∗𝑡₀<0, (4.10) thus 𝜓(𝑡) is also positive and bounded, and 𝜓^∗(𝑡) is negative and bounded, by virtue of 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (1.1) has two 𝜔-periodic solutions as follows: 𝜙(𝑡)=−(𝑛−1)^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝜒(𝑡)−()𝑑𝑠,=−𝑛−1^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.11) and 𝜙(𝑡₀)∈Ω₁, 𝜙(𝑡)∈Ω₂, 𝜒(𝑡₀)∈Ω₃, 𝜒(𝑡)∈Ω₄.
By (1.1), we set 𝑓(𝑡,𝑥)=∑^𝑛_𝑘=0𝑎_𝑘(𝑡)𝑥^𝑛−𝑘, thus 𝑓(𝑡,𝑥) is continuous on 𝑅×𝑅, and 𝑓_𝑥(𝑡,𝑥) is also continuous on 𝑅×𝑅, hence (1.1) satisfies the principle of existence and uniqueness of differential equation, therefore, (1.1) has no singular solution.
From the variable transformation of 𝑢(𝑡)=𝑦^1−𝑛(𝑡), it requires 𝑦≠0, but from (4.4), we know 𝑦=0 is indeed a solution of (4.4), since 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, hence 𝜂(𝑡)=∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 is a periodic solution of (1.1). Substitute 𝜂(𝑡)=∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 into (1.1), it just satisfies (1.1), so 𝜂(𝑡)=∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 is indeed a periodic particular solution of (1.1), therefore, we have proved that (1.1) has three 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), and 𝜒(𝑡).
(ii) According to the theories of linear differential equation, we know that the unique solution 𝑢(𝑡) of (4.5) with positive initial value 𝑢(𝑡₀)=𝑢₀ is given by 𝑢(𝑡)=𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠. (4.12) Since 𝑎_𝑛−1(𝑡)=(𝑛−1)𝑎₀(𝑡)^𝑡_𝛾𝑎_𝑛(𝜏)𝑑𝜏^𝑛−1, (4.13) thus 𝑎_𝑛−1(𝑡) is an 𝜔-periodic function on 𝑅, and by (4.7), we know 𝑎_𝑛−1(𝑡)≤0, thus (𝑎_𝑛−1)_𝐿<0.
Following we prove that there is a positive number 𝑇>𝑡₀, such that ()𝑎𝑛−1^0𝑀𝑎_𝑛−1𝐿(1−𝜀)≤𝑢(𝑡)≤𝑢₀−2𝜔𝑎_0𝐿1−∫^𝜔₀𝑎_𝑛−1(𝜏)𝑑𝜏, (4.14) as 𝑡>𝑇, here 𝜀 is any small positive number and 𝜀<1.
From (4.12), given any positive number 𝑢₀>0, by the condition 𝑎₀(𝑡)<0 of Theorem 4.1, it follows that 𝑢(𝑡)>0; following we first prove that 𝑢(𝑡) has upper bound on [𝑡₀,+∞)𝑢(𝑡)=𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠≤𝑢₀−(𝑛−1)^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠. (4.15) According to Lemma 2.3, it follows that 𝑢(𝑡)≤𝑢₀−(𝑛−1)^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠=𝑢₀−(𝑛−1)∫_𝑡^𝑡+𝜔𝑎₀(𝑠)𝑒^{∫𝑡𝑠−𝜔𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}=𝑢₀−𝜔(𝑛−1)𝑎₀(𝜉)𝑒^{∫𝑡𝜉−𝜔𝑎𝑛−1(𝜏)𝑑𝜏}1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏},(𝑡<𝜉<𝑡+𝜔)=𝑢₀−𝜔(𝑛−1)𝑎₀(𝜉)𝑒^{∫𝑡𝜉𝑎𝑛−1(𝜏)𝑑𝜏}𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}≤𝑢₀−𝜔(𝑛−1)𝑎₀(𝜉)𝑒^{∫𝜔0|𝑎𝑛−1(𝜏)|𝑑𝜏}𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}=𝑢₀−𝜔(𝑛−1)𝑎₀(𝜉)𝑒^{∫𝜔0(|𝑎𝑛−1(𝜏)|+𝑎𝑛−1(𝜏))𝑑𝜏}1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}=𝑢₀−𝜔(𝑛−1)𝑎₀(𝜉)1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}≤𝑢₀−𝜔(𝑛−1)𝑎_0𝐿1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏}. (4.16) So 𝑢(𝑡) has upper bound on [𝑡₀,+∞).
Secondly, we prove that 𝑢(𝑡)≥(𝑛−1)(𝑎_0𝑀/(𝑎_𝑛−1)_𝐿)(1−𝜀) as 𝑡>𝑇𝑢(𝑡)=𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}()𝑑𝑠≥−𝑛−1^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}()𝑑𝑠≥−𝑛−1^𝑡_𝑡0𝑎_0𝑀𝑒^{(𝑎𝑛−1)𝐿(𝑡−𝑠)}()−𝑎𝑑𝑠=−𝑛−1^0𝑀𝑎_𝑛−1𝐿1−𝑒^{(𝑎𝑛−1)𝐿(𝑡−𝑡0)}=()𝑎𝑛−1^0𝑀𝑎_𝑛−1𝐿1−𝑒^{(𝑎𝑛−1)𝐿(𝑡−𝑡0)}. (4.17) Since lim_𝑡→+∞𝑒^{(𝑎𝑛−1)𝐿(𝑡−𝑡0)}=0, for all 𝜀>0(𝜀<1), there must be 𝑇>𝑡₀, when 𝑡>𝑇, we have 𝑒^{(𝑎𝑛−1)𝐿(𝑡−𝑡0)}<𝜀, hence when 𝑡>𝑇, we can get 𝑢(𝑡)≥()𝑎𝑛−1^0𝑀𝑎_𝑛−1𝐿(1−𝜀), (4.18) therefore, we have proved that there is a positive number 𝑇>𝑡₀, as 𝑡>𝑇, it follows that ()𝑎𝑛−1^0𝑀𝑎_𝑛−1𝐿(1−𝜀)≤𝑢(𝑡)≤𝑢₀−𝜔(𝑛−1)𝑎_0𝐿1−𝑒^{∫𝜔0𝑎𝑛−1(𝜏)𝑑𝜏},(𝑡>𝑇), (4.19) under the transformation of 𝑢(𝑡)=𝑦^1−𝑛(𝑡), 𝑦(𝑡)=𝑥(𝑡)−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, we can get the following.
