Browse > Article

STRONG CONVERGENCE OF A NEW ITERATIVE ALGORITHM FOR AVERAGED MAPPINGS IN HILBERT SPACES  

Yao, Yonghong (Department of Mathematics, Tianjin Polytechnic University)
Zhou, Haiyun (Department of Mathematics, Shijiazhuang Mechanical Engineering College)
Chen, Rudong (Department of Mathematics, Tianjin Polytechnic University)
Publication Information
Journal of applied mathematics & informatics / v.28, no.3_4, 2010 , pp. 939-944 More about this Journal
Abstract
Let H be a real Hilbert space. Let T : $H\;{\rightarrow}\;H$ be an averaged mapping with $F(T)\;{\neq}\;{\emptyset}$. Let {$\alpha_n$} be a real numbers in (0, 1). For given $x_0\;{\in}\;H$, let the sequence {$x_n$} be generated iteratively by $x_{n+1}\;=\;(1\;-\;{\alpha}_n)Tx_n$, $n\;{\geq}\;0$. Assume that the following control conditions hold: (i) $lim_{n{\rightarrow}{\infty}}\;{\alpha}_n\;=\;0$; (ii) $\sum^{\infty}_{n=0}\;{\alpha}_n\;=\;{\infty}$. Then {$x_n$} converges strongly to a fixed point of T.
Keywords
Averaged mapping; non-expansive mapping; fixed point; iterative algorithm; strong convergence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 C. Byrne, Iterative oblique projection onto convex sets and the split feasibility problem, Inverse Problems 18(2002), 441-453.   DOI   ScienceOn
2 H.K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc. 2(2002), 240-256.
3 C. Byrne, A unified treatment of some iterative algorithms in signal processing and image reconstruction, Inverse Problems 20 (2004), 103-120.   DOI   ScienceOn
4 M. Aslam Noor, General variational inequalities and nonexpansive mappings, Journal of Mathematical Analysis and Applications 331 (2007), 810-822.   DOI   ScienceOn
5 M. Aslam Noor, Y. Yao, Three-step iterations for variational inequalities and nonexpansive mappings, Applied Mathematics and Computation 190 (2007), 1312-1321.   DOI   ScienceOn
6 Y. Yao, J.C. Yao, H. Zhou, Approximation methods for common fixed points of infinite countable family of nonexpansive mappings, Computers and Mathematics with Applications 53 (2007), 1380-1389.   DOI   ScienceOn
7 Y. Yao, Y.C. Liou, R. Chen, A general iterative method for an infinite family of nonexpansive mappings, Nonlinear Analysis. 69 (2008), 1644-1654.   DOI   ScienceOn
8 M. Aslam Noor, Z. Huang, Some resolvent iterative methods for variational inclusions and nonexpansive mappings, Applied Mathematics and Computation 194 (2007), 267-275.   DOI   ScienceOn
9 X. Qin, M. Aslam Noor, General WienerCHopf equation technique for nonexpansive mappings and general variational inequalities in Hilbert spaces, Applied Mathematics and Computation 201 (2008), 716-722.   DOI   ScienceOn
10 Y. Yao, M. Aslam Noor, On viscosity iterative methods for variational inequalities, Journal of Mathematical Analysis and Applications 325 (2007), 776-787.   DOI   ScienceOn
11 M. Aslam Noor, General Variational Inequalities, Appl. Math. Letetrs 1 (1988), 119-121.   DOI   ScienceOn
12 Y. Yao, M. Aslam Noor, Convergence of three-step iterations for asymptotically nonexpansive mappings, Applied Mathematics and Computation 187 (2007), 883-892.   DOI   ScienceOn
13 M. Aslam. Noor, Z. Huang, Three-step methods for nonexpansive mappings and variational inequalities, Applied Mathematics and Computation 187 (2007), 680-685.   DOI   ScienceOn
14 Z. Huang, M. Aslam Noor, Some new unified iteration schemes with errors for nonexpansive mappings and variational inequalities, Applied Mathematics and Computation 194 (2007), 135-142.   DOI   ScienceOn
15 M. Aslam Noor, K. Inayat Noor and Th. M. Rassais, Some aspects of variational inequalities, J. Comput. Appl. Math. 47 (1993), 285-312.   DOI   ScienceOn
16 M. Aslam Noor, New approximation schemes for general variational inequalities, J. Math. Anal. Appl. 251 (2000), 217-229.   DOI   ScienceOn
17 N. Shahzad, Approximating fixed points of non-self nonexpansive mappings in Banach spaces, Nonlinear Anal. 61(2005), 1031-1039.   DOI   ScienceOn
18 H. Zegeye and N. Shahzad, Viscosity approximation methods for a common fixed point of finite family of nonexpansive mappings, Appl. Math. Comput. 191 (2007), 155-163.   DOI   ScienceOn
19 B. Halpern, Fixed points of nonexpansive maps, Bull. Am. Math. Soc. 73 (1967), 957-961.   DOI
20 M. Aslam Noor, Some developments in general variational inequalities, Appl. Math. Computations 152 (2004), 199-277.   DOI   ScienceOn
21 R. Wittmann, Appoximation of fixed points of nonexpansive mappings, Arch. Math. 58 (1992), 486-491.   DOI
22 H.K. Xu, Relaxed projections, averaged mappings and image recovery, the Proceedings of the International Conference on Fixed Point Theory and Its Applications, Yokohama Publishers 2004,275-292.
23 H.K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Aust. Math. Soc. 65(2002), 109-113.   DOI
24 T. Suzuki, A sufficient and necessary condition for Halpern-type strong convergence to fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 135(2007), 99-106.
25 C. I. Podilchuk and R. J. Mammone, Image recovery by convex projections using a least squares constraint, J. Opt. Soc. Am. 7 (1990), 517-521.
26 Y. Yao and R. Chen, Convergence to common fixed points of averaged' mappings without commutativity assumption in Hilbert spaces, Nonlinear Anal. 67(2007), 1758-1763.   DOI   ScienceOn