Weak and strong singularities problems to Liénard equation

This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation: x″+f(x(t))x′(t)+a(t)x=b(t)xα+e(t),\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ x''+f\bigl(x(t)\bigr)x'(t)+a(t)x= \frac{b(t)}{x^{\alpha }}+e(t), $$\end{document} where the external force e(t)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$e(t)$\end{document} may change sign, α is a constant and α>0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\alpha >0$\end{document}. The novelty of the present article is that for the first time we show that weak and strong singularities enables the achievement of a new existence criterion of positive periodic solution through an application of the Manásevich–Mawhin continuation theorem. Recent results in the literature are generalized and significantly improved, and we give the existence interval of periodic solution of this equation. At last, two examples and numerical solution (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem.


Introduction
The main purpose of this paper is to consider the existence of a periodic solution for the Liénard equation with weak and strong singularities of repulsive type, where a, b ∈ C(R, (0, +∞)) are ω-periodic functions, f ∈ C(R, R), the external force e ∈ C(R, R) is an ω-periodic function. Moreover, note that when f (x(t)) ≡ 0, Eq. (1.1) becomes x + a(t)x = b(t) x α + e(t). (1.2) In 1987, Lazer and Solimini [1] investigated the following second-order differential equation with singularity of repulsive type: and obtained the result that if the external force h(t) was continuous and ω-periodic, then for all α > 0 a positive periodic solution existed if and only if the external force h(t) has a positive mean value. We say the equation to obey the strong force condition if α ≥ 1 and the weak force condition if 0 < α < 1.
Among these papers, there have been published some results on Eq. (1.2) (see [5,6,8,10,17]). Chu et al. [10] in 2007 discussed the existence of a positive periodic solution for Eq. (1.2) if the external force e(t) ≥ 0 and a := max t∈[0,ω] |a(t)| < π 2 ω 2 . Their results were based on a nonlinear alternative principle of Leray-Schauder and are applicable to the case of a strong singularity and the case of a weak singularity. Afterwards, Torres [8] proved Eq. (1.2) in the cases of weak and strong singularities had at least one positive periodic solution if the external force e(t) > 0 and a < π 2 ω 2 . Moreover, the author obtained the result that there was one positive periodic solution for Eq. (1.2) in the case of a weak singularity if one of the following conditions holds: (i) e(t) ≡ 0 and a < π 2 ω 2 ; or (ii) e(t) < 0 and a < π 2 ω 2 . Wang [17] in 2010 improved the above result and presented a new assumption, which is weaker than the singular condition in [8]. The author obtained the result that Eq. (1.2) in the cases that weak and strong singularities have at least one positive periodic solution if and only if one of the following conditions holds: (i) e(t) ≥ 0 and a < π 2 ω 2 ; or (ii) e(t) < 0 and a < π 2 ω 2 . The proof of their results was based on the Krasnoselskii fixed point theorem in a cone. All the aforementioned results are related to Eq. (1.2) with the external force e(t) not changing sign. Naturally, a new question arises: how may Eq. (1.1) with weak and strong singularities work on the external force e(t) changing sign? In this paper, we fill the gap and provide sufficient conditions for the existence of a positive periodic solution for Eq. (1.1) with weak and strong singularities, where the external force e(t) may change sign, α is a constant and α > 0. By applications of the Manásevich-Mawhin continuation theorem [20, Theorem 3.1], we obtain the following conclusion. Theorem 1.1 Assume that the following conditions are satisfied: Then Eq. (1.1) has at least one positive periodic solution x with , .
Remark 1.1 The techniques used are quite different from that in [5,8,10,17] and our results are more general than those in [5,8,10,17] in two aspects. We first obtain the existence of a positive periodic solution for equation (1.1) with weak and strong singularities if the external force e(t) may change sign. Secondly, we give the existence interval of positive periodic solution of Eq. (1.1).
In the following, we consider the existence of a periodic solution for Eq. (1.1) without the external force e(t). Corollary 1.1 Assume that (H 2 ) holds. Furthermore, suppose the following conditions are satisfied: (H 1 ) e(t) = 0; , Obviously, the condition (H 3 ) (or (H 3 )) is hard restrictive for the existence of a positive periodic solution to Eq. (1.1). In the following, we study the existence of a positive periodic solution for Eq. (1.1) with strong singularity (i.e. α ≥ 1) if conditions (H 1 ) and (H 2 ) are satisfied.
Next, we investigate a family of (1.1) as follows: Using [20, Theorem 3.1], we obtain the following conclusion.

Lemma 2.2
Assume that there exist positive constants E 1 , E 2 , E 3 and E 1 < E 2 such that the following conditions hold: Then Eq. (1.1) has at least one positive periodic solution.
We investigate the existence of a periodic solution for Eq. (1.1) with weak and strong singularities.

Figure 1
The first picture shows the system response in the (x, y). The second shows the system response in the (t, x) over the time interval of 0 ≤ t ≤ 50. The initial conditions are x 0 = 5.52907, y 0 = 0, t 0 = 0

Figure 2
The first picture shows the system response in the (x, y). The second shows the system response in the (t, x) over the time interval of 0 ≤ t ≤ 50. The initial conditions are x 0 = 2, y 0 = 0, t 0 = 0

Conclusions
In this paper, applying an extension of the Manásevich-Mawhin continuation theorem, we investigate the existence of a periodic solution for Eq. (1.1), where the external force e(t) may change sign, the singular term b(t) x α satisfies weak and strong singularities of repulsive type. Besides, we give the existence interval of periodic solution of Eq. (1.1). At last, two examples and numerical solutions (phase portraits and time portraits of periodic solutions of the example) are given to show applications of the theorem. The techniques used of this paper are quite different from that in [5,8,10,17] and our results are more general than those in [5,8,10,17] in two aspects. We first obtain the existence of a positive periodic solution for Eq. (1.1) with weak and strong singularities if the external force e(t) may change sign. Secondly, we give the existence interval of positive periodic solution of Eq. (1.1).