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 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=βnxn+(1−βn)Sn(αnu+(1−αn)Tnxn) for finding the common fixed point of Sn and Tn, where uinC is an arbitrarily (but fixed) element in C, x0∈C arbitrarily, αn and βn are sequences in [0,1]. But in the case where u∉C, 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 0∈C or 0∉C.