Boundary point algorithms for minimum norm fixed points of nonexpansive mappings
Songnian He, Yang Cai-ping · Fixed Point Theory and Applications · 2014
Let H be a real Hilbert space and C be a closed convex subset of H. Let be a nonexpansive mapping with a nonempty set of fixed points . If , then Halpern’s iteration process cannot be used for finding a minimum norm fixed point of T since may not belong to C. To overcome this weakness, Wang and Xu introduced the iteration process for finding the minimum norm fixed point of T, where the sequence , arbitrarily and 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 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 , they are easy to implement in actual computing programs. MSC:47H09, 47H10, 65K10.