Nonlinear perturbations of some non-invertible differential operators

We consider perturbations, depending on a small parameter lambda, of a non-invertible differential operator having a nonnegative spectrum. Given a pair of lower and upper solutions, belonging to the kernel of the differential operator, without any prescribed order, we prove the existence of a solution, when lambda is sufficiently small. Our method of proof has the advantage of permitting a uniform choice of lambda for a whole class of functions. Applications are given in a variety of situations, ranging from ODE problems to equations of parabolic type, or involving the p-Laplacian operator.