If given the initial value 𝑥𝑡₀=𝑢₀^{−1/(𝑛−1)}+^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠∈Ω₁, (4.20) then the unique solution 𝑥(𝑡) of (1.1) is given by 𝑥(𝑡)=𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.21) if given the initial value 𝑥𝑡₀=−𝑢₀^{−1/(𝑛−1)}+^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠∈Ω₃, (4.22) then the unique solution 𝑥^∗(𝑡) of (1.1) is given by 𝑥^∗(𝑡)=−𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.23) and it is easy to know 𝑥(𝑡)∈Ω₂ and 𝑥^∗(𝑡)∈Ω₄, respectively.
Define a Lyapunov function as follows: 𝑉𝑡,𝑢,𝜑=𝑢−𝜑², (4.24) where 𝑢(𝑡) is the unique solution with the positive initial value 𝑢(𝑡₀)=𝑢₀ of (4.5), and 𝜑 is the unique positive 𝜔-periodic solution of (4.5), differentiating both sides of (4.24) along the solution of (4.5), we get 𝑑𝑉𝑡,𝑢,𝜑𝑑𝑡=2𝑢−𝜑𝑑𝑢−𝑑𝑡𝑑𝜑𝑑𝑡=2𝑎_𝑛−1(𝑡)𝑢−𝜑², (4.25) by (4.24), we have 𝑑𝑉𝑡,𝑢,𝜑𝑉𝑡,𝑢,𝜑=2𝑎_𝑛−1(𝑡)𝑑𝑡, (4.26) thus 𝑉𝑡,𝑢,𝜑=𝑉𝑡₀,𝑢𝑡₀,𝜑𝑡₀𝑒^{∫𝑡𝑡02𝑎𝑛−1(𝑠)𝑑𝑠}, (4.27) by (4.24), we can get 𝑢−𝜑²=𝑢𝑡₀−𝜑𝑡₀²𝑒^{∫𝑡𝑡02𝑎𝑛−1(𝑠)𝑑𝑠}, (4.28) by (4.28), that it follows 𝑢−𝜑=𝑢𝑡₀−𝜑𝑡₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝑠)𝑑𝑠}, (4.29) since 𝑥(𝑡)=𝑢^{−1/(𝑛−1)}(𝑡)+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠,𝜙(𝑡)=𝜑^{−1/(𝑛−1)}(𝑡)+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.30) it follows that 𝑢(𝑡)=1𝑥(𝑡)−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1(𝑡)=1,𝜑𝜙(𝑡)−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1, (4.31) substitute (4.31) into (4.29), it follows that 1𝑥(𝑡)−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1−1𝜙(𝑡)−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1=1𝑥𝑡₀−∫^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1−1𝜙𝑡₀−∫^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝑠)𝑑𝑠}. (4.32) According to mean value theorem, we can get −(𝑛−1)𝜉^−𝑛𝑥(𝑡)−𝜙(𝑡)=−(𝑛−1)𝜁^−𝑛𝑥𝑡₀−𝜙𝑡₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝑠)𝑑𝑠}, (4.33) where 𝑥(𝑡)−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠<𝜉<𝜙(𝑡)−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 (4.34) or 𝜙(𝑡)−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠<𝜉<𝑥(𝑡)−^𝑡_𝛾𝑎_𝑛(𝑠)𝑥𝑡𝑑𝑠,₀−^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠<𝜁<𝜙𝑡₀−^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠 (4.35) or 𝜙𝑡₀−^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠<𝜁<𝑥𝑡₀−^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠. (4.36) Since 𝜑(𝑡) is positive and bounded, and from (4.19), (4.31), we know that 𝑥(𝑡)−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠,𝜙(𝑡)−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠,𝑥𝑡₀−^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠,𝜙𝑡₀−^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠 (4.37) are positive and bounded as 𝑡>𝑇, hence 𝜉, 𝜁 are positive and bounded as 𝑡>𝑇, so we have 𝑥(𝑡)(𝑡=𝜉−𝜙)𝜁^𝑛𝑥𝑡₀−𝜙𝑡₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝑠)𝑑𝑠}, (4.38) by (4.8), it follows that ^𝑡_𝑡0𝑎_𝑛−1(𝑠)𝑑𝑠⟶−∞, (4.39) as 𝑡→+∞, and by (4.38), we can get that the 𝜔-periodic solution 𝜙(𝑡) of (1.1) is globally attractive in Ω₂.
(iii) The proof of the periodic solution 𝜒(𝑡) of (1.1) being globally attractive in Ω₄ is similar to that of the periodic solution 𝜙(𝑡) of (1.1) being globally attractive in Ω₂, so we omit it here.
This is the end of the proof of Theorem 4.1.

