A convergence theorem for asymptotic contractions

We show that to each asymptotic contraction T with a bounded orbit in a complete metric space X, there corresponds a unique point x _* such that all the iterates of T converge to x _*, uniformly on any bounded subset of X. If, in addition, some power of T is continuous at x _*, then x _* is a fixed point of T. Dedicated to Professor Felix E. Browder with admiration and respect