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The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings. (English) Zbl 0729.47052

Let C be a nonempty subset of a normed space. An operator T: $C\to C$ is called

(i) hemicontractive, if ${\parallel Tx-p\parallel }^{2}\le {\parallel x-p\parallel }^{2}+{\parallel x-Tx\parallel }^{2}$ for $x\in C$ and $p\in Fix\left(T\right);$

(ii) generalized contractive, if $\parallel Tx-Ty\parallel , $\parallel x-Tx\parallel$, $\parallel y-Ty\parallel$, $\parallel x-Ty\parallel$, $\parallel y-Tx\parallel \right\}$ for x,y$\in C$, $x\ne y·$

The author proves theorems on convergence of the sequence of Ishikawa iterates in the case where T is continuously mapping a compact and convex subset C of a Hilbert space into itself, and satisfied either (i) or (ii).

##### MSC:
 47H09 Mappings defined by “shrinking” properties 47J25 Iterative procedures (nonlinear operator equations)