Remark 4.2. In Theorem 4.1, since the transformation of 𝑢=𝑦^1−𝑛, it follows 𝑢>0, thus we suppose the initial value 𝑢(𝑡₀)>0.

Theorem 4.3. Consider (1.1), 𝑛 is an even number, and 𝑛≥2, 𝑎₀(𝑡), 𝑎_𝑛(𝑡) are continuous 𝜔-periodic functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑎_𝑛−1(𝑡)=(1−𝑛)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1,(𝐻₂)𝑎_𝑘(𝑡)=(−1)^𝑘𝐶^𝑘_𝑛𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑘, 𝑘=1,2,…,𝑛−2,(𝐻₃)𝑚(𝑎_𝑛(𝑡))=0, 𝑎_𝑛(𝑡)≢0,(𝐻₄)𝑎₀(𝑡)<0, ∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠≤0,
then
(i) equation (1.1) has two 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), and they can be written as follows: 𝜂(𝑡)=^𝑡_𝛾𝑎_𝑛(𝑠)𝜙(𝑡)=−()𝑑𝑠,𝑛−1^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠. (4.40) (ii) If given any initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥>∫^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠}, then the periodic solution 𝜙(𝑡) of (1.1) is globally attractive in Ω₂={𝑥∣𝑥>inf_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠}.

Proof. The proof of Theorem 4.3 is similar to that of Theorem 4.1, so we omit it here.

Theorem 4.4. Consider (1.1), 𝑛 is an even number, and 𝑛≥2, 𝑎₀(𝑡), 𝑎_𝑛(𝑡) are continuous 𝜔-periodic functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑎_𝑛−1(𝑡)=(1−𝑛)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1,(𝐻₂)𝑎_𝑘(𝑡)=(−1)^𝑘𝐶^𝑘_𝑛𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑘, 𝑘=1,2,…,𝑛−2,(𝐻₃)𝑚(𝑎_𝑛(𝑡))=0, 𝑎_𝑛(𝑡)≢0,(𝐻₄)𝑎₀(𝑡)>0, ∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠≥0,
then
(i) equation (1.1) has two 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), and they can be written as follows:𝜂(𝑡)=^𝑡_𝛾𝑎_𝑛(𝑠)𝜙(𝑡)=−()𝑑𝑠,𝑛−1^𝑡_−∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠. (4.41) (ii) If given any initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥<∫^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠}, then the periodic solution 𝜙(𝑡) of (1.1) is globally attractive in Ω₂={𝑥∣𝑥<sup_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠}.

