Locally Quasi-dissipative Evolution Equations and Applications to Delay Differential Equations

In this paper we discuss applications of the generation theory of nonlinear semigroups for evolution equations with locally quasi-dissipative operators in the sense of Kobayashi and Oharu to delay differential equations. First we introduce the definitions and notations that are necessary for discussing the general generation theory for abstract evolution equations. Then we describe the existence and uniqueness of mild solutions of evolution equations with a quasi-dissipative operator and develop an approximation theory of Trotter-type. Finally, we discuss the application of the general theory to nonlinear delay differential equations including the state-dependent delay case.