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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 \(\| Tx-p\|^ 2\leq \| x-p\|^ 2+\| x-Tx\|^ 2\) for \(x\in C\) and \(p\in Fix(T);\)
(ii) generalized contractive, if \(\| Tx-Ty\| <\max \{\| x- y\|\), \(\| x-Tx\|\), \(\| y-Ty\|\), \(\| x-Ty\|\), \(\| y-Tx\| \}\) for x,y\(\in C\), \(x\neq y.\)
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).

47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
47J25 Iterative procedures involving nonlinear operators
Full Text: DOI
[1] Qihou, Liu, On naimpally and Singh’s open questions, J. math. anal. appl., 124, 157-164, (1987) · Zbl 0625.47044
[2] Naimpally, S.A; Singh, K.L, Extensions of some fixed point theorems of rhoades, J. math. anal. appl., 96, 437-446, (1983) · Zbl 0524.47033
[3] Ishikawa, S, Fixed points by a new iterator method, (), 147-150 · Zbl 0286.47036
[4] Rhoades, B.E, Comments on two fixed iteration methods, J. math. anal. appl., 56, 741-750, (1976) · Zbl 0353.47029
[5] {\scLiu Qihou}, The convergence theorems of the sequence of Ishikawa iterates for quasicontractive mappings, J. Math. Anal. Appl., in press. · Zbl 0729.47052
[6] Rhoades, B.E, A comparison of various definitions of contractive mappings, Trans. amer. math. soc., 226, 257-290, (1977) · Zbl 0365.54023
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