Open Access Research

Large time behavior of a linear delay differential equation with asymptotically small coefficient

Mihály Pituk1* and Gergely Röst2

Author Affiliations

1 Department of Mathematics, University of Pannonia, P.O. Box 158, Veszprém, H-8201, Hungary

2 Bolyai Institute, University of Szeged, Aradi vértanúk tere 1, Szeged, H-6720, Hungary

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Boundary Value Problems 2014, 2014:114  doi:10.1186/1687-2770-2014-114


Dedicated to Professor Ivan Kiguradze

Published: 14 May 2014

Abstract

The linear delay differential equation <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M1">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M2">View MathML</a>, is considered, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M3">View MathML</a> and the coefficient <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M4">View MathML</a> is continuous and small in the sense that <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M5">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a>. It is shown that the large time behavior of the solutions can be described in terms of a special solution of the associated formal adjoint equation and the initial data. In the special case of the Dickman-de Bruijn equation, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M7','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M7">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M8">View MathML</a>, our result yields an explicit asymptotic representation of the solutions as <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/114/mathml/M6">View MathML</a>.

MSC: 34K06, 34K25, 11A51.

Keywords:
delay differential equation; formal adjoint equation; Dickman-de Bruijn equation; asymptotic behavior