On some inverse and nonlocal problems for operator differential equations and numerical methods for their solution

We study some nonlocal problems for operator differential equations of the form
$$\frac{dv}{dt} = Av + f(t)$$
with an unbounded operator A in vector spaces. For a wide classes of equations we obtain necessary
and sufficient conditions of the unique solvability and the formulas for solutions of such problems.
In the second part of the talk we present fast numerical algorithms for solution of these problems
for matrix equations with rank structured matrices.