In the paper, we study weak invariance of differential inclusions with non-fixed time impulses under compactness type assumptions. When the right-hand side is one sided Lipschitz an extension of the well known relaxation theorem is proved. In this case also necessary and sufficient condition for strong invariance of upper semi continuous systems are obtained. Some properties of the solution set...

A version of the fundamental mean-square convergence theorem is proved for stochastic differential equations (SDEs) in which coefficients are allowed to grow polynomially at infinity and which satisfy a one-sided Lipschitz condition. The theorem is illustrated on a number of particular numerical methods, including a special balanced scheme and fully implicit methods. The proposed special balanc...

We study the spreading of characteristics for a class of one-dimensional scalar conservation laws for which the ux function has exactly one point of innection. It is well-known that the characteristic speed satisses a one-sided Lipschitz estimate in the convex case. Using Dafermos' theory of generalized characteristics 3] we show that the characteristic speed in the nonconvex case satisses a HH...

Nonnegative measure-valued solutions of the continuity equation are uniquely determined by their initial condition, if the characteristic ODE associated to the velocity field has a unique solution. In this paper we give a partial extension of this result to signed measurevalued solutions, under a quantitative two-sided Osgood condition on the velocity field. Our results extend those obtained fo...

We introduce strong, and in many cases optimal, lower bounds for the number of queries required to nonadaptively test three fundamental properties of functions f : [n] → R on the hypergrid: monotonicity, convexity, and the Lipschitz property. Our lower bounds also apply to the more restricted setting of functions f : [n]→ R on the line (i.e., to hypergrids with d = 1), where they give optimal l...

Abstract In this paper, we study the existence and uniqueness of random periodic solution for a stochastic differential equation with one-sided Lipschitz condition (also known as monotonicity condition) convergence its numerical approximation via backward Euler–Maruyama method. The is shown limit pull-back flows SDE discretized SDE, respectively. We establish rate strong error method order 1/2.

Necessary conditions of optimality are derived for optimal control problems with pathwise state constraints, in which the dynamic constraint is modelled as a differential inclusion. The novel feature of the conditions is the unrestrictive nature of the hypotheses under which these conditions are shown to be valid. An Euler Lagrange type condition is obtained for problems where the multifunction...

Dissipative solutions have recently been studied as a generalized concept for weak of the complete Euler system. Apparently, these are expectations suitable measure valued solutions. Motivated from Feireisl et al. (Commun Partial Differ Equ 44(12):1285–1298, 2019), we impose one-sided Lipschitz bound on velocity component uniqueness criteria solution in Besov space $$B^{\alpha ,\infty }_{p}$$ ...