Proof. The proof of Theorem 4.4 is similar to that of Theorem 4.1, so we omit it here.
From Theorem 4.1, if 𝑛=3, it is easy for us to draw the following corollary.
Define Ω₁=𝑥∣𝑥>^𝑡0_𝛾𝑑(𝑠)𝑑𝑠,Ω₂=𝑥∣𝑥>inf_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑑(𝑠),Ω𝑑𝑠₃=𝑥∣𝑥<^𝑡0_𝛾𝑑(𝑠)𝑑𝑠,Ω₄=𝑥∣𝑥<sup_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑑(𝑠)𝑑𝑠, (4.42) where 𝑑(𝑡), 𝛾, 𝜔 are the same function and numbers as the following Corollary 4.5, and 𝑡₀ is any given initial time of (1.2).

Corollary 4.5. Consider Abel type differential equation (1.2), 𝑎(𝑡), 𝑑(𝑡) are 𝜔-periodic continuous functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑏(𝑡)=−3𝑎(𝑡)∫^𝑡_𝛾𝑑(𝑠)𝑑𝑠; (𝐻₂)𝑐(𝑡)=2𝑎(𝑡)(∫^𝑡_𝛾𝑑(𝑠)𝑑𝑠)²; (𝐻₃)𝑚(𝑑(𝑡))=0, 𝑑(𝑡)≢0;(𝐻₄)𝑎(𝑡)<0;
then
(i) equation (1.2) has three 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), 𝜒(𝑡), and they can be written as follows: 𝜂(𝑡)=^𝑡_𝛾𝑑(𝑠)𝜙(𝑡)=𝑑𝑠,−2^𝑡_−∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑐(𝜏)𝑑𝜏}𝑑𝑠^−1/2+^𝑡_𝛾𝑑(𝑠)𝜒(𝑡)𝑑𝑠,=−−2^𝑡_−∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑐(𝜏)𝑑𝜏}𝑑𝑠^−1/2+^𝑡_𝛾𝑑(𝑠)𝑑𝑠. (4.43) (ii) If given any initial value 𝑥(𝑡₀)∈Ω₁, then the periodic solution 𝜙(𝑡) of (1.2) is globally attractive in Ω₂. (iii) If given any initial value 𝑥(𝑡₀)∈Ω₃, then the periodic solution 𝜒(𝑡) of (1.2) is globally attractive in Ω₄.

From Theorems 4.3 and 4.4, if 𝑛=2, it is easy for us to draw the following corollaries.

Corollary 4.6. Consider Ricatti’s equation (3.10), 𝑎(𝑡), 𝑐(𝑡) are 𝜔-periodic continuous functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑏(𝑡)=−𝑎(𝑡)∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠; (𝐻₂)𝑚(𝑐(𝑡))=0, 𝑐(𝑡)≢0;(𝐻₃)𝑎(𝑡)<0, ∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠≤0;
then
(i) equation (3.10) has two 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), and they can be written as follows:𝜂(𝑡)=^𝑡_𝛾𝑐(𝑠)𝜙(𝑡)=−𝑑𝑠,_𝑡−∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑏(𝜏)𝑑𝜏}𝑑𝑠⁻¹+^𝑡_𝛾𝑐(𝑠)𝑑𝑠. (4.44) (ii) If given any initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥>∫^𝑡0_𝛾𝑐(𝑠)𝑑𝑠}, then the periodic solution 𝜙(𝑡) of (3.10) is globally attractive in Ω₂={𝑥∣𝑥>inf_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠}.

Corollary 4.7. Consider Ricatti’s equation (3.10) 𝑎(𝑡), 𝑐(𝑡) are 𝜔-periodic continuous functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑏(𝑡)=−𝑎(𝑡)∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠; (𝐻₂)𝑚(𝑐(𝑡))=0, 𝑐(𝑡)≢0;(𝐻₃)𝑎(𝑡)>0, ∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠≥0;
then,
(i) equation (3.10) has two 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), and they can be written as follows:𝜂(𝑡)=^𝑡_𝛾𝑐(𝑠)𝜙(𝑡)=−𝑑𝑠,_𝑡−∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑏(𝜏)𝑑𝜏}𝑑𝑠⁻¹+^𝑡_𝛾𝑐(𝑠)𝑑𝑠. (4.45) (ii) If given any initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥<∫^𝑡0_𝛾𝑐(𝑠)𝑑𝑠}, then the periodic solution 𝜙(𝑡) of (3.10) is globally attractive in Ω₂={𝑥∣𝑥<sup_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠}.

