Application of argument principle for determination of number of solutions to nonlinear integral equations related to problems with free phase

The subject of investigation is a class of nonlinear integral equations of the Hammerstein type with separated argument (phase) of unknown function, that arise particularly in the phase optimization problems. Earlier their solutions were represented by finite number of complex parameters that were determined from a system of transcendental equations. For determination of the number of solutions to this system the argument principle is used. Numerical results are presented for the particular case related to the Fourier transform.