Fixed point theorems for nonexpansive mappings on nonconvex sets in UCED Banach spaces (Q1607940)

This article deals with self-mappings \(T:C\to C\) where \(C\) is a finite union of weakly compact convex subsets in a Banach space \(X\) which is uniformly convex in every direction (the latter means that, for any \(z\), \(\| z\|=1\) and \(\varepsilon>0\), \(\text{inf}\{1-\|(x+y)/2 \|:\| x\|,\| y\|\leq 1\), \(x-y=\lambda z\), \(|\lambda |\geq \varepsilon\}>0.)\) The authors prove the following two results: (I) Each asymptotically regular or \(\lambda\)-firmly nonexpansive mapping \(T:C\to C\) has a fixed point, and (II) If \(\{T_i\}_{i\in I}\) is any compatible family of strongly nonexpansive self-mappings of such a \(C\) and the graphs of \(T_i\), \(i\in I\), have a nonempty intersection, then \(T_i \), \(i\in I\), have a common fixed point in \(C\). Recall that a nonexpansive mapping \(T:C\to C\) is called asymptotically regular if \(\lim_{n\to \infty}\| T^nx-T^{n+1} x\|=0\), strongly nonexpansive if \(\| x_n-y_n \|-\| Tx_n-Ty_n \|\to 0\), \(x_n, y_n\in C\), implies \((x_n-y_n) -(Tx_n-Ty_n)\to 0\); \(T:C\to C\) is said to be \(\lambda\)-firmly nonexpansive \((\lambda \in(0,1))\) if \(\| Tx-Ty|\leq \|(1-\lambda)(x-y)+ \lambda(Tx- Ty)\|\), \(x,y\in C\).