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A new iteration method for nonexpansive mappings and monotone mappings in Hilbert spaces. (English) Zbl 1187.47049

Summary: We introduce a new composite iterative scheme by the viscosity approximation method for nonexpansive mappings and monotone mappings in a Hilbert space. It is proved that the sequence generated by the iterative scheme converges strongly to a common point of the set of fixed points of a nonexpansive mapping and the set of solutions of a variational inequality for an inverse-strongly monotone mapping, which is a solution of a certain variational inequality. Our results substantially develop and improve the corresponding results of J.-M.Chen, L.-J.Zhang and T.-G.Fan [J. Math.Anal.Appl.334, No.2, 1450–1461 (2007; Zbl 1137.47307) and H.Iiduka and W.Takahashi [Nonlinear Anal., Theory Methods Appl.61, No.3 (A), 341–350 (2005; Zbl 1093.47058)]. Essentially, a new approach for finding the fixed points of nonexpansive mappings and solutions of variational inequalities for monotone mappings is provided.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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References:

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