iterative scheme based on boundary point method for common fixed‎ ‎point of strongly nonexpansive sequences

let $c$ be a nonempty closed convex subset of a real hilbert space $h$. let ${s_n}$ and ${t_n}$ be sequences of nonexpansive self-mappings of $c$, where one of them is a strongly nonexpansive sequence. k. aoyama and y. kimura introduced the iteration process $x_{n+1}=beta_nx_n+(1-beta_n)s_n(alpha_nu+(1-alpha_n)t_nx_n)$ for finding the common fixed point of ${s_n}$ and ${t_n}$, where $uin c$ is an arbitrarily (but fixed) element in $c$, $x_0in c$arbitrarily, ${alpha_n}$ and ${beta_n}$ are sequences in $[0,1]$. but in the case where $unotin c$, the iterative scheme above becomes invalid because $x_n$ may not belong to $c$. to overcome this weakness, a new iterative scheme based on the thought of boundary point method is proposed and the strong convergence theorem is proved. as a special case, we can find the minimum-norm common fixed point of ${s_n}$ and ${t_n}$ whether $0in c$ or $0notin c$.