Asymptotic Comparison of the Solutions of Linear Time-Delay Systems with Point and Distributed Lags with Those of Their Limiting Equations

and Applied Analysis 3 A finite subset of j consecutive positive integers starting with 1 is denoted by j : {1, 2, . . . , j}. The set R−h : −h, 0 ∪ R0 will be used to define the solution of functional differential equations on R0 including its initial condition in −h, 0 . C i R0 ,Rm0 is the set of m-vector real functions of class C i and definition domain R0 and PC i R0 ,Rm0 is the set of m-vector real functions in C i−1 R0 ,Rm0 whose ith derivative is piecewise continuous. Similar sets of functions are defined when the ranges are complex as C i R0 ,Cm0 and PC i R0 ,Cm0 . For the delayed system, T : 0,∞ → L X is the inverse Laplace transform of the resolvent mapping ̂ T s , which is holomorphic where it exists, with X being the real Banach space of n-vector real functions endowedwith the supremumnorm on their definition domain defined for any such a complex or real vector function φ of definition domain D by |φ|α supτ∈D (‖φ τ ‖α ) · ‖ · ‖α denoting any vector or induced matrix norm, that is, α 1 for the 1-norm, α ∞ for the ∞-norm, and so forth. Similar notations are used for the corresponding matrix-induced norms. In particular, α 2 stands for 2 or spectral vector and corresponding induced matrix norms, which coincides with the Euclidean norm for vectors. The Euclidean or Froebenius norm is denoted by the unsubscripted symbol ‖ · ‖ so that ‖ · ‖ ‖ · ‖2 for vectors but not for matrices. In the case of vectors, the Euclidean norm coincides with its 2-norm. The unsubscripted symbol | · | is used for absolute values of real, complex, and integer numbers, as usual. It is said that the delays associated with Volterra-type dynamics are infinitely distributed because the contribution of the delayed dynamics is made under an integral over 0,∞ as t → ∞, that is, x t − τ − hi acts on the dynamics of x t from τ 0 to τ t for finite t and as t → ∞. Dom H is the definition domain of the operator H and sp A is the spectrum i.e., the set of distinct eigenvalues of the square matrix A. The matrix measure of the normdependent complex-valued matrix A is defined by κα A : limδ→ 0 ‖In δA‖α − δ /δ ≥ Reλi A , for allλi ∈ sp A . Also, ¬, ∨, ∧ are logic symbols for negation, disjunction, and conjuction of logic propositions. 2. Problem Statement and Basic First Results Consider the following linear nominal functional differential systems with point and, in general, both Volterra-type and finite distributed delays: