The purpose of this paper is to study sufficient and necessary conditions for finite-step iterative sequences with mean errors for a finite family of asymptotically quasi-nonexpansive and type mappings in Banach spaces to converge to a common fixed point. The results presented in this paper improve and extend the recent ones announced by Ghost-Debnath, Liu, Xu and Noor, Chang, Shahzad et al., Shahzad and Udomene, Chidume et al., and all the others.
Banach Space Differential Geometry Computational Biology Type Mapping Common Fixed Point
Alber YaI, Chidume CE, Zegeye H: Approximating fixed points of total asymptotically nonexpansive mappings.Fixed Point Theory & Applications 2006, 2006: 20 pages.MathSciNetCrossRefMATHGoogle Scholar
Chang S-S, Kim JK, Kang SM: Approximating fixed points of asymptotically quasi-nonexpansive type mappings by the Ishikawa iterative sequences with mixed errors.Dynamic Systems and Applications 2004,13(2):179–186.MathSciNetMATHGoogle Scholar
Chang S-S, Lee HWJ, Cho YJ: On the convergence of finite steps iterative sequences for asymptotically nonexpansive mappings.Dynamics of Continuous, Discrete & Impulsive Systems. Series A. Mathematical Analysis 2004,11(4):589–600.MathSciNetMATHGoogle Scholar
Chidume CE, Zegeye H, Shahzad N: Convergence theorems for a common fixed point of a finite family of nonself nonexpansive mappings.Fixed Point Theory & Applications 2005,2005(2):233–241. 10.1155/FPTA.2005.233MathSciNetCrossRefMATHGoogle Scholar
Ghosh MK, Debnath L: Convergence of Ishikawa iterates of quasi-nonexpansive mappings.Journal of Mathematical Analysis and Applications 1997,207(1):96–103. 10.1006/jmaa.1997.5268MathSciNetCrossRefMATHGoogle Scholar
Liu QH: Iterative sequences for asymptotically quasi-nonexpansive mappings.Journal of Mathematical Analysis and Applications 2001,259(1):1–7. 10.1006/jmaa.2000.6980MathSciNetCrossRefMATHGoogle Scholar
Liu QH: Iterative sequences for asymptotically quasi-nonexpansive mappings with error member.Journal of Mathematical Analysis and Applications 2001,259(1):18–24. 10.1006/jmaa.2000.7353MathSciNetCrossRefMATHGoogle Scholar
Liu QH: Iteration sequences for asymptotically quasi-nonexpansive mapping with an error member of uniform convex Banach space.Journal of Mathematical Analysis and Applications 2002,266(2):468–471. 10.1006/jmaa.2001.7629MathSciNetCrossRefMATHGoogle Scholar
Petryshyn WV, Williamson TE Jr.: Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings.Journal of Mathematical Analysis and Applications 1973, 43: 459–497. 10.1016/0022-247X(73)90087-5MathSciNetCrossRefMATHGoogle Scholar
Shahzad N, Udomene A: Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces.Fixed Point Theory & Applications 2006, 2006: 10 pages.MathSciNetCrossRefMATHGoogle Scholar
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.1993.1309MathSciNetCrossRefMATHGoogle Scholar
Xu BL, Noor MA: Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces.Journal of Mathematical Analysis and Applications 2002,267(2):444–453. 10.1006/jmaa.2001.7649MathSciNetCrossRefMATHGoogle Scholar
Zhou YY, Chang S-S: Convergence of implicit iteration process for a finite family of asymptotically nonexpansive mappings in Banach spaces.Numerical Functional Analysis and Optimization 2002,23(7–8):911–921. 10.1081/NFA-120016276MathSciNetCrossRefMATHGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.