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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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