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

Andrei Borisovich1 and Wacław Marzantowicz2

 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

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.