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Modified Mann iterative algorithms by hybrid projection methods for nonexpansive semigroups and mixed equilibrium problems

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Języki publikacji
EN
Abstrakty
EN
In this paper, we propose a modified Mann iterative algorithm by two hybrid projection methods for finding a common element of the set of fixed points of nonexpansive semigroups and the set of solutions of a mixed equilibrium problem in a real Hilbert space. Then, we obtain interesting and new strong convergence theorems for the sequences generated by these processes by using the hybrid projection methods in the mathematical programming. The results presented in this paper extend and improve the corresponding one by Nakajo and Takahashi [J. Math. Anal. Appl. 279 (2003), 372-379].
Wydawca
Rocznik
Strony
259--273
Opis fizyczny
Bibliogr. 23 poz.
Twórcy
autor
autor
  • Department of Mathematics and Statistics, Faculty of Science and Agricultural Technology, Rajamangala University of Technology Lanna Tak, Tak 63000, Thailand, p.katchang@hotmail.com
Bibliografia
  • [1] J. B. Baillon and H. Brezis, Une remarque sur le comportement asymptotique des semi-groupes non lineaires, Houston J. Math. 2 (1976), 5-7.
  • [2] E. Blum and W. Oettli, From optimization and variational inequalities to equilibrium problems, Math. Student 63 (1994), 123-145.
  • [3] F. R. Browder, Convergence of approximates to fixed points of nonexpansive nonlinear mappings in Banach spaces, Arch. Ration. Mech. Anal. 24 (1967), 82-90.
  • [4] R. E. Brück, A simple proof of the mean ergodic theorem for nonlinear contractions in Banach spaces, Israeli Math. 32 (1979), no. 2-3, 107-116.
  • [5] R. S. Burachik, J. O. Lopes and B. F. Svaiter, An outer approximation method for the variational inequality problem, SIAMJ. Control Optim. 43 (2005), 2071-2088.
  • [6] L. C. Ceng and J. C. Yao, A hybrid iterative scheme for mixed equilibrium problems and fixed point problems, J. Comput. Appl. Math. 214 (2008), 186-201.
  • [7] R. Chen and H. He, Viscosity approximation of common fixed points of nonexpan-sive semigroups in Banach spaces, Appl. Math. Lett. 20 (2007), 751-757.
  • [8] F. Cianciaruso, G. Marino and L. Muglia, Iterative methods for equilibrium and fixed point problems for nonexpansive semigroups in Hubert spaces, J. Optim. Theory Appl. 146 (2010), 491-509.
  • [9] F. Cianciaruso, G. Marino, L. Muglia and Y. Yao, A hybrid projection algorithm for finding solutions of mixed equilibrium problem and variational inequality problem, Fixed Point Theory Appl. (2010), article ID 383740.
  • [10] P. L. Combettes and S. A. Hirstoaga, Equilibrium programming in Hubert spaces, J. Nonlinear Convex Anal. 6(2005), 117-136.
  • [11] A. Genel and J. Lindenstrass, An example concerning fixed points, Israel J. Math. 22(1975), 81-86.
  • [12] K. Goebel and W. A. Kirk, Topics in Metric Fixed Point Theory, Cambridge University Press, Cambridge. 1990.
  • [13] B. Halpern, Fixed points of nonexpansive maps, Bull. Amer. Math. Soc. (N.S.j 73 (1976), 957-961.
  • [14] W. R. Mann, Mean value methods in iteration, Proc. Amer. Math. Soc. 4 (1953), 506-510.
  • [15] K. Nakajo and W. Takahashi, Strong convergence theorems for nonexpansive mappings and nonexpansive semigroups, J. Math. Anal. Appl. 279 (2003), 372-379.
  • [16] Z. Opial, Weak convergence of the sequence of successive approximations for non-expansive mappings, Bull. Amer. Math. Soc. 73 (1967), 595-597.
  • [17] J.-W. Peng and J.-C. Yao, Strong convergence theorems of iterative scheme based on the extragradient method for mixed equilibrium problems and fixed point problems, Math. Comput. Modelling 49 (2009), 1816-1828.
  • [18] T. Shimizu and W. Takahashi, Strong convergence to common fixed points of families of nonexpansive mappings, J. Math. Anal. Appl. 211 (1997), 71-83.
  • [19] T. Suzuki, On strong convergence to common fixed points of nonexpansive semigroups in Hubert spaces, Proc. Amer. Math. Soc. 131 (2002), 2133-2136.
  • [20] W. Takahashi, Nonlinear Functional Analysis, Yokohama Publishers, Yokohama, 2000.
  • [21] W. Takahashi, Y Takeuchi and R. Kubota, Strong convergence theorems by hybrid methods for families of nonexpansive mappings in Hubert spaces, J. Math. Anal. Appl. 341 (2008), 276-286.
  • [22] W. Takahashi and M. Toyoda, Weak convergence theorems for nonexpansive mappings and monotone mappings, J. Optim. Theory Appl. 118 (2003), 417-428.
  • [23] H. K. Xu, A strong convergence theorem for contraction semigroups in Banach spaces, Bull. Aust. Math. Soc. 72 (2005), 371-379.
Typ dokumentu
Bibliografia
Identyfikator YADDA
bwmeta1.element.baztech-article-LODD-0002-0061
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