Inequalities for Solutions of Linear Differential Equations in a Banach Space and Integro–Differential Equations

The chapter presents a survey of the recent results of the author on solution estimates for the linear differential equation du(t)∕dt = A(t)u(t) with a bounded operator A(t) in a Banach space satisfying various conditions. These estimates give us sharp stability conditions as well as upper and lower bounds for the evolution operator. Applications to integro-differential equations are also discussed. In particular, we consider equations with differentiable in t operators, equations with the Lipschitz property, equations in the lattice normed spaces, and equations with the generalized Lipschitz property. In addition, we investigate integrally small perturbations of autonomous equations. In appropriate situations our stability conditions are formulated in terms of the commutators of the coefficients of the considered equations. A significant part of these results has been generalized in the available literature to equations with unbounded operators. Some results presented in the chapter are new.