Theorem 4.8. Consider (1.1), 𝑛 is an odd number, and 𝑛>2, 𝑎₀(𝑡), 𝑎_𝑛(𝑡) are continuous 𝜔-periodic functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑎_𝑛−1(𝑡)=(𝑛−1)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1; (𝐻₂)𝑎_𝑘(𝑡)=(−1)^𝑘𝐶^𝑘_𝑛𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑘, 𝑘=1,2,…,𝑛−2;(𝐻₃)𝑚(𝑎_𝑛(𝑡))=0, 𝑎_𝑛(𝑡)≢0;(𝐻₄)𝑎₀(𝑡)>0;
then,
(i) equation (1.1) has three 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), 𝜒(𝑡), and they can be written as follows:𝜂(𝑡)=^𝑡_𝛾𝑎_𝑛(𝑠)𝜙(𝑡)=()𝑑𝑠,𝑛−1_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝜒(𝑡))𝑑𝑠,=−(𝑛−1_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠. (4.46) (ii) if given any initial value 𝑥(𝑡₀)∈Ω₁∪Ω₂, then the periodic solution 𝜂(𝑡) of (1.1) is globally attractive in Ω₃∪Ω₄, whereΩ₁=𝑥∣^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠≤𝑥<𝜑𝑡₀^{−1/𝑛−1}+^𝑡0_𝛾𝑎_𝑛(𝑠),Ω𝑑𝑠₂=𝑥∣^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠≥𝑥>−𝜑𝑡₀^{−1/𝑛−1}+^𝑡0_𝛾𝑎_𝑛(𝑠),Ω𝑑𝑠₃=𝑥∣inf_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠≤𝑥<sup_{[]𝑡∈0,𝜔}𝜑(𝑡)^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠),Ω𝑑𝑠₄=𝑥∣sup_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠≥𝑥>inf_{[]𝑡∈0,𝜔}−𝜑(𝑡)^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠),𝜑(𝑡)=()𝑑𝑠𝑛−1_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠. (4.47)

Remark 4.9. Where 𝜑(𝑡) above is the same function as that of the following formula (4.54), and 𝑡₀ is any given initial time of (1.1).

