Open Access

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

  • Ljubomir B. Ćirić1Email author,
  • Jeong S. Ume2 and
  • Siniša N. Ješić3
Journal of Inequalities and Applications20062006:81045

DOI: 10.1155/JIA/2006/81045

Received: 2 February 2006

Accepted: 22 July 2006

Published: 18 October 2006

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]

Authors’ Affiliations

(1)
Faculty of Mechanical Engineering, University of Belgrade
(2)
Department of Applied Mathematics, Changwon National University
(3)
Faculty of Electrical Engineering, University of Belgrade

References

  1. Ćirić LB: On some nonexpansive type mappings and fixed points. Indian Journal of Pure and Applied Mathematics 1993,24(3):145–149.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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.MATHMathSciNetGoogle Scholar
  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.2005.04.007MathSciNetView ArticleMATHGoogle Scholar
  5. Hadžić O: A random fixed point theorem for multivalued mappings of Ćirić's type. Matematički Vesnik 1979,3(16)(31)(4):397–401.MATHGoogle Scholar
  6. Hanš O: Reduzierende zufällige Transformationen. Czechoslovak Mathematical Journal 1957,7(82):154–158.MathSciNetMATHGoogle Scholar
  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. Itoh S: A random fixed point theorem for a multivalued contraction mapping. Pacific Journal of Mathematics 1977,68(1):85–90.MathSciNetView ArticleMATHGoogle Scholar
  9. Kubiaczyk I: Some fixed point theorems. Demonstratio Mathematica 1976,9(3):507–515.MathSciNetMATHGoogle Scholar
  10. Kubiak T: Fixed point theorems for contractive type multivalued mappings. Mathematica Japonica 1985,30(1):89–101.MathSciNetMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  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/S0002-9939-1988-0954994-0MathSciNetView ArticleMATHGoogle Scholar
  13. Papageorgiou NS: Random fixed point theorems for multifunctions. Mathematica Japonica 1984,29(1):93–106.MathSciNetMATHGoogle Scholar
  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/S0002-9939-1986-0840638-3MathSciNetView ArticleMATHGoogle Scholar
  15. Rhoades BE: A generalization of a fixed point theorem of Bogin. Mathematics Seminar Notes, Kobe University 1978,6(1):1–7.MathSciNetMATHGoogle Scholar
  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/S0161171284000466MathSciNetView ArticleMATHGoogle Scholar
  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)90104-8MathSciNetView ArticleMATHGoogle Scholar
  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/S0002-9939-1985-0796453-1MathSciNetView ArticleMATHGoogle Scholar
  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 ApplicationsGoogle Scholar
  20. Shahzad N, Latif A: A random coincidence point theorem. Journal of Mathematical Analysis and Applications 2000,245(2):633–638. 10.1006/jmaa.2000.6772MathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar
  22. Špaček A: Zufällige Gleichungen. Czechoslovak Mathematical Journal 1955,5(80)(80):462–466.MATHGoogle Scholar
  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.1994.1256MathSciNetView ArticleMATHGoogle Scholar
  24. Wagner DH: Survey of measurable selection theorems. SIAM Journal on Control and Optimization 1977,15(5):859–903. 10.1137/0315056MathSciNetView ArticleMATHGoogle Scholar
  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.MathSciNetMATHGoogle Scholar

Copyright

© Ljubomir B. Ćirić et al. 2006

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.