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Ishikawa iterations for equilibrium and fixed point problems for nonexpansive mappings in Hilbert spaces. (English) Zbl 1175.47059

Let \(H\) be a Hilbert space and let \(G:H\times H\rightarrow \mathbb{R}\) be a bifunction. It is said that \(x\in H\) is an equilibrium point if \(G(x,y)\geq 0\) for all \(y\in H\). Denote by \(EP(G)\) the set of all equilibrium points of \(G\). Let also \(T:H\rightarrow H\) be a nonexpansive mapping and denote by Fix\(\,(T)\) the set of all fixed points of \(T\). Suppose \(G\) and \(T\) are such that \(EP(G)\cap \text{Fix}\,(T)\neq \emptyset\).
The authors introduce a viscosity type Ishikawa iteration procedure in order to compute an element \(z\in EP(G)\cap \text{Fix}\,(T)\), which is in fact the unique solution of the variational inequality
\[ \left\langle (A-\lambda f)z,x-z\right\rangle \geq 0 \,\,\,\forall x\in EP(G)\cap \text{Fix}\,(T), \]
where \(f:H\rightarrow H\) is a \(\alpha\)-contraction and \(A\) is a strongly positive linear bounded selfadjoint operator on \(H\) into itself.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
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