Retarded functional differential equations: a global point of view

Abstract

This work deals with some of the fundamental aspects of retarded functional differential equations (RFDE's) on a differentiable manifold. We start off by giving a solution of the Cauchy initial value problem for a RFDE on a manifold X. Conditions for the existence of global solutions are given. Using a Riemannian structure on the manifold X, a RFDE may be pulled back into a vector field on the state space of paths on X. This demonstrates a relationship between vector fields and RFDE's by giving a natural embedding of the RFDE's on X as a submodule of the module o* vector fields on the state space. For a given RFDE it is shown that a global solution may level out asymptotically to an equilibrium path. Each differentiable RFDE on a Riemannian manifold linearizes in a natural way, thus generating a semi-flow on the tangent bundle to the state space. Sufficient conditions are given to smooth out the orbits and to obtain the stable bundle theorem for the semi-flow There are examples of RFDE's on a Riemannian manifold. These include the vector fields, the differential delay equations, the delayed Cartan development and equations of Levin-Nohel type. The retarded heat equation on a compact manifold provides an example of a partial RFDE on a function space. We conclude by making suggestions for further research