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.

Abstract

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, l^p 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 {t_n } of the parameters, we show that the sequence {x_n } implicitly defined by

x_n = $\displaystyle \alpha _{n}^{}$ u + (1 - $\displaystyle \alpha _{n}^{}$ )T(t_n )x_n

for all n $\ge$ 1 converges strongly to a member of Fix( $\mathcal {F}$ ).

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MathSciNet: MR2199638

Z'blatt-MATH: 1095.47016

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ISSN 0004-9727

Bulletin of the Australian Mathematical Society