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Zbl 1198.65100
Zegeye, Habtu; Ofoedu, Eric U.; Shahzad, Naseer
Convergence theorems for equilibrium problem, variational inequality problem and countably infinite relatively quasi-nonexpansive mappings.
(English)
[J] Appl. Math. Comput. 216, No. 12, 3439-3449 (2010). ISSN 0096-3003

Let $E$ be a real Banach space and $C\subset E$ closed, convex and nonempty. A mapping $A:D(A)\subset E \rightarrow E'$ is said to be $\gamma$-inverse strongly monotone if there exists $\gamma >0$ such that $$(Ax-Ay,x-y)\geq\gamma \parallel Ax-Ay\parallel^{2},\quad \forall x,y\in D(A).$$ The authors introduce an iterative process of the type: $$x_{n-1}=\Pi_{C_{n-1}}(x_{0}); \quad x_{0}\in C_{0}(\equiv C) \quad {\text{arbitrary}} \quad n\geq 0,$$ 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.
[Erwin Schechter (Moers)]
MSC 2000:
*65J15 Equations with nonlinear operators (numerical methods)
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J25 Methods for solving nonlinear operator equations (general)
47J40 Equations with hysteresis operators
65K15
47H09 Mappings defined by "shrinking" properties

Keywords: equilibrium problems; strong convergence; variational inequalities; common fixed points; iterative methods; Banach space; quasi-nonexpensive mappings;

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