2006 / September

Approximating fixed points of nonexpansive mappings

Stevo Stević

Iteration method, fixed point, nonexpansive mapping, sequence, Iteration method, fixed point, nonexpansive mapping, sequence

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437-450

Let $D$ be a subset of a normed space $X$ and $T: D \to X$ be a nonexpansive mapping. In this paper we consider the following iteration method which generalizes Ishikawa iteration process:
$x_{n+1} = t^{(1)}_n T(t^{(2)}_n T(\dots T(t^{(k)}_n Tx_n + (1 - t^{(k)}_n)x_n + u^{(k)}_n) + \dots)$
$+ (1-t^{(2)}_n)x_n + u^{(2)}_n) + (1 - t^{(1)}_n)x_n + u^{(1)}_n $,
$n=1, 2, 3 \dots$, where $0 \le t^{(i)}_n \le 1$ for all $n \ge 1$ and $i = 1, \dots, k$, and sequence {$x_n$} and {$u^{(i)}_n$}, $i=1, \dots, k$, are in $D$.
We improve several results in [2], concerning approximation of fixed points of $T$.

47H10

2005-02-05