About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 349305, 6 pages
http://dx.doi.org/10.1155/2013/349305
Research Article

On Fixed Point Theory of Monotone Mappings with Respect to a Partial Order Introduced by a Vector Functional in Cone Metric Spaces

1School of Statistics, Jiangxi University of Finance and Economics, Nanchang 330013, China
2Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330013, China

Received 19 November 2012; Revised 11 January 2013; Accepted 23 January 2013

Academic Editor: Micah Osilike

Copyright © 2013 Zhilong Li and Shujun Jiang. 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.

Linked References

  1. R. P. Agarwal and M. A. Khamsi, “Extension of Caristi's fixed point theorem to vector valued metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 1, pp. 141–145, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. J. Caristi, “Fixed point theorems for mappings satisfying inwardness conditions,” Transactions of the American Mathematical Society, vol. 215, pp. 241–251, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. S. H. Cho, J. S. Bae, and K. S. Na, “Fixed point theorems for multivalued contractive mappings and multivalued Caristi type mappings in cone metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 133, 2012. View at Publisher · View at Google Scholar
  4. W. A. Kirk and J. Caristi, “Mappings theorems in metric and Banach spaces,” Bulletin de l'Académie Polonaise des Sciences, vol. 23, no. 8, pp. 891–894, 1975. View at MathSciNet
  5. W. A. Kirk, “Caristi's fixed point theorem and metric convexity,” Colloquium Mathematicum, vol. 36, no. 1, pp. 81–86, 1976. View at Zentralblatt MATH · View at MathSciNet
  6. J. R. Jachymski, “Caristi's fixed point theorem and selections of set-valued contractions,” Journal of Mathematical Analysis and Applications, vol. 227, no. 1, pp. 55–67, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. J. S. Bae, “Fixed point theorems for weakly contractive multivalued maps,” Journal of Mathematical Analysis and Applications, vol. 284, no. 2, pp. 690–697, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. T. Suzuki, “Generalized Caristi's fixed point theorems by Bae and others,” Journal of Mathematical Analysis and Applications, vol. 302, no. 2, pp. 502–508, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. Y. Q. Feng and S. Y. Liu, “Fixed point theorems for multi-valued contractive mappings and multi-valued Caristi type mappings,” Journal of Mathematical Analysis and Applications, vol. 317, no. 1, pp. 103–112, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. A. Khamsi, “Remarks on Caristi's fixed point theorem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 1-2, pp. 227–231, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. Z. Li, “Remarks on Caristi's fixed point theorem and Kirk's problem,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 12, pp. 3751–3755, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. Z. Li and S. Jiang, “Maximal and minimal point theorems and Caristi's fixed point theorem,” Fixed Point Theory and Applications, vol. 2011, article 103, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  13. I. Altun and V. Rakočević, “Ordered cone metric spaces and fixed point results,” Computers & Mathematics with Applications, vol. 60, no. 5, pp. 1145–1151, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. H. K. Nashine, Z. Kadelburg, and S. Radenović, “Coincidence and fixed point results in ordered G-cone metric spaces,” Mathematical and Computer Modelling, vol. 57, no. 3-4, pp. 701–709, 2013. View at Publisher · View at Google Scholar
  15. D. R. Kurepa, “Tableaux ramifiés d'ensembles, Espaces pseudo-distanciés,” Comptes Rendus de l'Académie des Sciences, vol. 198, pp. 1563–1565, 1934.
  16. P. P. Zabrejko, “K-metric and K-normed linear spaces: survey,” Collectanea Mathematica, vol. 48, no. 4–6, pp. 825–859, 1997. View at MathSciNet
  17. Z. Kadelburg, S. Radenović, and V. Rakočević, “A note on the equivalence of some metric and cone metric fixed point results,” Applied Mathematics Letters, vol. 24, no. 3, pp. 370–374, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. S. Janković, Z. Kadelburg, and S. Radenović, “On cone metric spaces: a survey,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 7, pp. 2591–2601, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. Z. Kadelburg, M. Pavlović, and S. Radenović, “Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,” Computers & Mathematics with Applications, vol. 59, no. 9, pp. 3148–3159, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. M. A. Alghamdi, S. H. Alnafei, S. Radenović, and N. Shahzad, “Fixed point theorems for convex contraction mappings on cone metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 9-10, pp. 2020–2026, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. I. Arandjelović, Z. Kadelburg, and S. Radenović, “Boyd-Wong-type common fixed point results in cone metric spaces,” Applied Mathematics and Computation, vol. 217, no. 17, pp. 7167–7171, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. H. Alnafei, S. Radenović, and N. Shahzad, “Fixed point theorems for mappings with convex diminishing diameters on cone metric spaces,” Applied Mathematics Letters, vol. 24, no. 12, pp. 2162–2166, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. W. Shatanawi, V. Ć. Rajić, S. Radenović, and A. Al-Rawashdeh, “Mizoguchi-Takahashi-type theorems in tvs-cone metric spaces,” Fixed Point Theory and Applications, vol. 2012, article 106, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  24. S. Radenović and Z. Kadelburg, “Quasi-contractions on symmetric and cone symmetric spaces,” Banach Journal of Mathematical Analysis, vol. 5, no. 1, pp. 38–50, 2011. View at MathSciNet
  25. X. Zhang, “Fixed point theorems of monotone mappings and coupled fixed point theorems of mixed monotone mappings in ordered metric spaces,” Acta Mathematica Sinica, vol. 44, no. 4, pp. 641–646, 2001 (Chinese). View at Zentralblatt MATH · View at MathSciNet
  26. X. Zhang, “Fixed point theorems of multivalued monotone mappings in ordered metric spaces,” Applied Mathematics Letters, vol. 23, no. 3, pp. 235–240, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. Z. Li, “Fixed point theorems in partially ordered complete metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 1-2, pp. 69–72, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. L. G. Huang and X. Zhang, “Cone metric spaces and fixed point theorems of contractive mappings,” Journal of Mathematical Analysis and Applications, vol. 332, no. 2, pp. 1468–1476, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. Y. H. Du, “Total order minihedral cones,” Journal of Systems Science and Mathematical Sciences, vol. 8, no. 1, pp. 19–24, 1988. View at Zentralblatt MATH · View at MathSciNet
  30. K. Deimling, Nonlinear Functional Analysis, Springer, Berlin, Germany, 1985. View at MathSciNet