SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 1246-1274, January 2021.
In this paper, we investigate a generalized nonlinear quasi-hemivariational inequality (QHI) involving a multivalued map in a Banach space. Under general assumptions, by using a fixed point theorem combined with the theory of nonsmooth analysis and the Minty technique, we prove that the set of solutions for the hemivariational inequality associated to the QHI problem is nonempty, bounded, closed, and convex. Then, we prove the existence of a solution to QHI. Furthermore, an optimal control problem governed by QVI is introduced, and a solvability result for the optimal control problem is established. Finally, an approximation of an elastic contact problem with the constitutive law involving a convex subdifferential inclusion is studied as an illustrative application, in which approximate contact boundary conditions are described by a multivalued version of the normal compliance contact condition with frictionless effect and a frictional contact law with the slip dependent coefficient of friction.

中文翻译：

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非线性拟半混杂不等式：存在与最优控制

SIAM控制与优化杂志，第59卷，第2期，第1246-1274页，2021年1月。
在本文中，我们研究了涉及Banach空间中多值映射的广义非线性拟半不等式（QHI）。在一般假设下，通过使用不动点定理，非光滑分析理论和Minty技术，我们证明了与QHI问题相关的半变分不等式的解集是非空的，有界的，封闭的和凸的。然后，我们证明存在QHI解决方案。此外，引入了由QVI控制的最优控制问题，并建立了该最优控制问题的可解性结果。最后，研究了弹性接触问题与包含凸次微分包含的本构律的近似，作为一个说明性应用，