Positive periodic solution for indefinite singular Li\xe9nard equation with p-Laplacian

Abstract

The efficient conditions guaranteeing the existence of positive T-periodic solution to the p-Laplacian–Liénard equation \t\t\t(ϕp(x′(t)))′+f(x(t))x′(t)+α1(t)g(x(t))=α2(t)xμ(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}$$\\bigl(\\phi _{p}\\bigl(x'(t)\\bigr) \\bigr)'+f \\bigl(x(t)\\bigr)x'(t)+\\alpha _{1}(t)g\\bigl(x(t)\\bigr)= \\frac{ \\alpha _{2}(t)}{x^{\\mu }(t)}, $$\\end{document} are established in this paper. Here phi _{p}(s)=|s|^{p-2}s, p>1, alpha _{1},alpha _{2}in L([0,T],{R}) , fin C({R}_{+},{R}) ({R} _{+} stands for positive real numbers) with a singularity at x=0, g(x) is continuous on (0;+infty ), μ is a constant with mu >0, the signs of alpha _{1} and alpha _{2} are allowed to change. The approach is based on the continuation theorem for p-Laplacian-like nonlinear systems obtained by Manásevich and Mawhin in (J. Differ. Equ. 145:367–393, 1998).