A new hybrid projection algorithm for system of equilibrium problems and variational inequality problems and two finite families of quasi-\(\phi\)-nonexpansive mappings. (English) Zbl 1263.49007

Phuangphoo, Pongrus; Kumam, Poom

Abstr. Appl. Anal. 2013, Article ID 107296, 13 p. (2013).

Summary: We introduce a modified Mann’s iterative procedure by using the hybrid projection method for finding the common solution of a system of equilibrium problems for a finite family of bifunctions satisfying certain conditions, the common solution of fixed-point problems for two finite families of quasi-\(\phi\)-nonexpansive mappings, and the common solution of variational inequality problems for a finite family of continuous monotone mappings in a uniformly smooth and strictly convex real Banach space. Then, we prove a strong convergence theorem of the iterative procedure generated by some mild conditions. Our result presented in this paper improves and generalizes some well-known results in the literature.

MSC:

49J40	Variational inequalities
90C33	Complementarity and equilibrium problems and variational inequalities (finite dimensions) (aspects of mathematical programming)
47H09	Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47H10	Fixed-point theorems

Keywords:

modified Mann’s iterative procedure; hybrid projection algorithm; equilibrium problems; variational inequalities; quasi-\(\phi\)-nonexpansive mappings

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