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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 $\left\{{\alpha }_{n}\right\}$ in $\left(0,1\right)$, the modified Mann algorithm generates a sequence $\left\{{x}_{n}\right\}$ via the formula: ${x}_{n+1}={\alpha }_{n}{x}_{n}+\left(1-{\alpha }_{n}\right){T}^{n}{x}_{n}$, $n\ge 0$. It is proved that if the control sequence $\left\{{\alpha }_{n}\right\}$ is chosen so that $\kappa +\delta <{\alpha }_{n}-\delta$ for some $\delta \in \left(0,1\right)$, then $\left\{{x}_{n}\right\}$ 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:
 47J25 Iterative procedures (nonlinear operator equations) 47H09 Mappings defined by “shrinking” properties 65J15 Equations with nonlinear operators (numerical methods)