On some boundary value problems for second order nonlinear differential equations

We investigate two boundary value problems for the second order differential equation with p-Laplacian (a(t)P(x′))′ = b(t)F(x), t in I = [0,1), where a, b are continuous positive functions on I. We give necessary and sufficient conditions which guarantee the existence of a unique (or at least one) positive solution, satisfying one of the following two boundary conditions: i) x(0) = c > 0, lim x(t) = 0; ii) x′(0) = d < 0, lim x(t) = 0.