Proof. (i) Since the conditions (𝐻₁), (𝐻₂) hold, (1.1) can be written as follows: 𝑑𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠=𝑑𝑡𝑥−^𝑡_𝛾𝑎_𝑛(𝑠)×𝑑𝑠−𝑎₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1+𝑎₀(𝑡)𝑥−^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1. (4.48)
Let 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.48) can be changed as follows: 𝑑𝑦𝑑𝑡=𝑦−𝑎₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1+𝑎₀(𝑡)𝑦^𝑛−1. (4.49) This is the Bernoulli-type equation, set 𝑢(𝑡)=𝑦^1−𝑛(𝑡), the Bernoulli-type equation (4.49) becomes 𝑑𝑢=()𝑎𝑑𝑡𝑛−1₀(𝑡)^𝑡_𝑟𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1𝑢−(𝑛−1)𝑎₀(𝑡),𝑑𝑢𝑑𝑡=𝑎_𝑛−1(𝑡)𝑢−(𝑛−1)𝑎₀(𝑡). (4.50) By (𝐻₃), (𝐻₄), it follows that _𝛾^𝑡+𝜔𝑎_𝑛(𝑠)𝑑𝑠=^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠+_𝑡^𝑡+𝜔𝑎_𝑛(𝑠)=𝑑𝑠_𝑡𝛾𝑎_𝑛(𝑠)𝑑𝑠+^𝜔₀𝑎_𝑛(𝑠)=𝑑𝑠_𝑡𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.51) thus ∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 is an 𝜔-periodic function, therefore, (𝑛−1)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1 is also an 𝜔-periodic function and 𝑎_𝑛−1(𝑡)=(𝑛−1)𝑎₀(𝑡)^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠^𝑛−1≥0, (4.52) since 𝑎₀(𝑡)>0, 𝑎_𝑛(𝑡)≠0 (not identically equal to zero), thus we have 𝑚𝑎_𝑛−1(𝑡)>0, (4.53) according to Lemma 2.3, it follows that (4.50) has a unique 𝜔-periodic solution as follows: 𝜑(𝑡)=(𝑛−1)_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠, (4.54) from (4.54), it is easy to know that 𝜑(𝑡)>0, since 𝜑(𝑡) is periodic on 𝑅, it is bounded on 𝑅, so it is positive and bounded, since 𝑢(𝑡)=𝑦^1−𝑛(𝑡), and by Lemmas 2.1 and 2.2, it follows that (4.49) has two 𝜔-periodic solutions as follows: 𝜓(𝑡)=(𝑛−1)_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)},𝜓𝑡₀,𝜓>0_∗(𝑡)=−(𝑛−1)_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)},𝜓^∗𝑡₀<0, (4.55) thus 𝜓(𝑡) is also positive and bounded, and 𝜓^∗(𝑡) is negative and bounded, by virtue of 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (1.1) has two 𝜔-periodic solutions as follows: 𝜙(𝑡)=(𝑛−1)_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝜒(𝑡)()𝑑𝑠,=−𝑛−1_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠. (4.56) By (1.1), we set 𝑓(𝑡,𝑥)=∑^𝑛_𝑘=0𝑎_𝑘(𝑡)𝑥^𝑛−𝑘, thus 𝑓(𝑡,𝑥) is continuous on 𝑅×𝑅, and 𝑓_𝑥(𝑡,𝑥) is also continuous on 𝑅×𝑅, hence (1.1) satisfies the principle of existence and uniqueness of differential equation, therefore, (1.1) has no singular solution.
From the variable transformation of 𝑢(𝑡)=𝑦^1−𝑛(𝑡), it requires 𝑦≠0, but from (4.49), we know 𝑦=0 is indeed a solution of (4.49), since 𝑦(𝑡)=𝑥−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, hence 𝜂(𝑡)=∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 is a periodic solution of (1.1). Substitute 𝜂(𝑡)=∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 into (1.1), it just satisfies (1.1), so 𝜂(𝑡)=∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 is indeed a periodic particular solution of (1.1), therefore, we have proved that (1.1) has three 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡) and 𝜒(𝑡).
(ii) According to the theories of linear differential equation, we know that the unique solution 𝑢(𝑡) of (4.50) with the positive initial value 𝑢(𝑡₀)=𝑢₀ is given by 𝑢(𝑡)=𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠=𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}𝑢₀−𝜑𝑡₀+𝜑(𝑡). (4.57)
If given the initial value 𝑢₀≥𝜑(𝑡₀), then 𝑢(𝑡)≥𝜑(𝑡)>0.
Remark 4.10. Since the transformation of 𝑢(𝑡)=𝑦^1−𝑛(𝑡), it requires 𝑢(𝑡)>0, in addition, 𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}=𝑒^{(𝑛−1)∫𝑡𝑡0𝑎0(𝜏)(∫𝜏𝛾𝑎𝑛(𝜃)𝑑𝜃)𝑛−1𝑑𝜏}⟶+∞ (4.58) as 𝑡→+∞, that is to say, it requires 𝑢₀≥𝜑(𝑡₀).
Under the transformation of 𝑢(𝑡)=𝑦^1−𝑛(𝑡), 𝑦(𝑡)=𝑥(𝑡)−∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, we can get the following.
If given the initial value 𝑥𝑡₀=𝑢₀^{−1/(𝑛−1)}+^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠<𝜑𝑡₀^{−1/(𝑛−1)}+^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠∈Ω₁, (4.59)then the unique solution 𝑥(𝑡) of (1.1) is given by 𝑥(𝑡)=𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠, (4.60) and it is easy for us to know that 𝑥(𝑡)∈Ω₃.
If given the initial value 𝑥𝑡₀=−𝑢₀^{−1/(𝑛−1)}+^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠>−𝜑𝑡₀^{−1/(𝑛−1)}+^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠∈Ω₂; (4.61)then the unique solution 𝑥(𝑡) of (1.1) is given by 𝑥(𝑡)=−𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠 (4.62) it is easy for us to know that 𝑥(𝑡)∈Ω₄.
Therefore, we can draw the following conclusion.
If given the initial value 𝑥(𝑡₀)∈Ω₁∪Ω₂, then 𝑥(𝑡)∈Ω₃∪Ω₄.
Next, we prove that the periodic solution 𝜂(𝑡) of (1.1) is globally attractive in Ω₃∪Ω₄ if given the initial value 𝑥(𝑡₀)∈Ω₁∪Ω₂. 𝑥(𝑡)−𝜂(𝑡)=±𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}=𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}−(𝑛−1)^𝑡_𝑡0𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}=𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}𝑢₀−𝜑𝑡₀+𝜑(𝑡)^{−1/(𝑛−1)}=𝑥𝑡₀−𝜂𝑡₀𝑥𝑡₀−𝜂𝑡₀^𝑛−1𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}𝑢₀−𝜑𝑡₀(𝑡)=𝑥𝑡+𝜑₀−𝜂_0𝑛−1𝑡1/𝑢₀𝑒^{∫𝑡𝑡0𝑎𝑛−1(𝜏)𝑑𝜏}𝑢₀−𝜑𝑡₀(𝑡,+𝜑) (4.63) by (4.53), it follows that ^𝑡_𝑡0𝑎_𝑛−1(𝑠)𝑑𝑠⟶+∞, (4.64) as 𝑡→+∞, from (4.63), we can get that the 𝜔-periodic solution 𝜂(𝑡) of (1.1) is globally attractive in Ω₃∪Ω₄.
This is the end of the proof of Theorem 4.8.

