Skip to content


  • Research Article
  • Open Access

Application of the Banach Fixed-Point Theorem to the Scattering Problem at a Nonlinear Three-Layer Structure with Absorption

Fixed Point Theory and Applications20102010:439682

  • Received: 29 August 2009
  • Accepted: 23 April 2010
  • Published:


A method based on the Banach fixed-point theorem is proposed for obtaining certain solutions (TE-polarized electromagnetic waves) of the Helmholtz equation describing the reflection and transmission of a plane monochromatic wave at a nonlinear lossy dielectric film situated between two lossless linear semiinfinite media. All three media are assumed to be nonmagnetic and isotropic. The permittivity of the film is modelled by a continuously differentiable function of the transverse coordinate with a saturating Kerr nonlinearity. It is shown that the solution of the Helmholtz equation exists in form of a uniformly convergent sequence of iterations of the equivalent Volterra integral equation. Numerical results are presented.


  • Integral Equation
  • Electromagnetic Wave
  • Differential Geometry
  • Differentiable Function
  • Computational Biology

Publisher note

To access the full article, please see PDF

Authors’ Affiliations

Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu, 90014, Finland
Department of Physics, University of Osnabrück, Barbarastraße 7, D-49069 Osnabrück, Germany


© V. S. Serov et al. 2010

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.