Normal structure and orbital fixed point conditions
The main aim of this survey paper is to stress that the ``old'' concepts of normal structure (for a Banach space \(X\)) and diminishing orbital diameter (for a selfmap \(T\)) are strongly related to a number of recent similar concepts. The following is the main result in this exposition. Theorem. Let \(X\) be a Banach space, \(D\) be a weakly compact subset of \(X\) and let the selfmap \(T:D\to D\) be such that i)\ \(T\) is nonexpansive with respect to orbits ii)\ \(\inf_{m\in N}\limsup_n ||T^m(x)-T^n(x)||< \operatorname{diam}(O(x))\), \(\forall x\in D\), \(x\neq Tx\). Then, \(T\) has a fixed point in \(D\). Further aspects occasioned by these developments are also discussed.
scientific article; zbMATH DE number 6866175
scientific article