Fixed point theorems for multivalued noncompact acyclic mappings

Patrick Michael Fitzpatrick and Walter Volodymyr Petryshyn

Vol. 54 (1974), No. 2, 17–23

DOI: 10.2140/pjm.1974.54.17

Abstract

Let X be a Frechet space, D a closed convex subset of X, and T : D → 2^X an upper semicontinuous multivalued acyclic mapping. Using the Eilenberg-Montgomery Theorem and the earlier results of the authors, it is first shown that if W ⊃ T(D) and f : W → D a single-valued continuous mapping such that fT : D → 2^X is Φ-condensing, then fT has a fixed point. This result is then used to obtain various fixed point theorems for acyclic Φ-condensing mappings T : D → 2^X under the Leray-Schauder boundary conditions in case D = Int(D) and under the outward and /or inward type conditions in case Int(D) = Φ.

Mathematical Subject Classification 2000

Primary: 47H10

Milestones

Received: 16 May 1973

Published: 1 October 1974