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


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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