We introduce several new kinds of inferior and superior limits and corresponding kinds of semicontinuity of a set-valued map. Together with the known concepts of semicontinuity, they can be used to have a clearer insight of local behaviors of maps. Then we investigate all major semicontinuity properties of solution maps to a general quasivariational inclusion. Consequences are derived for sever...

In this article, we establish necessary conditions for optimality in stochastic control of systems governed by stochastic differential equations with nonsmooth coefficients. The approach used is based on the approximation of the nonsmooth coefficient by smooth one which generate a sequence of smooth control problems. Ekeland’s variational principle is then applied to obtain a sequence of nearly...

The problem of disorder seeks to determine a stopping time which is as close as possible to the unknown time of ’disorder’ when the observed process changes its probability characteristics. We give a partial answer to this question for some special cases of Lévy processes and present a complete solution of the Bayesian and variational problem for a compound Poisson process with exponential jump...

We shall establish the existence and non existence of solitons (travelling waves of finite energy) for a Benney-Luke equation of higher order, which includes models for long water waves with small amplitude. Following a variational approach, solitons are characterized as critical points of the action functional. Existence of solitons follows by the ConcentrationCompactness principle by P.-L. Li...

We extend the classical inverse and implicit function theorems, the implicit function theorems of Lyusternik and Graves, and the results of Clarke and Pourciau to the situation when the given function is not smooth, but it has a convex strict prederivative whose measure of noncompactness is smaller than its measure of surjectivity. The proof of the main results requires Banach’s open mapping th...

We show that some measures suuering from the so-called Renormalization Group pathologies satisfy a variational principle and that the corresponding limit of the pressure, with boundary conditions in a set of measure 1, is proportional to the pressure of the Ising model.

The variational principle states that if a differentiable functional F attains its minimum at some point zi, then F’(C) = 0; it has proved a valuable tool for studying partial differential equations. This paper shows that if a differentiable function F has a finite lower bound (although it need not attain it), then, for every E > 0, there exists some point u( where 11 F’(uJj* < l , i.e., its de...