Iteration processes related to nonexpanding mappings

Abstract

The following theorem is proved: if V is a nonexpanding mapping of a convex compactum X in a Banach space into itself, then the iteration sequence x_n+1 − αx_n + (1 − α)Vx_n (n = 0, 1, 2,...; 0 < α < 1) converges for each initial condition x₀ε χ to a fixed point of the mapping V and, moreover, we have the estimate

$$||x_{n + 1} - x_n || = 0\left( {\frac{1}{{\ln n}}} \right).$$