About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2014 (2014), Article ID 123613, 8 pages
http://dx.doi.org/10.1155/2014/123613
Research Article

Seminormal Structure and Fixed Points of Cyclic Relatively Nonexpansive Mappings

1Department of Mathematics, Ayatollah Boroujerdi University, Boroujerd, Iran
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 14 September 2013; Accepted 10 December 2013; Published 16 January 2014

Academic Editor: Calogero Vetro

Copyright © 2014 Moosa Gabeleh and Naseer Shahzad. 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. M. S. Brodskii and D. P. Milman, “On the center of a convex set,” Doklady Akademii Nauk SSSR, vol. 59, pp. 837–840, 1948 (Russian). View at MathSciNet
  2. W. A. Kirk, “A fixed point theorem for mappings which do not increase distances,” The American Mathematical Monthly, vol. 72, pp. 1004–1006, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, vol. 28 of Cambridge Studies in Advanced Mathematics, Cambridge University Press, Cambridge, UK, 1990. View at Publisher · View at Google Scholar · View at MathSciNet
  4. M. A. Khamsi and W. A. Kirk, “Pure and applied mathematics,” in An Introduction to Metric Spaces and Fixed Point Theory, pp. 303–304, Wiley-Interscience, New York, NY, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  5. A. Amini-Harandi, “Best proximity points theorems for cyclic strongly quasi-contraction mappings,” Journal of Global Optimization, vol. 56, no. 4, pp. 1667–1674, 2012. View at Publisher · View at Google Scholar
  6. W. A. Kirk, P. S. Srinivasan, and P. Veeramani, “Fixed points for mappings satisfying cyclical contractive conditions,” Fixed Point Theory, vol. 4, no. 1, pp. 79–89, 2003. View at Zentralblatt MATH · View at MathSciNet
  7. A. A. Eldred, W. A. Kirk, and P. Veeramani, “Proximal normal structure and relatively nonexpansive mappings,” Studia Mathematica, vol. 171, no. 3, pp. 283–293, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. 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 Zentralblatt MATH · View at MathSciNet
  9. A. Abkar and M. Gabeleh, “Best proximity points for asymptotic cyclic contraction mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 18, pp. 7261–7268, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  10. M. A. Al-Thagafi and N. Shahzad, “Convergence and existence results for best proximity points,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 10, pp. 3665–3671, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. C. Di Bari, T. Suzuki, and C. Vetro, “Best proximity points for cyclic Meir-Keeler contractions,” Nonlinear Analysis: Theory, Methods & Applications, vol. 69, no. 11, pp. 3790–3794, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. Espínola, “A new approach to relatively nonexpansive mappings,” Proceedings of the American Mathematical Society, vol. 136, no. 6, pp. 1987–1995, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. A. Fernández-León, “Existence and uniqueness of best proximity points in geodesic metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 73, no. 4, pp. 915–921, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Gabeleh, “Best proximity points for weak proximal contractions,” Malaysian Mathematical Sciences Society, In press.
  15. M. Gabeleh, “Proximal weakly contractive and proximal nonexpansive non-self-mappings in metric and Banach spaces,” Journal of Optimization Theory and Applications, vol. 158, no. 2, pp. 615–625, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. M. Gabeleh, “Best proximity points: global minimization of multivalued non-self mappings,” Optimization Letters, 2013. View at Publisher · View at Google Scholar
  17. C. Mongkolkeha and P. Kumam, “Best proximity point theorems for generalized cyclic contractions in ordered metric spaces,” Journal of Optimization Theory and Applications, vol. 155, no. 1, pp. 215–226, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. B. Piątek, “On cyclic Meir-Keeler contractions in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 1, pp. 35–40, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  19. 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 Publisher · View at Google Scholar · View at MathSciNet
  20. S. S. Basha and N. Shahzad, “Best proximity point theorems for generalized proximal contractions,” Fixed Point Theory and Applications, vol. 2012, article 42, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. S. Sadiq Basha, N. Shahzad, and R. Jeyaraj, “Best proximity points: approximation and optimization,” Optimization Letters, vol. 7, no. 1, pp. 145–155, 2013. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. S. S. Basha, N. Shahzad, and R. Jeyaraj, “Best proximity point theorems for reckoning optimal approximate solutions,” Fixed Point Theory and Applications, vol. 2012, article 202, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. M. A. Al-Thagafi and N. Shahzad, “Best proximity pairs and equilibrium pairs for Kakutani multimaps,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 3, pp. 1209–1216, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. M. A. Al-Thagafi and N. Shahzad, “Best proximity sets and equilibrium pairs for a finite family of multimaps,” Fixed Point Theory and Applications, vol. 2008, Article ID 457069, 10 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  25. A. Abkar and M. Gabeleh, “Proximal quasi-normal structure and a best proximity point theorem,” Journal of Nonlinear and Convex Analysis, vol. 14, pp. 653–659, 2013.
  26. T. C. Lim, “A fixed point theorem for families on nonexpansive mappings,” Pacific Journal of Mathematics, vol. 53, pp. 487–493, 1974. View at Publisher · View at Google Scholar · View at MathSciNet
  27. G. S. R. Kosuru and P. Veeramani, “A note on existence and convergence of best proximity points for pointwise cyclic contractions,” Numerical Functional Analysis and Optimization, vol. 32, no. 7, pp. 821–830, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet