RATE OF CONVERGENCE UNDER WEAK CONTRACTIVENESS CONDITIONS

We introduce a new class of selfrnaps T of metric spaces, which generalizes the weakly Zamfirescu maps (and therefore weakly contraction maps, weakly Kannan maps, weakly Chatterjea maps and quasi-contraction maps with constant h < 1/2). We give an explicit Cauchy rate for the Picard iteration sequences {T(n)x(0)}(n is an element of N) for this type of maps, and show that if the space is complete, then all Picard iteration sequences converge to the unique fixed point of T. Our Cauchy rate depends on the space (X, d), the map T, and the starting point x(0) is an element of X only through an upper bound b >= d(x(0),Tx(0)) and certain moduli theta, mu for the map, but is otherwise fully uniform. As a step on the way to proving our fixed point, result we also calculate a modulus of uniqueness for this type of maps.