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TWO GENERAL ITERATION SCHEMES FOR MULTI-VALUED MAPS IN HYPERBOLIC SPACES

  • Received : 2015.08.06
  • Published : 2016.10.31

Abstract

In this paper, we introduce two general iteration schemes with bounded error terms and prove some theorems related to the strong and ${\Delta}$-convergence of these iteration schemes for multi-valued maps in a hyperbolic space. The results which are presented here extend and improve some well-known results in the current literature.

Keywords

References

  1. M. Basarir and A. Sahin, On the strong and ${\Delta}$-convergence of new multi-step and S-iteration processes in a CAT(0) space, J. Inequal. Appl. 2013 (2013), Article ID 482, 13 pages. https://doi.org/10.1186/1029-242X-2013-13
  2. M. Bridson and A. Haefliger, Metric Spaces of Non-Positive Curvature, Springer, Berlin, 1999.
  3. S. S. Chang, Y. J. Cho, and H. Zhou, Demi-closed principle and weak convergence problems for asymptotically nonexpansive mappings, J. Korean Math. Soc. 38 (2001), no. 6, 1245-1260.
  4. W. Cholamjiak and S. Suantai, Approximation of common fixed points of two quasi-nonexpansive multi-valued maps in Banach spaces, Comput. Math. Appl. 61 (2011), no. 4, 941-949. https://doi.org/10.1016/j.camwa.2010.12.042
  5. S. Dhompongsa and B. Panyanak, On ${\Delta}$-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), no. 10, 2572-2579. https://doi.org/10.1016/j.camwa.2008.05.036
  6. H. Fukhar-ud-din and M. A. A. Khan, Convergence analysis of a general iteration schema of nonlinear mappings in hyperbolic spaces, Fixed Point Theory Appl. 2013 (2013), Article ID 238, 18 pages. https://doi.org/10.1186/1687-1812-2013-18
  7. H. Fukhar-ud-din, A. R. Khan, and M. Ubaid-ur-rehman, Ishikawa type algorithm of two multi-valued quasi-nonexpansive maps on nonlinear domains, Ann. Funct. Anal. 4 (2013), no. 2, 97-109. https://doi.org/10.15352/afa/1399899528
  8. K. Goebel and W. A. Kirk, Iteration processes for nonexpansive mappings, in: S. P. Singh, S. Thomeier, B. Watson (eds.), Topological methods in nonlinear functional analysis (Toronto, Ont., 1982), 115-123, Contemp. Math., 21, Amer. Math. Soc., Providence, RI, 1983.
  9. K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Marcel Dekker, New York, 1984.
  10. F. Gu and Z. He, Multi-step iterative process with errors for common fixed points of a finite family of nonexpansive mappings, Math. Commun. 11 (2006), no. 1, 47-54.
  11. F. Gursoy, V. Karakaya, and B. E. Rhoades, Data dependence results of new multi-step and S-iterative schemes for contractive-like operators, Fixed Point Theory Appl. 2013 (2013), Article ID 76, 12 pages.
  12. S. H. Khan, H. Fukhar-ud-din, and A. Kalsoom, Common fixed points of two multivalued nonexpansive maps by a one-step implicit algorithm in hyperbolic spaces, Mat. Vesnik 66 (2014), no. 4, 397-409.
  13. A. R. Khan, H. Fukhar-ud-din, and M. A. A. Khan, An implicit algorithm for two finite families of nonexpansive maps in hyperbolic spaces, Fixed Point Theory Appl. 2012 (2012), Article ID 54, 12 pages. https://doi.org/10.1186/1687-1812-2012-12
  14. U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2004), no. 1, 89-128. https://doi.org/10.1090/S0002-9947-04-03515-9
  15. L. Leustean, Nonexpansive iterations in uniformly convex W-hyperbolic spaces, in: A. Leizarowitz, B. S. Mordukhovich, I. Shafrir, A. Zaslavski (eds.), Nonlinear analysis and optimization I. Nonlinear analysis, 193-210, Contemp. Math., 513, Amer. Math. Soc., Providence, RI, 2010.
  16. T. C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182. https://doi.org/10.1090/S0002-9939-1976-0423139-X
  17. B. Panyanak, On the Ishikawa iteration processes for multivalued mappings in some CAT(${\kappa}$) spaces, Fixed Point Theory Appl. 2014 (2014), Article ID 1, 9 pages. https://doi.org/10.1186/1687-1812-2014-9
  18. W. Phuengrattana and S. Suantai, On the rate of convergence of Mann, Ishikawa, Noor and SP-iterations for continuous functions on an arbitrary interval, J. Comput. Appl. Math. 235 (2011), no. 9, 3006-3014. https://doi.org/10.1016/j.cam.2010.12.022
  19. T. Puttasontiphot, Mann and Ishikawa iteration schemes for multivalued mappings in CAT(0) spaces, Appl. Math. Sci. 4 (2010), no. 61-64, 3005-3018.
  20. S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal. 15 (1990), no. 6, 537-558. https://doi.org/10.1016/0362-546X(90)90058-O
  21. S. Reich and A. J. Zaslavski, Genericity in Nonlinear Analysis, Springer, New York, 2014.
  22. T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal. 8 (1996), no. 1, 197-203. https://doi.org/10.12775/TMNA.1996.028
  23. W. Takahashi, A convexity in metric space and nonexpansive mappings. I, Kodai Math. Sem. Rep. 22 (1970), no. 2, 142-149. https://doi.org/10.2996/kmj/1138846111
  24. S. Thianwan, Common fixed points of new iterations for two asymptotically nonexpan-sive nonself-mappings in a Banach space, J. Comput. Appl. Math. 224 (2009), no. 2, 688-695. https://doi.org/10.1016/j.cam.2008.05.051
  25. I. Yildirim and M. Ozdemir, A new iterative process for common fixed points of finite families of non-self-asymptotically non-expansive mappings, Nonlinear Anal. 71 (2009), no. 3-4, 991-999. https://doi.org/10.1016/j.na.2008.11.017