Existence of periodic solution for fourth-order generalized neutral p-Laplacian differential equation with attractive and repulsive singularities

Abstract

In this paper, we investigate the existence of a positive periodic solution for the following fourth-order p-Laplacian generalized neutral differential equation with attractive and repulsive singularities: \t\t\t(φp(u(t)−c(t)u(t−δ(t)))″)″+f(u(t))u′(t)+g(t,u(t))=k(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}$$\\bigl(\\varphi_{p}\\bigl(u(t)-c(t)u\\bigl(t-\\delta(t)\\bigr) \\bigr)''\\bigr)''+f \\bigl(u(t)\\bigr)u'(t)+g\\bigl(t,u(t)\\bigr)=k(t), $$\\end{document} where g has a singularity at the origin. The novelty of the present article is that we show that attractive and repulsive singularities enable the achievement of a new existence criterion of a positive periodic solution through an application of coincidence degree theory. Recent results in the literature are generalized and significantly improved.