DOI QR코드

DOI QR Code

ON THE CONVERGENCE OF HYBRID PROJECTION METHODS FOR ASYMPTOTICALLY PSEUDOCONTRACTIVE MAPPINGS IN THE INTERMEDIATE SENSE

  • Cho, Sun-Young (Department of Mathematics Gyeongsang National University) ;
  • Kang, Shin-Min (Department of Mathematics and the RINS Gyeongsang National University) ;
  • Qin, Xiaolong (Department of Mathematics Hangzhou Normal University)
  • 투고 : 2010.03.08
  • 발행 : 2011.07.31

초록

In this paper, mappings which are asymptotically pseudo-contractive in the intermediate sense are considered based on a hybrid projection method. Strong convergence theorems of fixed points are established in the framework of Hilbert spaces.

키워드

참고문헌

  1. G. L. Acedo and H. K. Xu, Iterative methods for strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 67 (2007), no. 7, 2258-2271. https://doi.org/10.1016/j.na.2006.08.036
  2. F. E. Browder and W. V. Petryshyn, Construction of fixed points of nonlinear mappings in Hilbert space, J. Math. Anal. Appl. 20 (1967), 197-228. https://doi.org/10.1016/0022-247X(67)90085-6
  3. R. E. Bruck, T. Kuczumow, and S. Reich, Convergence of iterates of asymptotically non-expansive mappings in Banach spaces with the uniform Opial property, Colloq. Math. 65 (1993), no. 2, 169-179. https://doi.org/10.4064/cm-65-2-169-179
  4. A. Genel and J. Lindenstrass, An example concerning fixed points, Israel J. Math. 22 (1975), no. 1, 81-86. https://doi.org/10.1007/BF02757276
  5. K. Goebel and W. A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174. https://doi.org/10.1090/S0002-9939-1972-0298500-3
  6. I. Inchan and S. Plubtieng, Strong convergence theorems of hybrid methods for two asymptotically nonexpansive mappings in Hilbert spaces, Nonlinear Anal. Hybrid Syst. 2 (2008), no. 4, 1125-1135. https://doi.org/10.1016/j.nahs.2008.09.006
  7. T. H. Kim and H. K. Xu, Convergence of the modified Mann's iteration method for asymptotically strict pseudo-contractions, Nonlinear Anal. 68 (2008), no. 9, 2828-2836. https://doi.org/10.1016/j.na.2007.02.029
  8. Y. Kimura and W. Takahashi, Strong convergence of modi ed Mann iterations for asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 64 (2006), no. 5, 1140-1152. https://doi.org/10.1016/j.na.2005.05.059
  9. Y. Kimura and W. Takahashi, On a hybrid method for a family of relatively nonexpansive mappings in a Banach space, J. Math. Anal. Appl. 357 (2009), no. 2, 356-363. https://doi.org/10.1016/j.jmaa.2009.03.052
  10. W. A. Kirk, Fixed point theorems for non-Lipschitzian mappings of asymptotically non-expansive type, Israel J. Math. 17 (1974), 339-346. https://doi.org/10.1007/BF02757136
  11. 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
  12. G. Marino and H. K. Xu, Weak and strong convergence theorems for strict pseudo-contractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), no. 1, 336-346. https://doi.org/10.1016/j.jmaa.2006.06.055
  13. C. Martinez-Yanes and H. K. Xu, Strong convergence of the CQ method for fixed point iteration processes, Nonlinear Anal. 64 (2006), no. 11, 2400-2411. https://doi.org/10.1016/j.na.2005.08.018
  14. S. Y. Matsushita and W. Takahashi, A strong convergence theorem for relatively non-expansive mappings in a Banach space, J. Approx. Theory 134 (2005), no. 2, 257-266. https://doi.org/10.1016/j.jat.2005.02.007
  15. K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), no. 2, 372-379. https://doi.org/10.1016/S0022-247X(02)00458-4
  16. S. Plubtieng and K. Ungchittrakool, Strong convergence of modified Ishikawa iteration for two asymptotically nonexpansive mappings and semigroups, Nonlinear Anal. 67 (2007), no. 7, 2306-2315. https://doi.org/10.1016/j.na.2006.09.023
  17. L. Qihou, Convergence theorems of the sequence of iterates for asymptotically demicon-tractive and hemicontractive mappings, Nonlinear Anal. 26 (1996), no. 11, 1835-1842. https://doi.org/10.1016/0362-546X(94)00351-H
  18. X. Qin, S. Y. Cho, and S. M. Kang, On hybrid projection methods for asymptotically $quasi-{\phi}-nonexpansive$ mappings, Appl. Math. Comput. 215 (2010), no. 11, 3874-3883. https://doi.org/10.1016/j.amc.2009.11.031
  19. X. Qin, Y. J. Cho, S. M. Kang, and M. Shang, A hybrid iterative scheme for asymptot- ically k-strict pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009), no. 5, 1902-1911. https://doi.org/10.1016/j.na.2008.02.090
  20. X. Qin, Y. J. Cho, S. M. Kang, and H. Zhou, Convergence theorems of common fixed points for a family of Lipschitz quasi-pseudocontractions, Nonlinear Anal. 71 (2009), no. 1-2, 685-690. https://doi.org/10.1016/j.na.2008.10.102
  21. X. Qin, S. Y. Cho, and J. K. Kim, Convergence results on asymptotically pseudocontractive mappings in the intermediate sense, Fixed Point Theory Appl. 2010 (2010), Article ID 186874.
  22. B. E. Rhoades, Comments on two xed point iteration methods, J. Math. Anal. Appl. 56 (1976), no. 3, 741-750. https://doi.org/10.1016/0022-247X(76)90038-X
  23. D. R. Sahu, H. K. Xu, and J. C. Yao, Asymptotically strict pseudocontractive mappings in the intermediate sense, Nonlinear Anal. 70 (2009), no. 10, 3502-3511. https://doi.org/10.1016/j.na.2008.07.007
  24. J. Schu, Iterative construction of fixed points of asymptotically nonexpansive mappings, J. Math. Anal. Appl. 158 (1991), no. 2, 407-413. https://doi.org/10.1016/0022-247X(91)90245-U
  25. W. Takahashi, Y. Takeuchi, and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hilbert spaces, J. Math. Anal. Appl. 341 (2008), no. 1, 276-286. https://doi.org/10.1016/j.jmaa.2007.09.062
  26. H. Zegeye and N. Shahzad, Strong convergence theorems for a finite family of nonex-pansive mappings and semigroups via the hybrid method, Nonlinear Anal. 72 (2010), no. 1, 325-329. https://doi.org/10.1016/j.na.2009.06.056
  27. H. Zhou, Demiclosedness principle with applications for asymptotically pseudo-contractions in Hilbert spaces, Nonlinear Anal. 70 (2009), no. 9, 3140-3145. https://doi.org/10.1016/j.na.2008.04.017