The L^p-version of the generalised Bohl-Perron principle for vector equations with delay
by Michael I. Gil'
International Journal of Dynamical Systems and Differential Equations (IJDSDE), Vol. 3, No. 4, 2011

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.

Online publication date: Sat, 24-Jan-2015

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