Monotone semiflows generated by functional differential equations.(English)Zbl 0612.34067

The author considers the following differential equation (1) $$x'(t)=f(x_ t)$$, where $$f: C([-r,0],{\mathbb{R}}^ n)\to {\mathbb{R}}^ n$$ and $$x_ t$$ is the element of $$C([-r,0],{\mathbb{R}}^ n)$$ defined by $$x_ t(s)=x(t+s)$$, -r$$\leq s\leq 0$$. Parallel with (1) the ordinary differential equation (2) $$x'(t)=F(x(t))$$, where $$F(x)=f(Ix)$$ and $$I: {\mathbb{R}}^ n\to C([-r,0],{\mathbb{R}}^ n)$$ is the canonical inclusion, is considered. The main result is as follows. Under suitable conditions, for a dense set of initial conditions for (1), the qualitative behavior of the solutions of (1) is the same as for (2).
Reviewer: R.R.Akhmerov

MSC:

 34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument) 37-XX Dynamical systems and ergodic theory 34C05 Topological structure of integral curves, singular points, limit cycles of ordinary differential equations
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References:

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