Bull. Austral. Math. Soc. 72(3) pp.371--379, 2005.

A strong convergence theorem for contraction semigroups in Banach spaces

Hong-Kun Xu

Received: 19th May, 2005

Supported in part by the National Research Foundation of South Africa.


We establish a Banach space version of a theorem of Suzuki . More precisely we prove that if X is a uniformly convex Banach space with a weakly continuous duality map (for example, lp for 1 < p < $ \infty $), if C is a closed convex subset of X, and if $ \mathcal {F}$ = $ \bigl \{$T(t) : t$ \ge $0$ \bigr \}$ is a contraction semigroup on C such that Fix($ \mathcal {F}$) $ \not =$$ \emptyset $, then under certain appropriate assumptions made on the sequences {$ \alpha _{n}^{}$} and {tn } of the parameters, we show that the sequence {xn } implicitly defined by
xn = $\displaystyle \alpha _{n}^{}$u + (1 - $\displaystyle \alpha _{n}^{}$)T(tn )xn
for all n$ \ge $1 converges strongly to a member of Fix($ \mathcal {F}$).

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(Metadata: XML, RSS, BibTeX) MathSciNet: MR2199638 Z'blatt-MATH: 1095.47016


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