Boundary Value Problems for Highly Nonlinear Inclusions Governed by Non-surjective Φ-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 Φ is a non-surjective homeomorphism and D is a generic positive continuous function.