Boundary-value problems for strongly nonlinear multivalued equations involving different $\Phi$-Laplacians

Abstract

We investigate the existence of solutions to the scalar differential inclusion $$ \qquad (D(x(t))\Phi(x'(t)))' \in G(t,x(t),x'(t)) \ \ \mbox{a.e. } t\in I=[0,T], $$ where $D(x)$ is a positive and continuous function, $G(t,x,x')$ is a Carathéodory multifunction and the increasing homeomorphism $\Phi$ can have a bounded domain of the type $(-a,a)$ or it can be the p-Laplacian operator. Using fixed-point techniques combined, in some cases, with the method of lower and upper solutions, we prove the existence of solutions satisfying various boundary conditions.