Unlocking of predicate: application to non-anticipating selections

We consider an approach to constructing a non-anticipating selection of a multivalued mapping; such a problem arises in control theory under conditions of uncertainty. The approach is called “unlocking of predicate” and consists in the reduction of finding the truth set of a predicate to searching fixed points of some mappings. Unlocking of predicate gives an extra opportunity to analyze the truth set and to build its elements with desired properties. This concept is used in many fields of mathematics: in differential equations and differential inclusions; in game theory, when studying the saddle points (see [1]) and the Nash equilibria (see [2, 3]); in dynamic games, when constructing the stable (weakly invariant) sets (see [4, 5]) and nonanticipating selections of multivalued mappings (see [6, 7]). However, in all the above cases “unlocking mappings” are presented as a ready–made product: a method for constructing an “unlocking mapping” has remained beyond the consideration. In this article, we outline how to build “unlocking mappings” for some general types of predicates: we give a formal definition of the predicate unlocking operation, the rules for the construction and calculation of “unlocking mappings” and their basic properties. As an illustration, we routinely construct two unlocking mappings for the predicate “be non-anticipating mapping” and then on this base we provide the expression for the greatest non-anticipating selection of a given multifunction. This work continues [8] where the procedure for the predicate "be Nash equilibrium"is presented.