Taiwanese Journal of Mathematics

Strong Convergence of Modified Iteration Processes for Relatively Asymptotically Nonexpansive Mappings

Tae-Hwa Kim and Wataru Takahashi

Full-text: Open access

Abstract

Ishikawa and Halpern’s iterations are modified to prove the strong convergence problems of such iteration processes for uniformly Lipschitzian mappings which are relatively asymptotically nonexpansive in Banach spaces, which extend the result due to Matsushita and Takahashi [J. Approx. Theory, 134 (2005), 257–266] for relatively nonexpansive mappings, and also some recent results due to Martinez-Yanez and Xu [Nonlinear Anal., 64 (2006), 2400–2411], and Kim and Xu [Nonlinear Anal., 64 (2006), 1140–1152] for nonexpansive mappings and asymptotically nonexpansive mappings, respectively, which are considered in the Hilbert space frameworks.

Article information

Source
Taiwanese J. Math., Volume 14, Number 6 (2010), 2163-2180.

Dates
First available in Project Euclid: 18 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1500406068

Digital Object Identifier
doi:10.11650/twjm/1500406068

Mathematical Reviews number (MathSciNet)
MR2742357

Zentralblatt MATH identifier
1226.47079

Subjects
Primary: 47H09: Contraction-type mappings, nonexpansive mappings, A-proper mappings, etc.
Secondary: 65J15: Equations with nonlinear operators (do not use 65Hxx)

Keywords
strong convergence modified Ishikawa's iteration modified Halpern's iteration relatively asymptotically nonexpansive mapping

Citation

Kim, Tae-Hwa; Takahashi, Wataru. Strong Convergence of Modified Iteration Processes for Relatively Asymptotically Nonexpansive Mappings. Taiwanese J. Math. 14 (2010), no. 6, 2163--2180. doi:10.11650/twjm/1500406068. https://projecteuclid.org/euclid.twjm/1500406068


Export citation

References

  • Ya. I. Alber, Metric and generalized projection operators in Banach spaces: properties and applications, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos (Ed.), Marcel Dekker, New York, 1996, pp. 15-50.
  • Ya. I. Alber and S. Reich, An iterative method for solving a class of nonlinear operator equations in Banach spaces, Pan-Amer. Math. J., 4 (1994), 39-54.
  • D. Butnariu, S. Reich and A. J. Zaslavski, Asymptotic behavior of relatively nonexpansive operators in Banach spaces, J. Appl. Anal., 7 (2001), 151-174.
  • D. Butnariu, S. Reich and A. J. Zaslavski, Weak convergence of orbits of nonlinear operators in reflexive Banach spaces, Numer. Funct. Anal. Optim., 24 (2003), 489- 508.
  • Y. Censor and S. Reich, Iterations of paracontractions and firmly nonexpansive operators with applications to feasibility and optimization, Optimization, 37 (1996), 323-339.
  • C. E. Chidume and S. A. Mutangadura, An example on the Mann iteration method for Lipschitz pseudocontractions, Proc. Amer. Math. Soc., 129 (2001), 2359-2363.
  • I. Cioranescu, Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990.
  • A. Genel and J. Lindenstrass, An example concerning fixed points, Israel J. Math., 22 (1975), 81-86.
  • K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc., 35 (1972), 171-174.
  • B. Halpern, Fixed points of nonexpanding maps, Bull. Amer. Math. Soc., 73 (1967), 957-961.
  • S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc., 44 (1974), 147-150.
  • S. Kamimura and W. Takahashi, Strong convergence of a proxiaml-type algorithm in a Banach space, SIAM J. Optim., 13 (2003), 938-945.
  • T. H. Kim and H. J. Lee, Strong convergence of modified iteration processes for relatively nonexpansive mappings in Banach Spaces, Kyungpook Math. J., 48 (2008), 685-703.
  • T. H. Kim and H. K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal., 64 (2006), 1140-1152.
  • P. L. Lions, Approximation de points fixes de contractions, C.R. Acad. Sci. Sèr. A-B Paris 284 (1977), 1357-1359.
  • W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4 (1953), 506-510.
  • C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal., 64 (2006), 2400-2411.
  • S. Matsushita and W. Takahashi, A strong convergence theorem for relatively nonexpansive mappings in a Banach space, J. Approx. Theory, 134 (2005), 257-266.
  • K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl., 279 (2003), 372-379.
  • S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl., 75 (1980), 287-292.
  • S. Reich, Review of Geometry of Banach Spaces, Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, 1990, Bull. Amer. Math. Soc., 26 (1992), 367-370.
  • S. Reich, A weak convergence theorem for the alternating method with Bregman distance, in: Theory and Applications of Nonlinear Operators of Accretive and Monotone Type, A. G. Kartsatos (Ed.), Marcel Dekker, New York, 1996, pp. 313-318.
  • N. Shioji and W. Takahashi, Strong convergence of approximated sequences for non- expansive mappings in Banach spaces, Proc. Amer. Math. Soc., 125 (1997), 3641-3645.
  • W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
  • R. Wittmann, Approximation of fixed points of nonexpansive mappings, Arch. Math., 58 (1992), 486-491.
  • H. K. Xu, Existence and convergence for fixed points of mappings of asymptotically nonexpansive type, Nonlinear Anal., 16 (1991), 1139-1146.
  • H. K. Xu, Iterative algorithms for nonlinear operators, J. London Math. Soc., 66 (2002), 240-256.