The Cauchy problem for an abstract second order ordinary differential equation

We prove the existence and uniqueness of a solution for the Cauchy problem for a linear abstract second order differential equation, obtain its representation, and prove that it is continuously dependent on the time at which the initial conditions are specified. Based on these results, we prove the existence and uniqueness of a solution of the Cauchy problem for a nonlinear abstract second order differential equation. This result is applied to show that the initial-boundary value problem for a nonlinear hyperbolic divergence structure equation has a unique solution.
Bibliography: 49 titles.