Atlas: On Verduyn Lunel's conjecture about small solutions by Gregory Derfel

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Equadiff 2003 - International Conference on Differential Equations
July 22-26, 2003
LUC Diepenbeek
Hasselt, Belgium

Organizers
Freddy Dumortier (Chair, LUC Diepenbeek), Henk Broer (Univ. Groningen), Jean-Pierre Gossez (Univ. Libre Bruxelles), Jean Mawhin (Univ. Cath. Louvain-la-Neuve), Andre Vanderbauwhede (Univ. Gent), Sjoerd Verduyn Lunel (Univ. Leiden)

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On Verduyn Lunel's conjecture about small solutions
by
Gregory Derfel
Ben-Gurion University, Israel
Coauthors: A. D. Myshkis (Moscow University of Communications, Moscow, Russia)

We discusse the asymptotics of the solutions for nonautonomous delay equation

\labelEq:1y'(t)=a(t)y(t-1), 0 <= t < \infty,

(\theequation)

where a(t) in C^\infty . Assuming that a(t) =/= 0 for all t in R₊ (though a(t) possibly tends to 0, as t --> \infty), we prove that under some additional conditions on a(t), every solution y(t) of (1), decreasing fast enough as t --> \infty, vanishes identically. We consider the case, when a(t) is an 1-periodic function with simple isolated zeros, as well. In this case, due to Verduyn Lunel, equation (1) has small (super-exponentially decreasing ) solution if and only if a(t) has a sign change. How small can such a small solution be? Under some additional conditions on a(t), we prove, that if

|y(t)| <= Cexp{-\gammatlnt}, t > T > 0,

for some C > 0 and \gamma > \gamma₀, then y(t) vanishes identically. Examples demonstrate that our results are sharp up to the constants .

Date received: May 30, 2003