# Article

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Keywords:
Hilbert space; nonexpansive mapping; strict pseudo-contraction; iterative algorithm; fixed point
Summary:
Let $K$ be a nonempty closed convex subset of a real Hilbert space $H$ such that $K\pm K\subset K$, $T\: K\rightarrow H$ a $k$-strict pseudo-contraction for some $0\leq k<1$ such that $F(T)=\{x\in K\: x=Tx\}\neq \emptyset$. Consider the following 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 self-adjoint operator, $f$ 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.
References:
[1] Acedo, G. L., Xu, H. K.: Iterative methods for strict pseudo-contractions in Hilbert spaces. Nonlinear Anal. 67 (2007), 2258-2271. DOI 10.1016/j.na.2006.08.036 | MR 2331876 | Zbl 1133.47050
[2] Browder, F. E.: Fixed point theorems for noncompact mappings in Hilbert spaces. Proc. Natl. Acad. Sci. USA 53 (1965), 1272-1276. DOI 10.1073/pnas.53.6.1272 | MR 0178324
[3] Browder, F. E.: Convergence of approximants to fixed points of nonexpansive nonlinear mappings in Banach spaces. Arch. Ration. Mech. Anal. 24 (1967), 82-90. DOI 10.1007/BF00251595 | MR 0206765 | Zbl 0148.13601
[4] Browder, F. E., Petryshyn, W. V.: Construction of fixed points of nonlinear mappings in Hilbert space. J. Math. Anal. Appl. 20 (1967), 197-228. DOI 10.1016/0022-247X(67)90085-6 | MR 0217658 | Zbl 0153.45701
[5] Halpern, B.: Fixed points of nonexpansive maps. Bull. Amer. Math. Soc. 73 (1967), 957-961. DOI 10.1090/S0002-9904-1967-11864-0 | MR 0218938
[6] Lions, P. L.: Approximation de points fixes de contractions. C.R. Acad. Sci. Paris Ser. A--B 284 (1977), A1357--A1359. MR 0470770 | Zbl 0349.47046
[7] Marino, G., Xu, H. K.: Weak and strong convergence theorems for $k$-strict pseudo-contractions in Hilbert spaces. J. Math. Anal. Appl. 329 (2007), 336-349. DOI 10.1016/j.jmaa.2006.06.055 | MR 2306805
[8] Marino, G., Xu, H. K.: A general iterative method for nonexpansive mappings in Hilbert spaces. J. Math. Anal. Appl. 318 (2006), 43-52. DOI 10.1016/j.jmaa.2005.05.028 | MR 2210870 | Zbl 1095.47038
[9] Moudafi, A.: Viscosity approximation methods for fixed points problems. J. Math. Anal. Appl. 241 (2000), 46-55. DOI 10.1006/jmaa.1999.6615 | MR 1738332 | Zbl 0957.47039
[10] Suzuki, T.: Strong convergence theorems for infinite families of nonexpansive mappings in general Banach spaces. Fixed Point Theory Appl. (2005), 103-123. MR 2172156 | Zbl 1123.47308
[11] Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. 58 (1992), 486-491. DOI 10.1007/BF01190119 | MR 1156581 | Zbl 0797.47036
[12] Xu, H. K.: An iterative approach to quadratic optimization. J. Optim. Theory Appl. 116 (2003), 659-678. DOI 10.1023/A:1023073621589 | MR 1977756 | Zbl 1043.90063
[13] Xu, H. K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. 66 (2002), 240-256. DOI 10.1112/S0024610702003332 | MR 1911872 | Zbl 1013.47032
[14] Xu, H. K.: Another control condition in an iterative method for nonexpansive mappings. Bull. Austral. Math. Soc. 65 (2002), 109-113. DOI 10.1017/S0004972700020116 | MR 1889384 | Zbl 1030.47036
[15] Xu, H. K.: Viscosity approximation methods for nonexpansive mappings. J. Math. Anal. Appl. 298 (2004), 279-291. DOI 10.1016/j.jmaa.2004.04.059 | MR 2086546 | Zbl 1061.47060
[16] Zhou, H.: Convergence theorems of fixed points for $k$-strict pseudo-contractions in Hilbert space. Nonlinear Analysis 69 (2008), 456-462. DOI 10.1016/j.na.2007.05.032 | MR 2426262

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