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Halpern iteration in CAT(κ) spaces

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Abstract

In this paper we show that an iterative sequence generated by the Halpern algorithm converges to a fixed point in the case of complete CAT(κ) spaces. Similar results for Hadamard manifolds were obtained in [Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)], but we study a much more general case. Moreover, we discuss the Halpern iteration procedure for set-valued mappings.

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References

  1. Goebel, K., Kirk, W. A.: Topics in Metric Fixed Point Theory, Cambridge Univ. Press, Cambridge, 1990

    Book  MATH  Google Scholar 

  2. Kirk, W. A., Sims, B.: Handbook of Metric Fixed Point Theory (W. A. Kirk and B. Sims eds.), Kluwer Academic Publishers, Dordrecht, 2001

    Google Scholar 

  3. Halpern, B.: Fixed points of nonexpansive maps. Bull. Amer. Math. Soc., 73, 957–961 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  4. Wittmann, R.: Approximation of fixed points of nonexpansive mappings. Arch. Math. (Basel), 58, 486–491 (1992)

    MathSciNet  MATH  Google Scholar 

  5. Suzuki, T.: A sufficient and necassary condition for Halpern-type strong convergence to fixed point of nonexpansive mappings. Proc. Amer. Math. Soc., 135, 99–106 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  6. Suzuki, T.: Reich’s problem concerning Halpern’s convergence. Arch. Math. (Basel), 92, 602–613 (2009)

    MathSciNet  MATH  Google Scholar 

  7. Chang, S. S., Yao, J. C., Kim, J. K., et al.: Iterative approximation to convex feasibility problems in Banach space. Fixed Point Theory Appl., 2007, 19pp. (2007)

  8. Chidume, C. E., Chidume, C. O.: Iterative approximation of fixed points of nonexpansive mappings. J. Math. Anal. Appl., 318, 288–295 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  9. Reich, S.: Strong convergence theorems for resolvents of accretive operators in Banach spaces. J. Math. Anal. Appl., 75, 287–292 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  10. Saejung, S.: Halpern’s iteration in CAT(0) spaces. Fixed Point Theory Appl., 2010, 13pp. (2010)

  11. Shioji, N., Takahashi, W.: Strong convergence of approximated sequences for nonexpansive mappings in Banach spaces. Proc. Amer. Math. Soc., 125, 3641–3645 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Xu, H.-K.: Iterative algorithms for nonlinear operators. J. London Math. Soc. (2), 66, 240–256 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  13. Zhang, S.-S., Joseph, W. W., Chan, C. K.: Approximation of nearest common fixed point of nonexpansive mappings in Hilbert spaces. Acta Mathematica Sinica, English Series, 23, 1889–1896 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  14. Alber, Y., Espínola, R., Lorenzo, P.: Strongly convergent approximations to fixed point of total asymptotically nnexpansive mappings. Acta Mathematica Sinica, English Series, 24, 1005–1022 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  15. Chen, R. D., He, H. M., Noor, M. A.: Modified Mann iterations for nonexpansive semigroups in Banach space. Acta Mathematica Sinica, English Series, 26, 193–202 (2010)

    Article  MathSciNet  Google Scholar 

  16. Zegeye, H., Shahzad, N.: Viscosity approximation methods for nonexpansive multimaps in Banach spaces. Acta Mathematica Sinica, English Series, 26, 1165–1176 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  17. Li, C., López, G., Martín-Márquez, V.: Iterative algorithms for nonexpansive mappings on Hadamard manifolds. Taiwanese J. Math., 14, 541–559 (2010)

    MathSciNet  MATH  Google Scholar 

  18. Bridson, M., Haefliger, A.: Metric Spaces of Non-positive Curvature, Springer-Verlag, Berlin, 1999

    MATH  Google Scholar 

  19. Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry and Nonexpansive Mappings, Pure Appl. Math., Marcel Dekker, Inc., New York-Basel, 1984

    MATH  Google Scholar 

  20. Espínola, R., Fernández-León, A.: CAT(κ)-spaces, weak convergence and fixed points. J. Math. Anal. Appl., 353, 410–427 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  21. Kirk, W. A.: Geodesic Geometry and Fixed Point Theory, Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), 195–225, Univ. Sevilla Secr. Publ., Seville, 2003

    Google Scholar 

  22. Aksoy, A. G., Khamsi, M. A.: A selection theorem in metric trees. Proc. Amer. Math. Soc., 134, 2957–2966 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  23. Espínola, R., Kirk, W. A.: Fixed point theorems in R-trees with applications to graph theory. Topology Appl., 153, 1046–1055 (2006)

    Article  MathSciNet  MATH  Google Scholar 

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  1. Institute of Mathematics, Silesian University of Technology, 44-100, Gliwice, Poland

    Bożena Piątek

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  1. Bożena Piątek
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Piątek, B. Halpern iteration in CAT(κ) spaces. Acta. Math. Sin.-English Ser. 27, 635–646 (2011). https://doi.org/10.1007/s10114-011-9312-7

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  • DOI: https://doi.org/10.1007/s10114-011-9312-7

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