Boundary value problems for some nonlinear systems with singular φ-laplacian

Systems of differential equations of the form (phi(u'))' = f(t, u, u') with phi a homeomorphism of the ball B-a subset of R-n on to R-n are considered, under various boundary conditions on a compact interval [0,T]. For nonhomogeneous Cauchy, terminal and some Sturm-Liouville boundary conditions including in particular the Dirichlet-Neumann and Neumann-Dirichlet conditions, existence of a solution is proved for arbitrary continuous right-hand sides f. For Neumann boundary conditions, some restrictions upon f are required, although, for Dirichlet boundary conditions, the restrictions are only upon phi and the boundary values. For periodic boundary conditions, both phi and f have to be suitably restricted. All the boundary value problems considered are reduced to finding a fixed point for a suitable operator in a space of functions, and the Schauder fixed point theorem or Leray-Schauder degree are used. Applications are given to the relativistic motion of a charged particle in some exterior electromagnetic field.