The super fixed point property for asymptotically nonexpansive mappings (Q2891277)

Motivated by the work of \textit{W. A. Kirk}, \textit{C. M. Yáñez} and \textit{S. S. Shin} [Nonlinear Anal., Theory Methods Appl. 33, No. 1, 1--12 (1998: Zbl 0940.47044)] and applying the Banach space ultrapower construction, the author shows that the super fixed point property for nonexpansive mappings is equivalent to the super fixed point property for asymptotically nonexpansive mappings in the intermediate sense. Moreover, some fixed point theorems for asymptotically nonexpansive mappings in the intermediate sense are obtained in the case where the underlying Banach space is uniformly nonsquare or uniformly noncreasy. Furthermore, the existence of a common fixed point for commuting families of asymptotically nonexpansive mappings in the intermediate sense is established.NEWLINENEWLINERecall that a Banach space \(X\) has the super fixed point property for a class \(\mathcal A\) of mappings if every Banach space which is finitely representable in \(X\) has the fixed point property for the class \(\mathcal A\), and a continuous mapping \(T:C\to C\) is asymptotically nonexpansive in the intermediate sense if NEWLINE\[NEWLINE\limsup_{n\to\infty} \sup_{x,y\in C}\;(\|T^nx-T^ny\|-\|x-y\|) \leq 0,NEWLINE\]NEWLINE where \(C\) is a nonempty, bounded, closed, and convex subset of \(X\).