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Algorithms of common solutions to quasi variational inclusion and fixed point problems. (English) Zbl 1196.47047

Summary: The purpose of this paper is to present an iterative scheme for finding a common element of the set of solutions to the variational inclusion problem with multivalued maximal monotone mapping and inverse-strongly monotone mappings and the set of fixed points of nonexpansive mappings in Hilbert space. Under suitable conditions, some strong convergence theorems for approximating this common elements are proved. The results presented in the paper not only improve and extend the main results in [G. M. Korpelevich, Ekonom. i Mat. Metody 12, 747–756 (1976; Zbl 0342.90044)], but also extend and replenish the corresponding results obtained by H. Iiduka and W. Takahashi [Nonlinear Anal., Theory Methods Appl. 61, No. 3 (A), 341–350 (2005; Zbl 1093.47058)], W. Takahashi and M. Toyoda [J. Optim. Theory Appl. 118, 417–428 (2003; Zbl 1055.47052)], N. Nadezhkina and W. Takahashi [J. Optim. Theory Appl. 128, No. 1, 191–201 (2006; Zbl 1130.90055)] and L.-C. Zeng and J.-C. Yao [Taiwanese J. Math. 10, No. 5, 1293–1303 (2006; Zbl 1110.49013)].

MSC:

47J25 Iterative procedures involving nonlinear operators
47J22 Variational and other types of inclusions
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H05 Monotone operators and generalizations
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