Journal of Inequalities and Applications
Volume 2006 (2006), Article ID 42908, 21 pages
doi:10.1155/JIA/2006/42908
Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part
1Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, Gdańsk 80-952, Poland
2Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Umultowska 87, Poznań 61-614, Poland
Received 8 February 2004; Revised 7 March 2004; Accepted 12 March 2004
Copyright © 2006 Andrei Borisovich and Wacław Marzantowicz. 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
We study nonlinear ODE problems in the complex Euclidean space,
with the right-hand side being polynomial with nonconstant
periodic coefficients. As the coefficients functions, we admit
only functions with vanishing Fourier coefficients for negative
indices. This leads to an existence theorem which relates the
number of solutions with the number of zeros of the averaged
right-hand side polynomial. A priori estimates of the norms of
solutions are based on the Wirtinger-Poincaré-type inequality.
The proof of existence theorem is based on the continuation method
of Krasnosielski et al., Mawhin et al., and the Leray-Schauder
degree. We give a few applications on the complex Riccati equation
and some others.