Abstract
Let (X, d) be a complete metric space and let f : X -> X satisfy in f {alpha(x, y)d(f(m)(x), f(m)(y)) : m is an element of J} <= Kd(x, y) for all x, y is an element of X and some K is an element of(0, 1) and alpha: X x X -> [0, infinity), where J is a set of positive integers. In this paper, we prove fixed point theorems for this mapping f. We also discuss the connection with tiling problems and give a titling proof of a fixed point theorem.