UDC 517.9 ON SOME SPECIAL TYPES OF e -APPROXIMATIONS FOR SET-VALUED MAPPINGS Yuri E. Gliklikh Voronezh State University Поступила в редакцию 23.03.2009 г. <...> 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 countable everywhere dense subset. <...> Key words: Upper semicontinuous set-valued mapping; convex closed values; aspheric closed values; e -approximations; point-wise convergence. 1. <...> INTRODUCTION The main aim of this paper is to show the existence of e -approximations of the upper semicontinuous set-valued finite-dimensional map that point-wise converge to a Borel measurable selector as e Ж0 . <...> Unlike the case of ordinary differential inclusions, such approximations are very much useful for investigation of stochastic differential inclusions. <...> Recall that e -approximations are proved to exist for upper semicontinuious set-valued map either with convex closed values or with aspheric closed ones (see below). <...> We consider both cases, but for convex-valued mappings we prove the existence of point-wise converging e -approximations on the entire domain while for aspheric-valued ones only on a certain countable everywhere dense subset. <...> More details can be found, e.g., [3,4] where in particular the proofs of many results, presented here, are given. © Gliklikh Yu. <...> Taking into account applications to stochastic differential inclusions, we consider two classes of set-valued mappings: those depending on points of phase space and those depending on curves but non-anticipating with respect to a special filtration generated by s - algebras of cylinder sets. <...> We prove the existence of point-wise converging sequences of e -approximation depending on points or non-anticipating with respect to the same filtration, respectively. <...> In Section 4 we construct a sequence of e -approximations for aspheric-valued mappings that point-wise converge to a selector on a countable everywhere dense subset X . <...> In this case for every point of X there exists a number such that for all greater numbers the values of all terms of the sequence at that point are stabilized (i.e., have the same value). <...> A BRIEF INTRODUCTION INTO THE THEORY <...>