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Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces. (English) Zbl 1116.47053

A projection Mann type iterative method, introduced in [K. Nakajo and W. Takahashi, J. Math. Anal. Appl. 279, No. 2, 372–379 (2003; Zbl 1035.47048)] and used there to approximate fixed points of nonexpansive mappings, is extended to a more general iterative method, appropriate for approximating fixed points of strict pseudocontractions. Let C be a nonempty closed convex subset of a real Hilbert space and T:CC be a k-strict pseudocontraction. In the present paper, the authors investigate the sequence {x n } generated by:

x 0 C,y n =α n x n +(1-α n )Tx n ,α n (0,1),C n =zC:y n -z 2 x n -z 2 +(1-α n )(k-α n )x n -Tx n 2 ,Q n =zC:x n -z,x 0 -x n 0,x n+1 =P C n Q n (x 0 ),

where P is the metric projection. They show that {x n } converges weakly to a fixed point of T (Theorem 3.1), or, respectively, {x n } converges strongly to P Fix(T) (x 0 ) (Theorem 4.1).

47J25Iterative procedures (nonlinear operator equations)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties