We study an abstract second order inclusion involving two nonlinear single-valued operators and a nonlinear multi-valued term. Our goal is to establish the existence of solutions to the problem by applying numerical scheme based on time discretization. We show that the sequence of approximate solution converges weakly to a solution of the exact problem. We apply our abstract result to a dynamic, second-order-in-time differential inclusion involving a Clarke subdifferential of a locally Lipschitz, possibly non-convex and non-smooth potential. In the two presented examples the Clarke subdifferential appears either in a source term or in a boundary term.

In this paper, we study the problem of the existence of solutions for a class of fractional semilinear evolution inclusions with non-convex right-hand side. We mainly focus on the existence of the extreme solution and the relationship of the solution sets between the original problem and the convexified problem. An example is provided to illustrate the application of the obtained theory.

This paper investigates the existence of solutions for boundary variational–hemivariational inequalities of elliptic type at resonance as well as at non-resonance. Using the notion of the generalized gradient of Clarke and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of variational–hemivariational inequalities of elliptic type.

This paper is devoted to the Dirichlet problem for quasilinear elliptic hemivariational inequalities at resonance as well as at non-resonance. Using Clarke's notion of the generalized gradient and the property of the first eigenfunction, we also build a Landesman–Lazer theory in the non-smooth framework of quasilinear elliptic hemivariational inequalities.