East Asian mathematical journal

The Youngnam Mathematical Society (영남수학회)
권호 표지
DOI QR코드

DOI QR Code

HYBRID MONOTONE PROJECTION ALGORITHMS FOR ASYMPTOTICALLY QUASI-PSEUDOCONTRACTIVE MAPPINGS

  • Wu, Changqun (SCHOOL OF BUSINESS AND ADMINISTRATION HENAN UNIVERSITY) ;
  • Cho, Sun-Young (DEPARTMENT OF MATHEMATICS GYEONGSANG NATIONAL UNIVERSITY)
  • Received : 2008.09.06
  • Accepted : 2009.08.11
  • Published : 2009.12.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we consider the hybrid monotone projection algorithm for asymptotically quasi-pseudocontractive mappings. A strong convergence theorem is established in the framework of Hilbert spaces. Our results mainly improve the corresponding results announced by [H. Zhou, Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009) 3140-3145] and also include Kim and Xu [T.H. Kim, H.K. Xu, Strong convergence of modified Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006) 1140-1152; Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions, Nonlinear Anal. 68 (2008) 2828-2836] as special cases.

Keywords

References

  1. K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Sco. 35 (1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3
  2. A. Genel and J. Lindenstrass, An example concerning fixed points, Israel J. Math. 22 (1975), 81-86. https://doi.org/10.1007/BF02757276
  3. I. Inchan and S. Plubtieng, Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 2 (2008), 1125-1135.
  4. 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. https://doi.org/10.1016/j.na.2005.05.059
  5. T. H. Kim and H. K. Xu, Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions, Nonlinear Anal. 68 (2008), 2828-2836. https://doi.org/10.1016/j.na.2007.02.029
  6. W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510. https://doi.org/10.1090/S0002-9939-1953-0054846-3
  7. G. Marino and H. K. Xu Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336-346. https://doi.org/10.1016/j.jmaa.2006.06.055
  8. C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), 2400-2411. https://doi.org/10.1016/j.na.2005.08.018
  9. K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372-379. https://doi.org/10.1016/S0022-247X(02)00458-4
  10. X. Qin, Y. Su and M. Shang, Strong convergence theorems for asymptotically nonex-pansive mappings by hybrid methods, Kyungpook Math. J. 48 (2008), 133-142 https://doi.org/10.5666/KMJ.2008.48.1.133
  11. X. Qin, Y. J. Cho, S. M. Kang and M. Shang, A hybrid iterative scheme for asymptotically k-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009), 1902-1911. https://doi.org/10.1016/j.na.2008.02.090
  12. X. Qin and Y. Su, Strong convergence theorems for relatively nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 1958-1965. https://doi.org/10.1016/j.na.2006.08.021
  13. X. Qin, Y. J. Cho, S. M. Kang, H. Zhou, Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions, Nonlinear Anal. 71 (2009), 685-690. https://doi.org/10.1016/j.na.2008.10.102
  14. S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 67 (1979), 274-276. https://doi.org/10.1016/0022-247X(79)90024-6
  15. J. Schu, Iteration constrction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), 107-413.
  16. Y. Su and X. Qin, Monotone CQ iteration processes for nonexpansive semigroups and maximal monotone operators, Nonlinear Anal. 68 (2008), 3657-3664. https://doi.org/10.1016/j.na.2007.04.008
  17. Y. Su and X. Qin, Strong convergence theorems for asymptotically nonexpansive mappings and asymptotically nonexpansive semigroups, Fixed Point Theory Appl. 2006 (2006), Art. ID 96215.
  18. H. Zhou, Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009), 3140-3145. https://doi.org/10.1016/j.na.2008.04.017