Open Access Research Article

Common Fixed Point Theorems for Hybrid Pairs of Occasionally Weakly Compatible Mappings Satisfying Generalized Contractive Condition of Integral Type

Mujahid Abbas12 and BE Rhoades1*

Author Affiliations

1 Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA

2 Department of Mathematics, Lahore University of Management Sciences, Lahore 54792, Pakistan

For all author emails, please log on.

Fixed Point Theory and Applications 2007, 2007:054101  doi:10.1155/2007/54101


The electronic version of this article is the complete one and can be found online at: http://www.fixedpointtheoryandapplications.com/content/2007/1/054101


Received: 29 January 2007
Accepted: 10 June 2007
Published: 10 September 2007

© 2007 Abbas and Rhoades

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.

We obtain several fixed point theorems for hybrid pairs of single-valued and multivalued occasionally weakly compatible maps defined on a symmetric space satisfying a contractive condition of integral type. The results of this paper essentially contain every theorem on hybrid and multivalued self-maps of a metric space as a special case.

http:// webciteErratumhttp://dx.doi.org/10.1155/2008/274793 webcite

References

  1. Sessa, S: On a weak commutativity condition of mappings in fixed point considerations. Publications de l'Institut Mathématique. 32(46), 149–153 (1982)

  2. Jungck, G: Compatible mappings and common fixed points. International Journal of Mathematics and Mathematical Sciences. 9(4), 771–779 (1986). Publisher Full Text OpenURL

  3. Jungck, G: Common fixed points for noncontinuous nonself maps on nonmetric spaces. Far East Journal of Mathematical Sciences. 4(2), 199–215 (1996)

  4. Jungck, G, Rhoades, BE: Fixed point theorems for occasionally weakly compatible mappings. Fixed Point Theory. 7(2), 287–296 (2006)

  5. Branciari, A: A fixed point theorem for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences. 29(9), 531–536 (2002). Publisher Full Text OpenURL

  6. Rhoades, BE: Two fixed-point theorems for mappings satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences. 2003(63), 4007–4013 (2003). Publisher Full Text OpenURL

  7. Aliouche, A: A common fixed point theorem for weakly compatible mappings in symmetric spaces satisfying a contractive condition of integral type. Journal of Mathematical Analysis and Applications. 322(2), 796–802 (2006). Publisher Full Text OpenURL

  8. Vijayaraju, P, Rhoades, BE, Mohanraj, R: A fixed point theorem for a pair of maps satisfying a general contractive condition of integral type. International Journal of Mathematics and Mathematical Sciences. 2005(15), 2359–2364 (2005). Publisher Full Text OpenURL

  9. Zhang, X: Common fixed point theorems for some new generalized contractive type mappings. Journal of Mathematical Analysis and Applications. 333(2), 780–786 (2007). Publisher Full Text OpenURL

  10. Nadler, SB Jr..: Multi-valued contraction mappings. Pacific Journal of Mathematics. 30, 475–488 (1969)

  11. Naidu, SVR: Fixed points and coincidence points for multimaps with not necessarily bounded images. Fixed Point Theory and Applications. 2004(3), 221–242 (2004). Publisher Full Text OpenURL

  12. Singh, SL, Mishra, SN: Coincidences and fixed points of nonself hybrid contractions. Journal of Mathematical Analysis and Applications. 256(2), 486–497 (2001). Publisher Full Text OpenURL

  13. Dube, LS: A theorem on common fixed points of multi-valued mappings. Annales de la Societé Scientifique de Bruxelles. 89(4), 463–468 (1975)

  14. Azam, A, Beg, I: Coincidence points of compatible multivalued mappings. Demonstratio Mathematica. 29(1), 17–22 (1996)

  15. Kamran, T: Common coincidence points of -weakly commuting maps. International Journal of Mathematics and Mathematical Sciences. 26(3), 179–182 (2001). Publisher Full Text OpenURL

  16. Hadžić, O: Common fixed point theorems for single-valued and multivalued mappings. Review of Research. Faculty of Science. Mathematics Series. 18(2), 145–151 (1988)

  17. Kaneko, H, Sessa, S: Fixed point theorems for compatible multi-valued and single-valued mappings. International Journal of Mathematics and Mathematical Sciences. 12(2), 257–262 (1989). Publisher Full Text OpenURL

  18. Kaneko, H: A common fixed point of weakly commuting multi-valued mappings. Mathematica Japonica. 33(5), 741–744 (1988)

  19. Chang, TH: Common fixed point theorems for multivalued mappings. Mathematica Japonica. 41(2), 311–320 (1995)

  20. Shrivastava, PK, Bawa, NPS, Nigam, SK: Fixed point theorems for hybrid contractions. Varāhmihir Journal of Mathematical Sciences. 2(2), 275–281 (2002). PubMed Abstract | Publisher Full Text OpenURL

  21. Taylor, LE: A contractive mapping without fixed points. Notices of the American Mathematical Society. 24, A-649 (1977)