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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
\[ x_{k+1}=(1-t_k)x_k+t_kT(x_k); \quad k\geq 0, \]
where \(x_0\in C\) and \((t_k)\subset \mathbb R^+\). Then (i) if \((x_k)\) remains in \(C\) and \(0< a\leq t_k\leq b< 1-p\), \(\forall k\), then \((x_k)\) converges weakly to an element of \(F(T)\); (ii) if in addition \(\langle x-Tx,h\rangle\leq 0\), for all \(x\in C\) and some \(h\in C\), \(h\neq 0\), then \((x_k)\) converges strongly to an element of \(F(T)\).

MSC:

65J15 Numerical solutions to equations with nonlinear operators
47J25 Iterative procedures involving nonlinear operators
47H10 Fixed-point theorems
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