This paper is concerned with the differential sensitivity analysis of variational inequalities in Banach spaces whose solution operators satisfy a generalized Lipschitz condition. We prove a sufficient criterion for the directional differentiability of the solution map that turns out to be also necessary for elliptic variational inequalities in Hilbert spaces (even in the presence of asymmetric bilinear forms, nonlinear operators and nonconvex functionals). Our method of proof is fully elementary. Moreover, our technique allows us to also study those cases where the variational inequality at hand is not uniquely solvable and where directional differentiability can only be obtained w.r.t. the weak or the weak-star topology of the underlying space. As tangible examples, we consider a variational inequality arising in elastoplasticity, the projection onto prox-regular sets, and a bang–bang optimal control problem.

中文翻译：

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局部Lipschitz连续解算子对变分不等式的差分敏感性分析

本文涉及Banach空间中变分不等式的微分灵敏度分析，其解决算符满足广义Lipschitz条件。我们证明了解图的方向可微性的充分标准，对于希尔伯特空间中的椭圆变分不等式（即使存在非对称双线性形式，非线性算子和非凸泛函），也证明是必要的。我们的证明方法是完全基础的。此外，我们的技术还允许我们研究那些手头的变化不等式不能唯一解决且只能通过下层空间的弱星型或弱星型拓扑获得方向微分的情况。作为具体的例子，我们考虑了由弹塑性引起的变分不等式，