Existence of positive solutions for periodic boundary value problem with sign-changing Green’s function

Abstract

We prove the existence of positive solutions for the boundary value problem

$$\begin{aligned} \left\{ \begin{array}{ll} y^{\prime \prime }+m^{2}y=\lambda g(t)f(y), &{}\quad 0\le t\le 2\pi , \\ y(0)=y(2\pi ), &{}\quad y^{\prime }(0)=y^{\prime }(2\pi ), \end{array} \right. \end{aligned}$$

for certain range of the parameter $\lambda >0$, where $m\in (1/2,1/2+\varepsilon )$ with $\varepsilon >0$ small, and f is superlinear or sublinear at $\infty$ with no sign-conditions at 0 assumed.