Loss of positivity in a nonlinear second order ordinary differential equations

We consider existence of nonzero solutions to the following boundary value problem u″(t)+f(t,u)=0,t∈(0,1),u′(0)=αu(ξ),u′(1)+βu(η)=0, where αα and ββ are positive parameters, 0≤ξ<η≤10≤ξ<η≤1. We prove that solutions lose positivity as the parameter αα or ββ increases. In particular, we study problems where the associated integral equation has a kernel that changes sign. The proof is based on the fixed point theorem in cones.