On the existence and nonexistence of positive solutions for nonlinear Sturm–Liouville boundary value problems ✩

In this paper the existence and nonexistence results of positive solutions are obtained for Sturm– Liouville boundary value problem − p(x)u + q(x)u = f (x,u), x ∈ (0, 1), au(0) −bp(0)u (0) = 0, cu(1) +dp(1)u (1) = 0, where p ∈ C1[0, 1], q ∈ C[0, 1], p(x) > 0, q(x) 0 for x ∈ [0, 1], f ∈ C([0, 1]×R+), a, b, c, d 0 are constants and satisfy (a +b)(c +d)>0. The discussion is based on the positivity estimation for the Green’s function of associated linear boundary value problem and the fixed point index theory in cones.  2004 Elsevier Inc. All rights reserved.