The Dirichlet BVP for the second order nonlinear ordinary differential equation at resonance

Abstract

Efficient su±cient conditions are established for the solvability of the Dirichlet problem

                 u"(t) = p(t)u(t) + f(t, u(t)) + h(t) for a ≤ t ≤ b,
                                  u(a) = 0, u(b) = 0,

where h, p ∈ L([a, b];R) and f ∈ K([a, b] x R;R); in the case where the linear problem

                          u"(t) = p(t)u(t), u(a) = 0, u(b) = 0

has nontrivial solutions.