Resonant systems with periodic nonlinearity
by
A. Canada
Department of Mathematical Analysis, University of Granada, Spain.
Coauthors: D. Ruiz, Department of Mathematical Analysis, University of Granada, Spain.

We report some recent results on boundary value problems for systems of equations whose nonlinear part involves periodic functions and such that the linear part has a one-dimensional solution space. This is the case, for example, of linearly coupled pendulum and the systems of equations arising in the theory of Josephson multipoint junctions. It should be mentioned here that since the corresponding linear problem has a one-dimensional solution space and we have more than one nonlinear term, this may cause some serious difficulties in order to study the range of the associated nonlinear operators. We shall deal with the existence and multiplicity of solutions using various methods of Nonlinear Analysis. First, we apply the alternative method (Liapunov-Schmidt reduction) which allows the presence of friction terms but requires some strong restrictions on the number of equations and on the nonlinearities. Afterwards, we shall consider the problem using methods of critical point theory. This will allow to deal with another situations where, for example, an arbitrary number of equations may be considered. Generally speaking, we prove that the presence of nontrivial nonlinearities causes a strict enlargement of the set of external disturbances for which the problem has solution, with respect to the linear case.

Date received: May 21, 2003

Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # calh-61.