About this Journal Submit a Manuscript Table of Contents
International Journal of Analysis
Volume 2013 (2013), Article ID 763261, 10 pages
http://dx.doi.org/10.1155/2013/763261
Research Article

Employing Common Limit Range Property to Prove Unified Metrical Common Fixed Point Theorems

1Department of Mathematics, Aligarh Muslim University, Aligarh 202 002, India
2Near Nehru Training Centre, H. No. 274, Nai Basti B-14, Bijnor, Uttar Pradesh 246 701, India

Received 29 January 2013; Accepted 4 April 2013

Academic Editor: Chuanxi Qian

Copyright © 2013 Mohammad Imdad and Sunny Chauhan. 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, “Sur les operations dans les ensembles abstraits et leur application aux equations integrales,” Fundamenta Mathematicae, vol. 3, pp. 133–181, 1922.
  2. W. A. Kirk, “Some recent results in metric fixed point theory,” Journal of Fixed Point Theory and Applications, vol. 2, no. 2, pp. 195–207, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. C. Semple and M. Steel, Phylogenetics, vol. 24 of Oxford Lecture Series in Mathematics and its Applications, Oxford University Press, Oxford, UK, 2003. View at MathSciNet
  4. L. B. Ćirić, “Generalized contractions and fixed-point theorems,” Publications de l'Institut Mathématique, vol. 12, no. 26, pp. 19–26, 1971. View at Zentralblatt MATH · View at MathSciNet
  5. L. Ćirić, A. Razani, S. Radenović, and J. S. Ume, “Common fixed point theorems for families of weakly compatible maps,” Computers & Mathematics with Applications, vol. 55, no. 11, pp. 2533–2543, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. M. Imdad and J. Ali, “Jungck's common fixed point theorem and E.A property,” Acta Mathematica Sinica, vol. 24, no. 1, pp. 87–94, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. M. Imdad, S. Chauhan, and Z. Kadelburg, “Fixed point theorems for mappings with common limit range property satisfying generalized (ψ,ϕ)-weak contractive conditions,” Mathematical Sciences, vol. 7, article 16, 2013. View at Publisher · View at Google Scholar
  8. M. Imdad and Q. H. Khan, “Six mappings satisfying a rational inequality,” Radovi Matematički, vol. 9, no. 2, pp. 251–260, 1999. View at Zentralblatt MATH · View at MathSciNet
  9. M. Imdad, M. S. Khan, and S. Sessa, “On some weak conditions of commutativity in common fixed point theorems,” International Journal of Mathematics and Mathematical Sciences, vol. 11, no. 2, pp. 289–296, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. Imdad, S. Kumar, and M. S. Khan, “Remarks on some fixed point theorems satisfying implicit relations,” Radovi Matematički, vol. 11, no. 1, pp. 135–143, 2002. View at Zentralblatt MATH · View at MathSciNet
  11. G. Jungck, “Commuting mappings and fixed points,” The American Mathematical Monthly, vol. 83, no. 4, pp. 261–263, 1976. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. M. S. Khan and M. Imdad, “A common fixed point theorem for a class of mappings,” Indian Journal of Pure and Applied Mathematics, vol. 14, no. 10, pp. 1220–1227, 1983. View at Zentralblatt MATH · View at MathSciNet
  13. R. P. Pant, “Discontinuity and fixed points,” Journal of Mathematical Analysis and Applications, vol. 240, no. 1, pp. 280–283, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. 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 · View at Zentralblatt MATH · View at MathSciNet
  15. B. E. Rhoades, “Some theorems on weakly contractive maps,” in Proceedings of the 3rd World Congress of Nonlinear Analysts, vol. 47, pp. 2683–2693, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  16. S. Sessa and Y. J. Cho, “Compatible mappings and a common fixed point theorem of Chang type,” Publicationes Mathematicae Debrecen, vol. 43, no. 3-4, pp. 289–296, 1993. View at Zentralblatt MATH · View at MathSciNet
  17. S. Sessa, M. S. Khan, and M. Imdad, “A common fixed point theorem with a weak commutativity condition,” Glasnik Matematicki Series III, vol. 21, no. 1, pp. 225–235, 1986. View at Zentralblatt MATH · View at MathSciNet
  18. W. Shatanawi, “Fixed point theorems for nonlinear weakly C-contractive mappings in metric spaces,” Mathematical and Computer Modelling, vol. 54, no. 11-12, pp. 2816–2826, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. V. Popa, “Some fixed point theorems for weakly compatible mappings,” Radovi Matematički, vol. 10, no. 2, pp. 245–252, 2001. View at MathSciNet
  20. J. Ali and M. Imdad, “An implicit function implies several contraction conditions,” Sarajevo Journal of Mathematics, vol. 4, no. 2, pp. 269–285, 2008. View at Zentralblatt MATH · View at MathSciNet
  21. S. Chauhan and S. Kumar, “Coincidence and ffxed points in fuzzy metric spaces using common property (E.A),” Kochi Journal of Mathematics, vol. 8, pp. 135–154, 2013.
