An iterative method for finding common solutions of equilibrium and fixed point problems. (English) Zbl 1141.47040

By combining the iterative scheme for a finite family of nonexpansive mappings due to S. Atsushiba and W. Takahashi [Indian J. Math. 41, No. 3, 435–453 (1999; Zbl 1055.47514)] and the iterative scheme for finding the solution of an equilibrium problem due to S. Plubtieng and R. Punpaeng [J. Math. Anal. Appl. 336, No. 1, 455–469 (2007; Zbl 1127.47053)], the authors present an iterative scheme for finding a common element of the set of solutions of an equilibrium problem and of the set of fixed points of a finite family of nonexpansive mappings in Hilbert space. They prove the strong convergence of the sequences generated by the proposed iterative scheme to a unique solution of a variational inequality. Such type of variational inequality provides the optimality condition for a minimization problem.


47J25 Iterative procedures involving nonlinear operators
47J20 Variational and other types of inequalities involving nonlinear operators (general)
49J40 Variational inequalities
47H10 Fixed-point theorems
Full Text: DOI


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