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Strong convergence theorems of $$k$$-strict pseudo-contractions in Hilbert spaces. (English) Zbl 1218.47115
Summary: Let $$K$$ be a nonempty closed convex subset of a real Hilbert space $$H$$ such that $$K\pm K\subset K$$, let $$T: K\rightarrow H$$ be a $$k$$-strict pseudo-contraction for some $$0\leq k<1$$ such that $$F(T)=\{x\in K\: x=Tx\}\neq \emptyset$$. Consider the iterative algorithm given by
$\forall x_1\in K,\quad x_{n+1}=\alpha _n\gamma f(x_n)+\beta _nx_n+((1-\beta _n)I-\alpha _n A)P_KSx_n,\quad n\geq 1,$
where $$S: K\rightarrow H$$ is defined by $$Sx=kx+(1-k)Tx$$, $$P_K$$ is the metric projection of $$H$$ onto $$K$$, $$A$$ is a strongly positive linear bounded selfadjoint operator, and $$f$$ is a contraction. It is proved that the sequence $$\{x_n\}$$ generated by the above iterative algorithm converges strongly to a fixed point of $$T$$, which solves a variational inequality related to the linear operator $$A$$. Our results improve and extend the results announced by many others.
MSC:
 47J25 Iterative procedures involving nonlinear operators 47H09 Contraction-type mappings, nonexpansive mappings, $$A$$-proper mappings, etc. 47H10 Fixed-point theorems
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References:
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