The convergence theorems of the sequence of Ishikawa iterates for hemicontractive mappings (Q806020)

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).