About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 672069, 8 pages
http://dx.doi.org/10.1155/2013/672069
Research Article

One-Local Retract and Common Fixed Point in Modular Metric Spaces

Department of Mathematics, King Abdulaziz University, P.O. Box 80203, Jeddah 21589, Saudi Arabia

Received 8 July 2013; Accepted 19 August 2013

Academic Editor: Mohamed A. Khamsi

Copyright © 2013 Afrah A. N. Abdou. 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. V. V. Chistyakov, “Modular metric spaces. I. Basic concepts,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 1, pp. 1–14, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  2. V. V. Chistyakov, “Modular metric spaces. II. Application to superposition operators,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 72, no. 1, pp. 15–30, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  3. H. Nakano, Modulared Semi-Ordered Linear Spaces, Maruzen, Tokyo, Japan, 1950. View at MathSciNet
  4. J. Musielak, Orlicz Spaces and Modular Spaces, vol. 1034 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 1983. View at MathSciNet
  5. W. Orlicz, Collected Papers, Part I, II, PWN Polish Scientific Publishers, Warsaw, Poland, 1988. View at MathSciNet
  6. L. Diening, Theoretical and numerical results for electrorheological fluids [Ph.D. thesis], University of Freiburg, Freiburg, Germany, 2002.
  7. M. Ruzicka, Electrorheological Fluids Modeling and Mathematical Theory, Springer, Berlin, Germany, 2002. View at Publisher · View at Google Scholar · View at MathSciNet
  8. P. Harjulehto, P. Hästö, M. Koskenoja, and S. Varonen, “The Dirichlet energy integral and variable exponent Sobolev spaces with zero boundary values,” Potential Analysis, vol. 25, no. 3, pp. 205–222, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. J. Heinonen, T. Kilpeläinen, and O. Martio, Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford University Press, Oxford, UK, 1993. View at MathSciNet
  10. M. A. Khamsi, W. M. Kozłowski, and S. Reich, “Fixed point theory in modular function spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 14, no. 11, pp. 935–953, 1990. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. M. A. Khamsi and W. M. Kozlowski, “On asymptotic pointwise nonexpansive mappings in modular function spaces,” Journal of Mathematical Analysis and Applications, vol. 380, no. 2, pp. 697–708, 2011. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. L. P. Belluce and W. A. Kirk, “Nonexpansive mappings and fixed-points in Banach spaces,” Illinois Journal of Mathematics, vol. 11, pp. 474–479, 1967. View at Zentralblatt MATH · View at MathSciNet
  13. F. E. Browder, “Nonexpansive nonlinear operators in a Banach space,” Proceedings of the National Academy of Sciences of the United States of America, vol. 54, pp. 1041–1044, 1965. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. R. E. Bruck,, “A common fixed point theorem for a commuting family of nonexpansive mappings,” Pacific Journal of Mathematics, vol. 53, pp. 59–71, 1974. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. 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
  16. K. Goebel and S. Reich, Reich Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, vol. 83 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1984. View at MathSciNet
  17. K. Goebel, T. Sekowski, and A. Stachura, “Uniform convexity of the hyperbolic metric and fixed points of holomorphic mappings in the Hilbert ball,” Nonlinear Analysis: Theory, Methods & Applications, vol. 4, no. 5, pp. 1011–1021, 1980. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. W. A. Kirk, “Fixed point theorems in CAT(0) spaces and -trees,” Fixed Point Theory and Applications, no. 4, pp. 309–316, 2004. View at Publisher · View at Google Scholar · View at MathSciNet
  19. N. Hussain and M. A. Khamsi, “On asymptotic pointwise contractions in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 71, no. 10, pp. 4423–4429, 2009. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. W. A. Kirk, “Fixed points of asymptotic contractions,” Journal of Mathematical Analysis and Applications, vol. 277, no. 2, pp. 645–650, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. W. A. Kirk, “Asymptotic pointwise contractions, plenary lecture,” in Proceedings of the 8th International Conference on Fixed Point Theory and Its Applications, pp. 16–22, ChiangMai University, Chiang Mai, Thailand, July 2007.
  22. W. A. Kirk and H.-K. Xu, “Asymptotic pointwise contractions,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 69, no. 12, pp. 4706–4712, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. J.-P. Penot, “Fixed point theorems without convexity,” Mémoires de la Société Mathématique de France, vol. 60, pp. 129–152, 1979, Analyse non convexe. View at Zentralblatt MATH · View at MathSciNet
  24. 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
  25. M. A. Khamsi, “On metric spaces with uniform normal structure,” Proceedings of the American Mathematical Society, vol. 106, no. 3, pp. 723–726, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. M. A. Khamsi, “One-local retract and common fixed point for commuting mappings in metric spaces,” Nonlinear Analysis: Theory, Methods & Applications A, vol. 27, no. 11, pp. 1307–1313, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. S. A. Al-Mezel, A. Al-Roqi, and M. A. Khamsi, “One-local retract and common fixed point in modular function spaces,” Fixed Point Theory and Applications, vol. 2012, article 109, 13 pages, 2012. View at Publisher · View at Google Scholar · View at MathSciNet
  28. M. A. Khamsi and W. A. Kirk, An Introduction to Metric Spaces and Fixed Point Theory, John Wiley, New York, NY, USA, 2001. View at Publisher · View at Google Scholar · View at MathSciNet
  29. W. M. Kozlowski, Modular Function Spaces, vol. 122 of Monographs and Textbooks in Pure and Applied Mathematics, Marcel Dekker, New York, NY, USA, 1988. View at MathSciNet
  30. A. A. N. Abdou and M. A. Khamsi, “On the fixed points of nonexpansive maps in modular metric spaces,” Preprint.
  31. J. B. Baillon, “Nonexpansive mappings and hyperconvex spaces,” Contemporary Mathematics, vol. 72, pp. 11–19, 1988.