A norm estimate for evolution operators of equations with the Lipschitz property in a Banach space

Abstract

In a Banach space we consider the equation $du(t)/dt=(A_0+B(t))u(t)$, where $A_0$ is a constant operator, generating a $C_0$-semigroup, and $B(t)$ is a bounded variable operator with the Lipschitz property. A norm estimate for the evolution operator of the considered equation is established. That estimate gives us exponential stability conditions. An illustrative example is given.