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Weak and strong convergence to fixed points of asymptotically nonexpansive mappings

Published online by Cambridge University Press:  17 April 2009

J. Schu
Affiliation:
RWTH Aachen, Lehrstuhl C für Mathematik, Templergraben 55, D-5100 Aachen, Federal Republic of Germany
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Abstract

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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.

Type
Research Article
Copyright
Copyright © Australian Mathematical Society 1991

References

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[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).CrossRefGoogle Scholar
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