Periodic value problems of differential systemson infinite-dimensional spaces and applications to differential geometry

Periodicity questions of differential systems on infinite-dimensional Hilbert spaces arestudied via a new methodology which is based on Fan-Knaster-Kuratowski-Mazurkiewicz theorem. We obtain in this way an alternative, to the classical fixed point theory, approach to the study of such type of problems with various applications on issues of mathematical analysis and differential geometry. Two examples of such applications are included.