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Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces. (English) Zbl 1192.47054

Summary: Consider on a real Hilbert space H a nonexpansive mapping T with a fixed point, a contraction f with coefficient 0<α<1, and two strongly positive linear bounded operators A,B with coefficients γ ¯(0,1) and β>0, respectively. Let 0<γα<β. We introduce a general iterative algorithm defined by

x n+1 :=(I-λ n+1 A)Tx n +λ n+1 [Tx n -μ n+1 (BTx n -γf(x n ))],n1,

with μ n μ(n), and prove the strong convergence of the iterative algorithm to a fixed point x ˜Fix(T)=:C which is the unique solution of the variational inequality (for short, VI(A-I+μ(B-γf),C)):[A-I+μ(B-γf)]x ˜,x-x ˜0,xC. On the other hand, assume C is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on H. We devise another iterative algorithm which generates a sequence {x n } from an arbitrary initial point x 0 H. The sequence {x n } is proven to converge strongly to an element of C which is the unique solution x * of the VI(A-I+μ(B-γf),C). Applications to constrained generalized pseudoinverses are included.


MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J22Variational and other types of inclusions
65J15Equations with nonlinear operators (numerical methods)
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