Eigenvalue criteria for existence of positive solutions to nonlinear boundary value problems

Abstract

Criteria are established for existence of positive solutions to the BVP Lu = f( t, u), αu(0)− βu′(0) = 0 = γu(1) + ( δu′(1), where Lu := −( pu′)′ + qu. Here p ϵ C 1[0, 1], p > 0, q ϵ C[0,1], q ≥ 0. Conditions are given in terms of the relative behavior of the quotient ( f(t,u)) u for u near 0 and ∞ with respect to the smallest positive eigenvalue of Lu = λu satisfying the boundary conditions.