Numerical Algorithms

, Volume 71, Issue 1, pp 25–39 | Cite as

Approximating fixed points of mappings satisfying condition (E) in Busemann space

Original Paper


It is well-known that in a Banach space, using the Ishikawa iterative process, one can find fixed points of nonexpansive mappings via asymptotic center’s method. In this paper, we obtain the fixed points of mappings satisfying so-called condition (E) in a uniformly convex Busemann space. Many known results in CAT (0) spaces are improved and extended by our results.


Asymptotic center Mappings satisfying condition (EIshikawa iterative process Uniformly convex Busemann space 

Mathematics Subject Classification (2010)

Primary: 47H10, 54H25 Secondary: 54E40 


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  1. 1.
    Busemann, H.: Spaces with nonpositive curvature. Acta Math. 80, 259–310 (1948)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    Dhompongsa, S., Inthakon, W., Kaewkhao, A.: Edelstein’s method and fixed point theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 350, 12–17 (2009)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Dhompongsa, S., Panyanak, B.: On Δ-convergence theorems in C A T(0) spaces. Comput. Math. Appl. 56, 2572–2579 (2008)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Espínola, R., Fernández-León, A.: C A T(k)-spaces, weak convergence and fixed points. J. Math. Anal. Appl. 353, 410–427 (2009)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Espínola, R., Fernández-León, A., Piatek, B.: Fixed points of single- and set-valued mappings in uniformly convex metric spaces with no metric convexity. Fixed Point Theory Appl. (2010). Article ID 169837Google Scholar
  6. 6.
    Foertsch, T., Lytchak, A., Schroeder, V.: Non-positive curvature and the ptolemy inequality. Int. Math. Res. Not. IMRN 22 (2007). doi: 10.1093/imrn/rnm100
  7. 7.
    García-Falset, J., Llorens-Fuster, E., Suzuki, T.: Fixed point theory for a class of generalized nonexpansive mappings. J. Math. Anal. Appl. 375, 185–195 (2011)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    Genel, A., Lindenstrass, J.: An example concerning fixed points. Israel J. Math. 22, 81–86 (1975)MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    Goebel, K., Kirk, W.A.: Topics in metric fixed point theory. Cambridge University Press, Cambridge (1990)CrossRefMATHGoogle Scholar
  10. 10.
    Khamsi, M.A., Khan, A.R.: Inequalities in metric spaces with applications. Nonlinear Anal. 74, 4036–4045 (2011)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Khan, S.H., Suzuki, T.: A Reich-type convergence theorem for generalized nonexpansive mappings in uniformly convex Banach spaces. Nonlinear Anal. 80, 211–215 (2013)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Kirk, W.A.: Geodesic geometry and fixed point theory. In: Girela, D., López, G., Villa, R. (eds.) Seminar of Mathematical Analysis, Proceedings, Universities of Malaga and Seville, Sept. 2002-Feb. 2003, pp 195–225. Universidad de Sevilla, Sevilla (2003)Google Scholar
  13. 13.
    Kirk, W.A.: Geodesic geometry and fixed point theory II. In: García-Falset, J., Llorens-Fuster, E., Sims, B. (eds.) Fixed Point Theory and its Applications, pp 113–142. Yokohama Publ., Yokohama (2004)Google Scholar
  14. 14.
    Kirk, W.A., Panyanak, B.: A concept of convergence in geodesic spaces. Nonlinear Anal. 68, 3689–3696 (2008)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Kohlenbach, U.: Some logical metatheorems with applications in functional analysis. Trans. Amer. Math. Soc. 357, 89–128 (2005)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Leuştean, L.: Nonexpansive iterations in uniformly convex W-hyperbolic spaces. In: Leizarowitz, A., Mordukhovich, B. S., Shafrir, I., Zaslavski A. (eds.) Nonlinear Analysis and Optimization I: Nonlinear Analysis, Contemporary Mathematics. AMS, vol. 513, pp 193–209 (2010)Google Scholar
  17. 17.
    Lim, T.C.: Remarks on some fixed point theorems. Proc. Amer. Math. Soc. 60, 179–182 (1976)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Nanjaras, B., Panyanak, B., Phuengrattana, W.: Fixed point theorems and convergence theorems for Suzuki-generalized nonexpansive mappings in C A T(0) spaces. Nonlinear Anal. Hybrid Syst. 4, 25–31 (2010)MathSciNetCrossRefMATHGoogle Scholar
  19. 19.
    Naraghirad, E., Wong, N.C., Yao, J.C.: Approximating fixed points of α-nonexpansive mappings in uniformly convex Banach spaces and C A T(0) spaces. Fixed Point Theory Appl. (2013). Article ID 57Google Scholar
  20. 20.
    Oppenheim, I.: Fixed point theorem for reflexive Banach spaces and uniformly convex non positively curved metric spaces. Math. Z., 1–13 (2012)Google Scholar
  21. 21.
    Panyanak, B., Laokul, T.: On the Ishikawa iteration process in C A T(0) spaces. Bull. Iranian Math. Soc. 37, 185–197 (2011)MathSciNetGoogle Scholar
  22. 22.
    Papadopoulos, A.: Metric spaces, Convexity and Nonpositive Curvature. IRMA Lectures in Mathematics and Theoretical Physics 6. European Mathematical Society (2005)Google Scholar
  23. 23.
    Phuengrattana, W.: Approximating fixed points of Suzuki-generalized nonexpansive mappings. Nonlinear Anal. Hybrid Syst. 5, 583–590 (2011)MathSciNetCrossRefMATHGoogle Scholar
  24. 24.
    Reich, S.: Weak convergence theorems for nonexpansive mappings in Banach spaces. J. Math. Anal. Appl. 67, 274–276 (1979)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Senter, H.F., Dotson, W.G.: Approximating fixed points of nonexpansive mappings. Proc. Amer. Math. Soc. 44, 375–380 (1974)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Sosov, E.N.: On analogues of weak convergence in a special metric space. Izv. Vyssh. Uchebn. Zaved. Mat. 5(2004), 84–89 (Russian); English transl. Russian Math. (Iz. VUZ) 48, 79–83 (2004)MathSciNetMATHGoogle Scholar
  27. 27.
    Suzuki, T.: Fixed point theorems and convergence theorems for some generalized nonexpansive mappings. J. Math. Anal. Appl. 340, 1088–1095 (2008)MathSciNetCrossRefMATHGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Mathematics, Khomeinishahr BranchIslamic Azad UniversityIsfahanIran

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