标题： 方向导数在一致凸和一致光滑的巴拿赫空间中的度量投影 显示英文标题

标题： Directional Differentiability of the Metric Projection in Uniformly Convex and Uniformly Smooth Banach Spaces

摘要： 设X为一个实的一致凸且一致光滑的巴拿赫空间，C为X中的非空闭凸子集。 设Pc从X到C表示(标准)度量投影算子。 在本文中，我们定义了Pc的Gateaux方向导数。 我们研究了Pc的Gateaux方向导数的一些性质。 特别是，如果C是一个闭球或一个闭的凸锥（包括适当的闭子空间），那么我们给出了Pc的方向导数的精确表达式。 最后，我们定义了p-q一致凸且一致光滑的巴拿赫空间的概念。 我们将证明，如果X是一个p-q一致凸且一致光滑的巴拿赫空间，那么对于X中的任何非空闭凸子集C，Pc在全空间X上是方向可微的。 本文的结果可以立即应用于希尔伯特空间。 显示英文摘要

摘要： Let X be a real uniformly convex and uniformly smooth Banach space and C a nonempty closed and convex subset of X. Let Pc from X to C denote the (standard) metric projection operator. In this paper, we define the Gateaux directional differentiability of Pc. We investigate some properties of the Gateaux directional differentiability of Pc. In particular, if C is a closed ball or a closed and convex cone (including proper closed subspaces), then, we give the exact representations of the directional derivatives of Pc. Finally, we define the concept of p-q uniformly convex and uniformly smooth Banach spaces. We will prove that if X is a p-q uniformly convex and uniformly smooth Banach space, then for any nonempty closed and convex subset C of X, Pc is directionally differentiable on the whole space X. The results in this paper can be immediately applied to Hilbert spaces.