Strong and â–³-Convergence of Modified Two-Step Iterations for Nearly Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

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G. S. Saluja, Strong and â–³-Convergence of Modified Two-Step Iterations for Nearly Asymptotically Nonexpansive Mappings in Hyperbolic Spaces, Int. J. Anal. Appl., 8 (1) (2015), 39-52.

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G. S. Saluja

Abstract

The aim of this article is to establish a â–³-convergence and some strong convergence theorems of modified two-step iterations for two nearly asymptotically nonexpansive mappings in the setting of hyperbolic spaces. Our results extend and generalize the previous work from the current existing literature.

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References

  1. M. Abbas, Z. Kadelburg and D.R. Sahu, Fixed point theorems for Lipschitzian type mappings in CAT(0) spaces, Math. Comput. Model. 55 (2012), 1418-1427.
  2. R.P. Agarwal, Donal O'Regan and D.R. Sahu, Iterative construction of fixed points of nearly asymptotically nonexpansive mappings, Nonlinear Convex Anal. 8(1) (2007), 61-79.
  3. I. Beg, An iteration scheme for asymptotically nonexpansive mappings on uniformly convex metric spaces, Nonlinear Anal. Forum, 6 (2001), 27-34.
  4. M.R. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Vol. 319 of Grundlehren der Mathematischen Wissenschaften, Springer, Berlin, Germany, 1999.
  5. S.S. Chang, L. Wang, H.W. Joesph Lee, C.K. Chan, L. Yang, Demiclosed principle and ∆-convergence theorems for total asymptotically nonexpansive mappings in CAT(0) spaces, Appl. Math. Comput. 219(5) (2012), 2611-2617.
  6. S. Dhompongsa and B. Panyanak, On 4-convergence theorem in CAT(0) spaces, Comput. Math. Appl. 56 (2008), 2572-2579.
  7. K. Goebel and W.A. Kirk, A fixed point theorem for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 35 (1972), 171-174.
  8. K. Goebel and W.A. Kirk, Iterations processes for nonexpansive mappings, Contemp. Math. 21 (1983), 115-123.
  9. M. Gromov, Hyperbolic groups. Essays in group theory (S. M. Gersten, ed). Springer Verlag, MSRI Publ. 8 (1987), 75-263.
  10. S. Ishikawa, Fixed points by a new iteration method, Proc. Amer. Math. Soc. 44 (1974), 147-150.
  11. S.H. Khan and M. Abbas, Strong and 4-convergence of some iterative schemes in CAT(0) spaces, Comput. Math. Appl. 61 (2011), 109-116.
  12. 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.
  13. W.A. Kirk, Krasnoselskii's iteration process in hyperbolic space, Numer. Funct. Anal. Optim 4 (1982), 371-381.
  14. W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), 3689-3696.
  15. U. Kohlenbach, Some logical metatheorems with applications in functional analysis, Trans. Amer. Math. Soc. 357 (2005), 89-128.
  16. T.C. Lim, Remarks on some fixed point theorems, Proc. Amer. Math. Soc. 60 (1976), 179-182.
  17. Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings, J. Math. Anal. Appl. 259 (2001), 1-7.
  18. Q.H. Liu, Iterative sequences for asymptotically quasi-nonexpansive mappings with error member, J. Math. Anal. Appl. 259 (2001), 18-24.
  19. W.R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
  20. B. Nanjaras and B. Panyanak, Demiclosed principle for asymptotically nonexpansive mappings in CAT(0) spaces, Fixed Point Theory Appl. 2010 (2010), Art. ID 268780.
  21. Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc. 73(1967), 591-597.
  22. M.O. Osilike, S.C. Aniagbosor, Weak and strong convergence theorems for fixed points of asymptotically nonexpansive mappings, Math. and Computer Modelling 32(2000), 1181-1191.
  23. S. Reich and I. Shafrir, Nonexpansive iterations in hyperbolic spaces, Nonlinear Anal.: TMA, Series A, Theory Methods, 15(6)(1990), 537-558.
  24. B.E. Rhoades, Fixed point iteration for certain nonlinear mappings, J. Math. Anal. Appl. 183(1994), 118-120.
  25. A. S ¸ahin and M. Ba ¸sarir, On the strong convergence of a modified S-iteration process for asymptotically quasi-nonexpansive mappings in a CAT(0) space, Fixed Point Theory Appl. 2013 (2013), Article ID 12.
  26. D.R. Sahu, Fixed points of demicontinuous nearly Lipschitzian mappings in Banach spaces, Comment. Math. Univ. Carolinae 46(4) (2005), 653-666.
  27. D.R. Sahu and I. Beg, Weak and strong convergence for fixed points of nearly asymptotically nonexpansive mappings, Int. J. Mod. Math. 3 (2008), 135-151.
  28. G.S. Saluja, Strong convergence theorem for two asymptotically quasinonexpansive mappings with errors in Banach space, Tamkang J. Math. 38(1) (2007), 85-92.
  29. G.S. Saluja, Convergence result of (L, α)-uniformly Lipschitz asymptotically quasi-nonexpansive mappings in uniformly convex Banach spaces, JËœna ¯n ¯abha 38 (2008), 41-48.
  30. H.F. Senter, W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc. 44 (1974), 375-380.
  31. N. Shahzad, A. Udomene, Approximating common fixed points of two asymptotically quasi-nonexpansive mappings in Banach spaces, Fixed Point Theory and Applications, 2006 (2006), Article ID 18909.
  32. T. Shimizu and W. Takahashi, Fixed points of multivalued mappings in certain convex metric spaces, Topol. Methods Nonlinear Anal. 8(1) (1996), 197-203.
  33. W. Takahashi, A convexity in metric spaces and nonexpansive mappings, Kodai Math. Semin. Rep. 22 (1970), 142-149.
  34. K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by the Ishikawa iteration process, J. Math. Anal. Appl. 178, 301-308, 1993.
  35. K.K. Tan and H.K. Xu, Fixed point iteration processes for asymptotically nonexpansive mappings, Proc. Amer. Math. Soc. 122(1994), 733-739.