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Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions. (English) Zbl 1220.47100
Summary: Let $C$ be a closed convex subset of a real Hilbert space $H$ and assume that $T$ is an asymptotically $\kappa$-strict pseudo-contraction on $C$ with a fixed point, for some $0\le \kappa<1$. Given an initial guess $x_0\in C$ and a real sequence $\{\alpha_n\}$ in $(0,1)$, the modified Mann algorithm generates a sequence $\{x_n\}$ via the formula: $x_{n+1}=\alpha_nx_n+(1-\alpha_n)T^nx_n$, $n\ge 0$. It is proved that if the control sequence $\{\alpha_n\}$ is chosen so that $\kappa+\delta<\alpha_n-\delta$ for some $\delta\in(0,1)$, then $\{x_n\}$ converges weakly to a fixed point of $T$. We also modify this iteration method by applying projections onto suitably constructed closed convex sets to get an algorithm which generates a strongly convergent sequence.

MSC:
47J25Iterative procedures (nonlinear operator equations)
47H09Mappings defined by “shrinking” properties
65J15Equations with nonlinear operators (numerical methods)
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References:
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