Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
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ITERATIVE ALGORITHMS WITH ERRORS FOR NONEXPANSIVE MAPPINGS IN BANACH SPACES

  • Published : 2006.11.30
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Abstract

The iterative algorithms with errors for nonexpansive mappings are investigated in Banach spaces. Strong convergence theorems for these algorithms are obtained. Our results improve the corresponding results in [5, 13-15, 23, 27-29, 32] as well as those in [1, 16, 19, 26] in framework of a Hilbert space.

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References

  1. H. H. Bauschke, The approximation of fixed points of compositions of nonexpansive mappings in Hilbert space, J. Math. Anal. Appl. 202 (1996), no. 1, 150-159 https://doi.org/10.1006/jmaa.1996.0308
  2. F. E. Browder, Convergence of approximations to fixed points of nonexpansive non-linear mappings in Banach spaces, Archs Ration. Mech. Anal. 24 (1967), 82-90 https://doi.org/10.1007/BF00251595
  3. F. E. Browder, Convergence theorems for sequences of nonlinear operators in Banach spaces, Math. Z. 100 (1967), 201-225 https://doi.org/10.1007/BF01109805
  4. R. E. Bruck, On the convex approximation property and the asymptotic behavior of nonlinear contractions in Banach spaces, Israel J. Math. 38 (1981), no. 4, 304-314 https://doi.org/10.1007/BF02762776
  5. Y. J. Cho, S. M. Kang, and H. Y. Zhou, Some control condition on iterative methods, Commun. Applied Nonlinear Anal. 12 (2005), no. 2, 27-34
  6. M. M. Day, Reflexive Banach space not isomorphic to uniformly convex spaces, Bull. Amer. Math. Soc. 47 (1941), 313-317 https://doi.org/10.1090/S0002-9904-1941-07451-3
  7. J. Diestel, Geometry of Banach Spaces, Lectures Notes in Math. 485, Springer-Verlag, Berlin, Heidelberg, 1975
  8. F. Deutsch and I. Yamada, Minimizing certain convex functions over the intersection of the fixed point sets of nonexpansive mappings, Numer. Funct. Ana. Optim. 19 (1998), no. 1-2, 33-56 https://doi.org/10.1080/01630569808816813
  9. K. Goebel and W. A. Kirk, Topics in metric fixed point theory, in 'Cambridge Studies in Advanced Mathematics,' Vol. 28, Cambridge Univ. Press, Cambridge, UK, 1990
  10. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York and Basel, 1984
  11. K. S. Ha and J. S. Jung, Strong convergence theorems for accretive operators in Banach spaces, J. Math. Anal. Appl. 147 (1990), no. 2, 330-339 https://doi.org/10.1016/0022-247X(90)90351-F
  12. B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. 73 (1967), 957-961 https://doi.org/10.1090/S0002-9904-1967-11864-0
  13. J. S. Jung, Iterative approaches to common fixed points of nonexpansive mappings in Banach spaces, J. Math. Anal. Appl. 302 (2005), no. 2, 509-520 https://doi.org/10.1016/j.jmaa.2004.08.022
  14. J. S. Jung, Y. J. Cho, and R. P. Agarwal, Iterative schemes with some control conditions for a family of finite nonexpansive mappings in Banach space, Fixed Point Theory Appl. 2005 (2005), no. 2, 125-135 https://doi.org/10.1155/FPTA.2005.125
  15. J. S. Jung and T. H. Kim, Convergence of approximate sequences for compositions of nonexpansive mappings in Banach spaces, Bull. Korean Math. Soc. 34 (1997), no. 1, 93-102
  16. P. L. Lions, Approximation de points fixes de contractions, C. R. Acad. Sci. Ser A-B, Paris 284 (1977), no. 21, 1357-1359
  17. L. S. Liu, Ishikawa and Mann iterative processes with errors for nonlinear strongly accretive mappings in Banach spaces, J. Math. Anal. Appl. 194 (1995), no. 1, 114-125 https://doi.org/10.1006/jmaa.1995.1289
  18. 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
  19. J. G. OHara, P. Oillay, and H. K. Xu, Iterative approaches to finding nearest common fixed points of nonexpansive mappings in Hilbert spaces, Nonlinear Anal. 54 (2003), no. 8, 1417-1426 https://doi.org/10.1016/S0362-546X(03)00193-7
  20. S. Reich, Strong convergence theorems for resolvents of accretive operators in Banach spaces, J. Math. Anal. Appl. 75 (1980), no. 1, 287-292 https://doi.org/10.1016/0022-247X(80)90323-6
  21. S. Reich, Approximating fixed points of nonexpansive mappings, Panamer. Math. J. 4 (1994), no. 2, 23-28
  22. T. Shimizu and W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997), no. 1, 71-83 https://doi.org/10.1006/jmaa.1997.5398