Asymptotic representation of solutions of the equation \(\dot y(t)=\beta(t)[y(t)-y(t-\tau(t))]\). (English) Zbl 0892.34067

The equation considered is the following differential difference equation \[ \dot{y}=\beta(t) [y(t)-y(t-\tau(t))], \tag{1} \] where \(\tau\in C(I_{-1},{\mathbb{R}}^+)\), \(I_{-1}=[t_{-1},\infty)\), \(t_{-1}=t_0-\tau(t_0)\), \(t_0\in{\mathbb{R}}\), \(t\mapsto t-\tau(t)\) is increasing, \(\tau(t)\geq\widetilde{\tau}\) for all \(t\), \(\widetilde{\tau}= \text{ const}>0\), \(\beta\in C([t_0,\infty),{\mathbb{R}}^+)\). The author proves theorems on the existence of solutions of (1) tending to infinity as \(t\to\infty\), and of solutions with the range in \([C_1,C_2]\) for any \(C_1<C_2\). Exponential estimates and comparison theorems are also discussed.


34K25 Asymptotic theory of functional-differential equations
34C11 Growth and boundedness of solutions to ordinary differential equations
34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
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