Hostname: page-component-5d59c44645-jb2ch Total loading time: 0 Render date: 2024-02-22T12:29:40.054Z Has data issue: false hasContentIssue false

Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

Published online by Cambridge University Press:  17 April 2009

J. Schu
RWTH Aachen, Lehrstuhl C für Mathematik, Templergraben 55, D-5100 Aachen, Federal Republic of Germany
Rights & Permissions [Opens in a new window]


Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

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.

Research Article
Copyright © Australian Mathematical Society 1991


[1]Bose, S.C., ‘Weak convergence to the fixed point of an asymptotically nonexpansive map’, Proc. Amer. Math. Soc. 68 (1978), 305308.Google Scholar
[2]Goebel, K. and Kirk, W.A., ‘A fixed point theorem for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 35 (1972), 171174.Google Scholar
[3]Górnicki, J., ‘Weak convergence theorems for asymptotically nonexpansive mappings in uniformly convex Banach spaces’, Comment. Math. Univ. Carolin. 30 (1989), 249252.Google Scholar
[4]Passty, G.B., ‘Construction of fixed points for asymptotically nonexpansive mappings’, Proc. Amer. Math. Soc. 84 (1982), 212216.Google Scholar
[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.Google Scholar
[6]Samanta, S.K., ‘Fixed point theorems in a Banach space satisfying Opial's condition’, J. Indian Math. Soc. 45 (1981), 251258.Google Scholar
[7]Schu, J., ‘Iterative construction of fixed points of asymptotically nonexpansive mappings’, J. Math. Anal. Appl. (to appear).Google Scholar
[8]Zeidler, E., Nonlinear Functional Analysis and its Applications I, Fixed-Point Theorems (Springer-Verlag, New York, Heidelberg, Tokyo, 1986).Google Scholar