We show the nonlinear ergodic theorem for monotone 1-Lipschitz mappings in uniformly convex spaces: if C is a bounded closed convex subset of an ordered uniformly convex space ( X , ∣·∣, ⪯), T : C → C a monotone 1-Lipschitz mapping and x ⪯ T ( x ), then the sequence of averages 1n∑i=0n−1Ti(x) $ \frac{1}{n}\sum\nolimits_{i=0}^{n-1}T^{i}(x) $ converges weakly to a fixed point of T . As a consequence, it is shown that the sequence of Picard’s iteration { T n ( x )} also converges weakly to a fixed point of T . The results are new even in a Hilbert space. The Krasnosel’skiĭ-Mann and the Halpern iteration schemes are studied as well.

中文翻译：

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一致凸空间中单调非膨胀映射的迭代方法

我们展示了均匀凸空间中单调1-Lipschitz映射的非线性遍历定理：如果C是有序均匀凸空间（X，∣·∣，⪯）的有界封闭凸凸子集，则T：C→C是单调1- Lipschitz映射和x⪯T（x），则平均值序列1n∑i = 0n-1Ti（x）$ \ frac {1} {n} \ sum \ nolimits_ {i = 0} ^ {n-1} T ^ {i}（x）$微弱地收敛到T的固定点。结果，表明了皮卡德迭代{T n（x）}的序列也微弱地收敛到T的固定点。即使在希尔伯特空间中，结果也是新的。还研究了Krasnosel'skiĭ-Mann和Halpern迭代方案。