Fixed point systems.

The problem stated in its most general terms is this. Given a space S and a mapping t of S into itself, under what circumstances is there or is there not a fixed point? By a fixed point we mean, simply, a point x in S for which tx*x. Most investigations are concerned with single valued continuous mappings and the trend has been to search for ever larger classes of spaces for which fixed point conditions can be derived. Studies have, however, been made notably by M. H. A. Newman (15) and P. A. Smith (17) of the question of periodic transformations i.e. transformations some power of which is the identity