Taiwanese Journal of Mathematics

Common Fixed Points of a Finite Family of Nonself Generalized Asymptotically Quasi-Nonexpansive Mappings

Shuechin Huang

Full-text: Open access

Abstract

Suppose that $C$ is a nonempty subset of a real Banach space $X$. In this article, we construct two types of iterative schemes with errors for a finite family $\{T_i\}_{i=1}^k$ of nonself generalized asymptotically quasi-nonexpansive mappings of $C$ into $X$. Furthermore, not only a necessary and sufficient condition for $\{x_n\}$ generated by each of those iterations to converge to a common fixed point of $\{T_i\}_{i=1}^k$ is obtained, but also the weak and strong convergence theorems of $\{x_n\}$ in uniformly convex Banach spaces are established as well.

Article information

Source
Taiwanese J. Math., Volume 15, Number 2 (2011), 745-772.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406233

Digital Object Identifier
doi:10.11650/twjm/1500406233

Mathematical Reviews number (MathSciNet)
MR2810180

Zentralblatt MATH identifier
05954243

Subjects
Primary: 47H10: Fixed-point theorems [See also 37C25, 54H25, 55M20, 58C30] 54H25: Fixed-point and coincidence theorems [See also 47H10, 55M20]

Keywords
nonself generalized asymptotically quasi-nonexpansive mapping Lipschitzian mapping retract retraction Opial property Kadec-Klee property demiclosedness semicompactness condition $(A)$

Citation

Huang, Shuechin. Common Fixed Points of a Finite Family of Nonself Generalized Asymptotically Quasi-Nonexpansive Mappings. Taiwanese J. Math. 15 (2011), no. 2, 745--772. doi:10.11650/twjm/1500406233. https://projecteuclid.org/euclid.twjm/1500406233


Export citation

References

  • A. G. Aksoy and M. A. Khamsi, Nonstandard Mathods in Fixed Point Theory, Springer-Verlag, New York, 1990.
  • C. E. Chidume and B. Ali, Weak and strong convergence theorems for finite families of asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 330 (2007), 377-387.
  • C. E. Chidume and B. Ali, Approximation of common fixed points for finite families of nonself asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 326 (2007), 960-973.
  • C. E. Chidume, E. U. Ofoedu and H. Zegeye, Strong and weak convergence theorems for asymptotically nonexpansive mappings, J. Math. Anal. Appl., 280 (2003), 364-374.
  • C. E. Chidume and N. Shahzad, Strong convergence of an implicit iteration process for finite family of nonexpansive mappings, Nonlinear Anal., 62(6) (2005), 1149-1156.
  • L. Deng and Q. Liu, Iterative scheme for nonself generalized asymptotically quasi-nonexpansive mappings, Applied Mathematics and Computation, 205 (2008), 317-324.
  • J. Diestel, Geometry of Banach Spaces-Selected Topics, Lecture Notes in Mathematics, vol. 485, Springer, New York, 1975.
  • H. Fukhar-ud-din and S. H. Khan, Convergence of iterates with errors of asymptotically quasi-nonexpansive mappings and applications, J. Math. Anal. Appl., 328 (2007), 821-829.
  • J. Garcia Falset, W. Kaczor, T. Kuczumow and S. Reich, Weak convergence theorems for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 43 (2001), 377-401.
  • K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge, England, 1990.
  • B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73(6) (1967), 957-961.
  • S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
  • A. R. Khan, A. A. Domlo and H. Fukhar-ud-din, Common fixed points Noor iteration for a finite family of asymptotically quasi-nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 341 (2008), 1-11.
  • H. Y. Lan, Common fixedpoint iterative processes with errors for generalized asymptotically quasi-nonexpansive mappings, Comput. Math. Appl., 52 (2006), 1403-1412.
  • Q. H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mapping with error member, J. Math Anal. Appl., 259 (2001), 18-24.
  • M. Maiti and B. Saha, Approximating fixed points of nonexpansive and generalized nonexpansive mappings, Int. J. Math. Math. Sci., 16(1) (1993), 81-86.
  • W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4,(1953), 506-510.
  • H. Oka, A nonlinear ergodic theorem for asymptotically nonexpansive mappings in Banach space, Proc. Japan Acad. Ser. A, 65 (1998), 284-287.
  • S. Plubtieng and R. Wangkeeree, Fixed point iteration for asymptotically quasi-nonexpansive mappings in Banach spaces, International Journal of Mathematics and Mathematical Sciences, 11 (2005), 1685-1692.
  • J. Quan, S. S. Chang and X. J. Long, Approximation common fixed point of asymptotically quasi-nonexpansive-type mappings by the finite steps iterative sequences, Fixed Point Theory and Applications, 2006 (2006), 1-8.
  • J. Schu, Weak and strong convergence to fixed points of asymptotically nonexpansive mappings, Bull. Austral. Math. Soc., 43 (1991), 153-159.
  • H. F. Senter and W. G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(2) (1974), 375-380.
  • N. Shahzad and H. Zegeye, Strong convergence of an implicit iteration process for a finite family of generalized asymptotically quasi-nonexpansive maps, Applied Mathematics and Computation, 189 (2007), 1058-1065.
  • Z. H. Sun, Strong convergence of an implicit iteration process for a finite family of asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl., 286 (2003), 351-358.
  • K. K. Tan and H. K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl., 178 (1993), 301-308.
  • K. K. Tan and X. Z. Yuan, Random fixed point theorems and approximation in cone, J. Math. Anal. Appl., 185 (1994), 378-390.
  • Y. X. Tian and C. D. Yang, Convergence theorems of three-step iterative scheme for a finite family of uniformly quasi-lipschitzian mappings in convex metric spaces, Fixed Point Theory and Applications, 2009 (2009), 1-12.
  • L. Wang, Strong and weak convergence theorems for common fixed points of nonself asymptotically nonexpansive mappings, J. Math. Anal. Appl., 323 (2006), 550-557.
  • B. Xu and M. A. Noor, Fixed-point iterations for asymptotically nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 267(2) (2002), 444-453.
  • H. K. Xu and R. G. Ori, An implicit iteration process for nonexpansive mappings, Numer. Funct. Anal. Optim., 22 (2001), 767-773.