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A general iterative algorithm for nonexpansive mappings in Hilbert spaces. (English) Zbl 1192.47064
Summary: Let H be a real Hilbert space. Suppose that T is a nonexpansive mapping on H with a fixed point, f is a contraction on H with coefficient 0<α<1, and F:HH is a k-Lipschitzian and η-strongly monotone operator with k>0, η>0. Let 0<μ<2η/k 2 , 0<γ<μη-μk 2 2/α=τ/α. We proved that the sequence {x n } generated by the iterative method x n+1 =α n γf(x n )+(I-μα n F)Tx n converges strongly to a fixed point x ˜Fix(T), which solves the variational inequality (γf-μF)x ˜,x-x ˜0, for xFix(T).
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H05Monotone operators (with respect to duality) and generalizations
47H06Accretive operators, dissipative operators, etc. (nonlinear)
47J20Inequalities involving nonlinear operators