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Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

  • J. Schu (a1)
Abstract

Let T be an asymptotically nonexpansive self-mapping of a closed bounded and convex subset of a uniformly convex Banach space which satisfies Opial's condition. It is shown that, under certain assumptions, the sequence given by xn+1 = αnTn(xn) + (1 - αn)xn converges weakly to some fixed point of T. In arbitrary uniformly convex Banach spaces similar results are obtained concerning the strong convergence of (xn) to a fixed point of T, provided T possesses a compact iterate or satisfies a Frum-Ketkov condition of the fourth kind.

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References
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[1]Bose S.C., ‘Weak convergence to the fixed point of an asymptotically nonexpansive map’, Proc. Amer. Math. Soc. 68 (1978), 305308.
[2]Goebel K. and Kirk W.A., ‘A fixed point theorem for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 35 (1972), 171174.
[3]Górnicki J., ‘Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces’, Comment. Math. Univ. Carolin. 30 (1989), 249252.
[4]Passty G.B., ‘Construction of fixed points for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 84 (1982), 212216.
[5]Petryshyn W.V. and Williamson T.E. Jr., ‘Strong and weak convergence of the sequence of successive approximations for quasi-nonexpansive mappings’, J. Math. Anal. Appl. 43 (1973), 459497.
[6]Samanta S.K., ‘Fixed point theorems in a Banach space satisfying Opial's condition’, J. Indian Math. Soc. 45 (1981), 251258.
[7]Schu J., ‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl. (to appear).
[8]Zeidler E., Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems (Springer-Verlag, New York, Heidelberg, Tokyo, 1986).
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Bulletin of the Australian Mathematical Society
  • ISSN: 0004-9727
  • EISSN: 1755-1633
  • URL: /core/journals/bulletin-of-the-australian-mathematical-society
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