Hemivariational Inequalities on Graphs

TL;DR

This paper introduces a new class of hemivariational inequalities on graphs involving nonmonotone nonlinearities, establishing existence and uniqueness of solutions for elliptic and parabolic types.

Contribution

It extends hemivariational inequality theory to graph settings with nonmonotone nonlinearities, providing foundational results on subdifferentiability and solution existence.

Findings

Proved subdifferentiability of nonconvex functionals on graphs

Established existence and uniqueness of solutions for elliptic inequalities

Demonstrated existence of solutions for parabolic inequalities

Abstract

In this paper, a new class of hemivariational inequalities is introduced. It concerns Laplace operator on locally finite graphs together with multivalued nonmonotone nonlinearities expressed in terms of Clarke's subdifferential. First of all, we state and prove some results on the subdifferentiability of nonconvex functionals defined on graphs. Thereafter, an elliptic hemivariational inequality on locally finite graphs is considered and the existence and uniqueness of its weak solutions are proved by means of the well-known surjectivity result for pseudomonotone mappings. In the end of this paper, we tackle the problem of hemivariational inequalities of parabolic type on locally finite graphs and we prove the existence of its weak solutions.