On the connectedness of the set of fixed points of a compact operator in the Fréchet space \(C^ m(\langle b,\infty),\mathbb{R}^ n\). (English) Zbl 0793.47055

The authors prove an Aronshejn-type result on the connectedness of the fixed point set of a nonlinear operator in a Fréchet space satisfying some technical conditions. The abstract theorem is illustrated by means of an application to an initial value problem for the functional- differential equation \(y'(t)=\omega(t)F(y(t)+c)\) in the Fréchet space \(C^ m([b,\infty),\mathbb{R}^ n)\) equipped with the usual “exhausting” sequence of seminorms.


47H10 Fixed-point theorems
46A04 Locally convex Fréchet spaces and (DF)-spaces
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