The method of upper-lower solutions for nonlinear second order differential inclusions

Abstract

In this paper we consider a second order differential inclusion driven by the ordinary p-Laplacian, with a subdifferential term, a discontinuous perturbation and nonlinear boundary value conditions. Assuming the existence of an ordered pair of appropriately defined upper and lower solutions φ and ψ respectively, using truncations and penalization techniques and results from nonlinear and multivalued analysis, we prove the existence of solutions in the order interval [ψ,φ] and of extremal solutions in [ψ,φ]. We show that our problem incorporates the Dirichlet, Neumann and Sturm–Liouville problems. Moreover, we show that our method of proof also applies to the periodic problem.