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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:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
54H25Fixed-point and coincidence theorems in topological spaces
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References:
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