Boundary value problems for some nonlinear systems with singular $\phi$-laplacian

Abstract.

Systems of differential equations of the form

$$(\phi(u^\prime))^\prime = f(t, u, u^\prime)$$

with $\phi$ a homeomorphism of the ball $B_a \subset {\mathbb{R}^{n}} \rm\,\,{onto}\,\, {\mathbb{R}^{n}}$ are considered, under various boundary conditions on a compact interval [0, T]. For non-homogeneous 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.