| Since Poincare introduced the subject in 1888 fixed point theory has always played a central role in the problems of analysis. This talk will include a historical survey of the development of fixed point theory in analysis with an emphasis on the evolution of metric methods. The metric theory begins with the classical result on contraction mappings found in Banach's 1922 thesis. The study of nonexpansive mappings, the limiting case in which the Lipschitz constant of the mappings is actually allowed to be 1, began in 1965 and many major developments have evolved since then. The talk will focus on the relationship between various fixed point properties for nonexpansive mappings and reflexivity of the underlying Banach space. A number of basic open questions will be discussed. |