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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 $C\subset E$ closed, convex and nonempty. A mapping $A:D\left(A\right)\subset E\to {E}^{\text{'}}$ is said to be $\gamma$-inverse strongly monotone if there exists $\gamma >0$ such that

$\left(Ax-Ay,x-y\right)\ge \gamma \parallel Ax-Ay{\parallel }^{2},\phantom{\rule{1.em}{0ex}}\forall x,y\in D\left(A\right)·$

The authors introduce an iterative process of the type:

${x}_{n-1}={{\Pi }}_{{C}_{n-1}}\left({x}_{0}\right);\phantom{\rule{1.em}{0ex}}{x}_{0}\in {C}_{0}\left(\equiv C\right)\phantom{\rule{1.em}{0ex}}\text{arbitrary}\phantom{\rule{1.em}{0ex}}n\ge 0,$

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 $\gamma$-inverse strongly monotone mapping in Banach spaces. The theorems of the paper improve, generalize, unify and extend several known results.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J25 Iterative procedures (nonlinear operator equations) 47J40 Equations with hysteresis operators 65K15 Numerical methods for variational inequalities and related problems 47H09 Mappings defined by “shrinking” properties