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Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings. (English) Zbl 1198.65100

Let E be a real Banach space and CE closed, convex and nonempty. A mapping A:D(A)EE ' is said to be γ-inverse strongly monotone if there exists γ>0 such that

(Ax-Ay,x-y)γAx-Ay 2 ,x,yD(A)·

The authors introduce an iterative process of the type:

x n-1 =Π C n-1 (x 0 );x 0 C 0 (C)arbitraryn0,

converging strongly to a common element of the set of common fixed points of the countably infinite family of closed relatively quasi-nonexpensive mappings, the solution set of a generalized equilibrium problem and the solution set of a variational inequality problem for a γ-inverse strongly monotone mapping in Banach spaces. The theorems of the paper improve, generalize, unify and extend several known results.

65J15Equations with nonlinear operators (numerical methods)
47H10Fixed point theorems for nonlinear operators on topological linear spaces
47J25Iterative procedures (nonlinear operator equations)
47J40Equations with hysteresis operators
65K15Numerical methods for variational inequalities and related problems
47H09Mappings defined by “shrinking” properties
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