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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:
  G. L. Acedo and H. K. Xu: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67 (2007), 2258–2271. · Zbl 1133.47050  F. E. Browder: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 53 (1965), 1272–1276. · Zbl 0125.35801  F. E. Browder: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24 (1967), 82–90. · Zbl 0148.13601  F. E. Browder and W. V. Petryshyn: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20 (1967), 197–228. · Zbl 0153.45701  B. Halpern: Fixed points of nonexpansive maps. Bull. Amer. Math. Soc. 73 (1967), 957–961. · Zbl 0177.19101  P. L. Lions: Approximation de points fixes de contractions. C.R. Acad. Sci. Paris Ser. A-B 284 (1977), A1357–A1359. · Zbl 0349.47046  G. Marino and H. K. Xu: Weak and strong convergence theorems for k-strict pseudocontractions in Hilbert spaces. J. Math. Anal. Appl. 329 (2007), 336–349. · Zbl 1116.47053  G. Marino and H. K. Xu: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318 (2006), 43–52. · Zbl 1095.47038  A. Moudafi: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 241 (2000), 46–55. · Zbl 0957.47039  T. Suzuki: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. (2005), 103–123. · Zbl 1123.47308  R. Wittmann: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58 (1992), 486–491. · Zbl 0797.47036  H. K. Xu: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116 (2003), 659–678. · Zbl 1043.90063  H. K. Xu: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66 (2002), 240–256. · Zbl 1013.47032  H. K. Xu: Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65 (2002), 109–113. · Zbl 1030.47036  H. K. Xu: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298 (2004), 279–291. · Zbl 1061.47060  H. Zhou: Convergence theorems of fixed points for k-strict pseudo-contractions in Hilbert space. Nonlinear Analysis 69 (2008), 456–462. · Zbl 1220.47139
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