Positivity 1: 319–330, 1997.
1997 Kluwer Academic Publishers. Printed in the Netherlands.
Stability of Semilinear Delay Equations with
at Graz, Institut f
ur Mathematik, Heinrichstraße 36, A-8010 Graz, Austria
Department of Mathematics and Computing, University of Veszpr
em, P.O.B. 158, H-8210
(Received: 11 September 1996; Accepted: 5 September 1997)
Abstract. We prove stability for a semilinear delay equation, whose nonlinearity is majorized by a
linear positive operator. The key ingredients are a spectral condition, positivity of solutions to the
linear problem, and lattice properties of the Banach space.
Mathematics Subject Classiﬁcation (1991): 34K20, 34K30
Keywords: delay equations, stability, positive solutions, spectral growth condition
In this paper we consider stability properties of the following abstract delay equa-
tion in a Banach lattice
B(d )x(t )+F(t; x(:))
are linear operators,
is nonlinear and depends also on the
history of the solution.
Our basic approach is to regard this equation as a perturbation of the linear
B(d )y (t )
and utilize positivity arguments. We assume that the linear problem has a posi-
tive integrable resolvent operator which amounts to admitting positive bounded
solutions for initial data
y (s) =
0. The nonlinearity
dominated by some bounded, positive linear operator
. Stability for the linear
This work is supported by Spezialforschungsbereich F 003, Optimierung und Kontrolle. It was
begun while W. D. was visiting the University of Veszpr
em. It is a pleasure to thank the University
em for its kind hospitality.
pips: 149860 MATHKAP
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