Czechoslovak Mathematical Journal

, Volume 57, Issue 1, pp 269–279 | Cite as

The characteristic of noncompact convexity and random fixed point theorem for set-valued operators

  • Poom Kumam
  • Somyot Plubtieng


Let (Ω, Σ) be a measurable space, X a Banach space whose characteristic of noncompact convexity is less than 1, C a bounded closed convex subset of X, KC(C) the family of all compact convex subsets of C. We prove that a set-valued nonexpansive mapping T: CKC(C) has a fixed point. Furthermore, if X is separable then we also prove that a set-valued nonexpansive operator T: Ω × CKC(C) has a random fixed point.


random fixed point set-valued random operator measure of noncompactness 


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  1. [1]
    J. P. Aubin and H. Frankowska: Set-valued Analysis. Birkhäuser, Boston, 1990.MATHGoogle Scholar
  2. [2]
    J. M. Ayerbe Toledano, T. Domínguez Benavides and G. López Acedo: Measures of Noncompactness in Metric Fixed Point Theory; Advances and Applications Topics in Metric Fixed Point Theory. Birkhauser-Verlag, Basel 99, 1997.Google Scholar
  3. [3]
    K. Deimling: Nonlinear Functional Analysis. Springer-Verlag, Berlin, 1974.Google Scholar
  4. [4]
    T. Domínguez Benavides and P. Lorenzo Ramírez: Fixed point theorem for multivalued nonexpansive mapping without uniform convexity. Abstr. Appl. Anal. 6 (2003), 375–386.CrossRefGoogle Scholar
  5. [5]
    T. Domínguez Benavides and P. Lorenzo Ramírez: Fixed point theorem for multivalued nonexpansive mapping satisfying inwardness conditions. J. Math. Anal. Appl. 291 (2004), 100–108.MATHCrossRefMathSciNetGoogle Scholar
  6. [6]
    T. Domínguez Benavides, G. Lopez Acedo and H. K. Xu: Random fixed point of set-valued operator. Proc. Amer. Math. Soc. 124 (1996), 838–838.CrossRefGoogle Scholar
  7. [7]
    K. Goebel and W. A. Kirk: Topics in metric fixed point theorem. Cambridge University Press, Cambridge, 1990.Google Scholar
  8. [8]
    S. Itoh: Random fixed point theorem for a multivalued contraction mapping. Pacific J. Math. 68 (1977), 85–90.MATHMathSciNetGoogle Scholar
  9. [9]
    W. A. Kirk: Nonexpansive mappings in product spaces, set-valued mappings, and k-uniform rotundity. Proceedings of the Symposium Pure Mathematics, Vol. 45, part 2, American Mathematical Society, Providence, 1986, pp. 51–64.Google Scholar
  10. [10]
    P. Lorenzo Ramírez: Some random fixed point theorems for nonlinear mappings. Nonlinear Anal. 50 (2002), 265–274.MATHCrossRefMathSciNetGoogle Scholar
  11. [11]
    P. Lorenzo Ramírez: Random fixed point of uniformly Lipschitzian mappings. Nonlinear Anal. 57 (2004), 23–34.MATHCrossRefMathSciNetGoogle Scholar
  12. [12]
    N. Shahzad and S. Latif: Random fixed points for several classes of 1-ball-contractive and 1-set-contractive random maps. J. Math. Anal. Appl. 237 (1999), 83–92.MATHCrossRefMathSciNetGoogle Scholar
  13. [13]
    K.-K. Tan and X. Z. Yuan: Some random fixed point theorems. Fixed Point Theory and Applications (K.-K. Tan, ed.). World Scientific, Singapore, 1992, pp. 334–345.Google Scholar
  14. [14]
    D.-H. Wagner: Survey of measurable selection theorems. SIAM J. Control Optim. 15 (1977), 859–903.MATHCrossRefGoogle Scholar
  15. [15]
    H. K. Xu: Some random fixed point for condensing and nonexpansive operators. Proc. Amer. Math. Soc. 110 (1990), 395–400.MATHCrossRefMathSciNetGoogle Scholar
  16. [16]
    H. K. Xu: Metric fixed point for multivalued mappings. Dissertationes Math. (Rozprawy Mat.) 389 (2000), 39.MathSciNetGoogle Scholar
  17. [17]
    H. K. Xu: A random theorem for multivalued nonexpansive operators in uniformly convex Banach spaces. Proc. Amer. Math. Soc. 117 (1993), 1089–1092.MATHCrossRefMathSciNetGoogle Scholar
  18. [18]
    H. K. Xu: Random fixed point theorems for nonlinear uniform Lipschitzian mappings. Nonlinear Anal. 26 (1996), 1301–1311.MATHCrossRefMathSciNetGoogle Scholar
  19. [19]
    H. K. Xu: Multivalued nonexpansive mappings in Banach spaces. Nonlinear Anal. 43 (2001), 693–706.MATHCrossRefMathSciNetGoogle Scholar
  20. [20]
    S. Reich: Fixed points in locally convex spaces. Math. Z. 125 (1972), 17–31.MATHCrossRefMathSciNetGoogle Scholar
  21. [21]
    X. Yuan and J. Yu: Random fixed point theorems for nonself mappings. Nonlinear Anal. 26 (1996), 1097–1102.MATHCrossRefMathSciNetGoogle Scholar

Copyright information

© Mathematical Institute, Academy of Sciences of Czech Republic 2007

Authors and Affiliations

  • Poom Kumam
    • 1
  • Somyot Plubtieng
    • 2
  1. 1.Department of Mathematics, Faculty of ScienceKing Mongkut’s University of Technology ThonburiBangkokThailand
  2. 2.Department of Mathematics, Faculty of ScienceNaresuan UniversityPhitsanulokThailand

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