Linear differential equations with several unbounded delays. (English) Zbl 1090.34595

Čermák, Jan

Arch. Math., Brno 36, Suppl., 421-427 (2000).

The author investigates the asymptotic behaviour of the solutions of the delay differential equation \[ \dot {x}(t)=-a(t)x(t)+b_1(t)x(\tau _1(t))+b_2(t)x(\tau _2(t)) {(E)} \] with the continuous coefficients \(a(t),\,b_1(t),\,b_2(t)\). Lags \(\tau _1(t)\), \(\tau _2(t)\) are assumed to be unbounded and commutable on \([t_0,\infty )\). The conditions, under which every solution of (E) satisfies \[ x(t)=O\Bigl ((\varphi (t))^\alpha \Bigr )\qquad \text{as}\;t\to \infty \] are introduced. Here \(\varphi (t)\) is the solution of system of Schröder’s functional equations \(\varphi (\tau _1(t))=\lambda _1\varphi (t)\), \(\varphi (\tau _2(t))=\lambda _2\varphi (t),\) \(\alpha \), \(\lambda _1\), \(\lambda _2\) are suitable real numbers.
The author also suggests generalization of the results to the equation with \(m\) delayed arguments.

Reviewer: Robert Mařík (Brno)

MSC:

34K25	Asymptotic theory of functional-differential equations
39B22	Functional equations for real functions

Keywords:

delay differential equation; functional equation; asymptotic behaviour of the solutions