A Convergence Criterion for a Class of Stationary Inclusions in Hilbert Spaces

Abstract

Here, we consider a stationary inclusion in a real Hilbert space X, governed by a set ofconstraints K, a nonlinear operator A, and an element f ∈ X. Under appropriate assumptions on thedata, the inclusion has a unique solution, denoted by u. We state and prove a covergence criterion,i.e., we provide necessary and sufficient conditions on a sequence {un} ⊂ X, which guarantee itsconvergence to the solution u. We then present several applications that provide the continuousdependence of the solution with respect to the data K, A and f on the one hand, and the convergenceof an associate penalty problem on the other hand. We use these abstract results in the study of africtional contact problem with elastic materials that, in a weak formulation, leads to a stationaryinclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis oftwo nonlinear elastic constitutive laws.