Browse > Article
http://dx.doi.org/10.7858/eamj.2011.27.1.001

STRONG CONVERGENCE THEOREMS FOR ASYMPTOTICALLY QUASI-NONEXPANSIVE MAPPINGS AND INVERSE-STRONGLY MONOTONE MAPPINGS  

He, Xin-Feng (College of Mathematics and Computer Hebei University)
Xu, Yong-Chun (Department of Mathematics Hebei North College)
He, Zhen (College of Mathematics and Computer Hebei University)
Publication Information
East Asian mathematical journal / v.27, no.1, 2011 , pp. 1-9 More about this Journal
Abstract
In this paper, we consider an iterative scheme for finding a common element of the set of fixed points of a asymptotically quasi nonexpansive mapping and the set of solutions of the variational inequality for an inverse strongly monotone mapping in a Hilbert space. Then we show that the sequence converges strongly to a common element of two sets. Using this result, we consider the problem of finding a common fixed point of a asymptotically quasi-nonexpansive mapping and strictly pseudocontractive mapping and the problem of finding a common element of the set of fixed points of a asymptotically quasi-nonexpansive mapping and the set of zeros of an inverse-strongly monotone mapping.
Keywords
Metric projection; inverse-strongly monotone mapping; asymptotically quasi-nonexpansive mapping; variational inequality; strong convergence;
Citations & Related Records
연도 인용수 순위
  • Reference
1 Z. Opial, Weak convergence ofthe sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 591-597.   DOI
2 W. Takahashi, Nonlinear complementarity problem and systems of convex inequalities, J. Optim. Theory Appl. 24 (1978), 493-508.
3 T. H. Kim and H. K. Xu, Remarks on asymptotically nonexpansive mappings, Nonlinear Anal. 41 (2000), 405-415.   DOI
4 Hideaki Iiduka and Wataru Takahashi, Strong convergence theorems for nonexpansive mappings and inverse-strongly monotone mappings, Nonlinear Anal. 61 (2005), 341-350.   DOI   ScienceOn
5 Tae-Hwa Kima and Hong-Kun Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006), 1140-1152.   DOI   ScienceOn
6 K. Goebel and W. A. Kirk, A Fixed Point Theorem for Asymptotically Nonexpansive Mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.   DOI   ScienceOn
7 Jong Kyu Kim, Young Man Nam and Jae Yull Sim, Convergence theorems of implicit iterative sequences for a finite family of asymptotically quasi-nonexpansive type mappings, Nonlinear Analysis 71 (2009), e2839-e2848.   DOI   ScienceOn
8 F. E. Browder, Nonlinear monotone operators and convex sets in Banach spaces, Bull. Amer. Math. Soc. 71 (1965), 780-785.   DOI
9 F. E. Browder, The fixed point theory of multi-valued mappings in topological vector spaces, Math. Ann. 177 (1968), 283-301.   DOI   ScienceOn
10 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.   DOI
11 R. E. Bruck, On the weak convergence of an ergodic iteration for the solution of vari- ational inequalities for monotone operators in Hilbert space, J. Math. Anal. Appl. 61 (1977), 159-164.   DOI
12 H. Iiduka, W. Takahashi and M. Toyoda, Approximation of solutions of variational inequalities for monotone mappings, Pan Amer. Math. J. 14 (2004), 49-61.
13 K. Nakajo and W. Takahashi, Strong and weak convergence theorems by an improved splitting method, Comm. Appl. Nonlinear Anal. 9 (2002), 99-107.
14 J. L. Lions and G. Stampacchia, Variational inequalities, Comm. Pure Appl. Math. 20 (1967), 493-517.   DOI
15 F. Liu and M. Z. Nashed, Regularization of nonlinear ill-posed variational inequalities and convergence rates, Set-Valued Anal. 6 (1998), 313-344.   DOI   ScienceOn