Remarks on the structure of the fixed-point sets of uniformly Lipschitzian mappings in uniformly convex Banach spaces
Let \(C\) be a nonempty, weakly compact, convex subset of a Banach space \(X\) and let \(T:C\to C\) be nonexpansive. If \(T\) has a fixed point in every nonempty, closed, bounded, convex set that it leaves invariant, \textit{R.\ E.\thinspace Bruck} [Trans.\ Am.\ Math.\ Soc.\ 179, 251--262 (1973; Zbl 0265.47043)] proved that the set \(F(T)\) of fixed points of \(T\) is a nonexpansive retract of \(C\). In the article under review, the author is able to relax the restriction on mapping \(T\), but at the cost of further restricting the Banach space. Recall that a map \(T:C\to C\) is uniformly \(k\)-Lipschitzian if \(\| T^nx -T^ny\| \leq k \| x-y \|\) for every \(x, y \in C\) and \(n\in \mathbb{N}\). The author proves that the fixed point set of a uniformly \(k\)-Lipschitzian self-map \(T\) of a nonempty, closed, bounded, convex set \(C\) in a uniform convex space is a nonexpansive retract of \(C\) if \(k<\alpha\), where \(\alpha\) is the unique solution of the equation \(\alpha^2\cdot \delta^{-1}_X (1 - 1/\alpha) \cdot \frac{1}{N(X)} = 1\) involving the modulus of convexity \(\delta_X\) and the normal structure coefficient \(N(X)\) of \(X\). In the setting that the uniformly convex space has a modulus of convexity of power type \(p\), the author derives another relationship involving \(p\) and \(N(X)\) that ensures that the set of fixed points of \(T\) is a nonexpansive retract of \(C\).
scientific article; zbMATH DE number 5555253
scientific article