×

Convergence theorems on asymptotically pseudocontractive mappings in the intermediate sense. (English) Zbl 1214.47072

Summary: A new nonlinear mapping is introduced. The convergence of Ishikawa iterative processes for the class of asymptotically pseudocontractive mappings in the intermediate sense is studied. Weak convergence theorems are established. A strong convergence theorem is also established without any compactness assumption by considering the so-called hybrid projection methods.

MSC:

47J25 Iterative procedures involving nonlinear operators
47H09 Contraction-type mappings, nonexpansive mappings, \(A\)-proper mappings, etc.
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] 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 · Zbl 0256.47045 · doi:10.1090/S0002-9939-1972-0298500-3
[2] 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. · Zbl 0849.47030
[3] Kirk WA: Fixed point theorems for non-Lipschitzian mappings of asymptotically nonexpansive type.Israel Journal of Mathematics 1974, 17: 339-346. 10.1007/BF02757136 · Zbl 0286.47034 · doi:10.1007/BF02757136
[4] Browder FE, Petryshyn WV: Construction of fixed points of nonlinear mappings in Hilbert space.Journal of Mathematical Analysis and Applications 1967, 20: 197-228. 10.1016/0022-247X(67)90085-6 · Zbl 0153.45701 · doi:10.1016/0022-247X(67)90085-6
[5] Marino G, Xu H-K: Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces.Journal of Mathematical Analysis and Applications 2007,329(1):336-346. 10.1016/j.jmaa.2006.06.055 · Zbl 1116.47053 · doi:10.1016/j.jmaa.2006.06.055
[6] Liu QH: Convergence theorems of the sequence of iterates for asymptotically demicontractive and hemicontractive mappings.Nonlinear Analysis: Theory, Methods & Applications 1996,26(11):1835-1842. 10.1016/0362-546X(94)00351-H · Zbl 0861.47047 · doi:10.1016/0362-546X(94)00351-H
[7] Chang, S-S; Huang, J.; Wang, X.; Kim, JK, Implicit iteration process for common fixed points of strictly asymptotically pseudocontractive mappings in Banach spaces, 12 (2008) · Zbl 1219.47129
[8] Kim T-H, Xu H-K: Convergence of the modified Mann’s iteration method for asymptotically strict pseudo-contractions.Nonlinear Analysis: Theory, Methods & Applications 2008,68(9):2828-2836. 10.1016/j.na.2007.02.029 · Zbl 1220.47100 · doi:10.1016/j.na.2007.02.029
[9] Qin X, Cho YJ, Kang SM, Shang M: A hybrid iterative scheme for asymptotically -strict pseudo-contractions in Hilbert spaces.Nonlinear Analysis: Theory, Methods & Applications 2009,70(5):1902-1911. 10.1016/j.na.2008.02.090 · Zbl 1309.47079 · doi:10.1016/j.na.2008.02.090
[10] Sahu DR, Xu H-K, Yao J-C: Asymptotically strict pseudocontractive mappings in the intermediate sense.Nonlinear Analysis: Theory, Methods & Applications 2009,70(10):3502-3511. 10.1016/j.na.2008.07.007 · Zbl 1221.47122 · doi:10.1016/j.na.2008.07.007
[11] 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 · Zbl 0734.47036 · doi:10.1016/0022-247X(91)90245-U
[12] Kim JK, Nam YM: Modified Ishikawa iterative sequences with errors for asymptotically set-valued pseudocontractive mappings in Banach spaces.Bulletin of the Korean Mathematical Society 2006,43(4):847-860. · Zbl 1130.47047 · doi:10.1007/BF02705939
[13] Rhoades BE: Comments on two fixed point iteration methods.Journal of Mathematical Analysis and Applications 1976,56(3):741-750. 10.1016/0022-247X(76)90038-X · Zbl 0353.47029 · doi:10.1016/0022-247X(76)90038-X
[14] Zhou H: Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces.Nonlinear Analysis: Theory, Methods & Applications 2009,70(9):3140-3145. 10.1016/j.na.2008.04.017 · Zbl 1207.47052 · doi:10.1016/j.na.2008.04.017
[15] 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 · Zbl 0895.47048 · doi:10.1006/jmaa.1993.1309
[16] Yanes CM, Xu H-K: Strong convergence of the CQ method for fixed point iteration processes.Nonlinear Analysis: Theory, Methods & Applications 2006,64(11):2400-2411. 10.1016/j.na.2005.08.018 · Zbl 1105.47060 · doi:10.1016/j.na.2005.08.018
[17] Kim T-H, Xu H-K: Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups.Nonlinear Analysis: Theory, Methods & Applications 2006,64(5):1140-1152. 10.1016/j.na.2005.05.059 · Zbl 1090.47059 · doi:10.1016/j.na.2005.05.059
[18] Qin X, Su Y, Shang M: Strong convergence theorems for asymptotically nonexpansive mappings by hybrid methods.Kyungpook Mathematical Journal 2008,48(1):133-142. · Zbl 1220.47114 · doi:10.5666/KMJ.2008.48.1.133
[19] Zhou H: Convergence theorems of fixed points for Lipschitz pseudo-contractions in Hilbert spaces.Journal of Mathematical Analysis and Applications 2008,343(1):546-556. 10.1016/j.jmaa.2008.01.045 · Zbl 1140.47058 · doi:10.1016/j.jmaa.2008.01.045
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.