On Dyson–Phillips type approach to differential equations with variable operators in a Banach space

Let A(t) (t≥ 0) be an unbounded variable operator on a Banach space X with a constant dense domain, and B(t) be a bounded operator in X. Assuming that the evolution operator U(t, s) (t≥ s) of the equation d x(t) / d t= A(t) x(t) is known we built the evolution operator U~ (t, s) of the equation d y(t) / d t= (A(t) + B(t)) y(t). Besides, we obtain C-norm estimates for the difference U~ (t, s) - U(t, s). We also discuss applications of the obtained estimates to stability of the considered equations. Our results can be considered as a generalization of the well-known Dyson–Phillips theorem for operator semigroups.