Existence of periodic solutions of a Liénard equation with a singularity of repulsive type
 Shiping Lu^{1},
 Yajiao Wang^{1}Email author and
 Yuanzhi Guo^{1}
Received: 5 April 2017
Accepted: 13 June 2017
Published: 27 June 2017
Abstract
Keywords
1 Introduction
2 Preliminary lemmas
Let \(C_{T}=x\in C(\mathbb{R},\mathbb{R}):x(t+T)=x(t)\) for all \(t\in\mathbb{R}\) with the norm defined by \(x_{\infty}=\max_{t\in [0,T]}x(t)\). For any Tperiodic solution \(h(t)\) with \(h \in C_{T}\), \(h_{+}(t)\) and \(h_{}(t)\) is denoted by \(\max\{(h(t),0)\}\) and \(\min\{ (h(t),0)\}\), respectively, and \(\overline{h}=\frac{1}{T}\int ^{T}_{0}h(s)\,ds\). Clearly, \(h(t)=h_{+}(t)h_{}(t)\) for all \(t\in \mathbb{R}\), and \(\overline{h}=\overline{h}_{+}\overline{h}_{}\). Furthermore, \(\\varphi\_{p}:=(\int_{0}^{T}\varphi(t)^{p}\,dt)^{\frac{1}{p}}\), \(p\in[1,+\infty)\), \(\varphi\in C_{T}\).
The following lemma is a corollary of Theorem 3.1 in [36].
Lemma 2.1
 1.for each \(\lambda\in(0,1]\), each possible positive Tperiodic solution x to the equationsatisfies the inequalities \(M_{0}< x(t)< M_{1}\) and \(x'(t)< M_{2}\) for all \(t\in[0,T]\);$$ u''+\lambda f(u)u'\lambda \frac{\alpha(t)}{u^{\mu}}=\lambda h(t), $$
 2.each possible solution \(x\in(0,+\infty)\) to the equationsatisfies the inequality \(M_{0} < x < M_{1}\);$$ \frac{\overline{\alpha}}{x^{\mu}}+\overline{h}=0 $$
 3.the inequality$$ \biggl(\frac{\overline{\alpha}}{M_{0}^{\mu}}+\overline{h}\biggr) \biggl(\frac{\overline {\alpha}}{M_{1}^{\mu}}+ \overline{h}\biggr) < 0 $$
Lemma 2.2
[29]
 (H_{1}):

\(\lim_{x\rightarrow0^{+}}\int_{x}^{1}f(s)\,ds=+\infty\).
 (H_{2}):

\((\frac{\overline{\alpha}}{\overline{h}})^{\frac{1}{\mu }}>T^{\frac{1}{2}}[\frac{T}{\pi}\h\_{2} +(T^{\frac{1}{2}}(\frac{\overline{\alpha}}{\overline{h}})^{\frac{1}{\mu }}\h\_{2})^{\frac{1}{2}}]\).
Remark 2.1
3 Main results
Theorem 3.1
If (H_{1}) holds, then equation (1.6) has a positive Tperiodic solution if and only if \(\overline{h}<0\).
Proof
Theorem 3.2
Suppose that \(\overline {h}<0\) and (H_{2}) holds. Then equation (1.6) has at least one positive Tperiodic solution.
Proof
Example 3.1
Example 3.2
Remark
The above two examples can neither be studied by using the results in [31, 32, 34] and [35], since \(f(x)\) in (3.19) and in (3.20) are all singular at \(x=0\), nor be studied by using the results in [33], since the restoring force terms of \(\frac{\sin^{2}t}{x^{\frac{2}{3}}}\) in (3.19) and \(\frac{\sin^{2}8t}{x^{\frac{3}{4}}}\) in (3.20) have weak singularities at \(x=0\).
Declarations
Acknowledgements
The authors thank the referees for valuable comments. This research is supported by the NSF of China (No. 11271197).
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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