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Zbl pre05883045
Shehu, Yekini
Iterative methods for fixed points and equilibrium problems. (Strong convergence theorems for family of relatively quasi nonexpansive mappings and systems of mixed equilibrium problems.)
(English)
[J] Ann. Funct. Anal. AFA 1, No. 2, 121-132, electronic only (2010). ISSN 2008-8752/e

Let $E$ be a uniformly convex real Banach space which is also uniformly smooth, and let $C$ be a nonempty closed convex subset of $E$. Consider the following: \par 1. An infinite family of closed relatively-quasi nonexpansive mappings $\{T_i\}_{i=1}^{\infty}$, $T_i:C\rightarrow C$, and denote by $Fix\,(T_i)$ the set of all fixed points of $T_i$, i.e., $$Fix\,(T_i)=\left\{x\in E:T_i(x)=x\right\}.$$ 2. A finite family of equilibrium bifunctions $\{F_j\}_{j=1}^{m}$, $F_j:C\times C\rightarrow \mathbb{R}$, which define the equilibrium problems: find $x\in C$ such that $F_j(x,y)\geq 0,\,\forall y\in C$, $j=1,2,\dots,m$. Denote by $EP(F_j),\,j=1,2,\dots,m$ the set of equilibrium points of these problems. \par In order to approximate an element of the set $$\cap_{k=1}^{m} EP(F_j) \cap \left(\cap_{i=1}^{\infty}Fix\,(T)\right),$$ a priori supposed to be nonempty, the authors introduce a projection type hybrid algorithm, which involves the duality mapping $J$ on $E$ and the operator $T$ defined by $T:=J^{-1}\left(\sum_{i=0}^{\infty}\zeta_i JT_i\right)$. The main result of the paper (Theorem 3.1) is a strong convergence theorem for this algorithm. \par Several corollaries are also obtained from the main result. \par No examples are given to illustrate that the hypotheses in Theorem 3.1 or its corollaries are feasible.
[Vasile Berinde (Baia Mare)]
MSC 2000:
*47J25 Methods for solving nonlinear operator equations (general)
47H09 Mappings defined by "shrinking" properties
47H10 Fixed point theorems for nonlinear operators on topol.linear spaces
47J20 Inequalities involving nonlinear operators

Keywords: Banach space; fixed point; equilibrium problem; closed relatively-quasi nonexpansive mapping; equilibrium bifunction; convergence theorem; generalized projection; duality mapping; hybrid iterative scheme

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