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Strong convergence of modified Mann iterations. (English) Zbl 1091.47055
Let X be a real Banach space with a norm · and let C be a nonempty, closed and convex subset of X. A mapping T:CC is nonexpansive provided that Tx-Tyx-y for all x,yC. Assume that T has at least one fixed point in C. The authors consider the following iteration sequence {x n } for T:x 0 =xC, y n =α n x n +(1-α n )Tx n , x n+1 =β n u+(1-β n )y n , where u is an arbitrary fixed element in C and {α n }, {β n } are two sequences in the interval (0,1) converging to 0 and such that α n =β n =. Moreover, |α n+1 -α n |<, |β n+1 -β n |<. Under the assumption that X is uniformly smooth, it is shown that the sequence {x n } converges strongly to a fixed point of T. An analogous result is proved for accretive operators.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
46B25Classical Banach spaces in the general theory of normed spaces
47H06Accretive operators, dissipative operators, etc. (nonlinear)