Fixed points of asymptotic contractions

Abstract Let φ : R + → R + be a contractive gauge function in the sense that φ is continuous, φ(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) ⩽φ n 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.