Abstract Cauchy problems

Abstract

If S(X→X) is the infinitesimal generator of a strongly continuous semigroup on a complex Banach space, the literature about such an S abounds with results on the existence of a unique solution of the Cauchy problem

$$ \left\{ \begin{gathered} u'\left( t \right) = Su\left( t \right) + f\left( t \right), t \in \mathbb{R}^ + , \hfill \\ u\left( {0^ + } \right) = x_0 . \hfill \\ \end{gathered} \right. $$

In this chapter we study the existence and uniqueness of classical, weak, and mild solutions to the Cauchy problem

$$ \left\{ \begin{gathered} u'\left( t \right) = Su\left( t \right) + f\left( t \right), 0 \ne t \in \mathbb{R}, \hfill \\ u\left( {0^ + } \right) - u\left( {0^ - } \right) = x_0 , \hfill \\ \end{gathered} \right. $$

where now S is an exponentially dichotomous operator. Furthermore, we characterize exponentially dichotomous operators as to their ability to lead to uniquely solvable Cauchy problems.