About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 680186, 5 pages
http://dx.doi.org/10.1155/2013/680186
Research Article

Best Proximity Points for Relatively -Continuous Mappings in Banach and Hyperconvex Spaces

1528 Rover Boulevard, Los Alamos, NM 87544, USA
2Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 21 May 2013; Accepted 9 August 2013

Academic Editor: Adrian Petrusel

Copyright © 2013 Jack Markin 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. 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
  2. A. A. Eldred, V. S. Raj, and P. Veeramani, “On best proximity pair theorems for relatively u-continuous mappings,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 12, pp. 3870–3875, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  3. 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
  4. S. S. Basha and N. Shahzad, “Best proximity point theorems for generalized proximal contractions,” Fixed Point Theory and Applications, vol. 2012, article 42, 9 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  5. S. S. 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 MathSciNet
  6. 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, 9 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  7. S. S. Basha, N. Shahzad, and R. Jeyaraj, “Common best proximity points: global optimal solutions,” Journal of Optimization Theory and Applications, vol. 148, no. 1, pp. 69–78, 2011. View at Publisher · View at Google Scholar · View at MathSciNet
  8. M. Gabeleh and N. Shahzad, “Existence and convergence theorems of best proximity points,” Journal of Applied Mathematics, vol. 2013, Article ID 101439, 6 pages, 2013. View at Publisher · View at Google Scholar
  9. 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
  10. S. Karpagam and S. Agrawal, “Existence of best proximity points of p-cyclic contractions,” Fixed Point Theory, vol. 13, no. 1, pp. 99–105, 2012. View at MathSciNet
  11. R. Sine, “Hyperconvexity and approximate fixed points,” Nonlinear Analysis: Theory, Methods & Applications, vol. 13, no. 7, pp. 863–869, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. R. Espínola and P. Lorenzo, “Metric fixed point theory on hyperconvex spaces: recent progress,” Arabian Journal of Mathematics, vol. 1, no. 4, pp. 439–463, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  13. F. Deutsch, V. Indumathi, and K. Schnatz, “Lower semicontinuity, almost lower semicontinuity, and continuous selections for set-valued mappings,” Journal of Approximation Theory, vol. 53, no. 3, pp. 266–294, 1988. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. J. T. Markin, “A selection theorem for quasi-lower semicontinuous mappings in hyperconvex spaces,” Journal of Mathematical Analysis and Applications, vol. 321, no. 2, pp. 862–866, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. M. A. Khamsi, “KKM and Ky Fan theorems in hyperconvex metric spaces,” Journal of Mathematical Analysis and Applications, vol. 204, no. 1, pp. 298–306, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. W. A. Kirk and S. S. Shin, “Fixed point theorems in hyperconvex spaces,” Houston Journal of Mathematics, vol. 23, no. 1, pp. 175–188, 1997. View at Zentralblatt MATH · View at MathSciNet
  17. V. S. Raj and P. Veeramani, “Best proximity pair theorems for relatively nonexpansive mappings,” Applied General Topology, vol. 10, no. 1, pp. 21–28, 2009. View at Zentralblatt MATH · View at MathSciNet
  18. W. A. Kirk, S. Reich, and P. Veeramani, “Proximinal retracts and best proximity pair theorems,” Numerical Functional Analysis and Optimization, vol. 24, no. 7-8, pp. 851–862, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. J. Markin and N. Shahzad, “Best approximation theorems for nonexpansive and condensing mappings in hyperconvex spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 70, no. 6, pp. 2435–2441, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. A. Abkar and M. Gabeleh, “Results on the existence and convergence of best proximity points,” Fixed Point Theory and Applications, vol. 2010, Article ID 386037, 10 pages, 2010. View at Zentralblatt MATH · View at MathSciNet
  21. A. Amini-Harandi, A. P. Farajzadeh, D. O'Regan, and R. P. Agarwal, “Coincidence point, best approximation, and best proximity theorems for condensing set-valued maps in hyperconvex metric spaces,” Fixed Point Theory and Applications, vol. 2008, Article ID 543154, 8 pages, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet