A Leray-Schauder alternative for Mönch maps on closed subsets of Fréchet spaces
The authors obtain an analogue of the famous Mönch fixed point theorem for maps defined on a closed convex subset of a quasicomplete, metrizable locally convex space which may have an empty interior, as well as a fixed point result of Leray-Schauder type in such a space. The presented results are illustrated by an existence result for first order systems of differential equations \[ y'(t)=g\bigl(t,y(t)\bigr),\quad t\in[0,T] \] \[ y(0)=a=(a_1,a_2,\dots)\in B=\prod_{n\in \mathbb{N}}B_n, \] where \([0,T]\) is a compact interval in \(\mathbb{R}\) and \((B_n,\|\cdot\|_n)\) is a Banach space for every \(n\in\mathbb{N}\). The right-hand side of the above problem satisfies a certain growth condition and a Mönch type condition formulated in terms of a certain measure of noncompactness.
scientific article; zbMATH DE number 1827609