Theorem 4.11. Consider (1.1), 𝑛 is an even number, and 𝑛≥2, 𝑎₀(𝑡), 𝑎_𝑛(𝑡) are continuous 𝜔-periodic functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑎_𝑛−1(𝑡)=(1−𝑛)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1; (𝐻₂)𝑎_𝑘(𝑡)=(−1)^𝑘𝐶^𝑘_𝑛𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑘, 𝑘=1,2,…,𝑛−2;(𝐻₃)𝑚(𝑎_𝑛(𝑡))=0, 𝑎_𝑛(𝑡)≢0;(𝐻₄)𝑎₀(𝑡)<0, ∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠≥0;
then
(i) equation (1.1) has two 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), and they can be written as follows:𝜂(𝑡)=^𝑡_𝛾𝑎_𝑛(𝑠)𝜙(𝑡)=()𝑑𝑠,𝑛−1_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠; (4.65) (ii) if given any initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥>∫^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠}, then the periodic solution 𝜂(𝑡) of (1.1) is globally attractive in Ω₂={𝑥∣𝑥>inf_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠}.

Proof. The Proof of Theorem 4.11 is similar to that of Theorem 4.8, so we omit it here.

Theorem 4.12. Consider (1.1), 𝑛 is an even number, and 𝑛≥2, 𝑎₀(𝑡), 𝑎_𝑛(𝑡) are continuous 𝜔-periodic functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:
(𝐻₁)𝑎_𝑛−1(𝑡)=(1−𝑛)𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑛−1; (𝐻₂)𝑎_𝑘(𝑡)=(−1)^𝑘𝐶^𝑘_𝑛𝑎₀(𝑡)(∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠)^𝑘, 𝑘=1,2,…,𝑛−2;(𝐻₃)𝑚(𝑎_𝑛(𝑡))=0, 𝑎_𝑛(𝑡)≢0;(𝐻₄)𝑎₀(𝑡)>0, ∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠≤0;
then
(i) equation (1.1) has two 𝜔-periodic solutions 𝜂(𝑡),𝜙(𝑡), and they can be written as follows: 𝜂(𝑡)=^𝑡_𝛾𝑎_𝑛(𝑠)𝜙(𝑡)=()𝑑𝑠,𝑛−1_𝑡^+∞𝑎₀(𝑠)𝑒^{∫𝑡𝑠𝑎𝑛−1(𝜏)𝑑𝜏}𝑑𝑠^{−1/(𝑛−1)}+^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠; (4.66) (ii) if given any initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥<∫^𝑡0_𝛾𝑎_𝑛(𝑠)𝑑𝑠}, then the periodic solution 𝜂(𝑡) of (1.1) is globally attractive in Ω₂={𝑥∣𝑥<sup_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑎_𝑛(𝑠)𝑑𝑠}.

Proof. The Proof of Theorem 4.12 is similar to that of Theorem 4.8, so we omit it here.

From Theorem 4.8, if 𝑛=3, it is easy for us to draw the following corollary.

