The theory of a transverse feedback system with a nonlinear regime of injection oscillations damping in circular accelerators and colliders is described. For this feedback loop the dependence between the kick value in the deflector and the beam deviation at the pick-up location is a nonlinear function. The beam dynamic nonlinear equation of the transverse coherent motion for deviation from the closed orbit has been solved using the approximation procedure based on the Krylov-Bogoliubov method for nonlinear differential equations. Approximate expressions for damping time and beam oscillation amplitude are analysed for different nonlinear transfer functions.