Fixed points of asymptotic contractions

Let φ :R+ →R+ be a contractive gauge function in the sense that φ is continuous, φ(s) < s for s >0, and if f :M →M satisfies d(f (x), f (y)) φ(d(x, y)) for all x, y in a complete metric space (M, d), then f always has a unique fixed point. It is proved that if T :M →M satisfies d T n(x), T n(y) φn d(x, y) , x,y∈ M, where each φn is continuous and φn →φ uniformly on the range of d, then T has a unique fixed point, and moreover all of the Picard iterates of T converge to this fixed point.  2003 Elsevier Science (USA). All rights reserved.