Tykhonov well-posedness of split problems

Abstract

AbstractIn (J. Optim. Theory Appl. 183:139–157, 2019) we introduced and studied the concept of well-posedness in the sense of Tykhonov for abstract problems formulated on metric spaces. Our aim of this current paper is to extend the results in (J. Optim. Theory Appl. 183:139–157, 2019) to a system which consists of two independent problems denoted by P and Q, coupled by a nonlinear equation. Following the terminology used in literature we refer to such a system as a split problem. We introduce the concept of well-posedness for the abstract split problem and provide its characterization in terms of metric properties for a family of approximating sets and in terms of the well-posedness for the problems P and Q, as well. Then we illustrate the applicability of our results in the study of three relevant particular cases: a split variational–hemivariational inequality, an elliptic variational inequality and a history-dependent variational inequality. We describe each split problem and clearly indicate the family of approximating sets. We provide necessary and sufficient conditions which guarantee the well-posedness of the split variational–hemivariational inequality. Moreover, under appropriate assumptions on the data, we prove the well-posedness of the split elliptic variational inequality as well as the well-posedness of the split history-dependent variational inequality. We illustrate our abstract results with various examples, part of them arising in contact mechanics.