A new result on the existence of periodic solutions for Liénard equations with a singularity of repulsive type

In this paper, the problem of the existence of a periodic solution is studied for the second order differential equation with a singularity of repulsive type \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$x''(t)+f\bigl(x(t)\bigr)x'(t)-g\bigl(x(t) \bigr)+\varphi(t)x(t)=h(t), $$\end{document}x″(t)+f(x(t))x′(t)−g(x(t))+φ(t)x(t)=h(t), where \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$g(x)$\end{document}g(x) is singular at \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$x=0$\end{document}x=0, φ and h are T-periodic functions. By using the continuation theorem of Manásevich and Mawhin, a new result on the existence of positive periodic solution is obtained. It is interesting that the sign of the function \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varphi(t)$\end{document}φ(t) is allowed to change for \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$t\in[0,T]$\end{document}t∈[0,T].


Introduction
The aim of this paper is to search for positive T-periodic solutions for a second order differential equation with a singularity in the following form: where f : [, ∞) → R is an arbitrary continuous function, g ∈ C((, +∞), (, +∞)), and g(x) is singular of repulsive type at x = , i.e., g(x) → +∞ as x →  + , ϕ, h : R → R are Tperiodic functions with h ∈ L  ([, T], R) and ϕ ∈ C([, T], R), and the sign of the function ϕ is allowed to change for t ∈ [, T].
The study of the problem of periodic solutions to scalar equations with a singularity began with work of Forbat and Huaux [, ], where the singular term in the equations models the restoring force caused by a compressed perfect gas (see [-] and the references therein). In the past years, many works used the methods, such as the approaches of critical point theory [-], the techniques of some fixed point theorems [-], and the approaches of topological degree theory, in particular, of some continuation theorems of Mawhin (see [, -]), to study the existence of positive periodic solutions for some second order ordinary differential equations with singularities. For example, in [], by using a fixed point theorem in cones, the existence of positive periodic solutions to equation (.) was investigated for the conservative case, i.e., f (x) ≡ . But the function ϕ(t) is required to be ϕ(t) ≥  for all t ∈ [, T]. The method of topological degree theory, together with the technique of upper and lower solutions, was first used by Lazer and Solimini in the pioneering paper [] for considering the problem of a periodic solution to a second order differential equations with singularities. Jebelean and Mawhin in [] considered the problem of a p-Laplacian Liénard equation of the form and where f : R → R is continuous, g : R × (, +∞) → R is an L  -Carathéodory function with T-periodic in the first argument, and it is singular at x = , i.e., g(t, x) is unbounded as x →  + . Different from the equation studied in [, ], which is only singular at x = , equation (.) is provided with both singularities at x = +∞ and at x = . In [], Wang further studied the existence of positive periodic solutions for a delay Liénard equation with a singularity of repulsive type In [, ], the following balance condition between the singular force at the origin and at infinity is needed.

Lemma  ([]) Let x be a continuous T-periodic continuously differential function. Then, for any
In order to study the existence of positive periodic solutions to equation (.), we list the following assumptions.
[H  ] The function ϕ(t) satisfies the following conditions: and and Now, we suppose that assumptions [H  ] and [H  ] hold, and we embed equation (.) into the following equation family with a parameter λ ∈ (, ]: and Proof If the conclusion does not hold, then, for each k > k  , there is a function u k ∈ Ω satisfying From the definition of Ω, we see and by using assumption [H  ], Since ϕ + (t) ≥  and ϕ -(t) ≥  for all t ∈ [, T], it follows from the integral mean value theorem that there is a point ξ ∈ [, T] such that which together with (.) yields On the other hand, by multiplying equation (.) with u k (t), and integrating it over the interval [, T], we obtain which together with the fact of g(x) >  for all x >  gives Substituting (.) and (.) into the above formula,

which is determined by assumption [H  ]. This gives
i.e., By the definition of k  , we see from (.) that (.) contradicts (.). This contradiction implies that the conclusion of Lemma  is true.

Theorem  Assume that
This implies u ∈ Ω. So by using Lemma  that there is a point t  ∈ [, T] such that and then Integrating (.) over the interval [, T], we have Since g(x) → +∞ as x →  + , we see from (.) that there is a point t  ∈ [, T] such that where γ < k * M is a positive constant, which is independent of λ ∈ (, ]. Similar to the proof of (.), we have which results in and then by (.), we have It follows from (.) that i.e., Now, if u attains its maximum over [, T] at t  ∈ [, T], then u (t  ) =  and we deduce from (.) that From (.), we see that It follows from (.) and (.) that which yields the estimate From (.) and (.), we get which gives This implies that condition  and condition  of Lemma  are satisfied. Also, we can deduce from Remark  that g(c) -φc +h > , for c ∈ (, m  ] and g(c) -φc +h < , for c ∈ [m  , +∞), which results in g(m  ) -φm  +h g(m  ) -φm  +h < .
So condition  of Lemma  holds. By using Lemma , we see that equation (.) has at least one positive T-periodic solution. The proof is complete.
Let us consider the equation