Approximating fixed points of nonexpansive type mappings. (Q5946264)

This article deals with Ishikawa approximations NEWLINE\[NEWLINEx_{n+1}= (1 -\alpha_n)x_n+ \alpha_nT\bigl((1-\beta_n) x_n+\beta_n Tx_n\bigr)NEWLINE\]NEWLINE for an operator \(T:D\to D\) \((D\) is a nonempty, closed and convex subset of a uniformly convex Banach space \(B\), \(0\leq\alpha_n \leq\beta_n\leq 1\), \(\lim_{n\to\infty} \beta_n=0\), \(\sum^\infty_{n=1} \alpha_n \beta_n= \infty)\), under the assumption that NEWLINE\[NEWLINE\| Tx-Ty\|\leq \max \left\{\beta \|-y\|,\frac {\| x-Tx\|+\| y-Ty\|} {2}, \frac {\| x-Ty\|+\| y-Tx\|} {2}\right\}NEWLINE\]NEWLINE or NEWLINE\[NEWLINE\| Tx-Ty\| \leq \max\left\{ \beta\| x-y\|,\frac {\| x-Tx\|+\| y-Ty \|} {2}, \| x-Ty\|,\| y-Tx\|\right\}NEWLINE\]NEWLINE \((0\leq\beta<1)\). In the first case, it is proved that \(\|(I-T((1-\mu)x+\mu Tx)x\|\geq f(d (x,\text{Fix}T))\), \(0\leq \mu\leq\beta\) for a nondecreasing function \(f:[0, \infty) \to[0,\infty)\), \(f(0)=0\), \(f(r)>0\) \((r>0)\) and, in the second one, \(\|(I-T((1-\mu)x +\mu Tx)x\|\geq kd(x, \text{Fix}\,T))\), \(0\leq\mu \leq\beta\) for a \(k<0\); in both cases, the Ishikawa approximations converge to the fixed point of \(T\).