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)Phi(p)(x'))' = b(t)F(x), t is an element of I = [0, infinity), 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 (t ->infinity) x(t) = 0; ii) x'(0) = d < 0, lim (t -> infinity) x(t) = O.