Skip to content


  • Research Article
  • Open Access

Fixed Points of Weakly Contractive Maps and Boundedness of Orbits

Fixed Point Theory and Applications20072007:020962

Received: 10 October 2006

Accepted: 31 January 2007

Published: 20 May 2007


We discuss weakly contractive maps on complete metric spaces. Following three methods of generalizing the Banach contraction principle, we obtain some fixed point theorems under some relatively weaker and more general contractive conditions.


  • Point Theorem
  • Differential Geometry
  • Fixed Point Theorem
  • Contractive Condition
  • Computational Biology


Authors’ Affiliations

Institute of Mathematics, Shantou University, Shantou, China
Institute of Mathematics, Guangxi University, Nanning, China


  1. Rhoades BE: A comparison of various definitions of contractive mappings. Transactions of the American Mathematical Society 1977, 226: 257–290.MathSciNetView ArticleMATHGoogle Scholar
  2. Collaço P, Silva JCE: A complete comparison of 25 contraction conditions. Nonlinear Analysis. Theory, Methods & Applications 1997,30(1):471–476. 10.1016/S0362-546X(97)00353-2MathSciNetView ArticleMATHGoogle Scholar
  3. Ciric LB: A generalization of Banach's contraction principle. Proceedings of the American Mathematical Society 1974,45(2):267–273.MathSciNetView ArticleMATHGoogle Scholar
  4. Fisher B: Quasi-contractions on metric spaces. Proceedings of the American Mathematical Society 1979,75(2):321–325.MathSciNetMATHGoogle Scholar
  5. Guseman LF Jr.: Fixed point theorems for mappings with a contractive iterate at a point. Proceedings of the American Mathematical Society 1970,26(4):615–618. 10.1090/S0002-9939-1970-0266010-3MathSciNetView ArticleMATHGoogle Scholar
  6. Walter W: Remarks on a paper by F. Browder about contraction. Nonlinear Analysis. Theory, Methods & Applications 1981,5(1):21–25. 10.1016/0362-546X(81)90066-3View ArticleMATHGoogle Scholar
  7. Jachymski JR, Schroder B, Stein JD Jr.: A connection between fixed-point theorems and tiling problems. Journal of Combinatorial Theory. Series A 1999,87(2):273–286. 10.1006/jcta.1998.2960MathSciNetView ArticleMATHGoogle Scholar
  8. Jachymski JR, Stein JD Jr.: A minimum condition and some related fixed-point theorems. Journal of the Australian Mathematical Society. Series A 1999,66(2):224–243. 10.1017/S144678870003932XMathSciNetView ArticleMATHGoogle Scholar
  9. Merryfield J, Rothschild B, Stein JD Jr.: An application of Ramsey's theorem to the Banach contraction principle. Proceedings of the American Mathematical Society 2002,130(4):927–933. 10.1090/S0002-9939-01-06169-XMathSciNetView ArticleMATHGoogle Scholar
  10. Merryfield J, Stein JD Jr.: A generalization of the Banach contraction principle. Journal of Mathematical Analysis and Applications 2002,273(1):112–120. 10.1016/S0022-247X(02)00215-9MathSciNetView ArticleMATHGoogle Scholar
  11. Boyd DW, Wong JSW: On nonlinear contractions. Proceedings of the American Mathematical Society 1969,20(2):458–464. 10.1090/S0002-9939-1969-0239559-9MathSciNetView ArticleMATHGoogle Scholar
  12. Jachymski J: A generalization of the theorem by Rhoades and Watson for contractive type mappings. Mathematica Japonica 1993,38(6):1095–1102.MathSciNetMATHGoogle Scholar
  13. Kirk WA: Fixed points of asymptotic contractions. Journal of Mathematical Analysis and Applications 2003,277(2):645–650. 10.1016/S0022-247X(02)00612-1MathSciNetView ArticleMATHGoogle Scholar
  14. Rakotch E: A note on contractive mappings. Proceedings of the American Mathematical Society 1962,13(3):459–465. 10.1090/S0002-9939-1962-0148046-1MathSciNetView ArticleMATHGoogle Scholar
  15. Jungck G: Fixed point theorems for semi-groups of self maps of semi-metric spaces. International Journal of Mathematics and Mathematical Sciences 1998,21(1):125–132. 10.1155/S0161171298000167MathSciNetView ArticleMATHGoogle Scholar
  16. Sharma BK, Thakur BS: Fixed point with orbital diametral function. Applied Mathematics and Mechanics 1996,17(2):145–148. 10.1007/BF00122309MathSciNetView ArticleMATHGoogle Scholar
  17. Xia DF, Xu SL: Fixed points of continuous self-maps under a contractive condition. Mathematica Applicata 1998,11(1):81–84.MathSciNetMATHGoogle Scholar


© J.-H. Mai and X.-H. Liu. 2007

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.