Boundary point algorithms for minimum norm fixed points of nonexpansive mappings

Abstract

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 x n+1 = (1 -t n )Tx n cannot be used for finding a minimum norm fixed point of T since x n may not belong to C. To overcome this weakness, Wang and Xu introduced the iteration process x n+1 = P C (1 -t n )Tx n for finding the minimum norm fixed point of T, where the sequence {t n } (0, 1), x 0 C arbitrarily and P C 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 P C 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 P C , they are easy to implement in actual computing programs.