Table of Contents Author Guidelines Submit a Manuscript
Abstract and Applied Analysis
Volume 2015 (2015), Article ID 165053, 7 pages
http://dx.doi.org/10.1155/2015/165053
Research Article

The Best Approximation Theorems and Fixed Point Theorems for Discontinuous Increasing Mappings in Banach Spaces

1 School of Mathematical Sciences, Qufu Normal University, Qufu, Shandong 273165, China
2 College of Information Sciences and Engineering, Shandong Agricultural University, Tai’an, Shandong 271018, China
3 Department of Mathematics and Statistics, Curtin University of Technology, Perth, WA 6845, Australia

Received 20 June 2014; Revised 30 September 2014; Accepted 30 September 2014

Academic Editor: Poom Kumam

Copyright © 2015 Dezhou Kong et al. 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. Y. Alber, “Generalized projection operators in Banach spaces: properties and applications,” in Proceedings of the Israel Seminar Ariel, Israel, Function Differential Equation, vol. 1, pp. 1–21, 1994.
  2. J. Li, “The generalized projection operator on reflexive Banach spaces and its applications,” Journal of Mathematical Analysis and Applications, vol. 306, no. 1, pp. 55–71, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  3. G. Isac, “On the order monotonicity of the metric projection operator,” in Approximation Theory, Wavelets and Applications, S. P. Singh, Ed., vol. 454 of NATO Science Series, pp. 365–379, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1995. View at Publisher · View at Google Scholar · View at MathSciNet
  4. H. Nishimura and E. A. Ok, “Solvability of variational inequalities on Hilbert lattices,” Mathematics of Operations Research, vol. 37, no. 4, pp. 608–625, 2012. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  5. J. Li and E. A. Ok, “Optimal solutions to variational inequalities on Banach lattices,” Journal of Mathematical Analysis and Applications, vol. 388, no. 2, pp. 1157–1165, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  6. K. Fan, “Extensions of two fixed point theorems of F. E. Browder,” Mathematische Zeitschrift, vol. 112, pp. 234–240, 1969. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  7. T. Lin and S. Park, “Approximation and fixed-point theorems for condensing composites of multifunctions,” Journal of Mathematical Analysis and Applications, vol. 223, no. 1, pp. 1–8, 1998. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  8. D. O'Regan and N. Shahzad, “Approximation and fixed point theorems for countable condensing composite maps,” Bulletin of the Australian Mathematical Society, vol. 68, no. 1, pp. 161–168, 2003. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  9. K. Tan and X. Yuan, “Random fixed-point theorems and approximation in cones,” Journal of Mathematical Analysis and Applications, vol. 185, no. 2, pp. 378–390, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  10. L. Liu, “Approximation theorems and fixed point theorems for various classes of 1-set-contractive mappings in Banach spaces,” Acta Mathematica Sinica, vol. 17, no. 1, pp. 103–112, 2001. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  11. L. Liu, “Random approximations and random fixed point theorems for random 1-set-contractive non-self-maps in abstract cones,” Stochastic Analysis and Applications, vol. 18, no. 1, pp. 125–144, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  12. L. Liu, “Some random approximations and random fixed point theorems for 1-set-contractive random operators,” Proceedings of the American Mathematical Society, vol. 125, no. 2, pp. 515–521, 1997. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  13. L. Liu, “Random approximations and random fixed point theorems in infinite-dimensional Banach spaces,” Indian Journal of Pure and Applied Mathematics, vol. 28, no. 2, pp. 139–150, 1997. View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet · View at Scopus
  14. I. Beg and N. Shahzad, “Random fixed points of random multivalued operators on Polish spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 20, no. 7, pp. 835–847, 1993. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  15. N. Shahzad, “Fixed point and approximation results for multimaps in S-KKM class,” Nonlinear Analysis: Theory, Methods & Applications, vol. 56, no. 6, pp. 905–918, 2004. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  16. 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 MathSciNet · View at Scopus
  17. A. Amini-Harandi, “Best and coupled best approximation theorems in abstract convex metric spaces,” Nonlinear Analysis: Theory, Methods & Applications, vol. 74, no. 3, pp. 922–926, 2011. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  18. D. Roux and S. P. Singh, “On some fixed point theorems,” International Journal of Mathematics and Mathematical Sciences, vol. 12, no. 1, pp. 61–64, 1989. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. L. Liu, “On approximation theorems and fixed point theorems for non-self-mappings in infinite-dimensional Banach spaces,” Journal of Mathematical Analysis and Applications, vol. 188, no. 2, pp. 541–551, 1994. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  20. L. Liu and X. Li, “On approximation theorems and fixed point theorems for non-self-mappings in uniformly convex Banach spaces,” Banyan Mathematical Journal, vol. 4, pp. 11–20, 1997. View at Google Scholar
  21. D. O'Regan, “Existence and approximation of fixed points for multivalued maps,” Applied Mathematics Letters, vol. 12, no. 6, pp. 37–43, 1999. View at Publisher · View at Google Scholar · View at MathSciNet · View at Scopus
  22. D. Guo and V. Lakshmikantham, Nonlinear Problems in Abstract Cones, Academic Press, New York, NY, USA, 1988. View at MathSciNet
  23. D. Guo, Y. Cho, and J. Zhu, Partial Ordering Methods in Nonlinear Problems, Nova Science Publishers, New York, NY, USA, 2004. View at MathSciNet
  24. P. Meyer-Nieberg, Banach Lattices, Universitext, Springer, New York, NY, USA, 1991. View at Publisher · View at Google Scholar · View at MathSciNet
  25. I. Cioranescu, “Geometry of Banach spaces,” in Duality Mappings and Nonlinear Problems, Kluwer Academic Publishers, Dordrecht, The Netherlands, 1990. View at Google Scholar
  26. W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and its Applications, Yokohama, Yokohama, Japan, 2000. View at MathSciNet
  27. D. Kong, L. Liu, and Y. Wu, “Best approximation and fixed-point theorems for discontinuous increasing maps in Banach lattices,” Fixed Point Theory and Applications, vol. 2014, article 18, 2014. View at Publisher · View at Google Scholar · View at Scopus