The equivalence between the Mann and Ishikawa iterations dealing with generalized contractions (Q884240)

The authors once again study and extend results about the equivalence between Mann and Ishikawa iterations using generalized contractions to obtain approximate fixed points of a map \(T\) satisfying the condition \(\| Tx-Ty\| \leq Q(M(x,y))\), where \(Q\) is a real-valued function satisfying (a) \(0 < Q(s) < s\) for each \(s > 0\) and \(Q(0)=0\); (b) \(Q\) is nondecreasing on \((0, \infty )\); (c) \(g(s):= \frac {s}{(s-Q(s))}\) is nonincreasing on \((0, \infty )\), and (d) \(M(x,y):= \max \{ \| x-y\|,\| x- Tx\|, \| y-Ty\|, \| x-Ty\|, \| y-Tx\|\}\).