Journal of Fixed Point Theory and Applications

, Volume 4, Issue 1, pp 57–75

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

Research Article

DOI: 10.1007/s11784-008-0072-7

Cite this article as:
Bereanu, C. & Mawhin, J. J. fixed point theory appl. (2008) 4: 57. doi:10.1007/s11784-008-0072-7


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.

Mathematics Subject Classification (2000).



Singular \(\phi\)-laplacianboundary value problems for systemsLeray–Schauder degreespecial relativity

Copyright information

© Birkhaeuser 2008

Authors and Affiliations

  1. 1.Département de MathématiqueUniversité Catholique de LouvainLouvain-la-NeuveBelgium