Multivalued Exponentiation Analysis. Part III: Forward Exponentials

Abstract

We continue our work on exponentiability of multivalued maps on Banach spaces. In Part I we studied the exponentiability of a map F : X ⇉ X by using a Maclaurin expansion approach. In Part II we studied the recursive exponentiation approach. Recursive exponentials are built by using the trajectories of a discrete-time evolution system governed by F. We now focus the attention on forward exponentiability. The forward exponential of F at the point x is defined as the Kuratowski limit $$e^{F}(x):=\lim\limits_{n\to\infty}\; \left(I+ n^{-1} F \right)^{n}(x).$$ This type of exponential arises in connection with the Euler discretization scheme for solving the first-order differential inclusion $\dot \psi (t)\in F(\psi (t))$ .