Pointwise completeness and degeneracy of functional differential equations in Banach spaces. I: General time delays (Q1105112)

The author studies the poinwise completeness and degeneracy for functional differential equations of the type \[ x'(t)=A_ 0x(t)+\sum^{m}_{i=1}A_ ix(t-h_ i)+\int^{0}_{- h}B(s)x(t+s)ds+f(t), \] \(t\geq 0\); \(x(0)=g^ 0\); \(x(s)=g(s)\), \(s<0\), where x(t)\(\in X\), X is a reflexive Banach space and \(A_ 0\) generates a \(C_ 0\)-semigroup. The concepts of exact and approximate pointwise completeness are defined as usual in terms of the attainable sets of the solutions. The degenerate set here is the orthogonal complement of the attainable set. Necessary and sufficient conditions for pointwise completeness as well as various characterizations of the degenerate set are given. The results are extensions of similar results of L. Weiss, F. Kappel and others for the finite-dimensional case \((X={\mathbb{R}}^ n)\) and of P. Charrier for the case when X is a Hilbert space.