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

Here, we consider a stationary inclusion in a real Hilbert space <i>X</i>, governed by a set of constraints <i>K</i>, a nonlinear operator <i>A</i>, and an element <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>X</mi></mrow></semantics></math></inline-formula>. Under appropriate assumptions on the data, the inclusion has a unique solution, denoted by <i>u</i>. We state and prove a covergence criterion, i.e., we provide necessary and sufficient conditions on a sequence <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mo>{</mo><msub><mi>u</mi><mi>n</mi></msub><mo>}</mo><mo>⊂</mo><mi>X</mi></mrow></semantics></math></inline-formula>, which guarantee its convergence to the solution <i>u</i>. We then present several applications that provide the continuous dependence of the solution with respect to the data <i>K</i>, <i>A</i> and <i>f</i> on the one hand, and the convergence of an associate penalty problem on the other hand. We use these abstract results in the study of a frictional contact problem with elastic materials that, in a weak formulation, leads to a stationary inclusion for the deformation field. Finally, we apply the abstract penalty method in the analysis of two nonlinear elastic constitutive laws.