Table of Contents Author Guidelines Submit a Manuscript
Journal of Function Spaces
Volume 2015, Article ID 470574, 11 pages
http://dx.doi.org/10.1155/2015/470574
Research Article

Some Results on Best Proximity Points of Cyclic Contractions in Probabilistic Metric Spaces

1Institute of Research and Development of Processes IIDP, Faculty of Science and Technology, University of the Basque Country, P.O. Box 644 de Bilbao Barrio Sarriena, Leioa, 48940 Bizkaia, Spain
2Department of Mathematics, ATILIM University, Incek, 06836 Ankara, Turkey
3Nonlinear Analysis and Applied Mathematics Research Group (NAAM), King Abdulaziz University, Jeddah 21589, Saudi Arabia

Received 12 October 2014; Accepted 30 December 2014

Academic Editor: Calogero Vetro

Copyright © 2015 Manuel De la Sen and Erdal Karapınar. 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. B. Schweizer and A. Sklar, Probabilistic Metric Spaces, North-Holland, Amsterdam, The Netherlands, 1983. View at MathSciNet
  2. E. Pap, O. Hadzic, and R. Mesiar, “A fixed point theorem in probabilistic metric spaces and an application,” Journal of Mathematical Analysis and Applications, vol. 202, no. 2, pp. 433–449, 1996. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  3. V. M. Sehgal and A. T. Bharucha-Reid, “Fixed points of contraction mappings on probabilistic metric spaces,” Theory of Computing Systems, vol. 6, no. 1, pp. 97–102, 1972. View at Google Scholar · View at MathSciNet
  4. B. Schweizer and A. Sklar, “Statistical metric spaces,” Pacific Journal of Mathematics, vol. 10, pp. 313–334, 1960. View at Publisher · View at Google Scholar · View at MathSciNet
  5. B. S. Choudhury, K. Das, and S. K. Bhandari, “Fixed point theorem for mappings with cyclic contraction in Menger spaces,” International Journal of Pure and Applied Sciences and Technology, vol. 4, no. 1, pp. 1–9, 2011. View at Google Scholar
  6. B. S. Choudhury, K. Das, and S. K. Bhandari, “Cyclic contraction result in 2-Menger space,” Bulletin of International Mathematical Virtual Institute, vol. 2, no. 1, pp. 223–234, 2012. View at Google Scholar · View at MathSciNet
  7. I. Beg and M. Abbas, “Fixed point and best approximation in Menger convex metric spaces,” Archivum Mathematicum (BRNO). Tomus, vol. 41, pp. 389–397, 2005. View at Google Scholar
  8. I. Beg, A. Latif, R. Ali, and A. Azam, “Coupled fixed points of mixed monotone operators on probabilistic Banach spaces,” Archivum Mathematicum, vol. 37, no. 1, pp. 1–8, 2001. View at Google Scholar · View at MathSciNet
  9. D. Miheţ, “Altering distances in probabilistic Menger spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 71, no. 7-8, pp. 2734–2738, 2009. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. D. Mihet, “A Banach contraction theorem in fuzzy metric spaces,” Fuzzy Sets and Systems, vol. 144, no. 3, pp. 431–439, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. S. Sedghi, B. S. Choudhury, and N. Shobe, “Strong common coupled fixed point result in fuzzy metric spaces,” Journal of Physical Science, vol. 17, pp. 1–9, 2013. View at Google Scholar · View at MathSciNet
  12. B. S. Choudhury and K. P. Das, “A new contraction principle in Menger spaces,” Acta Mathematica Sinica, vol. 24, no. 8, pp. 1379–1386, 2008. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. D. Gopal, M. Abbas, and C. Vetro, “Some new fixed point theorems in Menger PM-spaces with application to Volterra type integral equation,” Applied Mathematics and Computation, vol. 232, pp. 955–967, 2014. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  14. M. S. Khan, M. Swaleh, and S. Sessa, “Fixed point theorems by altering distances between the points,” Bulletin of the Australian Mathematical Society, vol. 30, no. 1, pp. 1–9, 1984. View at Publisher · View at Google Scholar · View at MathSciNet
