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

Abstract

Let $H$ be a real Hilbert space and $C\subset \mathrm{H\hspace{0.17em}}$a closed convex subset. Let $T:C\to C$ be a nonexpansive mapping with the nonempty set of fixed points $\text{F}\text{i}\text{x}(T)$. Kim and Xu (2005) introduced a modified Mann iteration $x_0=x\in C$, $y_n={\alpha }_n{x}_n+(1-{\alpha }_n)T{x}_n$, $x_{n+1}={\beta }_{n}u+(1-{\beta }_n)y_n$, where $u\in C$ is an arbitrary (but fixed) element, and $\{{\alpha }_n\}$ and $\{{\beta }_n\}$ are two sequences in $(0,1)$. In the case where $0\in C$, the minimum-norm fixed point of $T$ can be obtained by taking $u=0$. But in the case where $0\notin C$, 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 $\mathrm{\hspace{0.17em}T}$ 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.