Existence of periodic solutions of a Li\xe9nard equation with a singularity of repulsive type

Abstract

In this paper, the problem of positive periodic solutions is studied for the Liénard equation with a singularity of repulsive type, \t\t\tx″+f(x)x′−α(t)xμ=h(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}$$ x''+f(x)x'-\\frac{\\alpha(t)}{x^{\\mu}}=h(t), $$\\end{document} where f:(0,+infty)rightarrow R is continuous, α, h are continuous with T-periodic and alpha(t)ge0 for all tin R, mu in(0,+infty) is a constant. By means of a Manásevich-Mawhin’s continuation theorem, a sufficient and necessary condition is obtained for the existence of positive T-periodic solutions of the equation. The interesting point is that the weak singularity of restoring force frac{alpha(t)}{x^{mu}} at x=0 is allowed and f may have a singularity at x=0.