Fixed points of asymptotic contractions

Let phi:R+ --> R+ be a contractive gauge function in the sense that phi is continuous, phi(s) < s for s > 0, and if f : M --> M satisfies d (f (x), f (y)) less than or equal to phi (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(x), T(y)) less than or equal to phi(n) (d(x, y)), x, y is an element of M, where each phi(n) is continuous and phi(n) --> phi 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. (C) 2003 Elsevier Science (USA). All rights reserved.