Table of Contents Author Guidelines Submit a Manuscript
International Journal of Analysis
Volume 2014 (2014), Article ID 586096, 9 pages
http://dx.doi.org/10.1155/2014/586096
Research Article

Coupled Fixed Point Theorems for ()-Contractive Mixed Monotone Mappings in Partially Ordered Metric Spaces and Applications

1Department of Mathematics, Ahir College, Rewari 123401, India
2HAS Department, YMCAUST, Faridabad 121006, India
3Department of Mathematics, DCRUST, Murthal, Sonepat 131039, India

Received 14 November 2013; Accepted 22 January 2014; Published 18 March 2014

Academic Editor: Tohru Ozawa

Copyright © 2014 Manish Jain et al. 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. S. Banach, “Surles operations dans les ensembles et leur application aux equation sitegrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922. View at Google Scholar
  2. R. P. Agarwal, M. Meehan, and D. O'Regan, Fixed Point Theory and Applications, Cambridge University Press, 2001.
  3. R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, “Generalized contractions in partially orderedmetric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 109–116, 2008. View at Publisher · View at Google Scholar
  4. D. W. Boyd and J. S. W. Wong, “On nonlinear contractions,” Proceedings of the American Mathematical Society, vol. 20, pp. 458–464, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  5. A. Branciari, “A fixed point theorem for mappings satisfying a general contractive condition of integral type,” International Journal of Mathematics and Mathematical Sciences, vol. 29, no. 9, pp. 531–536, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  6. L. B. Cirić, “A generalization of Banach's contraction principle,” Proceedings of the American Mathematical Society, vol. 45, pp. 267–273, 1974. View at Google Scholar · View at Zentralblatt MATH
  7. J. Dugundji and A. Granas, Fixed Point Theory, Springer, New York, NY, USA, 2003.
  8. D. J. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, Boston, Mass, USA, 1988.
  9. A. Meir and E. Keeler, “A theorem on contraction mappings,” Journal of Mathematical Analysis and Applications, vol. 28, no. 2, pp. 326–329, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  10. J. J. Nieto and R. Rodríguez-López, “Contractive mapping theorems in partially ordered sets and applications to ordinary differential equations,” Order, vol. 22, no. 3, pp. 223–239, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at Scopus
  11. B. E. Rhoades, “A comparison of various definitions of contractive mappings,” Transactions of the American Mathematical Society, vol. 226, pp. 257–290, 1977. View at Publisher · View at Google Scholar
  12. D. R. Smart, Fixed Point Theorems, Cambridge University Press, London, UK, 1974.
  13. T. Suzuki, “Meir-Keeler contractions of integral type are still Meir-Keeler contractions,” International Journal of Mathematics and Mathematical Sciences, vol. 2007, Article ID 39281, 6 pages, 2007. View at Publisher · View at Google Scholar · View at Scopus
  14. T. Suzuki, “A generalized banach contraction principle that characterizes metric completeness,” Proceedings of the American Mathematical Society, vol. 136, no. 5, pp. 1861–1869, 2008. View at Publisher · View at Google Scholar · View at Scopus
  15. E. Zeidler, Nonlinear Functional Analysis and Its Applications I: Fixed-Point Theorems, Springer, Berlin, Germany, 1986.
  16. M. Turinici, “Abstract comparison principles and multivariable Gronwall-Bellman inequalities,” Journal of Mathematical Analysis and Applications, vol. 117, no. 1, pp. 100–127, 1986. View at Google Scholar · View at Scopus
  17. A. C. M. Ran and M. C. B. Reurings, “A fixed point theorem in partially ordered sets and some applications to matrix equations,” Proceedings of the American Mathematical Society, vol. 132, no. 5, pp. 1435–1443, 2004. View at Publisher · View at Google Scholar · View at Scopus
  18. J. J. Nieto and R. R. Lopez, “Existence and uniqueness of fixed point in partially ordered sets and applications to ordinary differential equations,” Acta Mathematica Sinica, vol. 23, no. 12, pp. 2205–2212, 2007. View at Publisher · View at Google Scholar · View at Scopus
  19. R. P. Agarwal, M. A. El-Gebeily, and D. O'Regan, “Generalized contractions in partially ordered metric spaces,” Applicable Analysis, vol. 87, no. 1, pp. 1–8, 2008. View at Publisher · View at Google Scholar
  20. I. Altun and H. Simsek, “Some fixed point theorems on ordered metric spaces and application,” Fixed Point Theory and Applications, vol. 2010, Article ID 621469, 17 pages, 2010. View at Publisher · View at Google Scholar
  21. J. Harjani and K. Sadarangani, “Generalized contractions in partially ordered metric spaces and applications to ordinary differential equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 3-4, pp. 1188–1197, 2010. View at Publisher · View at Google Scholar · View at Scopus
