Loss of positivity in a nonlinear second order ordinary differential equations

• DOI: 10.1016/j.na.2008.10.117
• Publication Date: 2008-11-14T12:14:50Z
• Authors: Hongxia Fan; Ruyun Ma
• Keywords: Kernel (algebra); Value (mathematics)

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 α and β are positive parameters, 0≤ξ<η≤1. We prove that solutions lose positivity as the parameter α or β 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.