Boundary value problems for highly nonlinear inclusions governed by non-surjective Phi-Laplacians

Combining fixed point theorems with the method of lower and upper solutions, we get the existence of solutions to the following nonlinear differential inclusion: \[ (D(x(t))\Phi(x'(t)))' \in G(t,x(t),x'(t)) \ \ \mbox{a.e. } t\in I=[0,T], \] satisfying various nonlinear boundary conditions, covering Dirichlet, Neumann and periodic problems. Here $\Phi$ is a non-surjective homeomorphism and $D$ is a generic positive continuous function.