POSITIVE PERIODIC SOLUTION FOR SECOND-ORDER NONLINEAR DIFFERENTIAL EQUATION WITH SINGULARITY OF ATTRACTIVE TYPE∗

This paper is devoted to investigate the following second-order nonlinear differential equation with singularity of attractive type x − a(t)x = f(t, x) + e(t), where the nonlinear term f has a singularity at the origin. By using the Green’s function of the linear differential equation with constant coefficient and Schauder’s fixed point theorem, we establish some existence results of positive periodic solutions.


Introduction
In this paper, we discuss the existence of positive periodic solutions of the following nonlinear differential equation with singularity x ′′ − a(t)x = f (t, x) + e(t), (1.1) where a(t) ∈ C(R, R + ) and e(t) ∈ L 1 (R) are ω-periodic functions, f (t, x) ∈ C(R × R + , R) is an ω-periodic function on t.
The nonlinear term f of equation (1.1) can be with a singularity at the origin, i.e.,

uniformly in t.
It is said that equation (1.1) is of attractive type (resp. repulsive type) if f (t, x) → −∞ (resp. f (t, x) → +∞) as x → 0 + . Since 1980s, there have been published many works in which singularity of differential equations is discussed. More concretely, in 1987, Lazer and Solimini [10] investigated the model equations with singularity where λ, ν, µ are positive constants and f is a continuous periodic functions with period ω. It is said that equation (1.2) has an attractive singularity, whereas equation (1.3) has a repulsive singularity. The authors provided the necessary and sufficient conditions for the existence of periodic solutions of equations (1.2) and (1.3). One of the common conditions to guarantee the existence of positive periodic solution is a so-called strong force condition (corresponds to the case λ ≥ 1 in equation (1.2)), see [1,6,7,17,18,20,23] and references therein. On the other hand, the existence of positive periodic solution of the singular differential equations has been established with a weak force condition (corresponds to the case 0 < λ < 1 in equation (1.2)), see [2,5,11,14]. During the last two decades, the study of the existence of positive periodic solutions for second-order differential equations with singularity of repulsive type has attracted the attention of many researchers (see [1, 2, 4-9, 11-19, 23]). For example, Torres [14] in 2007 investigated a kind of second order non-autonomous singular differential equation Applying Schauder's fixed point theorem, the author showed that the additional assumption of a weak singularity enabled the obtention of new criteria for the existence of positive periodic solutions. In 2010, Wang [17] discussed the existence and multiplicity of positive periodic solutions of singular systems (1.4) with superlinearity or sublinearity assumptions at infinity for an appropriately chosen parameter e(t). The proofs of their results are based on the Krasnoselskii fixed point theorem in cones. In 2014, Ma, Chen and He [11] improved above results and presented a new assumption which was weaker than the singular condition in [14]. All the aforementioned results are related to second-order differential equations with singularity of repulsive type. Naturally, a new question arises: how secondorder differential equation works on singularity of attractive type? In this paper, we try to establish the existence of a positive periodic solution of equation (1.1) by using the Green's function of the linear differential equation with constant coefficient and Schauder's fixed point theorem. This trick has been used to investigate a third-order singular differential equation with variable coefficients in [21]. Remark 1.1. As far as we know, the calculation of the Green's function of the second-order linear differential equation with variable coefficient is very complicated. In this paper, we will discuss the Green's function G(t, s) of the second-order linear differential equation with constant coefficient Obviously, attractive condition and repulsive condition are contradiction. Therefore, the above conditions in [11,14,17] are no long applicable to the proof of existence of positive periodic solution for equation (1.1) with singularity of attractive type. In this paper, we need find another conditions to overcome this problems.
From now on, we denote the essential supremum and infimum of the external force e(t) ∈ C[0, ω] by e * and e * . We define the function γ : R → R by which is the unique ω-periodic solution of equation (1.6) Obviously, γ(t) is closely depending on the external force e(t). We would like to emphasize that the value of γ(t) can influence the existence of a positive periodic solution of equation (1.1) with strong singularity or weak singularity. Specifically, the existence of a positive periodic solution of equation (1.1) with weak and strong singularities of attractive type if the following conditions satisfies: The existence of a positive periodic solution to equation (1.1) with weak singularity of attractive type if the following conditions satisfies: Moreover, we consider the existence of a positive periodic solution for equation (1.1) with attractive-repulsive singularities. The paper is organized as follows: In Section2, the Green's function for constant coefficients differential equation (1.5) will be given. Some useful properties for the Green's function are shown also. In Section 3, we consider the positive periodic solution of (1.1) with attractive singularities in three cases: γ * > 0, γ * = 0 and γ * < 0. Moreover, we also proved the existence of a positive periodic solution when (1.1) has an attractive-repulsive singularity. To conclude this introduction, we write and it is negative in a set of negative measure.

