Let C be a nonempty closed convex subset of a real Hilbert space H. Let {Sn} and {Tn} 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 xn+1=bnxn+(1-bn) Sn (anu+(1-an) Tnxn) for finding the common fixed point of {Sn} and {Tn}, where uÎC is an arbitrarily (but fixed) element in C, x0ÎC arbitrarily, {an} and {bn} are sequences in [0; 1]. But in the case where uÏC, the iterative scheme above becomes invalid because xn 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 {Sn} and {Tn} whether 0ÎC or 0ÏC.