Evolutionary Quasi-Variational-Hemivariational Inequalities I: Existence and Optimal Control

Shengda Zeng (), Dumitru Motreanu () and Akhtar A. Khan ()
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Shengda Zeng: Yulin Normal University
Dumitru Motreanu: Yulin Normal University
Akhtar A. Khan: Rochester Institute of Technology

Journal of Optimization Theory and Applications, 2022, vol. 193, issue 1, No 39, 950-970

Abstract: 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.

Keywords: Evolutionary quasi–variational–hemivariational inequality; surjectivity theorems; Kuratowski upper limit; optimal control.; 47J20; 58Exx; 35Kxx; 34H05; 49J52; 74B20 (search for similar items in EconPapers)
Date: 2022
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DOI: 10.1007/s10957-021-01963-3

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