Error estimates for projection-difference methods for differential equations with differentiable operators

We study the projection-difference methods for approximate solving the Cauchy problem for operator-differential equations with a leading self-adjoint operator $A(t)$ and a subordinate linear operator $K(t)$, whose definition domain is independent of $t$. Operators $A(t)$ and $K(t)$ are assumed to be sufficiently smooth. We obtain estimates for the rate of convergence of approximate solutions to the exact solution as well as those for fractional degrees of an operator similar to $A(0)$.