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Existence of Positive Periodic Solutions for the Liénard Differential Equations with Weakly Repulsive Singularity

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Abstract

In this paper we discuss the existence of positive T-periodic solutions for the following second order differential equation

$$\ddot{x}+f(x)\dot{x}+g(x)=c(t),$$

where f, g:ℝ+=(0,+∞)→ℝ are continuous functions, g has a repulsive singularity at the origin and c(t) is a continuous T-periodic function. The novelty of the present article is that for the first time we show that a weakly repulsive singularity enables the achievement of a new existence criterion of positive T-periodic solutions through a basic application of the coincidence degree theory. Recent results in the literature are generalized and significantly improved.

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Authors and Affiliations

  1. School of Mathematical Sciences, Beijing Normal University, Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing, 100875, People’s Republic of China

    Ziheng Zhang & Rong Yuan

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  1. Ziheng Zhang
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  2. Rong Yuan
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Correspondence to Ziheng Zhang.

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Zhang, Z., Yuan, R. Existence of Positive Periodic Solutions for the Liénard Differential Equations with Weakly Repulsive Singularity. Acta Appl Math 111, 171–178 (2010). https://doi.org/10.1007/s10440-009-9538-x

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  • DOI: https://doi.org/10.1007/s10440-009-9538-x

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