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

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 \(f:[0,\infty)\rightarrow R\) is an arbitrary continuous function, \(g\in C((0,+\infty), (0,+\infty))\), and \(g(x)\) is singular of repulsive type at \(x=0\), i.e., \(g(x)\rightarrow+\infty\) as \(x\rightarrow 0^{+}\), \(\varphi,h: R\rightarrow R\) are T-periodic functions with \(h\in L^{2}([0,T], R)\) and \(\varphi\in C([0,T], R)\), and the sign of the function φ is allowed to change for \(t\in[0,T]\).

The study of the problem of periodic solutions to scalar equations with a singularity began with work of Forbat and Huaux [1, 2], where the singular term in the equations models the restoring force caused by a compressed perfect gas (see [3–6] and the references therein). In the past years, many works used the methods, such as the approaches of critical point theory [7–12], the techniques of some fixed point theorems [13–15], and the approaches of topological degree theory, in particular, of some continuation theorems of Mawhin (see [6, 16–22]), to study the existence of positive periodic solutions for some second order ordinary differential equations with singularities. For example, in [15], by using a fixed point theorem in cones, the existence of positive periodic solutions to equation (1.1) was investigated for the conservative case, i.e., \(f(x)\equiv0\). But the function \(\varphi(t)\) is required to be \(\varphi(t)\ge0\) for all \(t\in [0,T]\). 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 [18] for considering the problem of a periodic solution to a second order differential equations with singularities. Jebelean and Mawhin in [6] considered the problem of a p-Laplacian Liénard equation of the form and where \(p>1\) is a constant, \(f: [0,+\infty)\rightarrow R\) is an arbitrary continuous function, \(h: R\rightarrow R\) is a T-periodic function with \(h\in L^{\infty}([0,T],R)\), \(g: (0,+\infty)\rightarrow (0,+\infty)\) is continuous, \(g(x)\rightarrow+\infty\) as \(x\rightarrow 0^{+}\). They extended the results of Lazer and Solimini in [16] to equation (1.2) and equation (1.3). For equation (1.3), the crucial condition is that the function \(g(x)\) is bounded, which means that equation (1.3) is not singular at \(x=+\infty\).

By using a continuation theorem of Mawhin, Zhang in [18] studied the problem of periodic solutions of the Liénard equation with a singularity of repulsive type, where \(f: {R}\rightarrow{R}\) is continuous, \(g: {R}\times(0,+\infty )\rightarrow{R}\) is an \(L^{2}\)-Carathéodory function with T-periodic in the first argument, and it is singular at \(x=0\), i.e., \(g(t,x)\) is unbounded as \(x\rightarrow0^{+}\). Different from the equation studied in [6, 16], which is only singular at \(x=0\), equation (1.4) is provided with both singularities at \(x=+\infty\) and at \(x=0\). In [19], Wang further studied the existence of positive periodic solutions for a delay Liénard equation with a singularity of repulsive type In [18, 19], the following balance condition between the singular force at the origin and at infinity is needed.

(h₁) There exist constants \(0< D_{1}< D_{2}\) such that if x is a positive continuous T-periodic function satisfying then From the proof of [18, 19], we see that the balance condition (h₁) is crucial for estimating a priori bounds of periodic solutions. Now, the question is how to investigate the existence of positive periodic solutions for the equations like equation (1.4) or equation (1.5) without the balance condition (h₁).

[H₁] The function \(\varphi(t)\) satisfies the following conditions: and

Now, we suppose that assumptions [H₁] and [H₂] hold, and we embed equation (1.1) into the following equation family with a parameter \(\lambda\in(0,1]\): Let and where \(A>0\) is a constant determined by assumption [H₂]. Clearly, \(M_{0}\) and \(A_{0}\) are all independent of \((\lambda,x)\in(0,1]\times \varOmega \). Let \(M>0\) be determined by assumption [H₂], then there is a positive integer \(k_{0}\) such that

Let us consider the equation where \(f : [0,+\infty)\rightarrow R\) is an arbitrary continuous function, \(\varphi,h: R\rightarrow R\) are T-periodic functions with \(h\in L^{1}([0,T], R)\) and \(\varphi\in C([0,T], R)\), and the sign of the function φ is allowed to change for \(t\in[0,T]\), \(\gamma \ge1\) is a constant. Corresponding to equation (1.1), \(g(x)=\frac{1}{x^{\gamma}}\). For this case, \(g(x)\rightarrow+\infty\) as \(x\rightarrow0^{+}\), and assumptions [H₂]-[H₃] are satisfied. Thus, by using Theorem 1, we have the following results.