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.

中文翻译：

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抽象二阶常微分方程的柯西问题

我们证明了线性抽象二阶微分方程柯西问题解的存在性和唯一性，获得了它的表示，并证明了它连续地取决于指定初始条件的时间。基于这些结果，我们证明了非线性抽象二阶微分方程Cauchy问题解的存在性和唯一性。应用该结果表明，非线性双曲型发散结构方程的初边值问题具有唯一的解。参考书目：49种。