Communications of the Korean Mathematical Society (대한수학회논문집)

Korean Mathematical Society (대한수학회)
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COMMON FIXED POINTS FOR SINGLE-VALUED AND MULTI-VALUED MAPPINGS IN COMPLETE ℝ-TREES

  • Phuengrattana, Withun (Department of Mathematics Faculty of Science and Technology Nakhon Pathom Rajabhat University) ;
  • Sopha, Sirichai (Department of Mathematics Faculty of Science and Technology Nakhon Pathom Rajabhat University)
  • Received : 2015.09.16
  • Published : 2016.07.31
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Abstract

The aim of this paper is to prove some strong convergence theorems for the modified Ishikawa iteration process involving a pair of a generalized asymptotically nonexpansive single-valued mapping and a quasi-nonexpansive multi-valued mapping in the framework of $\mathbb{R}$-trees under the gate condition.

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References

  1. N. Akkasriworn and K. Sokhuma, Convergence theorems for a pair of asymptotically and multivalued nonexpansive mapping in CAT(0) spaces, Commun. Korean Math. Soc. 30 (2015), no. 3, 177-189. https://doi.org/10.4134/CKMS.2015.30.3.177
  2. A. G. Aksoy and M. A. Khamsi, A selection theorem in metric trees, Proc. Amer. Math. Soc. 134 (2006), no. 10, 2957-2966. https://doi.org/10.1090/S0002-9939-06-08555-8
  3. S. M. A. Aleomraninejad, Sh. Rezapour, and N. Shahzad, Some fixed point results on a metric space with a graph, Topology Appl. 159 (2012), no. 3, 659-663. https://doi.org/10.1016/j.topol.2011.10.013
  4. 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
  5. M. Bridson and A. Haefliger, Metric Spaces of Non-positive Curvature, Springer, Berlin, 1999.
  6. S. Dhompongsa and B. Panyanak, On $\Delta$-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), no. 10, 2572-2579. https://doi.org/10.1016/j.camwa.2008.05.036
  7. R. Espinola and W. A. Kirk, Fixed point theorems in R-trees with applications to graph theory, Topology Appl. 153 (2006), no. 7, 1046-1055. https://doi.org/10.1016/j.topol.2005.03.001
  8. W. A. Kirk, Hyperconvexity of R-trees, Fund. Math. 156 (1998), no. 1, 67-72.
  9. W. A. Kirk, Fixed point theorems in CAT(0) spaces and $\mathbb{R}$-trees, Fixed Point Theory Appl. 2004 (2004), no. 4, 309-316.
  10. W. Laowang and B. Panyanak, Approximating fixed points of nonexpansive nonself mappings in CAT(0) spaces, Fixed Point Theory Appl. 2010 (2010), Article ID 367274, 11 pages.
  11. J. T. Markin, Fixed points, selections and best approximation for multivalued mappings in $\mathbb{R}$-trees, Nonlinear Anal. 67 (2007), no. 9, 2712-2716. https://doi.org/10.1016/j.na.2006.09.036
  12. W. Phuengrattana and S. Suantai, Existence theorems for generalized asymptotically nonexpansive mappings in uniformly convex metric spaces, J. Convex Anal. 20 (2013), no. 3, 753-761.
  13. T. Puttasontiphot, Mann and Ishikawa iteration schemes for multivalued mappings in CAT(0) spaces, Appl. Math. Sci. (Ruse) 4 (2010), no. 61-64, 3005-3018.
  14. S. Saejung, S. Suantai and P. Yotkaew, A note on "Common Fixed Point of Multistep Noor Iteration with Errors for a Finite Family of Generalized Asymptotically Quasi-Nonexpansive Mapping", Abstract and Applied Analysis 2009 (2009), Article ID 283461.
  15. K. Samanmit and B. Panyanak, On multivalued nonexpansive mappings in $\mathbb{R}$-trees, J. Appl. Math. 2012 (2012), Article ID 629149, 13 pages.
  16. N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. 71 (2009), no. 3-4, 838-844. https://doi.org/10.1016/j.na.2008.10.112
  17. K. Sokhuma and A. Kaewkhao, Ishikawa iterative process for a pair of single-valued and multivalued nonexpansive mappings in Banach spaces, Fixed Point Theory Appl. 2011 (2011), Article ID 618767.
  18. J. Tits, A Theorem of Lie-Kolchin for Trees, Contributions to Algebra: A Collection of Papers Dedicated to Ellis Kolchin, Academic Press, New York, 1977.
  19. R. T. Rockafellar and R. J.-B. Wets, Variational Analysis, Springer-Verlag, 2005.
  20. H. K. Xu, Another control condition in an iterative method for nonexpansive mappings, Bull. Austral. Math. Soc. 65 (2002), no. 1, 109-113. https://doi.org/10.1017/S0004972700020116

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