Fixed points of asymptotic contractions

AbstractLet 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 dTn(x),Tn(y)⩽φnd(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