  15. A. A. Eldred and P. Veeramani, “Existence and convergence of best proximity points,” Journal of Mathematical Analysis and Applications, vol. 323, no. 2, pp. 1001–1006, 2006. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. M. de la Sen, “Linking contractive self-mappings and cyclic Meir-Keeler contractions with Kannan self-mappings,” Fixed Point Theory and Applications, vol. 2010, Article ID 572057, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  17. M. de la Sen, R. P. Agarwal, and N. Nistal, “Non-expansive and potentially expansive properties of two modified p-cyclic self-maps in metric spaces,” Journal of Nonlinear and Convex Analysis, vol. 14, no. 4, pp. 661–686, 2013. View at Google Scholar · View at MathSciNet
  18. M. De la Sen and R. P. Agarwal, “Fixed point-type results for a class of extended cyclic self-mappings under three general weak contractive conditions of rational type,” Fixed Point Theory and Applications, vol. 2011, article 102, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  19. M. de la Sen, “On best proximity point theorems and fixed point theorems for p-cyclic hybrid self-mappings in Banach spaces,” Abstract and Applied Analysis, vol. 2013, Article ID 183174, 14 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  20. S. Karpagam and S. Agrawal, “Best proximity point theorems for p-cyclic Meir-Keeler contractions,” Fixed Point Theory and Applications, vol. 2009, Article ID 197308, 2009. View at Publisher · View at Google Scholar · View at MathSciNet
  21. T. Suzuki, “Some notes on Meir-Keeler contractions and L-functions,” Bulletin of the Kyushu Institute of Technology, no. 53, pp. 12–13, 2006. View at Google Scholar · View at MathSciNet
  22. C. Di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,” Nonlinear Analysis, Theory, Methods and Applications, vol. 69, no. 11, pp. 3790–3794, 2008. View at Publisher · View at Google Scholar · View at Scopus
  23. M. Derafshpour, S. Rezapour, and N. Shahzad, “On the existence of best proximity points of cyclic contractions,” Advances in Dynamical Systems and Applications, vol. 6, no. 1, pp. 33–40, 2011. View at Google Scholar · View at MathSciNet
  24. S. Rezapour, M. Derafshpour, and N. Shahzad, “Best proximity points of cyclic φ-contractions on reflexive Banach spaces,” Fixed Point Theory and Applications, vol. 2010, Article ID 946178, 7 pages, 2010. View at Google Scholar · View at MathSciNet
  25. M. A. Al-Thagafi and N. Shahzad, “Convergence and existence results for best proximity points,” Nonlinear Analysis: Theory, Methods and Applications, vol. 70, no. 10, pp. 3665–3671, 2009. View at Publisher · View at Google Scholar · View at Scopus
  26. W. Sanhan, C. Mongkolkeha, and P. Kumam, “Generalized proximal ψ-contraction mappings and best proximity points,” Abstract and Applied Analysis, vol. 2012, Article ID 896912, 19 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  27. M. de la Sen and E. Karapinar, “Best proximity points of generalized semicyclic impulsive self-mappings: applications to impulsive differential and difference equations,” Abstract and Applied Analysis, vol. 2013, Article ID 505487, 16 pages, 2013. View at Publisher · View at Google Scholar · View at MathSciNet
  28. E. Karapinar and I. M. Erhan, “Cyclic contractions and fixed point theorems,” Filomat, vol. 26, no. 4, pp. 777–782, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  29. X. L. Qin, S. Y. Cho, and L. Wang, “Algorithms for treating equilibrium and fixed point problems,” Fixed Point Theory and Applications, vol. 2013, article 308, 2013. View at Publisher · View at Google Scholar
  30. K. S. Kim, J. K. Kim, and W. H. Lim, “Convergence theorems for common solutions of various problems with nonlinear mapping,” Journal of Inequalities and Applications, vol. 2014, article 2, 2014. View at Publisher · View at Google Scholar · View at Scopus
  31. J. K. Kim, G. Li, Y. M. Nam, and K. . Kim, “Nonlinear ergodic theorems for almost-orbits of asymptotically nonexpansive type mappings in Banach spaces,” Bulletin of the Korean Mathematical Society, vol. 38, no. 3, pp. 587–603, 2001. View at Google Scholar · View at MathSciNet