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An iterative algorithm for solving fixed point problems, variational inequality problems and mixed equilibrium problems. (English) Zbl 1166.49013
Summary: We introduce an iterative algorithm for finding a common element of the set of solutions of a mixed equilibrium problem, the set of fixed points of an infinite family of nonexpansive mappings and the set of solutions of some variational inequality. Furthermore, we prove that the proposed iterative algorithm converges strongly to a common element of the above three sets. Our results extend and improve the corresponding results of L. C. Zeng and J. C. Yao [A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214, 186–201 (2008)], S. Takahashi and W. Takahashi [J. Math. Anal. Appl. 331, No. 1, 506–515 (2007; Zbl 1122.47056)] and many others.
MSC:
49J40Variational methods including variational inequalities
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47H09Mappings defined by “shrinking” properties
49M05Numerical methods in calculus of variations based on necessary conditions
90C25Convex programming
90C99Mathematical programming