The solution manifold and C1-smoothness for differential equations with state-dependent delay

[摘要] Let h>0, Usubset ofC(1)([-h,0], R-n) open, and f : U --> R-n continuously differentiable. If f satisfies two mild additional smoothness conditions then the set X = {phiis an element ofU: phi(0) =f (phi)} is a C-1-submanifold of codimension n in C-1([-h, 0], R-n) the maximal solutions x(phi) of the initial value problems x(t)-f(x(t)), x(0) = phiis an element ofX define a continuous semiflow F on X, and all operators F(t, (.)) are continuously differentiable. Their derivatives D2F(t,phi) are given in the usual way by solutions v to the variational equation along x(phi), with segments v(t) in the tangent spaces TF(t.phi)X. The additional conditions on f are motivated by properties of differential equations with state-dependent delay, and are verified for an example. (C) 2003 Elsevier Inc. All rights reserved.

[发布日期] 2003-11-20

[关键词] functional differential equation;state-dependent delay;invariant manifold;semiflow;linearization