Iteration processes related to nonexpanding mappings

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 – x_n + (1 – )Vx_n (n = 0, 1, 2,...; 0 < < 1) converges for each initial condition x₀ to a fixed point of the mapping V and, moreover, we have the estimate

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