Skip to main content

On random coincidence and fixed points for a pair of multivalued and single-valued mappings

Abstract

Let () be a Polish space, the family of all nonempty closed and bounded subsets of, and () a measurable space. A pair of a hybrid measurable mappings and, satisfying the inequality (1.2), are introduced and investigated. It is proved that if is complete,, are continuous for all,, are measurable for all, and for each, then there is a measurable mapping such that for all. This result generalizes and extends the fixed point theorem of Papageorgiou (1984) and many classical fixed point theorems.

[12345678910111213141516171819202122232425]

References

  1. 1.

    Ćirić LB: On some nonexpansive type mappings and fixed points. Indian Journal of Pure and Applied Mathematics 1993,24(3):145–149.

    MathSciNet  MATH  Google Scholar 

  2. 2.

    Ćirić LB: Nonexpansive type mappings and a fixed point theorem in convex metric spaces. Accademia Nazionale delle Scienze detta dei XL. Rendiconti. Serie V. Memorie di Matematica e Applicazioni. Parte I 1995, 19: 263–271.

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Ćirić LB: On some mappings in metric spaces and fixed points. Académie Royale de Belgique. Bulletin de la Classe des Sciences. 6e Série 1995,6(1–6):81–89.

    MATH  MathSciNet  Google Scholar 

  4. 4.

    Ćirić LB, Ume JS: Some common fixed point theorems for weakly compatible mappings. Journal of Mathematical Analysis and Applications 2006,314(2):488–499. 10.1016/j.jmaa.2005.04.007

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Hadžić O: A random fixed point theorem for multivalued mappings of Ćirić's type. Matematički Vesnik 1979,3(16)(31)(4):397–401.

    MATH  Google Scholar 

  6. 6.

    Hanš O: Reduzierende zufällige Transformationen. Czechoslovak Mathematical Journal 1957,7(82):154–158.

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Hanš O: Random operator equations. In Proceedings of the 4th Berkeley Symposium on Mathematical Statistics and Probability. Volume 2, part 1. University of California Press, California; 1961:185–202.

    Google Scholar 

  8. 8.

    Itoh S: A random fixed point theorem for a multivalued contraction mapping. Pacific Journal of Mathematics 1977,68(1):85–90.

    MathSciNet  Article  MATH  Google Scholar 

  9. 9.

    Kubiaczyk I: Some fixed point theorems. Demonstratio Mathematica 1976,9(3):507–515.

    MathSciNet  MATH  Google Scholar 

  10. 10.

    Kubiak T: Fixed point theorems for contractive type multivalued mappings. Mathematica Japonica 1985,30(1):89–101.

    MathSciNet  MATH  Google Scholar 

  11. 11.

    Kuratowski K, Ryll-Nardzewski C: A general theorem on selectors. Bulletin de l'Académie Polonaise des Sciences. Série des Sciences Mathématiques, Astronomiques et Physiques 1965, 13: 397–403.

    MathSciNet  MATH  Google Scholar 

  12. 12.

    Lin T-C: Random approximations and random fixed point theorems for non-self-maps. Proceedings of the American Mathematical Society 1988,103(4):1129–1135. 10.1090/S0002-9939-1988-0954994-0

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Papageorgiou NS: Random fixed point theorems for multifunctions. Mathematica Japonica 1984,29(1):93–106.

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Papageorgiou NS: Random fixed point theorems for measurable multifunctions in Banach spaces. Proceedings of the American Mathematical Society 1986,97(3):507–514. 10.1090/S0002-9939-1986-0840638-3

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Rhoades BE: A generalization of a fixed point theorem of Bogin. Mathematics Seminar Notes, Kobe University 1978,6(1):1–7.

    MathSciNet  MATH  Google Scholar 

  16. 16.

    Rhoades BE, Singh SL, Kulshrestha C: Coincidence theorems for some multivalued mappings. International Journal of Mathematics and Mathematical Sciences 1984,7(3):429–434. 10.1155/S0161171284000466

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Rockafellar RT: Measurable dependence of convex sets and functions on parameters. Journal of Mathematical Analysis and Applications 1969,28(1):4–25. 10.1016/0022-247X(69)90104-8

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Sehgal VM, Singh SP: On random approximations and a random fixed point theorem for set valued mappings. Proceedings of the American Mathematical Society 1985,95(1):91–94. 10.1090/S0002-9939-1985-0796453-1

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Shahzad N, Hussain N: Deterministic and random coincidence point results for-nonexpansive maps. to appear in Journal of Mathematical Analysis and Applications to appear in Journal of Mathematical Analysis and Applications

  20. 20.

    Shahzad N, Latif A: A random coincidence point theorem. Journal of Mathematical Analysis and Applications 2000,245(2):633–638. 10.1006/jmaa.2000.6772

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Singh SL, Mishra SN: On a Ljubomir Ćirić's fixed point theorem for nonexpansive type maps with applications. Indian Journal of Pure and Applied Mathematics 2002,33(4):531–542.

    MathSciNet  MATH  Google Scholar 

  22. 22.

    Špaček A: Zufällige Gleichungen. Czechoslovak Mathematical Journal 1955,5(80)(80):462–466.

    MATH  Google Scholar 

  23. 23.

    Tan K-K, Yuan X-Z: Random fixed-point theorems and approximation in cones. Journal of Mathematical Analysis and Applications 1994,185(2):378–390. 10.1006/jmaa.1994.1256

    MathSciNet  Article  MATH  Google Scholar 

  24. 24.

    Wagner DH: Survey of measurable selection theorems. SIAM Journal on Control and Optimization 1977,15(5):859–903. 10.1137/0315056

    MathSciNet  Article  MATH  Google Scholar 

  25. 25.

    Zhang SS, Huang N-J: On the principle of randomization of fixed points for set-valued mappings with applications. Northeastern Mathematical Journal 1991,7(4):486–491.

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Ljubomir B. Ćirić.

Rights and permissions

Open Access This article is distributed under the terms of the Creative Commons Attribution 2.0 International License (https://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Reprints and Permissions

About this article

Cite this article

Ćirić, L.B., Ume, J.S. & Ješić, S.N. On random coincidence and fixed points for a pair of multivalued and single-valued mappings. J Inequal Appl 2006, 81045 (2006). https://doi.org/10.1155/JIA/2006/81045

Download citation

Keywords

  • Measurable Mapping
  • Point Theorem
  • Measurable Space
  • Fixed Point Theorem
  • Polish Space
\