Periodic solutions of nonlinear problems with holomorphic type of the nonlinearity

We study nonlinear ODE problems in the complex plane, with the right hand side f(u, t) being complex analytic function of the space variable u with the coefficients being nonconstant periodic functions in the time variable t and having non-zero Fourier coefficients for non-negative indices only. Assuming first that , where h is holomorphic in the two variables, we drop out the local a priori estimate for the existence of T-periodic solutions of previous works. We prove that there exists r_o > 0, not depending of T, such that for every r < r₀ the equation has periodic solution. Next we show that the same still holds for the mentioned above class of nonlinear problems. This improves previous author results, also joint with A. Borisovich.