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MATHEMATICA BOHEMICA, Vol. 125, No. 3, pp. 355-364 (2000)
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# Asymptotic behaviour of solutions of some linear delay differential equations

## Jan Cermak

* Jan Cermak*, Department of Mathematics, Technical University of Brno, Technicka 2, 616 69 Brno, Czech Republic, e-mail: ` cermakh@mat.fme.vutbr.cz`

**Abstract:**
In this paper we investigate the asymptotic properties of all solutions of the delay differential equation $$y'(x)=a(x)y(\tau (x))+b(x)y(x),\qquad x\in I=[x_0,\infty ).$$ We set up conditions under which every solution of this equation can be represented in terms of a solution of the differential equation $$z'(x)=b(x)z(x),\qquad x\in I$$ and a solution of the functional equation $$|a(x)|\varphi (\tau (x))=|b(x)|\varphi (x),\qquad x\in I.$$

**Keywords:** asymptotic behaviour, differential equation, delayed argument, functional equation

**Classification (MSC2000):** 34K15, 34K25, 39B99

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