Let \(X=L_ p\) or \(X=\ell_ p\) (\(p \geq 2\)), \(K\subset X\) a nonempty closed bounded subset, and \(T: K\to K\) a strictly pseudocontractive Lipschitz map. Given any real sequence \((d_ n)\) in \((0,1)\) such that \(d_ n\to 0\) but \(\sum^{\infty}_{n=1} d_ n = \infty\), the author proves that the iterations \(x_{n+1}=(1-d_ n) x_ n + d_ n T x_ n\) converge to a (unique) fixed point of \(T\) in \(K\).