Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control
研究了一类含集值伪单调映射的非线性演化拟变分-半变分不等式，通过引入参数变分问题并利用Kluge集值不动点定理证明了解的存在性，进而探讨了抽象最优控制问题并证明了解的存在性与解集的弱序列紧性。
Abstract We study a nonlinear evolutionary quasi–variational–hemivariational inequality (in short, (QVHVI)) involving a set-valued pseudo-monotone map. The central idea of our approach consists of introducing a parametric variational problem that defines a variational selection associated with (QVHVI). We prove the solvability of the parametric variational problem by employing a surjectivity theorem for the sum of operators, combined with Minty’s formulation and techniques from the nonsmooth analysis. Then, an existence theorem for (QVHVI) is established by using Kluge’s fixed point theorem for set-valued operators. As an application, an abstract optimal control problem for the (QVHVI) is investigated. We prove the existence of solutions for the optimal control problem and the weak sequential compactness of the solution set via the Weierstrass minimization theorem and the Kuratowski-type continuity properties.