Existence results of partial differential mixed variational inequalities without Lipschitz continuity

The aim of this paper is to use order-theoretic approaches to investigate a class of partial differential mixed variational inequalities (PDMVIs, for short), which are receiving much attention recently. To this end, the order-preservation properties of proximity mapping are explored. Applying these order-preservation properties and combining with the Knaster-Tarski fixed point theorem, we prove the existence of solutions and the order-preservation properties of solution mapping for mixed variational inequalities on Hilbert lattices. Relying on the above obtained results as well as some order-theoretic techniques, the existence of mild solutions to PDMVI is established. In contrast to many existing results on PDMVIs, the order-theoretic approaches are new, and the results obtained in this paper do not require the Lipschitz continuity of the mapping associated to nonlinear evolution equation. (C) 2019 Elsevier Inc. All rights reserved.