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

Abstract

A general theory for pointwise completeness and degeneracy of functional differential equations in infinite dimensional spaces is presented. For the basis of the theory, a fundamental solution and a retarded resolvent are introduced and the fact that the Laplace transform of the fundamental solution is the retarded resolvent is proved. The concepts of exact and approximate pointwise completeness are defined by the (attainable) sets of all possible mild solutions of the equations and are investigated in the framework of linear operator theory. A necessary and sufficient condition and a negative result for exact pointwise completeness are given. The degenerate space is defined by the orthogonal complement of the attainable set and various characterizations of the space in terms of fundamental solution and/or coefficient operators are established with some examples. Interesting characterizations of the full degenerate set in terms of retarded resolvent and generalized eigenfunctions, which extend the well-known finite-dimensional degeneracy conditions, are also established.