Corollary 4.13. Consider Abel type differential equation (1.2), 𝑎(𝑡), 𝑑(𝑡) are 𝜔-periodic continuous functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑏(𝑡)=−3𝑎(𝑡)∫^𝑡_𝛾𝑑(𝑠)𝑑𝑠; (𝐻₂)𝑐(𝑡)=2𝑎(𝑡)(∫^𝑡_𝛾𝑑(𝑠)𝑑𝑠)²; (𝐻₃)𝑚(𝑑(𝑡))=0, 𝑑(𝑡)≢0;(𝐻₄)𝑎(𝑡)>0;
then
(i) equation (1.2) has three 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), 𝜒(𝑡), and they can be written as follows: 𝜂(𝑡)=^𝑡_𝛾𝑑(𝑠)𝜙(𝑡)=𝑑𝑠,2_𝑡^+∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑐(𝜏)𝑑𝜏}𝑑𝑠^−1/2+^𝑡_𝛾𝑑(𝑠)𝜒(𝑡)=2𝑑𝑠,_𝑡+∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑐(𝜏)𝑑𝜏}𝑑𝑠^−1/2+^𝑡_𝛾𝑑(𝑠)𝑑𝑠. (4.67) (ii) If given any initial value 𝑥(𝑡₀)∈Ω₁∪Ω₂, then the periodic solution 𝜂(𝑡) of (1.2) is globally attractive in Ω₃∪Ω₄, whereΩ₁=𝑥∣^𝑡0_𝛾𝑑(𝑠)𝑑𝑠≤𝑥<𝜑𝑡₀^−1/2+^𝑡0_𝛾𝑑(𝑠),Ω𝑑𝑠₂=𝑥∣^𝑡0_𝛾𝑑(𝑠)𝑑𝑠≥𝑥>−𝜑𝑡₀^−1/2+^𝑡0_𝛾𝑑(𝑠),Ω𝑑𝑠₃=𝑥∣inf_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑑(𝑠)𝑑𝑠≤𝑥<sup_{[]𝑡∈0,𝜔}𝜑(𝑡)^−1/2+^𝑡_𝛾𝑑(𝑠),Ω𝑑𝑠₄=𝑥∣sup_{[]𝑡∈0,𝜔}^𝑡_𝛾𝑑(𝑠)𝑑𝑠≥𝑥>inf_{[]𝑡∈0,𝜔}−𝜑(𝑡)^−1/2+^𝑡_𝛾𝑑(𝑠),𝜑(𝑡)𝑑𝑠=2_𝑡^+∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑐(𝜏)𝑑𝜏}𝑑𝑠. (4.68)

From Theorem 4.11 and 4.6, if 𝑛=2, it is easy for us to draw the following corollaries.

Corollary 4.14. Consider Riccati’s equation (3.10), 𝑎(𝑡), 𝑐(𝑡) are 𝜔-periodic continuous functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑏(𝑡)=−𝑎(𝑡)∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠; (𝐻₂)𝑚(𝑐(𝑡))=0, 𝑐(𝑡)≢0;(𝐻₃)𝑎(𝑡)<0, ∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠≥0;
then
(i) equation (3.10) has two 𝜔-periodic solutions 𝜂(𝑡),𝜙(𝑡), and they can be written as follows: 𝜂(𝑡)=^𝑡_𝛾𝑐(𝑠)𝜙(𝑡)=𝑑𝑠,_𝑡+∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑏(𝜏)𝑑𝜏}𝑑𝑠⁻¹+^𝑡_𝛾𝑐(𝑠)𝑑𝑠. (4.69) (ii) If given the initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥>∫^𝑡0_𝛾𝑐(𝑠)𝑑𝑠}, then the periodic solution 𝜂(𝑡) of (3.10) is globally attractive in Ω₂={𝑥∣𝑥>inf_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠}.

Corollary 4.15. Consider Riccati’s equation (3.10), 𝑎(𝑡), 𝑐(𝑡) are 𝜔-periodic continuous functions defined on 𝑅, if there is a constant 𝛾 such that the following conditions hold:(𝐻₁)𝑏(𝑡)=−𝑎(𝑡)∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠; (𝐻₂)𝑚(𝑐(𝑡))=0, 𝑐(𝑡)≢0;(𝐻₃)𝑎(𝑡)>0, ∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠≤0;
then
(i) equation (3.10) has two 𝜔-periodic solutions 𝜂(𝑡), 𝜙(𝑡), and they can be written as follows:𝜂(𝑡)=^𝑡_𝛾𝑐(𝑠)𝜙(𝑡)=𝑑𝑠,_𝑡+∞𝑎(𝑠)𝑒^{∫𝑡𝑠𝑏(𝜏)𝑑𝜏}𝑑𝑠⁻¹+^𝑡_𝛾𝑐(𝑠)𝑑𝑠. (4.70) (ii) If given any initial value 𝑥(𝑡₀)∈Ω₁={𝑥∣𝑥<∫^𝑡0_𝛾𝑐(𝑠)𝑑𝑠}, then the periodic solution 𝜂(𝑡) of (3.10) is globally attractive in Ω₂={𝑥∣𝑥<sup_{𝑡∈[0,𝜔]}∫^𝑡_𝛾𝑐(𝑠)𝑑𝑠}.

5. Concluding Remarks

Generally, just as Ricatti’s equation, (1.1) is nonintegrable, however in the present paper, we have got a sufficient condition which guarantees the integrability and have presented some exact periodic solutions of nonlinear equation if the coefficients of (1.1) satisfy certain conditions; moreover, we get that the periodic solutions are globally attractive in certain conditions. These conclusions will have certain applications.

Acknowledgments

The authors are highly grateful to the reviewers for their constructive comments. This work is supported by the National Nature Science Foundation of China (no. 11171135) and Jiangsu Province Innovation Project of Graduate Education (2011, no. 1221190037).