A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings - Songnian He; Wenlong Zhu (Abstract and Applied Analysis)
Let H be a real Hilbert space and C⊂H a closed convex subset. Let T:C→C be a nonexpansive mapping with the nonempty set of fixed points Fix(T). Kim and Xu (2005) introduced a modified Mann iteration x0=x∈C, yn=αnxn+(1−αn)Txn, xn+1=βnu+(1−βn)yn, where u∈C is an arbitrary (but fixed) element, and {αn} and {βn} are two sequences in (0,1). In the case where 0∈C, the minimum-norm fixed point of T can be obtained by taking u=0. But in the case where 0∉C, this iteration process becomes invalid because xn may not belong to C. In order to overcome this weakness, we introduce a new modified Mann iteration by boundary point method (see Section 3 for details) for finding the minimum norm fixed point of  T and prove its strong convergence under some assumptions. Since our algorithm does not involve the computation of the metric projection PC, which is often used so that the strong convergence is guaranteed, it is easy implementable. Our results improve and extend the results of Kim, Xu, and some others.