On a new class of abstract mixed variational–hemivariational problems

A new mixed variational–hemivariational problem is introduced, a(u,v−u)+b(v−u,λ)+φ(v)−φ(u)+J0(Tu;Tv−Tu)≥(f,v−u)X, b(u,μ−λ)−ψ(μ)+ψ(λ)−Ψ0(Rλ;Rμ−Rλ)≤0, (v∈X, μ∈Λ⊆Y), focusing on its well-posedness. We are looking for a pair solution (u,λ)∈X×Λ working into Hilbert spaces framework. Some particular cases with applications in mechanics highlighted. special attention payed to case leading penalty term. In order illustrate the theory, we provide examples such that X,Y,Λ,a,b,φ,ψ,J,Ψ,R,T fulfill hypotheses.