A Modified Mann Iteration by Boundary Point Method for Finding Minimum-Norm Fixed Point of Nonexpansive Mappings

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, y_n = α_nx_n + (1 − α_n)Tx_n, x_n+1 = β_nu + (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.