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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 $F(T)$ in $C$ fulfilling (a) $T$ is $p$-demicontractive on $C$, (b) $I-T$ is demiclosed at zero. Let $(x_k)$ be the Mann-type iterative process $$x_{k+1}=(1-t_k)x_k+t_kT(x_k); \quad k\ge 0,$$ where $x_0\in C$ and $(t_k)\subset \Bbb R^+$. Then (i) if $(x_k)$ remains in $C$ and $0< a\le t_k\le b< 1-p$, $\forall k$, then $(x_k)$ converges weakly to an element of $F(T)$; (ii) if in addition $\langle x-Tx,h\rangle\le 0$, for all $x\in C$ and some $h\in C$, $h\ne 0$, then $(x_k)$ converges strongly to an element of $F(T)$.

65J15Equations with nonlinear operators (numerical methods)
47J25Iterative procedures (nonlinear operator equations)
47H10Fixed-point theorems for nonlinear operators on topological linear spaces
Full Text: DOI
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