Some asymptotic fixed point theorems

By an asymptotic fixed point theorem we mean a theorem in functional analysis in which the existence of fixed points of a map f f is established with the aid of assumptions on the iterates f n {f^n} of f f . We prove below some new theorems of this type, and we obtain as corollaries results of F. E. Browder, G. Darbo, R. L. Frum-Ketkov, W. A. Horn and others. We also state a number of conjectures about fixed point theorems at the end of the paper.