Nonlinear evolutionary systems driven by quasi‐hemivariational inequalities

Abstract

This paper is devoted to the study of the differential systems in arbitrary Banach spaces that are obtained by mixing nonlinear evolutionary equations and generalized quasi‐hemivariational inequalities (EEQHVI). We start by showing that the solution set of the quasi‐hemivariational inequality associated to problem EEQHVI is nonempty, closed, and convex. Furthermore, we establish upper semicontinuity and measurability properties for this solution set. Then, based on them, we prove the existence of solutions for problem EEQHVI and the compactness of the set of corresponding trajectories of EEQHVI. These statements extend previous results in several directions, for instance, by dropping the boundedness requirement for the set of constraints and substantially relaxing monotonicity hypotheses.

Convergence of a generalized penalty method for variational–hemivariational inequalities

“…Moreover, a standard proof shows that there exist r 0 > 0 and a solution u * to the problem (3.4) with r = r 0 such that u * V < r 0 (see the proof of [ 17 ,Theorem 3.4]). Let w ∈ K be arbitrary and t ∈ (0, 1) be small enough.…”

A class of fractional differential hemivariational inequalities with application to contact problem

“…In fact, this contact condition without the bonding field has been treated in many papers, see, e.g., [15,34,35,44,45]. The initial displacement is given in (30). For more details on the mathematical theory of contact mechanics, we refer to [33,37,44,45].…”

“…We now focus on the variational formulation of the contact problem (22)- (30). We suppose in what follows that (u, σ) are smooth functions on Q which solve (22)- (30).…”

“…We suppose in what follows that (u, σ) are smooth functions on Q which solve (22)- (30). For any v ∈ V fixed, we multiply equilibrium equation (23) …”

“…t ∈ (0, T ). Finally, using conditions (28)- (30) and the last inequality, we obtain the following variational formulation of Problem 17. + c τ ).…”