Existence and Multiplicity of Solutions for a Periodic Hill's Equation with Parametric Dependence and Singularities

Abstract

We deal with the existence and multiplicity of solutions for the periodic boundary value problem $x\text{'}\text{'}\left(t\right)+a\left(t\right)x\left(t\right)=\lambda g\left(t\right)f\left(x\right)+c\left(t\right)$, $x\left(0\right)=x\left(T\right)$, $x\text{'}\left(0\right)=x\text{'}\left(T\right)$, where $\lambda$ is a positive parameter. The function $f:(0,\infty )\to (0,\infty )$ is allowed to be singular, and the related Green's function is nonnegative and can vanish at some points.