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Asymptotic behaviour for an almost-orbit of nonexpansive semigroups in Banach spaces

Published online by Cambridge University Press:  17 April 2009

Jong Kyu Kim  and
Gang Li
Jong Kyu Kim
Affiliation:
Department of Mathematics, Kyungnam University, Masan, Kyungnam 631-701, Korea, e-mail: [email protected]
Gang Li
Affiliation:
Department of Mathematics, Yangzhou University, Yangzhou, 225002, Peoples Republic of China e-mail: [email protected]
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Abstract

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In this paper, by using the technique of product nets, we are able to prove a weak convergence theorem for an almost-orbit of right reversible semigroups of nonexpansine mappings in a general Banach space X with Opial's condition. This includes many well known results as special cases. Let C be a weakly compact subset of a Banach space X with Opial's condition. Let G be a right reversible semitopological semigroup,  = {T (t): t ∈ G} a nonexpansive semigroup on C, and u (·) an almost-orbit of . Then {u (t): t ∈ G} is weakly convergent (to a common fixed point of ) if and only if it is weakly asymptotically regular (that is, {u (ht) − u (t)} converges to 0 weakly for every h ∈ G).

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 61 , Issue 2 , April 2000 , pp. 345 - 350
Copyright
Copyright © Australian Mathematical Society 2000

References

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