A non-convex setup for multivalued differential equations driven by oblique subgradients

Abstract

The paper proves the existence and uniqueness of the solution for the following multivalued deterministic variational inequality with oblique subgradients, considered in a non-convex domain: {x′(t)+H(t,x(t))∂−φ(x(t))∋g(t,x(t)),t≥0,x(0)=x0, where ∂−φ stands for the Fréchet subdifferential of the semiconvex function φ and the matrix application x↦H(⋅,x) is a Lipschitz mapping. The presence of the oblique reflection brought by the term H∂−φ leads to the use of different techniques comparing to the cases of standard reflection in non-convex domains or oblique reflection in convex domains. The last section of the article is focused on the qualitative analysis of a non-convex Skorohod problem, with generalized reflection and, as applications, stochastic variational inequalities driven by oblique Fréchet subgradients are addressed.