Second-order differential equation with indefinite and repulsive singularities

TL;DR

This paper studies a second-order differential equation with both indefinite and repulsive singularities, establishing conditions for positive periodic solutions using Green's function and fixed point theory, and providing numerical examples.

Contribution

First analysis of differential equations with both indefinite and repulsive singularities, extending existing results with new sufficient conditions for solutions.

Findings

Established existence of positive periodic solutions under new conditions

Extended previous results on differential equations with singularities

Provided numerical simulations illustrating theoretical results

Abstract

This paper concerns a second-order differential equation with indefinite and repulsive singularities. It is the first time to study differential equation containing both indefinite and repulsive singularities simultaneously. A set of sufficient conditions are obtained for the existence of positive periodic solutions. The theoretical underpinnings of this paper are the positivity of Green's function and fixed point theorem in cones. Our results improve and extend the results in previous literatures. Finally, three examples and their numerical simulations (phase diagrams and time diagrams of periodic solutions) are given to show the effectiveness of our conclusions.