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Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. (English) Zbl 1035.47048

Let C be a nonempty closed convex subset of a real Hilbert space and T:CC be a nonexpansive mapping. In the present paper, the authors investigate the sequence {x n } generated by:

x 0 =xC,y n =α n x n +(1-α n )Tx n ,α n [0,a),a[0,1),C n =zC: y n -z x n -z,Q n =z C : ( 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 strongly to P Fix(T) (x 0 ) by the hybrid method which is used in mathematical programming and obtain a strong convergence theorem for a family of nonexpansive mappings in a real Hilbert space.

47H20Semigroups of nonlinear operators
47H09Mappings defined by “shrinking” properties
49M37Methods of nonlinear programming type in calculus of variations