Lower and Upper Solutions Method for Nonlinear Second-order Differential Equations Involving a $\Phi-$Laplacian Operator

Abstract

In this paper, we consider the following nonlinear second-order differential equations: $-(\Phi(u'(t)))' = f (t, u(t), u'(t)) + \Xi (u(t)) \text { a.e on } \Omega = [0, T]$ with a discontinuous perturbation and multivalued boundary conditions. The nonlinear differential operator is not necessarily homogeneous and incorporates as a special case the one-dimensional p-Laplacian. By combining lower and upper solutions method, a fixed point theorem for multifunction and theory of monotone operators, we show the existence of solutions and existence of extremal solutions in the order interval $[\alpha, \beta]$ where $\alpha$ and $\beta$ are assumed respectively an ordered pair of lower and upper solutions of the problem. Moreover, we show that our method of proof also applies to the periodic problem.