On a class of generalized saddle-point problems arising from contact mechanics

Abstract

Abstract In the present paper we consider a class of generalized saddle-point problems described by means of the following variational system: a ( u , v − u ) + b ( v − u , λ ) + j ( v ) − j ( u ) + J ( u , v ) − J ( u , u ) ≥ ( f , v − u ) X , b ( u , μ − λ ) − ψ ( μ ) + ψ ( λ ) ≤ 0 , $$\begin{aligned} &a(u,v-u)+b(v-u,\lambda )+j(v)-j(u)+J(u,v)-J(u,u)\geq (f,v-u)_{X}, \\ &b(u,\mu -\lambda )-\psi (\mu )+\psi (\lambda )\leq 0, \end{aligned}$$ ( v ∈ K ⊆ X $v\in K\subseteq X$ , μ ∈ Λ ⊂ Y $\mu \in \Lambda \subset Y$ ), where ( X , ( ⋅ , ⋅ ) X ) $(X,(\cdot,\cdot )_{X})$ and ( Y , ( ⋅ , ⋅ ) Y ) $(Y,(\cdot,\cdot )_{Y})$ are Hilbert spaces. We use a fixed-point argument and a saddle-point technique in order to prove the existence of at least one solution. Then, we obtain uniqueness and stability results. Subsequently, we pay special attention to the case when our problem can be seen as a perturbed problem by setting ψ ( ⋅ ) = ϵ ψ ¯ ( ⋅ ) $\psi (\cdot )=\epsilon \bar{\psi}(\cdot )$ ( ϵ > 0 ) $(\epsilon >0)$ . Then, we deliver a convergence result for ϵ → 0 $\epsilon \to 0$ , the case ψ ≡ 0 $\psi \equiv 0$ appearing like a limit case. The theory is illustrated by means of examples arising from contact mechanics, focusing on models with multicontact zones.

Keywords

• Fixed point
• Saddle point
• Perturbed problem
• Convergence result
• Multicontact zones