Existence and relaxation results for second order multivalued systems (Q2043330)

The authors study a second order multivalued boundary value problem of the form \[ \begin{cases} a(u'(t))'\in A(u(t))+F(t,u(t),u'(t))& \text{for a.\,a.\,}t\in T=[0,b]\\ u\in \text{BC},& \end{cases} \] where BC stands for Dirichlet, Neumann, periodic or mixed boundary conditions. The nonlinear system above is driven by a general nonhomogeneous differential operator with a reaction in which one has the combined effects of a maximal monotone term \(A(x)\) and of a multivalued perturbation \(F(t,x,y)\) which can be convex or nonconvex valued. The authors consider the cases where \(D(A)\neq \mathbb{R}^N\) and \(D(A)=\mathbb{R}^N\) and prove existence and relaxation theorems. Applications to differential variational inequalities and control systems are also discussed.