Open Access

Positive oriented periodic solutions of the first-order complex ODE with polynomial nonlinear part

Journal of Inequalities and Applications20062006:42908

DOI: 10.1155/JIA/2006/42908

Received: 8 February 2004

Accepted: 12 March 2004

Published: 16 January 2006

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.

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Authors’ Affiliations

(1)
Institute of Mathematics, University of Gdańsk
(2)
Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań

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© Hindawi Publishing Corporation. 2006

This article is published under license to BioMed Central Ltd. 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.