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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 $\overline\gamma\in (0,1)$ and $\beta>0$, respectively. Let $0<\gamma\alpha<\beta$. We introduce a general iterative algorithm defined by $$x_{n+1}:=(I-\lambda_{n+1}A)Tx_n+\lambda_{n+1}[Tx_n-\mu_{n+1}(BTx_n-\gamma f(x_n))],\quad \forall n\ge 1,$$ with $\mu_n\to\mu(n\to \infty)$, and prove the strong convergence of the iterative algorithm to a fixed point $\widetilde x\in\text{Fix}(T)=:C$ which is the unique solution of the variational inequality (for short, $\text{VI}(A-I+\mu(B-\gamma f),C)): \langle[A-I+\mu(B-\gamma f)]\widetilde x,x-\widetilde x\rangle\ge 0,\quad \forall x\in C$. 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 $\{x_n\}$ from an arbitrary initial point $x_0\in H$. The sequence $\{x_n\}$ is proven to converge strongly to an element of $C$ which is the unique solution $x^*$ of the $\text{VI}(A-I+\mu(B-\gamma f),C)$. Applications to constrained generalized pseudoinverses are included.

47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
47J22Variational and other types of inclusions
65J15Equations with nonlinear operators (numerical methods)
Full Text: DOI
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