Vol. 34, No. 3, 1970

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Coincidences and fixed points of multifunctions into trees

Helga Schirmer

Vol. 34 (1970), No. 3, 759–767
Abstract

The main purpose of this paper is to find conditions on an upper semi-continuous (usc) multifunction φ from a compact Hausdorff space X onto a tree T so that it has a coincidence with any multifunction ψ : X T which is either continuous or usc and connected-valued. It is shown that it is sufficient (but not necessary) that φ be either open or monotone. This result contains as special cases known conditions for coincidence producing single-valued maps onto trees as well as known fixed point theorems for multifunctions on trees. It is used to obtain a new result on fixed points, namely that any composite of an usc and connected-valued and a continuous multifunction of a tree into itself has a fixed point. All proofs make use of the order-theoretic characterization of trees by L. E. Ward, Jr.

Mathematical Subject Classification
Primary: 54.85
Milestones
Received: 21 November 1969
Revised: 25 March 1970
Published: 1 September 1970
Authors
Helga Schirmer