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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 $$ \dot{y}=\beta(t) [y(t)-y(t-\tau(t))], \tag1$$ where $\tau\in C(I_{-1},{\bbfR}^+)$, $I_{-1}=[t_{-1},\infty)$, $t_{-1}=t_0-\tau(t_0)$, $t_0\in{\bbfR}$, $t\mapsto t-\tau(t)$ is increasing, $\tau(t)\geq\widetilde{\tau}$ for all $t$, $\widetilde{\tau}= \text{ const}>0$, $\beta\in C([t_0,\infty),{\bbfR}^+)$. 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.

34K25Asymptotic theory of functional-differential equations
34C11Qualitative theory of solutions of ODE: growth, boundedness
34K99Functional-differential equations
Full Text: DOI
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