Semidynamical Systems in Banach Space

Material fundamental to the existence and qualitative behavior of partial differential equations and differential delay equations (to name just two areas) are developed in this chapter. The general formulation is an evolution equation in a Banach space. The work of Crandall and Liggett on the nonlinear version of the Hille — Yosida — Phillips theorem for linear semigroups has spawned an elegant analysis of nonlinear evolution equations in Banach spaces. One special feature of the semigroup generation theorem (linear or nonlinear) is that we obtain a representation of the solutions to du/dt + Au = 0 in terms of the operator A. The classical approach was to establish the existence, uniqueness, and continuous dependence of the solutions of the particular partial differential equation, for example, and then demonstrate that the solutions generate a semigroup. This was essentially the approach we also took in Chapter IV.