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Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions. (English) Zbl 1220.47100
Summary: Let C be a closed convex subset of a real Hilbert space H and assume that T is an asymptotically κ-strict pseudo-contraction on C with a fixed point, for some 0κ<1. Given an initial guess x 0 C and a real sequence {α n } in (0,1), the modified Mann algorithm generates a sequence {x n } via the formula: x n+1 =α n x n +(1-α n )T n x n , n0. It is proved that if the control sequence {α n } is chosen so that κ+δ<α n -δ for some δ(0,1), then {x n } converges weakly to a fixed point of T. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)