  We consider the Cauchy problem for a second-order nonlinear evolution
equation in a Hilbert space. This equation represents the abstract
generalization of the Ball integro-differential equation. The general nonlinear
case with respect to terms of the equation which include a square of a norm of
a gradient is considered. A three-layer semi-discrete scheme is proposed in
order to find an approximate solution. In this scheme, the approximation of
nonlinear terms that are dependent on the gradient is carried out by using an
integral mean. We show that the solution of the nonlinear discrete problem and
its corresponding difference analogue of a first-order derivative is uniformly
bounded. For the solution of the corresponding linear discrete problem, it is
obtained high-order a priori estimates by using two-variable Chebyshev
polynomials. Based on these estimates we prove the stability of the nonlinear
discrete problem. For smooth solutions, we provide error estimates for the
approximate solution. An iteration method is applied in order to find an
approximate solution for each temporal step. The convergence of the iteration
process is proved.
