Strong nonlinear functional-differential variational inequalities: problems without initial conditions

Abstract

Problems without initial conditions for evolution equations and variational inequalities appear in the modeling of different non-stationary processes within many fields of science, such as ecology, economics, physics, cybernetics, etc., if these processes started a long time ago and initial conditions do not affect them in the actual time moment. Thus, we can assume that the initial time is minus infinity. In the case of linear and weakly nonlinear evolution equations and variational inequalities, standard initial conditions should be replaced with the behavior of the solution as the time variable goes to minus infinity. However, for some strongly nonlinear evolution equations and variational inequalities, this problem has a unique solution in the class of functions without behavior restriction as the time variable goes to minus infinity. In this study, the correctness of the problem without initial conditions for such types of variational inequalities from a new class, or more precisely, for sub-differential inclusions with functionals, is investigated. Moreover, estimates of solutions are obtained. The results are new and mostly theoretical.