Let H be a real Hilbert space. Let T : $H;{ ightarrow};H$ be an averaged mapping with $F(T);{ eq};{emptyset}$. Let {$alpha_n$} be a real numbers in (0, 1). For given $x_0;{in};H$, let the sequence {$x_n$} be generated iteratively by $x_{n+1};=;(1;-;{alpha}_n)Tx_n$, $n;{geq};0$. Assume that the following control conditions hold: (i) $lim_{n{ ightarrow}{infty}};{alpha}_n;=;0$; (ii) $sum^{infty}_{n=0};{alpha}_n;=;{infty}$. Then {$x_n$} converges strongly to a fixed point of T.