Two Step Modified Ishikawa Iteration Scheme for Multi-Valued Mappings in CAT(0) Space
Main Article Content
Abstract
The aim of this paper is to prove some strong convergence theorems for the modified Ishikawa iteration scheme involving quasi-nonexpansive multi-valued mappings in the framework of CAT(0) spaces.
Article Details
References
- N.A. Assad and W.A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math., 43(3)(1972),553-562.
- V. Berinde, Iterative approximation of fixed points, in: Lecture Notes in Mathematics, Vol. 1912, Springer, Berlin, 2007.
- M. Bridson and A. Haefliger, Metric spaces of non-positive curvature, Vol. 319 of Fundamental principles of Mathematical Sciences, Springer-Verlag, Berlin, Germany, 1999.
- K. S. Brown, Buildings, Springer-Verlag, New York, 1989.
- F. Bruhat and J. Tits, Groupes r ´eductifs sur un corps local, Publications Mathematiques de LTHES,41(1)1972,5-251.
- P. Chaoha and A. Phon-on, A note on fixed sets in CAT(0) spaces, J. Math. Anal. Appl., 320(2)(2006), 983-987.
- S. Dhompongsa, A. Kaewkhao and B. Panyanak, Lim's theorems for multi-valued mappings in CAT(0) spaces, J. Math. Anal. Appl., 312(2)(2005), 478-487.
- S. Dhompongsa and B. Panyanak, On ∆-convergence theorems in CAT(0) spaces, Comput. Math. Appl., 56(2008), 2572-2579.
- H. Fujiwara, K. Nagano and T. Shioya, Fixed point sets of parabolic isometries of CAT(0) spaces, Commentarii Mathematici Helvetici, 81(2)(2006), 305-335.
- K. Goebel and R. Reich, Uniform convexity, Hyperbolic Geometry and Nonexpansive mappings, vol. 83 of Monographs and Textbooks in Pure & Applied Mathematics, Marcel Dekker Inc., New York, 1984.
- S. Ishikawa, Fixed point and iteration of a nonexpansive mapping in a Banach space, Proc. Amer. Math. Soc., 59(1976), 65-71.
- W. A. Kirk, Geodesic geometry and fixed point theory, In Seminar of Mathematical Analysis (Malaga/Seville, 2002/2003), vol. 64(2003) of Colecc. Abierta, pp. 195-225, Univ. Sevilla Serc. Publ., Seville.
- W.A. Kirk, Fixed point theorems in CAT(0) spaces and R-trees, Fixed Point Theory & Application, vol. 2004(4)(2004),309-316.
- W.A. Kirk and B. Panyanak, A concept of convergence in Geodesic spaces, Nonlinear Analysis: TMA, 68(12), 3689-3696.
- W.A. Kirk, Geodesic geometry and fixed point theory II, International conference on Fixed Point Theory & Applications, Yokohama Publ. Japan, (2004), 113-142.
- W. Laowang and B. Panyanak, Srong and ∆-convergence theorems for multi-valued mappings in CAT(0) spaces, J.Inequal. Appl., Vol. 2009, Article ID 730132, 16 pages.
- L. Leustean, A quadratic rate of asymptotic regularity for CAT(0) spaces, J. Math. Anal. Appl., 325(1)(2007), 386-399.
- W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc., 4(1953), 506-510.
- S. B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math., 30(1969), 475-487.
- Z. Opial, Weak convergence of the sequence of successive approximations for nonexpansive mappings, Bull. Amer. Math. Soc.,73(1967), 591-597.
- B. Panyanak, Mann and Ishikawa iterative processes for multi-valued mappings in Banach spaces, Comput. Math. Appl., 54(6)(2007), 872-877.
- S. Reich, Weak convergence theorems for nonexpansive mappings in Banach spaces, J. Math. Anal. Appl., 67(1979), 274-276.
- K.P.R. Sastry and G.V.R. Babu, Convergence of Ishikawa iterates for a multivalued mapping with fixed point, Czechoslovak Math. J., 55(1)(2005), 817-399.
- H.F. Senter and W.G. Dotson, Approximating fixed points of nonexpansive mappings, Proc. Amer. Math. Soc., 44(1974), 375-380.
- N. Shahzad and J. Markin, Invariant approximations for commuting mappings in CAT(0) and hyperconvex spaces, J. Math. Anal. Appl., 337(2)(2008), 1457-1464.
- N. Shahzad and H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Analysis, 71(2009), no. 3-4, 838-844.
- C. Shiau, K.K. Tan and C.S. Wang, Quasi-nonexpansive multi-valued maps and selections, Fund. Math., 87(1975), 109-119.
- Y. Song and H. Wang, Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Analysis: Theory, Method & Applications, 70(4)(2009), 1547- 1556.
- Y. Song and H. Wang, Erratum to ”Mann and Ishikawa iterative processes for multi-valued mappings in Banach spaces” [Comput. Math. Appl. 54(2007),872-877], Comput. Math. Appl., 55(12)(2008), 2999-3002.
- K.K. Tan and H.K. Xu, Approximating fixed points of nonexpansive mappings by Ishikawa iteration process, J. Math. Anal. Appl., 178(1993), 301-308.