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

Abstract

Let A(t) $(t\ge 0)$ be an unbounded variable operator on a Banach space ${\mathcal {X}}$ with a constant dense domain, and B(t) be a bounded operator in ${\mathcal {X}}$. Assuming that the evolution operator U(t, s) $(t\ge s)$ of the equation $\mathrm{d}x(t)/\mathrm{d}t=A(t)x(t)$ is known we built the evolution operator $\tilde{U}(t,s)$ of the equation $\mathrm{d}y(t)/\mathrm{d}t=(A(t)+B(t))y(t)$. Besides, we obtain C-norm estimates for the difference $\tilde{U}(t,s)-U(t,s)$. 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.