A stationary point theorem characterizing metric completeness
We give a stationary point theorem for some set-valued mappings on a metric space. The existence of fixed points of such mappings characterizes the metric completeness. Our result easily yields the order-theoretic Cantor theorem of Granas and Horvath, and the famous Ekeland variational principle.
Existence and uniqueness of endpoints of closed set-valued asymptotic contractions in metric spaces
Abstract Investigations of problems of set-valued asymptotic fixed point theory of set-valued dynamic systems are initiated. The concepts of the set-valued asymptotic contractions of set-valued dynamic systems T : M → 2 M in metric space M are introduced and conditions guaranteeing the existence and uniqueness of endpoints v ∈ M of set-valued dynamic systems of set-valued asymptotic contractions T : M → 2 M and convergence to v of all generalized sequences of iterations of T are established. Examples and remarks show a fundamental difference between our results and the well-known ones.
The Globalization problem for inner automorphisms and Skolem-Noether theorems
We investigate the circumstances under which an inner automorphism of a ring with local units can be built from “local” information. Specifically, we consider three natural “inner” type properties for an automorphism of a ring with local units. We show that every inner automorphism is locally inner but the converse is false, even if the automorphism is “piecewise” inner. On the other hand, we construct a large class of rings for which every locally inner automorphism is actually inner. Finally we obtain some Skolem-Noether type Theorems for infinite matrix and triangular matrix rings.