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Convergence on composite iterative schemes for nonexpansive mappings in Banach spaces. (English) Zbl 1203.47053

Let E be a reflexive Banach space with a uniformly Gateaux differentiable norm. Suppose that every weakly compact convex subset of E has the fixed point property for nonexpansive mappings. Let C be a nonempty closed convex subset of E, f:CC be a contractive mapping (or a weakly contractive mapping), and T:CC be a nonexpansive mapping with nonempty fixed point set F(T). Let the sequence {x n } be generated by the following composite iterative scheme:

y n =λ n f(x n )+(1-λ n )Tx n ,x n+1 =(1-β n )y n +β n Ty n ,n0·

It is proved that {x n } converges strongly to a point in F(T), which is a solution of a certain variational inequality, provided that the sequence {λ n }(0,1) satisfies lim n λ n =0 and n=1 λ n =, {β n }[0,a) for some 0<a<1, and the sequence {x n } is asymptotically regular.

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