Fuzzy power sets and fuzzy implication operators

The theory of fuzzy power sets, which has hitherto been insufficiently developed, is shown very naturally to require the use of a fuzzy implication operator (Section 1). Six such operators are gathered from the literature on multiple-valued logic (Section 2), and their effects on fuzzy power-set theory are compared throughout the rest of the paper. After certain fundamental definitions of set characteristics (Section 3), the six operators are carried in parallel while working out basic aspects of power-set theory. Among these are the properties of the set-inclusion relation and the set-equivalence relation (Section 4), two distinct concepts of disjointness (Section 5), questions of consistency in the relations between a set and its complement (Section 6), and a very concrete theorem on a difference among the operators with regard to the derivation of crisp conclusions from fuzzy premises (Section 7). Finally (Section 8), emphasis is placed on the dependence of the choice of operators upon the purposes the user has in hand.     

Invariant sets and Lyapunov pairs for differential inclusions with maximal monotone operators

We give different conditions for the invariance of closed sets with respect to differential inclusions governed by a maximal monotone operator defined on Hilbert spaces, which is subject to a Lipschitz continuous perturbation depending on the state. These sets are not necessarily weakly closed as in [3], [4], while the invariance criteria are still written by using only the data of the system. So, no need to the explicit knowledge of neither the solution of this differential inclusion, nor the semi-group generated by the maximal monotone operator. These invariant/viability results are next applied to derive explicit criteria for a-Lyapunov pairs of lower semi-continuous (not necessarily weakly-lsc) functions associated to these differential inclusions. The lack of differentiability of the candidate Lyapunov functions and the consideration of general invariant sets (possibly not convex or smooth) are carried out by using techniques from nonsmooth analysis.     

New fixed point theorems for a family of mappings and applications to problems on sets with convex sections

Some new fixed point theorems for a family of mappings are obtained and applied to problems on sets with convex sections that were first studied by Ky Fan.