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

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 φφ and ψψ 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][ψ,φ] and of extremal solutions in [ψ,φ][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.