Periodic Solutions of Nonlinear Problems with Positive Oriented Periodic Coefficients

We study nonlinear ODE problems in the complex Euclidean space, with the right hand side being a complex analytic function of the space variable z with nonconstant periodic coefficients in the time variable t. 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 function, and finally gives a theorem of the existence of periodic solution which originates from infinity. The work generalizes and extends previous results of the author, joint with A. Borisovich, for the polynomial case.