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

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 C1[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.