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.

翻译：我们考虑Hilbert 空间中二阶非线性进化方程式的突变问题。 这个方程式代表了Ball Integro- 差异方程式的抽象概括化。 考虑了包含一个梯度标准方形的方程式条件的普通非线性非线性案例。 提出了一个三层半分解方案, 以找到近似的解决办法。 在这个方案中, 依赖梯度的非线性术语的近似值是通过使用一个整体平均值来进行的。 我们显示非线性离散问题的解决方案及其对应的一阶衍生物的类似差异是统一的。 对于相应的线性离散性问题,它通过使用两种可变的Chebyshev 聚义体获得高排序的预估值。 基于这些估算, 我们证明了非线性离心问题的稳定性。 关于光滑的解决方案, 我们为近似的解决方案提供了错误估计值。 我们采用循环法是为了找到每个时间步骤的近似解决方案。 迭接过程的趋同性得到了证明。