Abstract
This paper is devoted to an investigation of the existence of a positive periodic solution for the following singular Liénard equation: x″+f(x(t))x′(t)+a(t)x=b(t)xα+e(t),\\documentclass[12pt]{minimal}\t\t\t\t\\usepackage{amsmath}\t\t\t\t\\usepackage{wasysym}\t\t\t\t\\usepackage{amsfonts}\t\t\t\t\\usepackage{amssymb}\t\t\t\t\\usepackage{amsbsy}\t\t\t\t\\usepackage{mathrsfs}\t\t\t\t\\usepackage{upgreek}\t\t\t\t\\setlength{\\oddsidemargin}{-69pt}\t\t\t\t\\begin{document}$$ x''+f\\bigl(x(t)\\bigr)x'(t)+a(t)x= \\frac{b(t)}{x^{\\alpha }}+e(t), $$\\end{document} 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.
Highlights
1 Introduction The main purpose of this paper is to consider the existence of a periodic solution for the Liénard equation with weak and strong singularities of repulsive type, b(t)
The Poincaré–Birkhoff twist theorem [2,3,4], Schauder’s fixed point theorem [5,6,7,8], the Leray–Schauder alternative principle [9,10,11], coincidence degree theory [12,13,14,15],the Krasnoselskii fixed point theorem in cones [16, 17] and Leray–Schauder degree theory [18, 19] have been employed to discuss the existence of a positive periodic solution of singular equations
We study the existence of a positive periodic solution for Eq (1.1) with strong singularity (i.e. α ≥ 1) if conditions (H1) and (H2) are satisfied
Summary
The Poincaré–Birkhoff twist theorem [2,3,4], Schauder’s fixed point theorem [5,6,7,8], the Leray–Schauder alternative principle [9,10,11], coincidence degree theory [12,13,14,15],the Krasnoselskii fixed point theorem in cones [16, 17] and Leray–Schauder degree theory [18, 19] have been employed to discuss the existence of a positive periodic solution of singular equations Among these papers, there have been published some results on Eq (1.2) (see [5, 6, 8, 10, 17]). Chu et al [10] in 2007 discussed the existence of a positive periodic solution for Eq (1.2) if the external force e(t) ≥ 0 and a
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