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Weak and strong singularities problems to Liénard equation
Boundary Value Problems ( IF 1.7 ) Pub Date : 2020-08-28 , DOI: 10.1186/s13661-020-01441-1
Yun Xin , Guixin Hu

This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation: $$ x''+f\bigl(x(t)\bigr)x'(t)+a(t)x= \frac{b(t)}{x^{\alpha }}+e(t), $$ where the external force $e(t)$ may change sign, α is a constant and $\alpha >0$ . 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.

中文翻译:

Liénard方程的弱奇异性和强奇异性问题

本文致力于研究以下奇异Liénard方程的正周期解的存在:$$ x''+ f \ bigl(x(t)\ bigr)x'(t)+ a(t)x = \ frac {b(t)} {x ^ {\ alpha}} + e(t),$$,其中外力$ e(t)$可能会改变符号,α为常数,$ \ alpha> 0 $ 。本文的新颖之处在于,我们首次证明了弱奇异性和强奇异性通过应用Manásevich-Mawhin连续定理可以实现正周期解的新存在准则。文献中的最新结果得到了概括和显着改进,我们给出了该方程周期解的存在区间。最后,
更新日期:2020-08-28
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