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Boundary layer phenomena for differential-delay equations with state- dependent time lags. I. (English) Zbl 0763.34056
The paper deals with the differential-delay equation (1) $$\varepsilon x'(t)=-x(t)+f(x(t-r(x(t)))$$, $$r(x)>0$$. Existence, regularity and monotonicity properties of solutions are investigated. The so-called slowly oscillating periodic solutions are the goal of the paper. A variety of new mathematical phenomena, which are not present in the case $$r(x)\equiv\text{const}\geq 0$$, arise for (1). The limiting behaviour of (1) when $$\varepsilon\to+0$$, “sawtooth” periodic solutions, qualitative properties of such solutions are considered. The equation describes some mathematical models of electrodynamics and blood cell production.

##### MSC:
 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 34K10 Boundary value problems for functional-differential equations 34C15 Nonlinear oscillations and coupled oscillators for ordinary differential equations 34C25 Periodic solutions to ordinary differential equations 92C50 Medical applications (general)
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