Let (X, d) be a complete metric space and let f : X → X satisfy inf{α(x, y)d(f m (x), f m (y)) : m ∈ J} ≤ Kd(x, y) for all x, y ∈ X and some K ∈ (0, 1) and α : X × X → [0, ∞), 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.