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Zbl 1095.47038
Marino, Giuseppe; Xu, Hong-Kun
A general iterative method for nonexpansive mappings in Hilbert spaces.
(English)
[J] J. Math. Anal. Appl. 318, No. 1, 43-52 (2006). ISSN 0022-247X

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.
[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
54H25 Fixed-point theorems in topological spaces

Keywords: Hilbert space; nonexpansive mapping; fixed point; contraction; iterative method; projection; variational inequality; viscosity approximation

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