The present article is based on the Fermi Lectures I gave in May, 1985, at Scuola Normale Superiore, Pisa, in which I discussed various methods for solving the Cauchy problem for abstract nonlinear differential equations of evolution type. Here I present a detailed exposition of one of these methods, which deals with "elliptic-hyperbolic" equations in the abstract form and which has applications, among other things, to mixed initial-boundary value problems for certain nonlinear partial differential equations, such as elastodynamic and Schrödinger equations.