Fixed Point Theory and Applications
Volume 2010 (2010), Article ID 439682, 18 pages
Research Article

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

1Department of Mathematical Sciences, University of Oulu, P.O. Box 3000, Oulu 90014, Finland
2Department of Physics, University of Osnabrück, Barbarastraße 7, D-49069 Osnabrück, Germany

Received 29 August 2009; Accepted 23 April 2010

Academic Editor: Massimo Furi

Copyright © 2010 V. S. Serov et al. 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.