On the averaging method for quasilinear hyperbolic equations

A study was made on the averaging method, applied to hiperbolic equations in the Hilbert space. The selected class of equations includes as particular case the system of one-dimensional quasilinear equations with partial derivatives of hyperbolic type with the right part being periodical with respect to time variable. The version of averaging method, not considered earlier in publications, has been substantiated. The analogue of the first main N.N.Bogolyubov theorem about the approximate solution in finite, but as much as is desired long time interval, has been proved. The obtained statements are applied to investigation of induced vibrations of elastic string in nonlinear force field