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Iterative methods for strict pseudo-contractions in Hilbert spaces. (English) Zbl 1133.47050

This article deals with two iterative algorithms of finding a common fixed points for N strict pseudo-contractions {T i } i=1 N defined on a closed convex subset C of a real Hilbert space H (an operator T:CC is a strict pseudo-contraction, if there exists a constant 0k<1 such that Tx-Ty 2 x-y 2 +k(I-T)x-(I-T)y 2 ). The first algorithm, called parallel, is defined by the formula

x n+1 =α n x n +(1-α n ) i=1 N λ i (n) T i x n ,x 0 C,λ i (n) >0,λ 1 (n) ++λ N (n) =1;(1)

the second one, called cyclic, by the formula

x n+1 =α n x n +(1-α)T [n] x n ,x 0 C,T [n] =T i ,i=n(modN),1iN·(2)

The main results describe (provided that F= i=1 N Fix(T i )) conditions on the control sequence {α n } so that the approximations x n converge weakly to a common fixed point of {T i } i=1 N . At the end of the article, some modifications of algorithms (1) and (2) are proposed; it is proved that approximations x n for these modified algorithms converge strongly to P F x 0 , where P F is the nearest point projection from H onto F.

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