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A general iterative method for nonexpansive mappings in Hilbert spaces. (English) Zbl 1095.47038
Let $$H$$ be a Hilbert space, $$T:H\rightarrow H$$ a nonexpansive mapping, $$f:H\rightarrow H$$ a contraction and $$A$$ a strongly positive linear bounded operator. In order to approximate the unique solution $$x^{*}$$ of the variational inequality $\left\langle \left(A-\gamma f\right)x^{*}, x-x^{*}\right\rangle \geq 0,\quad x\in \text{Fix}(T),$ the authors presents two strong convergence theorems for: (1) a continuous path $$\{x_t\}$$ (Theorem 3.2) and (2) a general viscosity type iterative method $$\{x_n\}$$ of the form $x_{n+1}=\left(I-\alpha_n A\right) T x_n+\alpha_n \gamma f(x_n),\quad n\geq 0,$ where $$\{\alpha_n\}\subset [0,1]$$ is a sequence of real numbers satisfying appropriate conditions.

##### MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems 54H25 Fixed-point and coincidence theorems (topological aspects)
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