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This paper investigates the inverse problem of estimating a discontinuous parameter in a quasi-variational inequality involving multi-valued terms. We prove that a well-defined parameter-to-solution map admits weakly compact values under some quite general assumptions. The Kakutani-Ky Fan fixed point principle for multi-valued maps is the primary technical tool for this result. Inspired by the total variation regularization for estimating discontinuous parameters, we develop an abstract regularization framework for the inverse problem and provide a new existence result. The theoretical results are applied to identify a parameter in an elliptic mixed boundary value system with the $p$-Laplace operator, an implicit obstacle, and multi-valued terms involving convex subdifferentials and the generalized subdifferentials in the sense of Clarke.

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The goal of this paper is to study a comprehensive system called differential variational–hemivariational inequality which is composed of a nonlinear evolution equation and a time-dependent variational–hemivariational inequality in Banach spaces. Under the general functional framework, a generalized existence theorem for differential variational–hemivariational inequality is established by employing KKM principle, Minty’s technique, theory of multivalued analysis, the properties of Clarke’s subgradient. Furthermore, we explore a well-posedness result for the system, including the existence, uniqueness, and stability of the solution in mild sense. Finally, using penalty methods to the inequality, we consider a penalized problem-associated differential variational–hemivariational inequality, and examine the convergence result that the solution to the original problem can be approached, as a parameter converges to zero, by the solution of the penalized problem.

Abstract The goal of this paper is to discuss the well-posedness and the generalized well-posedness of a new kind of differential quasi-variational-hemivariational inequality (DQHVI) in Hilbert spaces. Employing these concepts, we explore the essential relation between metric characterizations and the well-posedness of DQHVI. Moreover, the compactness of the set of solutions for DQHVI is delivered, when problem DQHVI is well-posed in the generalized sense.