DOI QR코드

DOI QR Code

STRONG CONVERGENCE THEOREMS FOR INFINITE COUNTABLE NONEXPANSIVE MAPPINGS AND IMAGE RECOVERY PROBLEM

  • Yao, Yonghong ;
  • Liou, Yeong-Cheng
  • Published : 2008.11.01

Abstract

In this paper, we introduce an iterative scheme given by infinite nonexpansive mappings in Banach spaces. We prove strong convergence theorems which are connected with the problem of image recovery. Our results enrich and complement the recent many results.

Keywords

nonexpansive mapping;strong convergence;uniformly $G{\hat{a}}teaux$ differentiable norm;fixed point

References

  1. S. S. Chang, Some problems and results in the study of nonlinear analysis, Nonlinear Anal. 30 (1997), no. 7, 4197-4208 https://doi.org/10.1016/S0362-546X(97)00388-X
  2. S. S. Chang, Viscosity approximation methods for a finite family of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 323 (2006), no. 2, 1402-1416 https://doi.org/10.1016/j.jmaa.2005.11.057
  3. G. Das and J. P. Debata, Fixed points of quasinonexpansive mappings, Indian J. Pure Appl. Math. 17 (1986), no. 11, 1263-1269
  4. S. Kitahara and W. Takahashi, Image recovery by convex combinations of sunny nonexpansive retractions, Topol. Methods Nonlinear Anal. 2 (1993), no. 2, 333-342 https://doi.org/10.12775/TMNA.1993.046
  5. C. H. Morales and J. S. Jung, Convergence of paths for pseudocontractive mappings in Banach spaces, Proc. Amer. Math. Soc. 128 (2000), no. 11, 3411-3419 https://doi.org/10.1090/S0002-9939-00-05573-8
  6. J. G. O'Hara, P. Pillay, and H. K. Xu, Iterative approaches to convex feasibility problems in Banach spaces, Nonlinear Anal. 64 (2006), no. 9, 2022-2042 https://doi.org/10.1016/j.na.2005.07.036
  7. K. Shimoji and W. Takahashi, Strong convergence to common fixed points of infinite nonexpansive mappings and applications, Taiwanese J. Math. 5 (2001), no. 2, 387-404 https://doi.org/10.11650/twjm/1500407345
  8. Y. Song and R. Chen, Iterative approximation to common fixed points of nonexpansive mapping sequences in reflexive Banach spaces, Nonlinear Anal. 66 (2007), no. 3, 591-603 https://doi.org/10.1016/j.na.2005.12.004
  9. Y. Song, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings, Appl. Math. Comput. 180 (2006), no. 1, 275-287 https://doi.org/10.1016/j.amc.2005.12.013
  10. T. Suzuki, Strong convergence of Krasnoselskii and Mann's type sequences for oneparameter nonexpansive semigroups without Bochner integrals, J. Math. Anal. Appl. 305 (2005), no. 1, 227-239 https://doi.org/10.1016/j.jmaa.2004.11.017
  11. W. Takahashi, Fixed point theorems and nonlinear ergodic theorems for nonlinear semigroups and their applications, Proceedings of the Second World Congress of Nonlinear Analysts, Part 2 (Athens, 1996). Nonlinear Anal. 30 (1997), no. 2, 1283-1293
  12. W. Takahashi, Nonlinear Functional Analysis, Kindai-kagakusha, Tokyo, 1988
  13. W. Takahashi and K. Shimoji, Convergence theorems for nonexpansive mappings and feasibility problems, Math. Comput. Modelling 32 (2000), no. 11-13, 1463-1471 https://doi.org/10.1016/S0895-7177(00)00218-1
  14. W. Takahashi and T. Tamura, Limit theorems of operators by convex combinations of nonexpansive retractions in Banach spaces, J. Approx. Theory 91 (1997), no. 3, 386-397 https://doi.org/10.1006/jath.1996.3093
  15. H. K. Xu, An iterative approach to quadratic optimization, J. Optim. Theory Appl. 116 (2003), no. 3, 659-678 https://doi.org/10.1023/A:1023073621589
  16. H. Y. Zhou, L. Wei, and Y. J. Cho, Strong convergence theorems on an iterative method for a family of finite nonexpansive mappings in reflexive Banach spaces, Appl. Math. Comput. 173 (2006), no. 1, 196-212 https://doi.org/10.1016/j.amc.2005.02.049
  17. P. Kuhfittig, Common fixed points of nonexpansive mappings by iteration, Pacific J. Math. 97 (1981), no. 1, 137-139 https://doi.org/10.2140/pjm.1981.97.137

Cited by

  1. A strong convergence of a modified Krasnoselskii‐Mann method for non‐expansive mappings in Hilbert spaces vol.15, pp.2, 2010, https://doi.org/10.3846/1392-6292.2010.15.265-274