Relation between fixed point and asymptotical center of nonexpansive maps (Q623545)

Let \(C\) be a nonempty bounded closed convex subset of a Banach space \(X\). A sequence \(\{x_n\}\subset X\) is said to be an asymptotic center for a mapping \(TC\to X\) if, for each \(x\in C\), \[ \limsup_n\,\| Tx- x_n\|\leq \limsup_n\,\| x- x_n\|. \] We say that \(C\neq\varnothing\) has the fixed-point property for continuous mappings of \(C\) with asymptotic center if every continuous mapping \(T: C\to C\) admitting an asymptotic center has a fixed point. We say that \(C\) has Property (Z) if, for every bounded sequence \(\{x_n\}\subset X\setminus C\), the set \[ Z_a(C,\{x_n\})= \Biggl\{z\in C: \limsup_n\,\| x_n- z\|= \underset{x\in C}{}{\text{inf}}\,\Biggl\{\limsup_n\,\| x_n- x\|\Biggr\}\Biggr\} \] is a nonempty and compact subset of \(C\). The authors prove the following: Corollary 2.3. Let \(X\) be a Banach space and let \(C\) be a nonempty closed bounded and convex subset of \(JX\). \(C\) has the fixed point property for continuous mappings admitting an asymptotic enter iff \(C\) has Property (Z). Theorem 2.4. Let \(X\) be a Banach space and let \(C\) be a nonempty closed bounded and convex subset of \(X\) satisfying Property (Z). Let \(KC(C)\) denote the family of all nonempty compact convex subsets of \(C\). If \(T: C\to KC(C)\) is nonexpansive mapping, then \(T\) has a fixed point.