Approximation of a linear system of second-order differential equations

In a Hilbert space $H$ we consider the approximation by systems
\begin{equation} \frac{d^2u_1}{dt^2}=A_{11}u_1+A_{12}u_2+f_1,\quad\varepsilon\frac{d^2u_2}{dt_2}A_2u_1+A_{22}u_2+f_2,\quad\varepsilon>0,\tag{1} \end{equation}
of the semievolutionary system obtained from (1) when $\varepsilon=0$. Under certain conditions on the solutions of the Cauchy problem for system (1) and the existence of a bounded linear operator $A_{22}^{-1}$ we establish the convergence of the solutions $u^\varepsilon$ ($\varepsilon\to0$) to a solution of the corresponding problem for system (1) with $\varepsilon=0$. We also establish the uniform correctness of the Cauchy problem for the above system.