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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:CC 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 k T(x k );k0,

where x 0 C and (t k ) + . Then (i) if (x k ) remains in C and 0<at k b<1-p, k, then (x k ) converges weakly to an element of F(T); (ii) if in addition x-Tx,h0, for all xC and some hC, h0, then (x k ) converges strongly to an element of F(T).

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