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.
