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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 $$\lim_{n \to \infty} \prod_{k=0}^n [\frac\delta{1 - (1 + \delta)\alpha_k}]=0$$.

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