In volume I we developed the tools of "Multivalued Analysis. " In this
volume we examine the applications. After all, the initial impetus for
the development of the theory of set-valued functions came from its
applications in areas such as control theory and mathematical economics.
In fact, the needs of control theory, in particular the study of systems
with a priori feedback, led to the systematic investigation of
differential equations with a multi valued vector field (differential
inclusions). For this reason, we start this volume with three chapters
devoted to set-valued differential equations. However, in contrast to
the existing books on the subject (i. e. J. -P. Aubin - A. Cellina:
"Differential Inclusions," Springer-Verlag, 1983, and Deimling:
"Multivalued Differential Equations," W. De Gruyter, 1992), here we
focus on "Evolution Inclusions," which are evolution equations with
multi- valued terms. Evolution equations were raised to prominence with
the development of the linear semigroup theory by Hille and Yosida
initially, with subsequent im- portant contributions by Kato, Phillips
and Lions. This theory allowed a successful unified treatment of some
apparently different classes of nonstationary linear par- tial
differential equations and linear functional equations. The needs of
dealing with applied problems and the natural tendency to extend the
linear theory to the nonlinear case led to the development of the
nonlinear semigroup theory, which became a very effective tool in the
analysis of broad classes of nonlinear evolution equations.
