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Approximating fixed points of non-self nonexpansive mappings in Banach spaces. (English) Zbl 1089.47058

Let K be a nonempty closed convex subset of a real uniformly convex Banach space E, which is also a nonexpansive retract of E (P is a nonexpansive retraction of E onto K). Let {x n } be the sequence defined by x 1 =xK, x n+1 =P((1-α n )x n +α n TP[(1-β n )x n +β n Tx n ]), n1, where T:EK is a nonexpansive mapping and {α n }, {β n } are sequences in [ε,1-ε] for some ε(0,1). In the present paper, the author proves the following:

(1) If F(T) and the dual E * of E has the Kadec-Klee property, then {x n } convergence weakly to some fixed point of T.

(2) If F(T) and if there is a nondecreasing function f:[0,+)[0,+) with f(0)=0 and f(r)>0 for all r>0 such that for all xK, x-Txf(d(x,F(T))), then {x n } converges strongly to some fixed point of T.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties