Vol. 41, No. 1, 1972

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Semigroups with diminishing orbital diameters

Mo Tak Kiang

Vol. 41 (1972), No. 1, 143–151

In this paper, fixed point theorems for semigroups of selfmappings on a metric space (X, d) subject to conditions on the size of the orbits are considered.

The concepts of diminishing orbital diameters (d.o.d.) for semigroups of mappings on a metric space and that of convex diminishing orbital diameters (c.d.o.d.) for semigroups of mappings on a convex subset of a normed linear space are introduced. Also discussed are the concepts of linearly ordered semigroups and in particular those that are Archimedean at some of its members. Certain results of Belluce and Kirk concerning a single mapping satisfying d.o.d. are generalized. Also included are results on semigroups of self-mappings on a weakly compact, convex subset of a Banach space.

Mathematical Subject Classification 2000
Primary: 54H15
Secondary: 47H10
Received: 12 July 1971
Published: 1 April 1972
Mo Tak Kiang