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BY-NC-ND 4.0 license Open Access Published by De Gruyter Open Access February 28, 2018

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

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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.

MSC 2010: 34G10; 34D05; 34D20

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Received: 2017-05-24
Accepted: 2018-01-26
Published Online: 2018-02-28

© 2018

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

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