Some hemivariational inequalities in the Euclidean space

The purpose of this paper is to study the existence of weak solutions for some classes of hemivariational problems in the Euclidean space ℝd (d ≥ 3). These hemivariational inequalities have a variational structure and, thanks to this, we are able to find a non-trivial weak solution for them by using variational methods and a non-smooth version of the Palais principle of symmetric criticality for locally Lipschitz continuous functionals, due to Krawcewicz and Marzantowicz. The main tools in our approach are based on appropriate theoretical arguments on suitable subgroups of the orthogonal group O(d) and their actions on the Sobolev space H1(ℝd). Moreover, under an additional hypotheses on the dimension d and in the presence of symmetry on the nonlinear datum, the existence of multiple pairs of sign-changing solutions with different symmetries structure has been proved. In connection to classical Schrödinger equations a concrete and meaningful example of an application is presented.

A General Approximation Approach for the Simultaneous Treatment of Integral and Discrete Operators

In this paper, we give a unitary approach for the simultaneous study of the convergence of discrete and integral operators described by means of a family of linear continuous functionals acting on functions defined on locally compact Hausdorff topological groups. The general family of operators introduced and studied includes very well-known operators in the literature. We give results of uniform convergence and modular convergence in the general setting of Orlicz spaces. The latter result allows us to cover many other settings as the {L^{p}}-spaces, the interpolation spaces, the exponential spaces and many others.

TRAJECTORY-BASED MODELS, ARBITRAGE AND CONTINUITY

In a nonprobabilistic setting, we prove general trajectory-based models to have no free lunch with vanishing risk. The main ingredient is a local continuity requirement on the final portfolio value considered as a functional on the trajectory space. This is shown to be a natural assumption by establishing that a large class of practical trading strategies, defined by means of trajectory-based stopping times, give rise to locally continuous functionals. The theory is applied to two specific trajectory models of practical interest. The established results are then used to derive no free lunch results for nonsemimartingale stochastic models.