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Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups
Fixed Point Theory and Applications volume 2006, Article number: 96215 (2006)
Abstract
Strong convergence theorems are obtained from modified Halpern iterative scheme for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, respectively. Our results extend and improve the recent ones announced by Nakajo, Takahashi, Kim, Xu, and some others.
References
Genel A, Lindenstrauss J: An example concerning fixed points. Israel Journal of Mathematics 1975,22(1):81–86. 10.1007/BF02757276
Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972,35(1):171–174. 10.1090/S0002-9939-1972-0298500-3
Halpern B: Fixed points of nonexpanding maps. Bulletin of the American Mathematical Society 1967, 73: 957–961. 10.1090/S0002-9904-1967-11864-0
Kim T-H, Xu H-K: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups. Nonlinear Analysis 2006,64(5):1140–1152. 10.1016/j.na.2005.05.059
Lin P-K, Tan K-K, Xu H-K: Demiclosedness principle and asymptotic behavior for asymptotically nonexpansive mappings. Nonlinear Analysis 1995,24(6):929–946. 10.1016/0362-546X(94)00128-5
Lions P-L: Approximation de points fixes de contractions. Comptes Rendus de l'Académie des Sciences de Paris, Série. A-B 1977,284(21):A1357-A1359.
Martinez-Yanes C, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes. Nonlinear Analysis 2006,64(11):2400–2411. 10.1016/j.na.2005.08.018
Nakajo K, Takahashi W: Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups. Journal of Mathematical Analysis and Applications 2003,279(2):372–379. 10.1016/S0022-247X(02)00458-4
Reich S: Weak convergence theorems for nonexpansive mappings in Banach spaces. Journal of Mathematical Analysis and Applications 1979,67(2):274–276. 10.1016/0022-247X(79)90024-6
Reich S: Strong convergence theorems for resolvents of accretive operators in Banach spaces. Journal of Mathematical Analysis and Applications 1980,75(1):287–292. 10.1016/0022-247X(80)90323-6
Reich S: Approximating fixed points of nonexpansive mappings. Panamerican Mathematical Journal 1994,4(2):23–28.
Tan K-K, Xu H-K: Fixed point theorems for Lipschitzian semigroups in Banach spaces. Nonlinear Analysis 1993,20(4):395–404. 10.1016/0362-546X(93)90144-H
Wittmann R: Approximation of fixed points of nonexpansive mappings. Archiv der Mathematik 1992,58(5):486–491. 10.1007/BF01190119
Youla D: Mathematical theory of image restoration by the method of convex projections. In Image Recovery Theory and Application. Edited by: Stark H. Academic Press, Florida; 1987:29–77.
Youla D: On deterministic convergence of iteration of relaxed projection operators. Journal of Visual Communication and Image Representation 1990,1(1):12–20. 10.1016/1047-3203(90)90013-L
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Su, Y., Qin, X. Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups. Fixed Point Theory Appl 2006, 96215 (2006). https://doi.org/10.1155/FPTA/2006/96215
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DOI: https://doi.org/10.1155/FPTA/2006/96215