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Boundary point method and the Mann-Dotson algorithm for non-self mappings in Banach spaces. (English) Zbl 1485.47102

Summary: 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 \(h_{C,T} = \max\{\lambda \in [0,1] : [(1-\lambda)x + \lambda Tx] \in C\}\), 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 \(\sum_{n \in \mathbb{N}} \alpha_{n} < \infty\) and weak convergence if \(\sum_{n \in \mathbb{N}} \alpha_{n} = \infty\).

MSC:

47J25 Iterative procedures involving nonlinear operators
47H05 Monotone operators and generalizations
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