An elementary topological approach to solvability of nonlinear operator equations
by
Alfonso Vignoli
Università degli Studi di Roma "Tor Vergata". Dipartimento di Matematica - Roma, Italia
Coauthors: Massimo Furi (Università degli Studi di Firenze - Dipartimento di Matematica Applicata)

Let X and Y be topological spaces and f, h be continuous maps from X into Y. Assume, moreover, that h is locally compact. A map h from a topological space X into a topological space Y is said to be locally compact, if for any x belonging to X there exists a neighbourhood U of x such that f(U) is a relatively compact subset of Y. Clearly, if X is a locally compact topological space and h is continuous, then h is locally compact.

Definition. The equation f(x) = h(x) is called hyper-solvable if the equation f(x) = H(x, 1) is solvable for any conditionally compact homotopy H(t, x) from the product of X with the unit interval I into Y.

A continuation principle for hyper-solvable equations is available. The following result is a reformulation of both the Brower and Schauder fixed point theorems.

Proposition. Let X be a metrizable ANR and U an open subset of X. Let p be a point of X. Then, the equation x=p is hyper-solvable if and only if p belongs to U.

Corollary. Let X = Y = a finite dimensional Euclidean space. Let f, from the closure of the open bounded subset U of X, be nonvanishing on the boundary of U. Assume that the Brower topological degree deg (f, U, 0) is different from zero. Then the equation f(x)=0 is hyper-solvable.

An analogous statement holds in infinite dimensional Banach spaces, replacing Brower by Leray-Schauder topological degree.

Date received: May 26, 1999

Copyright © 1999 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cacl-21.