Title: The L^p-version of the generalised Bohl-Perron principle for vector equations with delay

Authors: Michael I. Gil'

Addresses: Department of Mathematics, Ben Gurion University of the Negev, P.O. Box 653, Beer-Sheva 84105, Israel

Abstract: We consider the equation •y = Ey, where Ey(t) = fη0d_τR(t, τ)y(t − τ) (t ≥ 0) with an n × n-matrix-valued function R(t, τ). It is proved that, if for a p ≥ 1, the non-homogeneous equation •x = Ex + f with the zero initial condition, for any f ∈ L^p, has a solution x ∈ L^p, then the considered homogeneous equation is exponentially stable. By that result, sharp stability conditions are derived for vector functional differential equations |close| to autonomous ones and for equations with small delays.

Keywords: functional differential equations; linear equations; exponential stability; Bohl-Perron; vector equations; delay.

DOI: 10.1504/IJDSDE.2011.042940

International Journal of Dynamical Systems and Differential Equations, 2011 Vol.3 No.4, pp.448 - 458

Published online: 24 Jan 2015 *

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