A new one-step iterative process for approximating common fixed points of a countable family of quasi-nonexpansive multi-valued mappings in CAT(0) spaces

Document Type : Research Paper

Authors

1 Department of Mathematics‎, ‎Faculty of Science‎, ‎Chiang Mai University‎, ‎Chiang Mai 50200‎, ‎Thailand.

2 Department of Mathematics‎, ‎Faculty of Science and Technology‎, ‎Nakhon Pathom Rajabhat University‎, ‎Nakhon Pathom 73000‎, ‎Thailand‎, ‎and‎ Research Center for Pure and Applied Mathematics‎, ‎Research and‎ ‎Development Institute‎, ‎Nakhon Pathom Rajabhat University‎, ‎Nakhon‎ ‎Pathom 73000‎, ‎Thailand.

Abstract

‎In this paper‎, ‎we propose a new one-step iterative process for a‎ ‎countable family of quasi-nonexpansive multi-valued mappings in a‎ ‎CAT(0) space‎. ‎We also prove strong and $Delta$-convergence theorems‎ ‎of the proposed iterative process under some control conditions‎. ‎Our‎ ‎main results extend and generalize many results in the literature.

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References

A. Abkar and M. Eslamian, Convergence theorems for a finite family of generalized nonexpansive multivalued mappings in CAT(0) spaces, Nonlinear Anal. 75 (2012), no. 4, 1895--1903.
A. Abkar and M. Eslamian, Geodesic metric spaces and generalized nonexpansive multivalued mappings, Bull. Iranian Math. Soc. 39 (2013), no. 5, 993--1008.
M. Bridson and A. Haeiger, Metric Spaces of Non-positive Curvature, Springer-Verlag, Berlin, 1999.
P. Chanthorn and P. Chaoha, Fixed point sets of set-valued mappings, Fixed Point Theory Appl. 2015 (2015), no. 56, 16 pages.
P. Chaoha and A. Phon-on, A note on fixed point sets in CAT(0) spaces, J. Math. Anal. Appl. 320 (2006), no. 2, 983--987.
W. Cholamjiak and S. Suantai, Approximation of common fixed points of two quasinonexpansive multi-valued maps in Banach spaces, Comput. Math. Appl. 61 (2011), no. 4, 941--949.
S. Dhompongsa, A. Kaewkhao and B. Panyanak, Lim's theorems for multivalued mappings in CAT(0) spaces, J. Math. Anal. Appl. 312 (2005), no. 2, 478--487.
S. Dhompongsa, A. Kaewkhao and B. Panyanak, On Kirk's strong convergence theorem for multivalued nonexpansive mappings on CAT(0) spaces, Nonlinear Anal. 75 (2012) 459--468.
S. Dhompongsa, W.A. Kirk and B. Panyanak, Nonexpansive set-valued mappings in metric and Banach spaces, J. Nonlinear Convex Anal. 8 (2007), no. 1, 35--45.
S. Dhompongsa, W.A. Kirk and B. Sims, Fixed points of uniformly Lipschitzian mappings, Nonlinear Anal. 65 (2006), no. 4, 762--772.
S. Dhompongsa and B. Panyanak, On Δ-convergence theorems in CAT(0) spaces, Comput. Math. Appl. 56 (2008), no. 10, 2572--2579.
M. Eslamian and A. Abkar, One-step iterative process for a finite family of multivalued mappings, Math. Comp. Model. 54 (2011), no. 1-2, 105--111.
J. Garcia-Falset, E. Lorens-Fuster and T. Suzuki, Fixed point theory for a class of generalized nonexpansive mappings, J. Math. Anal. Appl. 375 (2011), no. 1, 185--195.
A. Geletu, Introduction to Topological Spaces and Set-Valued Maps, Lecture Notes in Math. Institute of Mathematics, Department of Operations Research & Stochestics, Ilmenau University of Technology, 2006.
K. Goebel and S. Reich, Uniform Convexity, Hyperbolic Geometry, and Nonexpansive Mappings, Monographs and Textbooks in Pure and Applied Mathematics 83, Marcel Dekker Inc. New York, 1984.
W.A. Kirk, Geodesic geometry and fixed point theory, in: Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), pp. 195--225, Colecc. Abierta, 64, Univ. Sevilla Secr. Publ., Seville, 2003.
W.A. Kirk, Geodesic geometry and fixed point theory II, in: International Conference on Fixed Point Theory and Applications, pp. 113--142, Yokohama Publ., Yokohama, 2004.
W.A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory Appl. 2004 (2004), no. 4, 309--316.
W.A. Kirk and B. Panyanak, A concept of convergence in geodesic spaces, Nonlinear Anal. 68 (2008), no. 12, 3689--3696.
L. Leuştean, A quadratic rate of asymptotic regularity for CAT(0)-spaces, J. Math. Anal. Appl. 325 (2007), no. 1, 386--399.
J.T. Markin, Continuous dependence of fixed point sets, Proc. Amer. Math. Soc. 38 (1973) 545--547.
B. Panyanak, Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces, Comput. Math. Appl. 54 (2007), no. 6, 872--877.
K.P.R. Sastry and G.V.R. Babu, Convergence of Ishikawa iterates for a multivalued mapping with a fixed point, Czechoslovak Math. J. 55(130) (2005), no. 4, 817--826.
Y. Song and H. Wang, Erratum to Mann and Ishikawa iterative processes for multivalued mappings in Banach spaces", Comput. Math. Appl. 55 (2008), no. 12, 2999--3002.
R.T. Rockafellar and R.J.B. Wets, Variational Analysis, Springer-Verlag, 2005.
H.K. Xu, On weakly nonexpansive and *-nonexpansive multivalued mappings, Math. Japonica. 36 (1991) 441--445.
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.
H. Zegeye and N. Shahzad, Viscosity approximation methods for nonexpansive multimaps in Banach spaces, Acta Math. Sin. (Engl. Ser.) 26 (2010), no. 6, 1165--1176.
Bulletin of the Iranian Mathematical Society
Volume 43, Issue 5
October 2017
Pages 1127-1141

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History

  • Receive Date: 03 July 2015
  • Revise Date: 21 March 2016
  • Accept Date: 05 May 2016

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