Existence and Compactness of Solution Semiflows

The purpose of this chapter is to establish the existence and compactness of solution semiflows defined by a class of semilinear functional differential equations. We will start with the case where the linear operator generates a C₀-semigroup of bounded linear operators and the nonlinear term satisfies a global Lipschitz condition. We will then indicate how to relax this global Lipschitz condition by imposing compactness on the linear semigroup. In the final section, we will derive a class of semilinear functional differential equations of neutral type from a continuous array of coupled lossless transmission lines and we will obtain the basic existence-uniqueness result for such a class of equations.