×

A viscosity approximation method for finding common solutions of variational inclusions, equilibrium problems, and fixed point problems in Hilbert spaces. (English) Zbl 1186.47075

Summary: We introduce an iterative method for finding a common element of the set of common fixed points of a countable family of nonexpansive mappings, the set of solutions of a variational inclusion with set-valued maximal monotone mapping, and inverse strongly monotone mappings and the set of solutions of an equilibrium problem in Hilbert spaces. Under suitable conditions, some strong convergence theorems for approximating these common elements are proved. The results presented in the paper improve and extend the main results of J. W. Peng, Y. Wang, D. S. Shyu and J.-C. Yao [J. Inequal. Appl. 2008, Article ID 720371 (2008; Zbl 1161.65050)] and many others.

MSC:

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

Citations:

Zbl 1161.65050
PDF BibTeX XML Cite
Full Text: DOI EuDML

References:

[1] Combettes, PL; Hirstoaga, SA, Equilibrium programming in Hilbert spaces, Journal of Nonlinear and Convex Analysis, 6, 117-136, (2005) · Zbl 1109.90079
[2] Zeng, LC; Schaible, S; Yao, JC, Iterative algorithm for generalized set-valued strongly nonlinear mixed variational-like inequalities, Journal of Optimization Theory and Applications, 124, 725-738, (2005) · Zbl 1067.49007
[3] Plubtieng, S; Punpaeng, R, A new iterative method for equilibrium problems and fixed point problems of nonexpansive mappings and monotone mappings, Applied Mathematics and Computation, 197, 548-558, (2008) · Zbl 1154.47053
[4] Chang, S-S; Joseph Lee, HW; Chan, CK, A new method for solving equilibrium problem fixed point problem with application to optimization, Nonlinear Analysis: Theory, Methods & Applications, 70, 3307-3319, (2009) · Zbl 1198.47082
[5] Colao, V; Marino, G; Xu, H-K, An iterative method for finding common solutions of equilibrium and fixed point problems, Journal of Mathematical Analysis and Applications, 344, 340-352, (2008) · Zbl 1141.47040
[6] Plubtieng, S; Punpaeng, R, A general iterative method for equilibrium problems and fixed point problems in Hilbert spaces, Journal of Mathematical Analysis and Applications, 336, 455-469, (2007) · Zbl 1127.47053
[7] Rockafellar, RT, Monotone operators and the proximal point algorithm, SIAM Journal on Control and Optimization, 14, 877-898, (1976) · Zbl 0358.90053
[8] Adly, S, Perturbed algorithms and sensitivity analysis for a general class of variational inclusions, Journal of Mathematical Analysis and Applications, 201, 609-630, (1996) · Zbl 0856.65077
[9] Peng, J-W; Wang, Y; Shyu, DS; Yao, J-C, Common solutions of an iterative scheme for variational inclusions, equilibrium problems, and fixed point problems, No. 2008, 15, (2008) · Zbl 1161.65050
[10] Aoyama, K; Kimura, Y; Takahashi, W; Toyoda, M, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Analysis: Theory, Methods & Applications, 67, 2350-2360, (2007) · Zbl 1130.47045
[11] Blum, E; Oettli, W, From optimization and variational inequalities to equilibrium problems, The Mathematics Student, 63, 123-145, (1994) · Zbl 0888.49007
[12] Marino, G; Xu, H-K, A general iterative method for nonexpansive mappings in Hilbert spaces, Journal of Mathematical Analysis and Applications, 318, 43-52, (2006) · Zbl 1095.47038
[13] Xu, H-K, Iterative algorithms for nonlinear operators, Journal of the London Mathematical Society, 66, 240-256, (2002) · Zbl 1013.47032
[14] Brézis H: Opérateurs Maximaux Monotones et Semi-Groupes de Contractions dans les Espaces de Hilbert. North-Holland, Amsterdam, The Netherlands; 1973:vi+183. · Zbl 0252.47055
[15] Lemaire, B, Which fixed point does the iteration method select?, No. 452, 154-167, (1997), Berlin, Germany · Zbl 0882.65042
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.