Advertisement

Fixed point, approximate fixed point and Kantorovich-Rubinstein maximum principle in convex metric spaces

  • Ismat Beg
  • Mujahid Abbas
Article

Abstract

We obtain necessary conditions for the existence of fixed point and approximate fixed point of nonexpansive and asymptotically nonexpansive maps defined on a closed bounded convex subset of a uniformly convex complete metric space and study the structure of the set of fixed points. We construct Mann type iterative sequences in convex metric space and study its convergence. As a consequence of fixed point results, we prove best approximation results. We also prove Kantorovich-Rubinstein maximum principle in convex metric spaces.

Keywords

Fixed point Convex metric space Uniformly convex metric space Best approximation Kantorovich-Rubinstein maximum principle 

Mathematics Subject Classification (2000)

47H09 47H10 54H25 49J35 30C80 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Alber, Ya., Delabriere, S.: Principles of weakly contractive maps in Hilbert spaces. In: Gohberg, I., Lyubich, Yu. (eds.) New Results in Operator Theory. Advances and Appl., vol. 98, pp. 7–22. Birkhauser, Basel (1997) Google Scholar
  2. 2.
    Alspach, D.E.: A fixed point free nonexpansive map. Proc. Am. Math. Soc. 82, 423–424 (1981) CrossRefMathSciNetzbMATHGoogle Scholar
  3. 3.
    Aronszajn, N., Panitchpakdi, P.: Extension of uniformly continuous transformations and hyper convex metric spaces. Pac. J. Math. 6, 405–439 (1956) MathSciNetGoogle Scholar
  4. 4.
    Baillon, J.B., Schoneberg, R.: Asymptotic normal structure and fixed points of nonexpansive mappings. Proc. Am. Math. Soc. 81, 257–264 (1981) CrossRefMathSciNetzbMATHGoogle Scholar
  5. 5.
    Beg, I., Azam, A.: Fixed points of asymptotically regular multivalued mappings. J. Aust. Math. Soc. Ser. A 53(3), 313–326 (1992) MathSciNetzbMATHGoogle Scholar
  6. 6.
    Beg, I., Azam, A.: Common fixed points for commuting and compatible maps. Discuss. Math. Differ. Incl. 16, 121–135 (1996) MathSciNetzbMATHGoogle Scholar
  7. 7.
    Beg, I., Khan, A.R., Hussain, N.: Approximation of *-nonexpansive random multivalued operators on Banach spaces. J. Aust. Math. Soc. 76, 51–66 (2004) MathSciNetzbMATHCrossRefGoogle Scholar
  8. 8.
    Benavides, T., Ramirez, P.: Structure of the fixed point set and common fixed points of asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 12, 3549–3557 (2001) CrossRefGoogle Scholar
  9. 9.
    Blumenthal, L.M.: Distance Geometry. Clarendon, Oxford (1953) zbMATHGoogle Scholar
  10. 10.
    Brodskii, M.S., Milman, D.P.: On the center of convex sets. Dokl. Akad. Nauk SSSR 59, 837–840 (1948) MathSciNetGoogle Scholar
  11. 11.
    Browder, F.E.: Nonexpansive nonlinear operators in Banach space. Proc. Natl. Acad. Sci. USA 54, 1041–1044 (1965) CrossRefMathSciNetzbMATHGoogle Scholar
  12. 12.
    Gacki, H.: An application of the Kantorovich-Rubinstein maximum principle in the stability theory of Markov operators. Bull. Pol. Acad. Sci. Math. 46, 215–223 (1998) MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gacki, H., Lasota, A.: A nonlinear version of the Kantorovich-Rubinstein maximum principle. Nonlinear Anal. 52, 117–125 (2003) CrossRefMathSciNetzbMATHGoogle Scholar
  14. 14.
    Goebel, K., Kirk, W.A.: A fixed point theorem for asymptotically nonexpansive mappings. Proc. Am. Math. Soc. 35(1), 171–174 (1972) CrossRefMathSciNetzbMATHGoogle Scholar
  15. 15.
    Goebel, K., Reich, S.: Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings. Dekker, New York (1984) zbMATHGoogle Scholar
  16. 16.
    Goebel, K., Kirk, W.A.: Topics in Metric Fixed Point Theory. Cambridge University Press, Cambridge (1990) zbMATHGoogle Scholar
  17. 17.
    Göhde, D.: Zum Prinzip der Kontraktiven Abbildung. Math. Nachr. 30, 251–258 (1965) CrossRefMathSciNetzbMATHGoogle Scholar
  18. 18.
    Habiniak, L.: Fixed point theorem and invariant approximation. J. Approx. Theory 56, 241–244 (1989) CrossRefMathSciNetzbMATHGoogle Scholar
  19. 19.
    Khalil, R.: Extreme points of the unit ball of Banach spaces. Math. Rep. Toyama Univ. 4, 41–45 (1981) MathSciNetzbMATHGoogle Scholar
  20. 20.
    Khalil, R.: Best approximation in metric spaces. Proc. Am. Math. Soc. 103, 579–586 (1988) CrossRefMathSciNetzbMATHGoogle Scholar
  21. 21.
    Kirk, W.A.: Fixed point theory for nonexpansive mappings I & II. Lectures Notes in Math., vol. 886, pp. 484–504. Springer, Berlin (1981). Also Contemp. Math. 18, 121–140 (1983) Google Scholar
  22. 22.
    Kirk, W.A.: Nonexpansive mappings and asymptotic regularity. Nonlinear Anal. 40, 323–332 (2000) CrossRefMathSciNetGoogle Scholar
  23. 23.
    Kirk, W.A.: An application of a generalized Krasnoselskii-Ishikawa iteration Process. In: Fixed point Theory and Applications (Marseille 1989), pp. 261–268. Longman, Harlow (1991) Google Scholar
  24. 24.
    Lasota, A., Traple, J.: An application of the Kantorovich-Rubinstein maximum principle in the Tjon-Wu equation. J. Differ. Equ. 159, 578–596 (1999) CrossRefMathSciNetzbMATHGoogle Scholar
  25. 25.
    Menger, K.: Untersuchungen über allegemeine Metrik. Math. Ann. 100, 75–163 (1928) CrossRefMathSciNetzbMATHGoogle Scholar
  26. 26.
    Rhoades, B.E.: Some theorems on weakly contractive maps. Nonlinear Anal. 47, 2683–2693 (2001) CrossRefMathSciNetzbMATHGoogle Scholar
  27. 27.
    Simmons, G.F.: Topology and Modern Analysis. McGraw-Hill, New York (1963) zbMATHGoogle Scholar
  28. 28.
    Singer, I.: Best Approximation in Normed Linear Spaces by Elements of Linear Subspaces. Springer, New York (1970) zbMATHGoogle Scholar
  29. 29.
    Singh, S.P.: An application of a fixed point theorem to approximation theory. J. Approx. Theory 25, 89–90 (1979) CrossRefMathSciNetzbMATHGoogle Scholar

Copyright information

© KSCAM and Springer-Verlag 2008

Authors and Affiliations

  1. 1.Centre for Advanced Studies in MathematicsLahore University of Management SciencesLahorePakistan

Personalised recommendations