Abstract
In this paper, we consider a p-Laplacian singular Rayleigh equation with time-dependent deviating argument \t\t\t(φp(x′(t)))′+f(t,x′(t))+g(t,x(t−σ(t)))=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}$$\\bigl(\\varphi_{p}\\bigl(x'(t)\\bigr)\\bigr)'+f \\bigl(t,x'(t)\\bigr)+g\\bigl(t,x\\bigl(t-\\sigma(t)\\bigr)\\bigr)=e(t), $$\\end{document} where g has an attractive singularity at x=0. Using the Manásevich–Mawhin continuation theorem, we prove that the equation has at least one T-periodic solution.
Highlights
In the past years, researchers paid much attention to investigating the problem of periodic solutions for second-order equations with singularities
In 2014, Wang [2] investigated the existence of positive periodic solutions of the following Liénard equation with singularity and deviating argument: x (t) + f x(t) x (t) + g t, x(t – σ ) = 0, (1.2)
In this paper, applying the Manásevich–Mawhin continuation theorem, we consider the existence of positive periodic solutions for the following Rayleigh equation with attractive singularity and time-dependent deviating argument: φp x (t) + f t, x (t) + g t, x t – σ (t) = e(t), (1.3)
Summary
Researchers paid much attention to investigating the problem of periodic solutions for second-order equations with singularities (see [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16]). In 2014, Wang [2] investigated the existence of positive periodic solutions of the following Liénard equation with singularity and deviating argument: x (t) + f x(t) x (t) + g t, x(t – σ ) = 0, (1.2)
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