×

Fix-finite approximation of n-valued multifunctions. (English) Zbl 0537.55005

A multifunction \(\phi\) : \(X\to Y\) is called n-valued if \(\phi\) (x) consists, for all \(x\in X\), of n points. it is shown that an n-valued continuous multifunction from a compact polyhedron to itself has an arbitrarily close approximation by an n-valued continuous multifunction which has only finitely many fixed points. The proof includes multivalued analogues of the simplicial approximation theorem and the Hopf construction. A basic tool is the splitting lemma, which shows that n- valued continuous multifunctions are locally equivalent to n single- valued continuous multifunctions, and implies that every n-valued continuous multifunction from a compact, Hausdorff, path connected and simply connected space with the fixed point property to itself has at least n fixed points.

MSC:

55M20 Fixed points and coincidences in algebraic topology
54H25 Fixed-point and coincidence theorems (topological aspects)
54C60 Set-valued maps in general topology
PDFBibTeX XMLCite
Full Text: DOI EuDML