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  • Research Article
  • Open Access

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

Journal of Inequalities and Applications20062006:42908

  • Received: 8 February 2004
  • Accepted: 12 March 2004
  • Published:


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.


  • Periodic Solution
  • Euclidean Space
  • Existence Theorem
  • Fourier Coefficient
  • Riccati Equation


Authors’ Affiliations

Institute of Mathematics, University of Gdańsk, ul. Wita Stwosza 57, Gdańsk, 80-952, Poland
Faculty of Mathematics and Computer Science, Adam Mickiewicz University of Poznań, ul. Umultowska 87, Poznań, 61-614, Poland


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© Hindawi Publishing Corporation. 2006

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