Mann type implicit iteration approximation for multivalued mappings in Banach spaces (Q1958013)

Let \(K\) be a nonempty compact convex subset of a uniformly convex Banach space \(E\) and let \(T\) be a multivalued nonexpansive mapping. For the implicit iterates \(x_0\in K\), \(x_n=\alpha_n x_{n-1}+(1-\alpha_n)y_n\), \(y_n\in Tx_n\), \(n\geq 1\), we prove that \(\{x_n\}\) converges strongly to a fixed point of \(T\) under some suitable conditions. Our results extend corresponding ones and revise a gap in the work of \textit{B.\,Panyanak} [Comput.\ Math.\ Appl.\ 54, No.\,6, 872--877 (2007; Zbl 1130.47050)].