From the Fan-KKM principle to extended real-valued equilibria and to variational-hemivariational inequalities with application to nonmonotone contact problems

Abstract

Abstract This paper starts off by the celebrated Knaster–Kuratowski–Mazurkiewicz principle in the formulation by Ky Fan. We provide a novel variant of this principle and build an existence theory for extended real-valued equilibrium problems with general, then monotone and pseudomonotone bifunctions. We develop our existence theory first in general topological vector spaces, then in reflexive Banach spaces, where we investigate the issue of coerciveness for existence on unbounded sets. Thereafter we use the Clarke generalized differential calculus for locally Lipschitz functions and derive existence results for nonlinear variational-hemivariational inequalities and hemivariational quasivariational inequalities. As application, we treat a unilateral contact problem in solid mechanics with nonmonotone friction.

Keywords

• Extended real-valued equilibria
• Pseudomonotone bifunction
• Hemivariational inequality
• Unilateral contact problem
• Non-monotone friction