Summary: It is proved that a recent theorem of

*Liu Qihou* [Nonlinear Anal., Theory Methods Appl. 26, No. 11, 1835-1842 (1996;

Zbl 0861.47047)] concerning the iterative approximation of fixed points of demicontractive mappings in Hilbert spaces can be extended to the much more general

$q$-uniformly smooth Banach spaces,

$1<q<\infty $, and to a more general iteration method.