Existence of positive periodic solutions for Liénard equation with a singularity of repulsive type

In this paper, the existence of positive periodic solutions is studied for Liénard equation with a singularity of repulsive type,


Introduction
Since singular equations have a wide range of application in physics, engineering, mechanics, and other subjects (see [1][2][3][4][5][6][7]), the periodic problem for a certain second order differential equation has attracted much attention from many researchers.In the past years, lots of papers (see [8][9][10][11][12][13][14]) were concerned with the problem of periodic solutions to the second order singular equation without the first derivative term, where and μ > 0 is a constant.Among these papers, we notice that the coefficient function ϕ(t) is required to be ϕ(t) ≥ 0 for a.e.t ∈ [0, T]. (1.2) This is because (1.2), together with other conditions, can ensure that the function G(t, s) ≥ 0 for (t, s) ∈ [0, T] × [0, T], where the G(t, s) is the Green function associated with the boundary value problem for Hill's equation x (t) + ϕ(t)x(t) = h(t), x(0) = x(T), x (0) = x (T).
The condition G(t, s) ≥ 0 for (t, s) ∈ [0, T] × [0, T] is crucial for obtaining the positive periodic solutions to (1.1) by means of some fixed point theorems on cones.Beginning with the paper of Habets-Sanchez [15], many works (see [16][17][18][19][20][21]) discussed the existence of a periodic solution for Liénard equations with singularities, where ϕ(t) and e(t) are T-periodic with ϕ, e ∈ L 1 [0, T], while γ is a constant with γ > 0. However, in those papers, the conditions of ϕ(t) ≥ 0 for a.e.t ∈ [0, T], the strong singularity γ ∈ [1, +∞), and f (x) being continuous on [0, +∞) are needed.To the best of our knowledge, there are fewer papers dealing with the equation where the function f (x) possesses a singularity at x = 0. We find that Hakl, Torres, and Zamora in [22] considered the periodic problem for the singular equation of repulsive type, where , and the sign of ϕ(t) can change, while f ∈ C((0, +∞), R) may be singular at x = 0 and g ∈ C((0, +∞), R) has a repulsive singularity at x = 0, i.e., lim x→0 + g(x) = -∞.By using Schauder's fixed point theorem, some results on the existence of positive T-periodic solutions were obtained.However, the strong singularity condition 1 0 g(s)ds = -∞ is also required.In a recent paper [23], the authors consider the periodic problem to (1.4) for the special case g(x) = 1 x γ , where γ ∈ (0, +∞).But, in [23], the function ϕ(t) is required to satisfy ϕ(t) ≥ 0 a.e.t ∈ [0, T] for the case μ > 1 (see Theorem 3.1, [23]).Motivated by this, in the present paper, we continue to study the periodic problem for the singular equation, where f , ϕ are as same as those in (1.4); μ > 0 and γ > 0 are constants, e is a T-periodic function with e ∈ L 1 ([0, T], R), and T 0 e(s)ds = 0.By means of a continuation theorem of coincidence degree principle developed by Manásevich and Mawhin, as well as the techniques of a priori estimates, some new results on the existence of positive periodic solutions are obtained.The interesting point in this paper is that the function f (x) has a singularity at x = 0, the sign of ϕ(t) is allowed to change, and μ, γ ∈ (0, +∞).Compared with [22], we allow the singular term 1 x γ to have a weak singularity, i.e., γ ∈ (0, 1).Also, for the case of μ > 1, the sign of ϕ(t) is allowed to change, which is essentially different from the condition ϕ(t) ≥ 0 for a.e.t ∈ [0, T] in [23].

Essential definitions and lemmas
Throughout this paper, let Clearly, C T is a Banach space.For any T-periodic function x(t), we denote x = 1 T T 0 x(s)ds, x + (t) = max{x(t), 0}, and x -= -min{x(t), 0}.Thus, x(t) = x + (t)x -(t) for all t ∈ R, and x = x + -x -.Furthermore, for each u ∈ C T , let u p = (
(2.39) Thus, by the assumptions of (2.6), (2.35), and (2.36), and according to the proof of Lemma 2.6, we have which together with (2.39) yields (2.41) On the other hand, assumption (2.37) gives that there exits a constant γ 3 > 0 such that

Example
In this section, we present two examples to demonstrate the main results.
Remark 4.3 In (4.5), since μ = 3 2 > 1 and ϕ(t) = 1 + 2 cos t is a sign-changing function, the result of Example 4.2 can be obtained neither by using the main results of [23], nor by using the theorems of [23].In this sense, the theorems of the present paper are new results on the existence of positive periodic solutions for singular Liénard equations.