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Picard iteration converges faster than Mann iteration for a class of quasi-contractive operators. (English) Zbl 1153.47055
The article deals with fixed points of mappings in metric spaces. A mapping \(T: X \to X\) (\(X\) is a metric space with a metric \(d\)) is called quasi-\(\varphi\)-contractive if \[ d(Tx,Ty)\leq\varphi(d(x,y))+L m(x,y)\quad\text{for some }L\geq 0, \] where \(m(x,y)=\min\{d(x,Tx),d(y,Ty),d(x,Ty),d(y,Tx)\}\). Here, \(\varphi\) is a function satisfying two conditions: (i) \(\varphi\) is monotone increasing, and (ii) \((\varphi^{(n)}(t))_{n=0}^\infty\) converges to \(0\) for all \(t \geq 0\). It is proved that a quasi-\(\varphi\)-mapping in a complete metric space \((X,d)\) has a unique fixed point and the Picard iterations \(x_{n+1} = Ax_n\), \(n = 0,1,2,\dots\), converge to this fixed point. If \(X\) is a closed convex subset of a Banach space, one can consider the Mann iterations \[ x_{n+1} = (1 - \alpha_n)x_n + \alpha_nTx_n,\quad n = 0,1,2,\dots, \] with \(\alpha_n \in [0,1]\) and \(\sum_{n=0}^\infty \alpha_n = \infty\). If \(\varphi(t) = \delta t\), \(0 \leq \delta < 1\), it is proved that Picard iterations converge faster than Mann iterations, provided that \(\alpha_n < \frac1{1 + \delta}\) for all \(n = 0,1,2,\dots\) and \(\lim_{n \to \infty} \prod_{k=0}^n [\frac\delta{1 - (1 + \delta)\alpha_k}]=0\).

47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
54H25 Fixed-point and coincidence theorems (topological aspects)