Loss of positivity in a nonlinear second order ordinary differential equations

Abstract

We consider existence of nonzero solutions to the following boundary value problem u ″ ( t ) + f ( t , u ) = 0 , t ∈ ( 0 , 1 ) , u ′ ( 0 ) = α u ( ξ ) , u ′ ( 1 ) + β u ( 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.