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Periodic solutions of a planar delay equation. (English) Zbl 0719.34125

The problem of existence of nonconstant periodic solutions of the planar delay differential equation \((1)\quad \dot x(t)=-x(t)+\alpha F(x(t-1))\) (where \(x=col(x_ 1,x_ 2)\in {\mathbb{R}}^ 2\), \(F=col(F_ 1,F_ 2)\) is a \(C^ 3\) map from \({\mathbb{R}}^ 2\) into itself and \(\alpha >0\) is a real parameter) is considered. The method used in the article comes from a widely known idea, which consists in finding a cone in phase space which maps into itself under a certain oprator defined by the flow in the phase space. The nontrivial fixed points of that operator will correspond to the periodic solutions of equation (1).

MSC:

34K99 Functional-differential equations (including equations with delayed, advanced or state-dependent argument)
34C25 Periodic solutions to ordinary differential equations
47H10 Fixed-point theorems
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References:

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