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


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).

34B15 34B16 34C25 47H11 78A35 


Singular \(\phi\)-laplacian boundary value problems for systems Leray–Schauder degree special relativity 


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Copyright information

© Birkhaeuser 2008

Authors and Affiliations

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

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