Suppose that Q is a $$\hbox {weak}^{*}$$ weak ∗ compact convex subset of a dual Banach space with the Radon–Nikodým property. We show that if ( S , Q ) is a nonexpansive and norm-distal dynamical system, then there is a fixed point of S in Q and the set of fixed points is a nonexpansive retract of Q . As a consequence we obtain a nonlinear extension of the Bader–Gelander–Monod theorem concerning isometries in L -embedded Banach spaces. A similar statement is proved for weakly compact convex subsets of a locally convex space, thus giving the nonlinear counterpart of the Ryll-Nardzewski theorem.

中文翻译：

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围绕非线性 Ryll-Nardzewski 定理

假设 Q 是具有 Radon–Nikodým 性质的对偶 Banach 空间的 $$\hbox {weak}^{*}$$weak ∗ 紧致凸子集。我们证明，如果 ( S , Q ) 是一个非膨胀的范数远端动力系统，那么 Q 中有一个 S 的不动点，不动点的集合是 Q 的非膨胀收缩。因此，我们获得了关于 L 嵌入 Banach 空间中的等距的 Bader-Gelander-Monod 定理的非线性扩展。对于局部凸空间的弱紧凸子集，证明了类似的陈述，从而给出了 Ryll-Nardzewski 定理的非线性对应物。