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Viscosity approximation methods for nonexpansive multi-valued nonself mappings and equilibrium problems

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Języki publikacji
EN
Abstrakty
EN
In this paper, strong convergence theorems by the viscosity approximation method for nonexpansive multi-valued nonself mappings and equilibrium problems are established under some suitable conditions in a Hilbert space. The obtained results extend and improve the corresponding results existed in the literature.
Wydawca
Rocznik
Strony
382--395
Opis fizyczny
Bibliogr. 32 poz.
Twórcy
  • School of Science, University of Phayao, Phayao 56000, Thailand
  • School of Science, University of Phayao, Phayao 56000, Thailand
autor
  • Department of Mathematics, Faculty of Science Chiang Mai University, Chiang Mai 50200, Thailand
Bibliografia
  • [1] M. Abbas, S. H. Khan, A. R. Khan, and R.P. Agarwal, Common fixed points of two multivalued nonexpansive mappings by one-step iterative scheme, Appl. Math. Lett. 2010. doi: 10.1016/j.aml.2010.08.025
  • [2] R. P. Agarwal, D. O’Regan, D. R. Sahu, Fixed Point Theory for Lipschitzian-type Mappings with Applications, Springer, 2009.
  • [3] K. Aoyama, Y. Kimura, W. Takahashi, M. Toyoda, Approximation of common fixed points of a countable family of nonexpansive mappings in a Banach space, Nonlinear Anal. 67 (2007), 2350–2360.
  • [4] N. A. Assad, W. A. Kirk, Fixed point theorems for set-valued mappings of contractive type, Pacific J. Math. 43 (1972), 553–562.
  • [5] E. Blum, W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Stud. 63 (1994), 123–145.
  • [6] L.-C. Ceng, J.-C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186–201.
  • [7] P. Cholamjiak, A hybrid iterative scheme for equilibrium problems, variational inequality problems and fixed point problems in Banach spaces, Fixed Point Theory Appl. Volume 2009 (2009), Article ID 719360, 18 pages.
  • [8] W. Cholamjiak, S. Suantai, A hybrid method for a countable family of multivalued maps, equilibrium problems, and variational inequality problems, Discrete Dyn. Nat. Soc. Volume 2010 (2010), Article ID 349158, 14 pages.
  • [9] P. L. Combettes, S. A. Hirstoaga, Equilibrium programming in Hilbert spaces, J. Nonlinear Convex Anal. 6 (2005), 117–136.
  • [10] H. He, S. Liu, Y. J. Cho, An explicit method for systems of equilibrium problems and fixed points of infinite family of nonexpansive mappings, J. Comput. Appl. Math. 235 (2011), 4128–4139.
  • [11] N. Hussain, A. R. Khan, Applications of the best approximation operator to *-nonexpansive maps in Hilbert spaces, Numer. Funct. Anal. Optim. 24 (2003), 327–338.
  • [12] J. S. Jung, Convergence of approximating fixed pints for multivalued nonself-mappings in Banach spaces, Korean J. Math. 16 (2008), 215–231.
  • [13] P. Kumam, A hybrid approximation method for equilibrium and fixed point problems for a monotone mapping and a nonexpansive mapping, Nonlinear Anal.: Hybr. Syst. 2 (2008), 1245–1255.
  • [14] P. Kumam, A new hybrid iterative method for solution of equilibrium problems and fixed point problems for an inverse strongly monotone operator and a nonexpansive mapping, J. Appl. Math. Comput. 29 (2009), 263–280.
  • [15] G. Marino, H. K. Xu, Weak and strong convergence theorems for strict pseudocontractions in Hilbert spaces, J. Math. Anal. Appl. 329 (2007), 336–346.
  • [16] A. Moudafi, Viscosity approximation methods for fixed-point problems, J. Math. Anal. Appl. 241 (2000), 46–55.
  • [17] S. B. Nadler Jr., Multi-valued contraction mappings, Pacific J. Math. 30 (1969), 475–488.
  • [18] Z. Opial, Weak convergence of the sequence of successive approximation for nonexpansive mappings, Bull. Amer. Math. Soc. 73 (1967), 561–597.
  • [19] J.-W. Peng, Y.-C. Liou, J.-C. Yao, An iterative algorithm combining viscosity method with parallel method for a generalized equilibrium problem and strict pseudocontractions, Fixed Point Theory Appl., Volume 2009 (2009), Article ID 794178, 21 pages.
  • [20] P. Pietramala, Convergence of approximating fixed points sets for multivalued nonexpansive mappings, Comment. Math. Univ. Carolin. 32 (1991), 697–701.
  • [21] X. Qin, Y. J. Cho, S. M. Kang, Convergence theorems of common elements for equilibrium problems and fixed point problems in Banach spaces, J. Comput. Appl. Math. 225 (2009), 20–30.
  • [22] N. Shahzad, H. Zegeye, Strong convergence results for nonself multimaps in Banach spaces, Proc. Amer. Math. Soc. 136 (2008), 539–548.
  • [23] N. Shahzad, H. Zegeye, On Mann and Ishikawa iteration schemes for multi-valued maps in Banach spaces, Nonlinear Anal. 71 (2009), 838–844.
  • [24] Y. Shehu, Iterative approximation method for finite family of relatively quasi nonexpansive mappings and systems of equilibrium problems, J. Global Optim. (2010). doi: 10.1007/s10898-010-9619-4
  • [25] Y. Shehu, A new hybrid iterative scheme for countable families of relatively quasinonexpansive mappings and system of equilibrium problems, Int. J. Math. Math. Sci. (2011), Art. ID 131890, 23 pages, 2011.
  • [26] Y. Song, Y. J. Cho, Some note on Ishikawa iteration for multi-valued mappings, Bull. Korean Math. Soc. 48 (2011), 575–584.
  • [27] Y. Song, H. Wang, Convergence of iterative algorithms for multivalued mappings in Banach spaces, Nonlinear Anal. 70 (2009), 1547–1556.
  • [28] A. Tada, W. Takahashi, Strong convergence theorem for an equilibrium problem and a nonexpansive mapping, in: W. Takahashi. T. Tanaka (Eds.), Nonlinear Analysis and Convex Analysis, Yokohama Publishers, Yokohama (2005).
  • [29] W. Takahashi, Nonlinear Functional Analysis, Fixed Point Theory and Its Application, Yokohama-Publishers, Yokohama, Japan 2000.
  • [30] S. Takahashi, W. Takahashi, Viscosity approximation methods for equilibrium problems and fixed point problems in Hilbert spaces, J. Math. Anal. Appl. 331 (2007), 506–515.
  • [31] R. Wittmann, Approximation of fixed point of nonexpansive mappings, Arch. Math. 58 (1992), 486–491.
  • [32] H. Zegeye, N. Shahzad, Viscosity approximation methods for nonexpansive multimaps in Banach space, Acta Math. Sinica 26 (2010), 1165–1176.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-ea609bc6-1e98-47ca-8967-c84798ead1b6
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