Abstract We demonstrate that strong global asymptotic stability (GAS) of the origin for an upper semicontinuous difference inclusion is equivalent to the existence of a smooth Lyapunov function. This result is of interest in discrete-time because the robustness of the stability property is dependent on the existence of such a smooth Lyapunov function. We also propose a regularization that allow...

We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super–replication prices of game options with upper semicontinuous payoffs. We show that the super–replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. This type of results is also new for American...

For upper semicontinuous finite-dimensional set-valued mappings with either convex closed or aspheric closed values we prove the existence of special continuous e -approximations that point-wise converge to a Borel measurable selector of the set-valued mapping as e tends to zero. For convex-valued case the convergence holds on the entire domain while for aspheric-valued case — on a certain coun...

The paper considers upper semicontinuous behavior in distribution of sequences of random closed sets. Semiconvergence in distribution will be described via convergence in distribution of random variables with values in a suitable topological space. Convergence statements for suitable functions of random sets are proved and the results are employed to derive stability statements for random optim...

We give an extension to a nonconvex setting of the classical radial representation result for lower semicontinuous envelope of a convex function on the boundary of its effective domain. We introduce the concept of radial uniform upper semicontinuity which plays the role of convexity, and allows to prove a radial representation result for nonconvex functions. An application to the relaxation of ...

We investigate analyticity of joint spectra of Am-valued holomorphic mappings, where A denotes a complex Banach algebra. We show also that if K is an analytic set-valued function whose values are compact subsets of Cn and d is the transfinite diameter in Cn, then the upper-semicontinuous regularization of logd(X) is plurisubharmonic. Moreover, we give higher dimensional extensions of Aupetit's ...

In the paper we provide construction of a coincidence degree being homotopy invariant detecting existence solutions equations or inclusions form $Ax\in F(x)$, $x\in U$, where $A\colon D(A)\multimap E$ is an $m$-accretive operator in Banach space $ E$, $F\colon K\multimap weakly upper semicontinuous set-valued map constrained to open subset $U$ closed set $K\subset E$. Two different approaches w...

Admissibility is a technical condition that ensures that the fixed point index of the flow is well-defined [Gor06]. Recall that a flow is admissible if and only if it is upper semicontinuous (i.e. if the limit of a convergent sequence of trajectories is a trajectory) and if there exists h > 0 such that, for any point x, the set of trajectory segments S(x,h) starting at x and defined on the time...