A general iterative method for nonexpansive mappings in Hilbert spaces. (English) Zbl 1095.47038

Marino, Giuseppe; Xu, Hong-Kun

J. Math. Anal. Appl. 318, No. 1, 43-52 (2006).

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.

Reviewer: Vasile Berinde (Baia Mare)

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)

Keywords:

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

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