Study of Nonlinear Second-Order Differential Inclusion Driven by a Φ−Laplacian Operator Using the Lower and Upper Solutions Method

Abstract

In this paper, we study a second-order differential inclusion under boundary conditions governed by maximal monotone multivalued operators. These boundary conditions incorporate the classical Dirichlet, Neumann, and Sturm–Liouville problems. Our method of study combines the method of lower and upper solutions, the analysis of multivalued functions, and the theory of monotone operators. We show the existence of solutions when the lower solution $\sigma$ and the upper solution $\gamma$ are well ordered. Next, we show how our arguments of proof can be easily exploited to establish the existence of extremal solutions in the functional interval $\left[\sigma ,\gamma \right]$. We also show that our method can be applied to the periodic case.