Abstract
We prove a generalization of Brouwer's famous fixed point theorem to discontinuous maps. The main result shows that for discontinuous functions on a compact convex domainX one can always find a pointx ∈X such that ∥x−f(x)∥ is less than a certain measure of discontinuity. Applications to artificial neural nets, economic equilibria and analysis are given.
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Cromme, L.J., Diener, I. Fixed point theorems for discontinuous mapping. Mathematical Programming 51, 257–267 (1991). https://doi.org/10.1007/BF01586937
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DOI: https://doi.org/10.1007/BF01586937