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

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
Received: 21 November 1969
Revised: 25 March 1970
Published: 1 September 1970
Helga Schirmer