A Generalization of Ekeland's ϵ-Variational Principle and Its Borwein–Preiss Smooth Variant
A Generalization of Ekeland's ϵ-Variational Principle and Its Borwein–Preiss Smooth Variant
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DOI：
10.1006/jmaa.2000.6813
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发表时间：
2000-06
影响因子：
1.3
通讯作者：
Liao Yongxin;Shi Shu-zhong
中科院分区：
文献类型：
--
作者：
Liao Yongxin;Shi Shu-zhong
本文将Ekeland的广义变分原理及其Borwein-Preiss光滑变分原理推广到一个“规范型”的下半连续函数中，用它代替距离和范数。作为这一推广的应用，我们证明了：如果在Banach空间X上存在Lipschitz β-光滑“凸函数”，则X的开子集U上的每个连续凸函数在U中是稠密β-可微的。这推广了关于凸函数可微性的Borwein-Preiss定理。
Abstract We give a generalization of Ekeland's ϵ-Variational Principle and of its Borwein–Preiss smooth variant, replacing the distance and the norm by a “gauge-type” lower semi-continuous function. As an application of this generalization, we show that if on a Banach space X there exists a Lipschitz β-smooth “bump function,” then every continuous convex function on an open subset U of X is densely β-differentiable in U . This generalizes the Borwein–Preiss theorem on the differentiability of convex functions.