Convergence to attractors of nonexpansive set-valued mappings (Q2291841)

In this paper, the authors first give an alternative proof of the main result from the paper [\textit{E. Pustylnik} et al., Fixed Point Theory 13, No. 1, 165--172 (2012; Zbl 1329.54052)]. The main result of the paper is the following. Theorem. Let \((X,d)\) be a metric space, \(F\) a nonempty subset of it and \(T:X\multimap X\) be a set-valued nonexpansive mapping. Assume that \begin{itemize} \item[(a)] \(z\in T(z)\) for every \(z\in F\), \item[(b)] for each \(x\in X\) there exists a sequence \((x_i)\) such that \(x_0=x, x_{i+1}\in T(x_i)\), for \(i\in \mathbb{N}\) and \(\displaystyle\liminf_{i\to \infty}\rho(x_i,F)=0\). \end{itemize} Then, for each \(\delta>0\) and each \(x\in X\), there exist \(z\in F\) and a~sequence \((x_i)\) in \(X\) such that \(x_0=x, x_{i+1}\in T(x_i)\), for \(i\in \mathbb{N}\) and \(\rho(x_i,z)<\delta\) for all sufficiently large integers \(i\ge 0\).