Let C be a closed, convex and nonempty subset of a Banach space X. Let \({T : C \rightarrow X}\) be a nonexpansive inward mapping. We consider the boundary point map \({h_{C,T } : C \rightarrow \mathbb{R}}\) depending on C and T defined by \({h_{C,T} = {\rm max}\{\lambda \in [0,1] : [(1-\lambda)x + \lambda Tx] \in C\}}\), for all \({x \in C}\). Then for a suitable step-by-step construction of the control coefficients by using the function \({h_{C,T }}\), we show the convergence of the Mann-Dotson algorithm to a fixed point of T. We obtain strong convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} < \infty}\) and weak convergence if \({\sum\limits_{n \in \mathbb{N}} \alpha_{n} = \infty}\).

中文翻译：

---

Banach空间中非自映射的边界点方法和Mann-Dotson算法

令C为Banach空间X的封闭，凸且非空子集。令\（{T：C \ rightarrow X} \）为非扩张向内映射。我们认为，边界点映像\（{H_ {C，T}：C \ RIGHTARROW \ mathbb {R}} \）取决于Ç和Ť通过定义\（{H_ {C，T} = {\ RM MAX} \ {\ lambda \ in [0,1]：[（1- \ lambda）x + \ lambda Tx] \ in C \}} \））对于所有\（{x \ in C} \）。然后，通过使用函数\（{h_ {C，T}} \）逐步构建合适的控制系数，我们展示了Mann-Dotson算法对T的固定点的收敛性。如果我们获得强收敛\（{\ sum \ limits_ {n \ in \ mathbb {N}} \ alpha_ {n} <\ infty} \）和弱收敛性，如果\（{\\ sum \ limits_ {n \ in \ mathbb {N}} \ alpha_ {n} = \ infty} \）。