Approximating fixed points of total asymptotically nonexpansive mappings

Ya. I. Alber, C. E. Chidume, Habtu Zegeye

Tóm tắt

AbstractWe introduce a new class of asymptotically nonexpansive mappings and study approximating methods for finding their fixed points. We deal with the Krasnosel'skii-Mann-type iterative process. The strong and weak convergence results for self-mappings in normed spaces are presented. We also consider the asymptotically weakly contractive mappings.

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Tài liệu tham khảo

Alber YaI: On the solution of equations and variational inequalities with maximal monotone operators. Soviet Mathematics Doklady 1979,20(4):871–876.

Alber YaI: Metric and generalized projection operators in Banach spaces: properties and applications. In Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, Lecture Notes in Pure and Appl. Math.. Volume 178. Edited by: Kartsatos AG. Dekker, New York; 1996:15–50.

Alber YaI, Guerre-Delabriere S: Problems of fixed point theory in Hilbert and Banach spaces. Functional Differential Equations. Israel Seminar 1994, 2: 5–10 (1995).

Alber YaI, Guerre-Delabriere S: Principle of weakly contractive maps in Hilbert spaces. In New Results in Operator Theory and Its Applications, Oper. Theory Adv. Appl.. Volume 98. Birkhäuser, Basel; 1997:7–22.

Alber YaI, Guerre-Delabriere S: On the projection methods for fixed point problems. Analysis. International Mathematical Journal of Analysis and its Applications 2001,21(1):17–39.

Alber YaI, Iusem AN: Extension of subgradient techniques for nonsmooth optimization in Banach spaces. Set-Valued Analysis. An International Journal Devoted to the Theory of Multifunctions and its Applications 2001,9(4):315–335. 10.1023/A:1012665832688

Alber YaI, Iusem AN, Solodov MV: Minimization of nonsmooth convex functionals in Banach spaces. Journal of Convex Analysis 1997,4(2):235–255.

Alber YaI, Reich S: An iterative method for solving a class of nonlinear operator equations in Banach spaces. Panamerican Mathematical Journal 1994,4(2):39–54.

Alber YaI, Reich S, Yao J-C: Iterative methods for solving fixed-point problems with nonself-mappings in Banach spaces. Abstract and Applied Analysis 2003,2003(4):193–216. 10.1155/S1085337503203018

Bose SC: Weak convergence to the fixed point of an asymptotically nonexpansive map. Proceedings of the American Mathematical Society 1978,68(3):305–308. 10.1090/S0002-9939-1978-0493543-4

Bruck RE, Kuczumow T, Reich S: Convergence of iterates of asymptotically nonexpansive mappings in Banach spaces with the uniform Opial property. Colloquium Mathematicum 1993,65(2):169–179.

Chang SS, Cho YJ, Zhou H: Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings. Journal of the Korean Mathematical Society 2001,38(6):1245–1260.

Chidume CE: Nonexpansive mappings, generalizations and iterative algorithms. to appear in Nonlinear Anal

Chidume CE, Ofoedu EU, Zegeye H: Strong and weak convergence theorems for asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications 2003,280(2):364–374. 10.1016/S0022-247X(03)00061-1

Chidume CE, Shahzad N, Zegeye H: Strong convergence theorems for nonexpansive mappings in arbitrary Banach spaces. to appear in Nonlinear Anal

Diestel J: Geometry of Banach Spaces—Selected Topics, Lecture Notes in Mathematics. Volume 485. Springer, Berlin; 1975:xi+282.

Figiel T: On the moduli of convexity and smoothness. Polska Akademia Nauk. Instytut Matematyczny. Studia Mathematica 1976,56(2):121–155.

Goebel K, Kirk WA: A fixed point theorem for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1972, 35: 171–174. 10.1090/S0002-9939-1972-0298500-3

Górnicki J: Nonlinear ergodic theorems for asymptotically nonexpansive mappings in Banach spaces satisfying Opial's condition. Journal of Mathematical Analysis and Applications 1991,161(2):440–446. 10.1016/0022-247X(91)90343-X

Ishikawa S: Fixed points by a new iteration method. Proceedings of the American Mathematical Society 1974, 44: 147–150. 10.1090/S0002-9939-1974-0336469-5

Ishikawa S: Fixed points and iteration of a nonexpansive mapping in a Banach space. Proceedings of the American Mathematical Society 1976,59(1):65–71. 10.1090/S0002-9939-1976-0412909-X

Kim GE, Kim TH: Mann and Ishikawa iterations with errors for non-Lipschitzian mappings in Banach spaces. Computers & Mathematics with Applications. An International Journal 2001,42(12):1565–1570. 10.1016/S0898-1221(01)00262-0

Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type. Israel Journal of Mathematics 1974, 17: 339–346. 10.1007/BF02757136

Lindenstrauss J, Tzafriri L: Classical Banach Spaces. II. Function Spaces, Ergebnisse der Mathematik und ihrer Grenzgebiete. Volume 97. Springer, Berlin; 1979:x+243.

Mann W: Mean value methods in iteration. Proceedings of the American Mathematical Society 1953, 4: 506–510. 10.1090/S0002-9939-1953-0054846-3

Opial Z: Weak convergence of the sequence of successive approximations for nonexpansive mappings. Bulletin of the Australian Mathematical Society 1967, 73: 591–597. 10.1090/S0002-9904-1967-11761-0

Osilike MO, Aniagbosor SC: Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings. Mathematical and Computer Modelling 2000,32(10):1181–1191. 10.1016/S0895-7177(00)00199-0

Passty GB: Construction of fixed points for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1982,84(2):212–216. 10.1090/S0002-9939-1982-0637171-7

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

Rhoades BE: Fixed point iterations for certain nonlinear mappings. Journal of Mathematical Analysis and Applications 1994,183(1):118–120. 10.1006/jmaa.1994.1135

Schu J: Iterative construction of fixed points of asymptotically nonexpansive mappings. Journal of Mathematical Analysis and Applications 1991,158(2):407–413. 10.1016/0022-247X(91)90245-U

Schu J: Weak and strong convergence to fixed points of asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society 1991,43(1):153–159. 10.1017/S0004972700028884

Senter HF, Dotson WG Jr.: Approximating fixed points of nonexpansive mappings. Proceedings of the American Mathematical Society 1974, 44: 375–380. 10.1090/S0002-9939-1974-0346608-8

Tan K-K, Xu HK: A nonlinear ergodic theorem for asymptotically nonexpansive mappings. Bulletin of the Australian Mathematical Society 1992,45(1):25–36. 10.1017/S0004972700036972

Tan K-K, Xu HK: The nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach spaces. Proceedings of the American Mathematical Society 1992,114(2):399–404. 10.1090/S0002-9939-1992-1068133-2

Tan K-K, Xu HK: Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process. Journal of Mathematical Analysis and Applications 1993,178(2):301–308. 10.1006/jmaa.1993.1309

Tan K-K, Xu HK: Fixed point iteration processes for asymptotically nonexpansive mappings. Proceedings of the American Mathematical Society 1994,122(3):733–739. 10.1090/S0002-9939-1994-1203993-5

Xu HK: Existence and convergence for fixed points of mappings of asymptotically nonexpansive type. Nonlinear Analysis. Theory, Methods & Applications. An International Multidisciplinary Journal. Series A: Theory and Methods 1991,16(12):1139–1146. 10.1016/0362-546X(91)90201-B