Limit Properties of Solutions of Singular Second-Order Differential Equations

We discuss the properties of the differential equation u ''(t) = (a/t)u'(t) + f(t, u(t), u'(t)), a.e. on (0, T], where a is an element of R\{0}, and f satisfies the L(p)-Caratheodory conditions on [0, T] x R(2) for some p > 1. A full description of the asymptotic behavior for t -> 0+ of functions u satisfying the equation a.e. on (0, T] is given. We also describe the structure of boundary conditions which are necessary and sufficient for u to be at least in C(1)[0, T]. As an application of the theory, new existence and/or uniqueness results for solutions of periodic boundary value problems are shown.