Convergence of Modified S-Iteration to Common Fixed Points of Asymptotically Nonexpansive Mappings in Hyperbolic Spaces

Main Article Content

S. Jone Jayashree, A. Anthony Eldred

Abstract

In this paper, we provide some sufficient conditions for the strong convergence of an improved form of modified S-iteration for approximating common fixed points of two asymptotically nonexpansive mappings defined on a closed convex subset of a uniformly convex hyperbolic spaces.

Article Details

References

  1. K. Goebel, W.A. Kirk, A Fixed Point Theorem for Asymptotically Nonexpansive Mappings, Proc. Amer. Math. Soc. 35 (1972), 171–174.
  2. J. Schu, Weak and Strong Convergence to Fixed Points of Asymptotically Nonexpansive Mappings, Bull. Austral. Math. Soc. 43 (1991), 153–159. https://doi.org/10.1017/s0004972700028884.
  3. R.P. Agarwal, D. O’Regan, D.R. Sahu, Iterative Construction of Fixed Points of Nearly Asymptotically Nonexpansive Mappings, J. Nonlinear Convex Anal. 8 (2007), 61–79.
  4. C.E. Chidume, E.U. Ofoedu, H. Zegeye, Strong andWeak Convergence Theorems for Asymptotically Nonexpansive Mappings, J. Math. Anal. Appl. 280 (2003), 364–374. https://doi.org/10.1016/s0022-247x(03)00061-1.
  5. H. Fukhar-ud-din, S.H. Khan, Convergence of Iterates With Errors of Asymptotically Quasi-Nonexpansive Mappings and Applications, J. Math. Anal. Appl. 328 (2007), 821–829. https://doi.org/10.1016/j.jmaa.2006.05.068.
  6. L. Qihou, Iterative Sequences for Asymptotically Quasi-nonexpansive Mappings, J. Math. Anal. Appl. 259 (2001), 1–7. https://doi.org/10.1006/jmaa.2000.6980.
  7. W. Kaczor, Weak Convergence of Almost Orbits of Asymptotically Nonexpansive Commutative Semigroups, J. Math. Anal. Appl. 272 (2002), 565–574. https://doi.org/10.1016/s0022-247x(02)00175-0.
  8. D.R. Sahu, J.S. Jung, Fixed-Point Iteration Processes for Non-Lipschitzian Mappings of Asymptotically QuasiNonexpansive Type, Int. J. Math. Math. Sci. 2003 (2003), 2075–2081. https://doi.org/10.1155/s0161171203302182.
  9. K.K. Tan, H.K. Xu, Fixed Point Iteration Processes for Asymptotically Nonexpansive Mappings, Proc. Amer. Math. Soc. 122 (1994), 733–739.
  10. U. Kohlenbach, Some Logical Metatheorems With Applications in Functional Analysis, Trans. Amer. Math. Soc. 357 (2005), 89–128
  11. A. ¸Sahin, M. Ba¸sarır, On the Strong Convergence of a Modified S-Iteration Process for Asymptotically QuasiNonexpansive Mappings in a CAT (0) Space, Fixed Point Theory Appl. 2013 (2013), 12. https://doi.org/10.1186/1687-1812-2013-12.
  12. G.S. Saluja, Convergence of Modified S-Iteration Process for Two Asymptotically Quasi-Nonexpansive Type Mappings in CAT(0) Spaces, Demonstr. Math. 49 (2016), 107–118. https://doi.org/10.1515/dema-2016-0010.
  13. L. Leu¸stean, 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, American Mathematical Society, pp. 193–209, (2010).
  14. A.R. Khan, H. Fukhar-ud-din, M.A. Ahmad Khan, An Implicit Algorithm for Two Finite Families of Nonexpansive Maps in Hyperbolic Spaces, Fixed Point Theory Appl. 2012 (2012), 54. https://doi.org/10.1186/1687-1812-2012-54.
  15. K.K. Tan, H.K. Xu, Approximating Fixed Points of Nonexpansive Mappings by the Ishikawa Iteration Process, J. Math. Anal. Appl. 178 (1993), 301–308. https://doi.org/10.1006/jmaa.1993.1309.