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On the Mann-type iteration and the convex feasibility problem. (English) Zbl 1135.65027

Let $C$ be a closed convex subset of a Hilbert space $H$; and $T:C\to C$ be a nonlinear map with nonempty fixed point set $F\left(T\right)$ in $C$ fulfilling (a) $T$ is $p$-demicontractive on $C$, (b) $I-T$ is demiclosed at zero. Let $\left({x}_{k}\right)$ be the Mann-type iterative process

${x}_{k+1}=\left(1-{t}_{k}\right){x}_{k}+{t}_{k}T\left({x}_{k}\right);\phantom{\rule{1.em}{0ex}}k\ge 0,$

where ${x}_{0}\in C$ and $\left({t}_{k}\right)\subset {ℝ}^{+}$. Then (i) if $\left({x}_{k}\right)$ remains in $C$ and $0, $\forall k$, then $\left({x}_{k}\right)$ converges weakly to an element of $F\left(T\right)$; (ii) if in addition $〈x-Tx,h〉\le 0$, for all $x\in C$ and some $h\in C$, $h\ne 0$, then $\left({x}_{k}\right)$ converges strongly to an element of $F\left(T\right)$.

##### MSC:
 65J15 Equations with nonlinear operators (numerical methods) 47J25 Iterative procedures (nonlinear operator equations) 47H10 Fixed point theorems for nonlinear operators on topological linear spaces