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Hybrid viscosity-like approximation methods for nonexpansive mappings in Hilbert spaces. (English) Zbl 1192.47054

Summary: Consider on a real Hilbert space $H$ a nonexpansive mapping $T$ with a fixed point, a contraction $f$ with coefficient $0<\alpha <1$, and two strongly positive linear bounded operators $A,B$ with coefficients $\overline{\gamma }\in \left(0,1\right)$ and $\beta >0$, respectively. Let $0<\gamma \alpha <\beta$. We introduce a general iterative algorithm defined by

${x}_{n+1}:=\left(I-{\lambda }_{n+1}A\right)T{x}_{n}+{\lambda }_{n+1}\left[T{x}_{n}-{\mu }_{n+1}\left(BT{x}_{n}-\gamma f\left({x}_{n}\right)\right)\right],\phantom{\rule{1.em}{0ex}}\forall n\ge 1,$

with ${\mu }_{n}\to \mu \left(n\to \infty \right)$, and prove the strong convergence of the iterative algorithm to a fixed point $\stackrel{˜}{x}\in \text{Fix}\left(T\right)=:C$ which is the unique solution of the variational inequality (for short, $\text{VI}\left(A-I+\mu \left(B-\gamma f\right),C\right)\right):〈\left[A-I+\mu \left(B-\gamma f\right)\right]\stackrel{˜}{x},x-\stackrel{˜}{x}〉\ge 0,\phantom{\rule{1.em}{0ex}}\forall x\in C$. On the other hand, assume $C$ is the intersection of the fixed-point sets of a finite number of nonexpansive mappings on $H$. We devise another iterative algorithm which generates a sequence $\left\{{x}_{n}\right\}$ from an arbitrary initial point ${x}_{0}\in H$. The sequence $\left\{{x}_{n}\right\}$ is proven to converge strongly to an element of $C$ which is the unique solution ${x}^{*}$ of the $\text{VI}\left(A-I+\mu \left(B-\gamma f\right),C\right)$. Applications to constrained generalized pseudoinverses are included.

##### MSC:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 47H10 Fixed point theorems for nonlinear operators on topological linear spaces 47J22 Variational and other types of inclusions 65J15 Equations with nonlinear operators (numerical methods)
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