Estimates for Solutions of Differential Equations in a Banach Space via Commutators

Abstract

In a Banach space we consider the equation dx(t)/dt = (A + B(t))×(t) (t ≥ 0), where A is a constant bounded operator, and B(t) is a bounded variable operator.Norm estimates for the solutions of the considered equation are derived in terms of the commutator AB(t) − B(t)A. These estimates give us sharp stability conditions. Our results are new even in the finite dimensional case.We also discuss applications of the obtained results to a class of integro-differential equations.

Keywords

• banach space
• differential equation
• solution estimates
• stability
• integro-differential equation
• 34g10
• 34d05
• 34d20