  22. D. Gopal, M. Imdad, and C. Vetro, “Impact of common property (E.A.) on fixed point theorems in fuzzy metric spaces,” Fixed Point Theory and Applications, vol. 2011, Article ID 297360, 14 pages, 2011. View at Zentralblatt MATH · View at MathSciNet
  23. M. Imdad, J. Ali, and M. Hasan, “Common fixed point theorems in modified intuitionistic fuzzy metric spaces,” Iranian Journal of Fuzzy Systems, vol. 9, no. 5, pp. 77–92, 2012.
  24. M. Imdad, M. Tanveer, and M. Hasan, “Some common fixed point theorems in Menger PM spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 819269, 14 pages, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  25. H. K. Pathak, R. Rodríguez-López, and R. K. Verma, “A common fixed point theorem using implicit relation and property (E.A) in metric spaces,” Filomat, vol. 21, no. 2, pp. 211–234, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. V. Popa, M. Imdad, and J. Ali, “Using implicit relations to prove unified fixed point theorems in metric and 2-metric spaces,” Bulletin of the Malaysian Mathematical Sciences Society B, vol. 33, no. 1, pp. 105–120, 2010. View at Zentralblatt MATH · View at MathSciNet
  27. V. Popa, M. Imdad, and J. Ali, “Fixed point theorems for a class of mappings governed by strictly contractive implicit function,” Southeast Asian Bulletin of Mathematics, vol. 34, no. 5, pp. 941–952, 2010. View at Zentralblatt MATH · View at MathSciNet
  28. S. Sessa, “On a weak commutativity condition of mappings in fixed point considerations,” Publications de l'Institut Mathématique, vol. 32, no. 46, pp. 149–153, 1982. View at Zentralblatt MATH · View at MathSciNet
  29. G. Jungck, “Compatible mappings and common fixed points,” International Journal of Mathematics and Mathematical Sciences, vol. 9, no. 4, pp. 771–779, 1986. View at Publisher · View at Google Scholar · View at MathSciNet
  30. R. P. Pant, “Noncompatible mappings and common fixed points,” Soochow Journal of Mathematics, vol. 26, no. 1, pp. 29–35, 2000. View at Zentralblatt MATH · View at MathSciNet
  31. G. Jungck and B. E. Rhoades, “Fixed points for set valued functions without continuity,” Indian Journal of Pure and Applied Mathematics, vol. 29, no. 3, pp. 227–238, 1998. View at Zentralblatt MATH · View at MathSciNet
  32. P. P. Murthy, “Important tools and possible applications of metric fixed point theory,” Nonlinear Analysis: Theory, Methods & Applications, vol. 47, no. 5, pp. 3479–3490, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  33. M. Aamri and D. El Moutawakil, “Some new common fixed point theorems under strict contractive conditions,” Journal of Mathematical Analysis and Applications, vol. 270, no. 1, pp. 181–188, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  34. J.-x. Fang and Y. Gao, “Common fixed point theorems under strict contractive conditions in Menger spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 1, pp. 184–193, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. Y. Liu, J. Wu, and Z. Li, “Common fixed points of single-valued and multivalued maps,” International Journal of Mathematics and Mathematical Sciences, no. 19, pp. 3045–3055, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. W. Sintunavarat and P. Kumam, “Common fixed point theorems for a pair of weakly compatible mappings in fuzzy metric spaces,” Journal of Applied Mathematics, vol. 2011, Article ID 637958, 14 pages, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. S. L. Singh, B. D. Pant, and S. Chauhan, “Fixed point theorems in non-Archimedean Menger PM-spaces,” Journal of Nonlinear Analysis and Optimization, vol. 3, no. 2, pp. 153–160, 2012. View at MathSciNet
  38. M. Imdad, B. D. Pant, and S. Chauhan, “Fixed point theorems in Menger spaces using the (CLRST) property and applications,” Journal of Nonlinear Analysis and Optimization, vol. 3, no. 2, pp. 225–237, 2012. View at MathSciNet
  39. M. Imdad, J. Ali, and M. Tanveer, “Coincidence and common fixed point theorems for nonlinear contractions in Menger PM spaces,” Chaos, Solitons & Fractals, vol. 42, no. 5, pp. 3121–3129, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  40. G. S. Jeong and B. E. Rhoades, “Some remarks for improving fixed point theorems for more than two maps,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 9, pp. 1177–1196, 1997. View at Zentralblatt MATH · View at MathSciNet
  41. S. M. Kang and Y. P. Kim, “Common fixed point theorems,” Mathematica Japonica, vol. 37, no. 6, pp. 1031–1039, 1992. View at Zentralblatt MATH · View at MathSciNet
  42. S. P. Singh and B. A. Meade, “On common fixed point theorems,” Bulletin of the Australian Mathematical Society, vol. 16, no. 1, pp. 49–53, 1977. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet