Boundary point algorithms for minimum norm fixed points of nonexpansive mappings

Let H be a real Hilbert space and C be a closed convex subset of H. Let T:C→C be a nonexpansive mapping with a nonempty set of fixed points Fix(T). If 0∉C, then Halpern’s iteration process xn+1=(1−tn)Txn cannot be used for finding a minimum norm fixed point of T since xn may not belong to C. To overcome this weakness, Wang and Xu introduced the iteration process xn+1=PC(1−tn)Txn for finding the minimum norm fixed point of T, where the sequence {tn}⊂(0,1), x0∈C arbitrarily and PC is the metric projection from H onto C. However, it is difficult to implement this iteration process in actual computing programs because the specific expression of PC cannot be obtained, in general. In this paper, three new algorithms (called boundary point algorithms due to using certain boundary points of C at each iterative step) for finding the minimum norm fixed point of T are proposed and strong convergence theorems are proved under some assumptions. Since the algorithms in this paper do not involve PC, they are easy to implement in actual computing programs.