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

Colao, Vittorio; Marino, Giuseppe; Xu, Hong-Kun

J. Math. Anal. Appl. 344, No. 1, 340-352 (2008).

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.

Reviewer: Qamrul Hasan Ansari (Dhahran)

MSC:

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

Keywords:

equilibrium problem; fixed point; nonexpansive mapping; variational inequality; iterative algorithm

Citations:

Zbl 1055.47514; Zbl 1127.47053

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