  22. Z. Kadelburg, M. Pavlovi, and S. Radenovi, “Common fixed point theorems for ordered contractions and quasicontractions in ordered cone metric spaces,” Computers and Mathematics with Applications, vol. 59, no. 9, pp. 3148–3159, 2010. View at Publisher · View at Google Scholar · View at Scopus
  23. S. Radenovi and Z. Kadelburg, “Generalized weak contractions in partially ordered metric spaces,” Computers and Mathematics with Applications, vol. 60, no. 6, pp. 1776–1783, 2010. View at Publisher · View at Google Scholar · View at Scopus
  24. W. Sintunavarat, Y. J. Cho, and P. Kumam, “Common fixed point theorems for c-distance in ordered cone metric spaces,” Computers and Mathematics with Applications, vol. 62, no. 4, pp. 1969–1978, 2011. View at Publisher · View at Google Scholar · View at Scopus
  25. T. G. Bhaskar and V. Lakshmikantham, “Fixed point theorems in partially ordered metric spaces and applications,” Nonlinear Analysis, Theory, Methods and Applications, vol. 65, no. 7, pp. 1379–1393, 2006. View at Publisher · View at Google Scholar · View at Scopus
  26. N. V. Luong and N. X. Thuan, “Coupled fixed points in partially ordered metric spaces and application,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 3, pp. 983–992, 2011. View at Publisher · View at Google Scholar · View at Scopus
  27. V. Berinde, “Generalized coupled fixed point theorems for mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 18, pp. 7347–7355, 2011. View at Publisher · View at Google Scholar · View at Scopus
  28. V. Berinde, “Coupled fixed point theorems for φ-contractive mixed monotone mappings in partially ordered metric spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 75, no. 6, pp. 3218–3228, 2012. View at Publisher · View at Google Scholar · View at Scopus
  29. J. Harjani, B. Lpez, and K. Sadarangani, “Fixed point theorems for mixed monotone operators and applications to integral equations,” Nonlinear Analysis, Theory, Methods and Applications, vol. 74, no. 5, pp. 1749–1760, 2011. View at Publisher · View at Google Scholar · View at Scopus
  30. B. Samet, “Coupled fixed point theorems for a generalized Meir-Keeler contraction in partially ordered metric spaces,” Nonlinear Analysis, Theory, Methods and Applications, vol. 72, no. 12, pp. 4508–4517, 2010. View at Publisher · View at Google Scholar · View at Scopus
  31. W. Sintunavarat, Y. J. Cho, and P. Kumam, “Coupled fixed point theorems for weak contraction mapping under F-invariant set,” Abstract and Applied Analysis, vol. 2012, Article ID 324874, 15 pages, 2012. View at Publisher · View at Google Scholar
  32. W. Sintunavarat and P. Kumam, “Coupled best proximity point theorem in metric spaces,” Fixed Point Theory and Applications, vol. 2012, 93 pages, 2012. View at Publisher · View at Google Scholar
  33. W. Sintunavarat, Y. J. Cho, and P. Kumam, “Coupled fixed point theorems for contraction mapping induced by cone ball-metric in partially ordered spaces,” Fixed Point Theory and Applications, vol. 2012, 128 pages, 2012. View at Publisher · View at Google Scholar
  34. V. Berinde and M. Păcurar, “Coupled fixed point theorems for generalized symmetric Meir-Keeler contractions in ordered metric spaces,” Fixed Point Theory and Applications, vol. 2012, 115 pages, 2012. View at Publisher · View at Google Scholar
  35. M. Jain, K. Tas, S. Kumar, and N. Gupta, “Coupled common fixed points involving a (φ,ψ)-contractive condition for mixed g-monotone operators in partially ordered metric spaces,” Journal of Inequalities and Applications, vol. 2012, 285 pages, 2012. View at Publisher · View at Google Scholar
  36. M. Jain, K. Tas, B. E. Rhoades, and N. Gupta, “Coupled fixed point theorems for generalized symmetric contractions in partially ordered metric spaces and applications,” Journal of Computational Analysis and Applications, vol. 16, no. 3, pp. 438–4454, 2014. View at Google Scholar
  37. M. Jain, N. Gupta, C. Vetro, and S. Kumar, “Coupled fixed point theorems for symmetric (φ,ψ)-weakly contractive mappings in ordered partial metric spaces,” The Journal of Mathematics and Computer Sciences, vol. 7, no. 4, pp. 230–304, 2013. View at Google Scholar
  38. M. Jain, K. Tas, S. Kumar, and N. Gupta, “Coupled fixed point theorems for a pair of weakly compatible maps along with CLRg property in Fuzzy metric spaces,” Journal of Applied Mathematics, vol. 2012, Article ID 961210, 13 pages, 2012. View at Publisher · View at Google Scholar
  39. M. Jain, K. Tas, and N. Gupta, “Coupled common fixed point results involving (φ,ψ)-contractions in ordered generalized metric spaces with application to integral equations,” Journal of Inequalities and Applications, vol. 2013, 372 pages, 2013. View at Google Scholar
  40. M. Jain and K. Tas, “A unique coupled common fixed point theorem for symmetric (φ,ψ)-contractive mappings in ordered G-metric spaces with applications,” Journal of Applied Mathematics, vol. 2013, Article ID 134712, 13 pages, 2013. View at Publisher · View at Google Scholar