Purpose of the work — substantiation of the Krylov-Bogolyubov averaging method for nonlinear abstract hyperbolic equations with rapidly time-oscillating data and initial conditions (Cauchy problem) in a Banach space.

Methods. The Cauchy problem for an abstract hyperbolic equation with a large parameter, the role of which is played by a high frequency of data oscillations, is written in the form of an integral equation and, thus, a nonlinear integral operator is constructed based on the initial differential equation with initial conditions, depending on the large parameter and operating in the corresponding Banach space. The principle of mapping compression is applied to this operator (or rather, if necessary, to its square).

Results. The nonlinear hyperbolic equations with initial conditions in a Banach space considered in this paper serve as abstract analogues of classical semilinear hyperbolic equations with initial boundary conditions. The coefficient of the main stationary linear part of the abstract equation is the square of an unbounded operator generating a strongly continuous group, and the nonlinear part — is in a certain sense subordinate to the specified linear part. In the proposal of solvability of the averaged problem, it is proved that for large values of a high oscillation frequency, the perturbed problem is also solvable in a certain neighborhood of the solution of the averaged problem and the indicated solutions are asymptotically close.

Conclusion. It is established that the classical Krylov-Bogolyubov averaging method is applicable to nonlinear abstract hyperbolic equations.