Monotone operators and (firmly) nonexpansive mappings are fundamental objects in modern analysis and computational optimization. It was shown in 2012 that if finitely many firmly nonexpansive mappings have or “almost have” fixed points, then the same is true for compositions and convex combinations. More recently, sharp information about the minimal displacement vector of compositions and of convex combinations of firmly nonexpansive mappings was obtained in terms of the displacement vectors of the underlying operators. Using a new proof technique based on the Brezis–Haraux theorem and reflected resolvents, we extend these results from firmly nonexpansive to general averaged nonexpansive mappings. Various examples illustrate the tightness of our results.

单调运算符和（肯定）非扩展映射是现代分析和计算优化中的基本对象。2012年的研究表明，如果有限地有很多固定的非膨胀映射具有或几乎具有固定点，那么对于合成和凸组合也是如此。最近，根据底层算子的位移矢量，获得了有关合成的最小位移矢量和牢固非膨胀映射的凸组合的最小信息。使用基于Brezis-Haraux定理和反射分解体的新证明技术，我们将这些结果从稳固的非扩张映射扩展到一般平均非扩张映射。各种示例说明了我们结果的严格性。