  We study existence, uniqueness, and a limiting behaviour of solutions to an
abstract linear evolution equation in a scale of Banach spaces. The generator
of the equation is a perturbation of the operator which satisfies the classical
assumptions of Ovsyannikov's method by a generator of a C_0-semigroup acting in
each of the spaces of the scale. The results are (slightly modified) abstract
version of those considered in [Math. Models Methods Appl. Sci., 25, 2, 2015,
pp.343-370] for a particular equation. An application to a birth-and-death
stochastic dynamics in the continuum is considered.
