A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings. (27th March 2013) - He, Songnian; Zhu, Wenlong
Abstract : 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 F i x ( T ) . Kim and Xu (2005) introduced a modified Mann iteration x 0 = x ∈ C, y n = α n x n + ( 1 − α n ) T x n, x n + 1 = β n u + ( 1 − β n ) y n, where u ∈ C is an arbitrary (but fixed) element, and { α n } and { β n } 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 x n 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 P C, 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.