Open Access Research Article

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

VS Serov1*, HW Schüermann2 and E Svetogorova2

Author Affiliations

1 Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland

2 Department of Physics, University of Osnabrück, Barbarastraße 7, D-49069 Osnabrück, Germany

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Fixed Point Theory and Applications 2010, 2010:439682  doi:10.1155/2010/439682

The electronic version of this article is the complete one and can be found online at:

Received:29 August 2009
Accepted:23 April 2010
Published:30 May 2010

© 2010 The Author(s).

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.


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.

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