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Viscosity approximation with weak contractions for fixed point problem, equilibrium problem, and variational inequality problem. (English) Zbl 1204.41028

Summary: This paper proposes a modified iterative algorithm using a viscosity approximation method with a weak contraction. The purpose is to find a common element of the set of common fixed points of an infinite family of nonexpansive mappings and the set of a finite family of equilibrium problems that is also a solution to a variational inequality. Under suitable conditions, some strong convergence theorems are established in the framework of Hilbert spaces. The results presented in the paper improve and extend the corresponding results of V. Colao, G. L. Acedo and G. Marino [Nonlinear Anal., Theory Methods Appl. 71, No. 7–8, A, 2708–2715 (2009; Zbl 1175.47058)], S. Plubtieng and R. Punpaeng [J. Math. Anal. Appl. 336, No. 1, 455–469 (2007; Zbl 1127.47053)], V. Colao, G. Marino and H.-K. Xu [J. Math. Anal. Appl. 344, No. 1, 340–352 (2008; Zbl 1141.47040)], Y. Yao, Y.-C. Liou and J.-C. Yao [Fixed Point Theory Appl. 2007, Article ID 64363, 12 p. (2007; Zbl 1153.54024)], and others.

MSC:

41A65 Abstract approximation theory (approximation in normed linear spaces and other abstract spaces)
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J20 Variational and other types of inequalities involving nonlinear operators (general)
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