Strongly nonlinear multivalued, periodic problems with maximal monotone terms

Abstract

In this paper we study periodic, nonlinear, second-order differential inclusions, driven by the differential operator $$ x\rightarrow (\alpha(x)\|x'\|^{p-2}x')' $$ and involving a maximal monotone term $A$ and a multivalued nonlinearity $F(t,x)$ which satisfies the Hartman condition. We do not assume that $domA$ is all of $\mathbb{R}^{N}$, and so our formulation incorporates variational inequalities. Then we obtain partial generalizations. First, we allow $F$ to depend on $x'$, but for $p=2$ and for the scalar problem ($N=1$). Second, we assume a general multivalued, nonlinear differential operator $x\rightarrow \alpha(x,x')'$; the nonlinearity $F$ depends also on $x'$, but the boundary conditions are Dirichlet. Our methods are based on notions and techniques from multivalued analysis and from the theory of operators of monotone type.