Orbitally nonexpansive mappings (Q2814349)

Let \(X\) be a Banach space and let \(C\) be a nonempty, closed, convex, and bounded subset of \(X\). A~mapping \(T: C\to X\) is called nonexpansive if for every \(x,y\in X\) we have \(\|Tx-Ty\|\leq \|x-y\|\).NEWLINENEWLINEIn the paper under review, the author introduces a generalisation of the class of nonexpansive mappings; namely, the \textit{orbitally nonexpansive mappings}. According to Definition 3.1 of the paper, a mapping \(T:C\to C\) is said to be orbitally nonexpansive if, for every nonempty, closed, convex, and \(T\)-invariant subset \(D\) of \(C\), there exists some \(x_0\in D\) such that, for every \(x\in D\), we have NEWLINE\[NEWLINE\limsup_{n\to\infty}\|T^nx_0-Tx\|\leq \limsup_{n\to\infty}\|T^nx_0-x\|.NEWLINE\]NEWLINE It is proved that every nonexpansive mapping as well as every asymptotically regular \(L\)-type mapping is orbitally nonexpansive. Moreover, as the main result of this paper, it is proved that if \(K\) is a nonempty weakly compact convex subset of a Banach space with normal structure, then every orbitally nonexpansive mapping \(T:K\to K\) has a fixed point.