Summary: Inner estimation of the eigenvalue set and exponential series
solutions to differential inclusions
Dedicated to C. Lemarechal on the occasion of his 60th birthday.
We obtain inner estimations, around special eigenvalues, for the eigenvalue set of a properly
nonlinear closed convex process. We also consider a differential inclusion associated with a
general closed convex process and we construct smooth power series solutions of exponential
type for some initial states.
Key words: convex process, eigenvalues, differential inclusion.
Set-valued analysis is a flexible framework which permits to treat in a unified manner a wide
variety of applications, ranging from equilibrium problems in theoretical economics to the control
of dynamical systems. Although multivalued maps share some properties with their singlevalued
analogues, the set-valued structure gives rise to important differences in many aspects of the theory.
A particularly interesting multivalued concept is that of convex process on a vector space, that is,
a set-valued map whose graph is a convex cone containing the origin. This natural generalization
of a linear transformation was first introduced by Rockafellar [8, 9], and since his pioneering work