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: ${\left({\phi }_p{\left(u\right(t)-c(t\left)u(t-\delta (t\left)\right)\right)}^{″}\right)}^{″}+f\left(u\right(t\left)\right)u^{\text{'}}\left(t\right)+g(t,u(t\left)\right)=k\left(t\right),$ 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.

Keywords: Attractive and repulsive singular; Fourth-order; Generalized neutral operator; Positive periodic solution; p-Laplacian.