Journal of Applied Mathematics and Stochastic Analysis
Volume 13 (2000), Issue 1, Pages 41-50
doi:10.1155/S1048953300000058
Periodic solutions of systems with asymptotically even
nonlinearities
1Johann Wolfgang Goethe-Universität, Fachbereich Mathernatik, Frankfurt D-60054, Germany
2Institute of Information Transmission Problems, 19 Bolshoi Karetny Lane, Moscow 101447, Russia
Received 1 July 1998; Revised 1 January 1999
Copyright © 2000 Peter E. Kloeden and Alexander M. Krasnosel'skii. 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.
Abstract
New conditions of solvability based on a general theorem on the calculation of the index at infinity for vector fields that have degenerate principal
linear part as well as degenerate “next order” terms are obtained for the
2π-periodic problem for the scalar equation x″+n2x=g(|x|)+f(t,x)+b(t) with bounded g(u) and f(t,x)→0 as |x|→0. The result is also applied to the solvability of a two-point boundary value problem and to resonant problems for equations arising in control theory.