Constant coefficients differential equation
In this section, we discuss the Green's function of the differential equation with constant coefficients Solution of equation (2.2) is written as Solution of equation (2.3) is written as Therefore, we know that the solution of equation (2.1) is written as Denote then we can get

Variable coefficients differential equation
In this section, we consider variable coefficients differential equations Obviously, the calculation of the Green's function of equation (2.8) is very complicated. To overcome this difficuly, we will make a shift on the linear term.
then equation (2.8) can be rewritten as , the solution of equation (2.9) can be written in the form Since where we used the fact Define an operator P : X → X by Proof. By the Neumann expansion of P , we have Since T h(t) > 0 for any t, we get Noting that ∥T H∥ < 1, we get

Singularity of attractive type
In this section, we establish the existence of a positive periodic solution of equation (1.1) by applications of Schauder's fixed point theorem. Define an operator Q :  Let R be the positive constant and r := γ * , then we have R > r > 0, since R > γ * . Define then, Ω is a closed convex set. For any x ∈ Ω, t ∈ R, from equation (3.1), we deduce which show that (Qx)(t) is ω-periodic. Next we will prove Q(Ω) ⊂ Ω. In fact, for each x ∈ Ω and for all t ∈ [0, ω], from Lemma 2.1 and condition (H 1 ), we know that non-positive sign of the Green's function G(t, s) and the nonlinear term On the other hand, by Lemma 2.2, we see that since γ * > 0, we know γ(t) > 0, then ∥γ∥ = γ * . Therefore, by conditions (H 1 ) and (H 2 ), we get In conclusion, we see that Q(Ω) ⊂ Ω.
Next, we show that Q is completely continuous. According to equations (2.11), (2.13) and (3.1), we shall prove that T is completely continuous and H is a continuous bounded operator.
Firstly, we show that T is completely continuous. Let {h k } ∈ Ω be a convergent sequence of functions, such that h k (t) → h(t) as k → ∞. Since Ω is closed, for h ∈ Ω and t ∈ [0, ω], it is clear that Therefore, T is continuous. On the other hand, we deduce  From above analysis, we conclude that T H is completely continuous. From equation (3.1), we have Q is completely continuous. Therefore, the proof is finished by Schauder's fixed point theorem.
If γ * > 0, then there exists a positive constant µ 1 such that equation (1.1) has at least one positive periodic solution for each 0 ≤ µ ≤ µ 1 .

Proof.
Take then the condition (H 2 ) is satisfied. Next, we consider condition (H 1 ). In fact, In view of β < α, then we have µ < R β−α . As a consequence, the result holds for Proof. We follow the same strategy and notation as in the proof of Theorem 3.1.
Next we prove that Q(Ω) ⊂ Ω. For each x ∈ Ω and for all t ∈ [0, ω], by the non-positive sign of the Green's function G(t, s) and the nonlinear term f (t, x) we have, from condition (H 3 ), On the other hand, from Lemma 2.2, conditions (H 1 ) and (H 4 ), we obtain By above two inequalities, we have Q(Ω) ⊂ Ω. Therefore, by Schauder's fixed point theorem, our result is proved.

Corollary 3.4. Assume the following condition holds:
for all x > 0 and a.e. t.
Proof. Take x ρ and p(x) = 0, then conditions (H 1 ) and (H 3 ) are satisfied and the existence condition (H 4 ) becomes Note that Ψ * > 0, since 0 < ρ < 1, we choose R > 0 as large as possible so that equation (3.5) is satisfied and the proof is complete.
In the following, we investigate equation (1.1) with attractive-repulsive singularities.
If γ * = 0, then there exists a positive constant µ 2 such that equation (1.1) has at least one positive periodic solution for each 0 ≤ µ ≤ µ 2 .
Proof. Let R be the positive constant satisfying (H 5 ) and r = (Φ R ) * + γ * , then R > r > 0 since R > (Φ R ) * + γ * . Next we prove that Q(Ω) ⊂ Ω. For each x ∈ Ω and for all t ∈ [0, ω], by the non-positive sign of the Green's function G(t, s) and the nonlinear term f (t, x) we have, from conditions (H 3 ) and (H 5 ), On the other hand, from Lemma 2.2, conditions (H 1 ) and (H 5 ), we deduce Therefore, by (H 2 ) and (H 5 ), we have By above two inequalities, Q(Ω) ⊂ Ω. Therefore, by Schauder's fixed point theorem, our result is proven.
Corollary 3.6. Assume that condition (F 4 ) holds. If γ * ≤ 0 and then there exists a positive periodic solution of equation (1.1).
Next, we consider the condition (H 5 ) is also satisfied. Take R = M Ψ * m(r) ρ , then or equivalently, The function f (r) possesses a minimum at The condition (H 5 ) holds directly by the choice of R, and it would remain to prove that R = M Ψ * m(r0) ρ > r 0 . This is easily verified through elementary computations. In the following, we investigate equation (1.1) with attractive-repulsive singularities.