DOI QR코드

DOI QR Code

Approximation of Common Fixed Points of Two Strictly Pseudononspreading Multivalued Mappings in ℝ-Trees

PHUENGRATTANA, WITHUN

  • Received : 2013.12.26
  • Accepted : 2014.04.11
  • Published : 2015.06.23

Abstract

In this paper, we introduce and study a new multivalued mapping in $\mathbb{R}$-trees, called k-strictly pseudononspreading. We also introduce a new two-step iterative process for two k-strictly pseudononspreading multivalued mappings in $\mathbb{R}$-trees. Strong convergence theorems of the proposed iteration to a common fixed point of two k-strictly pseudononspreading multivalued mappings in $\mathbb{R}$-trees are established. Our results improve and extend the corresponding results existing in the literature.

Keywords

fixed point;$\mathbb{R}$-tree;strictly pseudononspreading mapping;convergence theorems

References

  1. A. G. Aksoy and M. A. Khamsi, A selection theorem in metric trees, Proc. Amer. Math. Soc., 134(2006), 2957-2966. https://doi.org/10.1090/S0002-9939-06-08555-8
  2. S. M. A. Aleomraninejad, Sh. Rezapour and N. Shahzad, Some fixed point results on a metric space with a graph, Topology Appl., 159(2012), 659-663. https://doi.org/10.1016/j.topol.2011.10.013
  3. A. Amini-Harandi and A. P. Farajzadeh, Best approximation, coincidence and fixed point theorems for set-valued maps in ${\mathbb{R}}$-trees, Nonlinear Anal., 71(2009), 1649-1653. https://doi.org/10.1016/j.na.2009.01.001
  4. I. Bartolini, P. Ciaccia and M. Patella, String matching with metric trees using an approximate distance, SPIR Lecture notes in Comput. Sci. 2476, Springer, Berlin, 1999.
  5. H. H. Bauschke and P. L. Combettes, Convex Analysis and Monotone Operator Theory in Hilbert Spaces, Springer, New York, NY, USA, 2011.
  6. M. Bestvina, ${\mathbb{R}}$-trees in topology, geometry, and group theory, in: Handbook of Geometric Topology, North-Holland, Amsterdam, 2002, pp. 55-91.
  7. S. Dhompongsa and B. Panyanak, On ${\Delta}$-convergence theorems in CAT(0) spaces, Comp. Math. Appl., 56(10)(2008), 2572-2579. https://doi.org/10.1016/j.camwa.2008.05.036
  8. R. Espinola, W. A. Kirk, Fixed point theorems in R-trees with applications to graph theory, Topology and its Applications, 153(2006), 1046-1055. https://doi.org/10.1016/j.topol.2005.03.001
  9. W. A. Kirk, Hyperconvexity of ${\mathbb{R}}$-trees, Fundamenta Mathematicae, 156(1)(1998), 67-72.
  10. W. A. Kirk, Fixed point theorems in CAT(0) spaces and ${\mathbb{R}}$-trees, Fixed Point Theory Appl., 2004(2004), 309-316.
  11. W. A. Kirk, Some recent results in metric fixed point theory, J. Fixed Point Theory Appl., 2(2007), 195-207. https://doi.org/10.1007/s11784-007-0031-8
  12. J. T. Markin, Fixed points, selections and best approximation for multivalued mappings in ${\mathbb{R}}$-trees, Nonlinear Anal., 67(2007), 2712-2716. https://doi.org/10.1016/j.na.2006.09.036
  13. S. B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math., 30(1969), 475-488. https://doi.org/10.2140/pjm.1969.30.475
  14. M. O. Osilike and F. O. Isiogugu, Weak and strong convergence theorems for nonspreading-type mappings in Hilbert spaces, Nonlinear Anal., 74(2011), 1814-1822. https://doi.org/10.1016/j.na.2010.10.054
  15. T. Puttasontiphot, Mann and Ishikawa iteration schemes for multivalued mappings in CAT(0) spaces, Applied Mathematical Sciences, 4(61)(2010), 3005-3018.
  16. K. Samanmit and B. Panyanak, On multivalued nonexpansive mappings in ${\mathbb{R}}$-trees, Journal of Applied Mathematics, Volume 2012, Article ID 629149, 13 pages.
  17. C. Semple and M. Steel, Phylogenetics, Oxford Lecture Ser. Math. Appl., vol. 24, Oxford Univ. Press, Oxford, 2003.
  18. N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal., 71(2009), 838-844. https://doi.org/10.1016/j.na.2008.10.112
  19. J. Tits, A Theorem of LieKolchin for Trees. Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977.
  20. K. Wlodarczyk, D. Klim and R. Plebaniak, Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces, J. Math. Anal. Appl., 328(2007), 46-57. https://doi.org/10.1016/j.jmaa.2006.05.029