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

Abstract

This paper investigates the relations between the particular eigensolutions of a limiting functional differential equation of any order, which is the nominal (unperturbed) linear autonomous differential equations, and the associate ones of the corresponding perturbed functional differential equation. Both differential equations involve point and distributed delayed dynamics including Volterra class dynamics. The proofs are based on a Perron-type theorem for functional equations so that the comparison is governed by the real part of a dominant zero of the characteristic equation of the nominal differential equation. The obtained results are also applied to investigate the global stability of the perturbed equation based on that of its corresponding limiting equation.

1. Introduction

Time-delay dynamic systems are an interesting field of research in dynamic systems and functional differential equations because of intrinsic theoretical interest because the formalism lies in that of functional differential equations, then infinite dimensional, and because of the wide range of applicability in modelling of physical systems, like transportation systems, queuing systems, teleoperated systems, war/peace models, biological systems, finite impulse response filtering, and so forth [1–4]. Important particular interest has been devoted to stability, stabilization, and model-matching of control systems where the object to be controlled possess delayed dynamics and the controller is synthesized incorporating delayed dynamics or its structure may be delay-free (see, e.g., [1, 4–14]). The properties are formulated as either being independent of or dependent on the sizes of the delays. An intrinsic problem which increases the analysis complexity is the presence of infinitely many characteristic zeros because of the functional nature of the dynamics. This fact generates difficulties in the closed-loop pole–placement problem compared to the delay-free case [14], as well as in the stabilization problem [2, 4–6, 8–11, 13, 15–20], including the case of singular time-delay systems where the solution is sometimes nonunique, and impulsive, because of the dynamics associated to a nilpotent matrix, [15]. The properties of the associated evolution operators have been investigated in [2, 6, 11]. Interesting recent results on infinite dimensional Banach spaces are given in [21–24]. In particular, the existence of periodic solutions of semilinear evolution equations with time lags is investigated in [21]. In [22], a class of linear impulsive periodic systems with time-varying generating operators on a Banach space is considered. The set of impulsive periodic motion controllers that are robust to parameter drift are synthesized for a given periodic motion. The research in [23] is devoted to investigate the existence and the global asymptotic stability of a periodic 𝑃𝐶-mild solution for the 𝑇-periodic logistic system with time-varying generating operators and 𝑇-0-periodic impulsive perturbations on Banach spaces. In [24], a close problem is solved based on a generalized Gronwall's lemma. In [25], the robust stability of a variational control problem is solved by providing the stability radius. Also, the approximation properties of the homogeneous system associated with a class of linear elliptic differential equations with periodic coefficients is investigated in [26].

This paper is devoted to obtain results relying on a comparison and an asymptotic comparison of the eigensolutions between a nominal (unperturbed) functional differential equation involving wide classes of delays and a perturbed version (describing the current dynamics) with some appropriate assumptions smallness in the limit on the perturbed functional differential equation. The nominal equation is defined as the limiting equation of the perturbed one since the parameters of the last one converge asymptotically to those of its limiting counterpart. The problem is of interest in practice since the perturbations related to a nominal model in dynamic systems very often occur during the transients while they are asymptotically vanishing in the steady-state or, in the most general worst case, they grow at a smaller rate than the solution of the nominal differential equation. In this context, the nominal differential equation may be viewed as the limiting equation of the perturbed one. The comparison between the solutions of the limiting differential equation and those of the perturbed one based on Perron-type results have been studied classically for ordinary differential equations and, more recently, for the case of functional equations [27–29]. Particular functional equations of interest in real-life problems are those involving both point and distributed delays, the last ones potentially include Volterra-type terms, [2, 5–7, 30].

Notation
The following sets are used through the manuscript:where 𝐑,𝐂, and 𝐙 are the sets of real, complex and integer numbers, respectively, The complex imaginary unity is 𝐢=√−1.

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₊,𝐑^𝑚0+) is the set of m-vector real functions of class 𝐶^(𝑖) and definition domain 𝐑₀₊ and PC^(𝑖)(𝐑₀₊,𝐑^𝑚0+) is the set of m-vector real functions in 𝐶^(𝑖−1)(𝐑₀₊,𝐑^𝑚0+) whose 𝑖th derivative is piecewise continuous. Similar sets of functions are defined when the ranges are complex as 𝐶^(𝑖)(𝐑₀₊,𝐂^𝑚0+) and PC^(𝑖)(𝐑₀₊,𝐂^𝑚0+).

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 endowed with the supremum norm on their definition domain defined for any such a complex or real vector function 𝜙 of definition domain 𝐷 by |𝜙|_𝛼=sup_𝜏∈𝐷(‖𝜙(𝜏)‖_𝛼)⋅ ‖⋅‖_𝛼 denoting any vector or induced matrix norm, that is, 𝛼=1 for the ℓ₁-norm, 𝛼=∞ for the ℓ_∞-norm, and so forth. Similar notations are used for the corresponding matrix-induced norms. In particular, 𝛼=2 stands for ℓ₂ (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 ℓ₂-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−ℎ^𝑖) 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 norm-dependent complex-valued matrix 𝐴 is defined by 𝜅_𝛼(𝐴)∶=lim_𝛿→0⁺(‖𝐼_𝑛+𝛿𝐴‖_𝛼−𝛿)/𝛿≥Re𝜆_𝑖(𝐴),forall𝜆_𝑖∈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: Equation (2.1) is the limiting equation of the perturbed equation (2.2), subject to (2.3), for 𝜙∈𝐶_𝑒(−ℎ) as 𝐶_𝑒(−ℎ)∶={𝜙=𝜙₁+𝜙₂∶𝜙₁∈𝐶(−ℎ),𝜙₂∈𝐵⁰(−ℎ)},𝜙(0)=𝑥₀ under the following technical hypothesis.

H.1: The initial conditions of both differential equations (2.1) and (2.2) are real n-vector functions 𝐶(−ℎ)∶={𝐶⁰([−ℎ,0],𝑋)} where [−ℎ,0], with 𝑋; that is, the set of continuous mappings from 𝜙_𝛼∶=|𝜙|_𝛼=Sup{‖𝜙(𝑡)‖_𝛼∶−ℎ≤𝑡≤0};‖⋅‖ into the Banach space 𝐂^𝑛 with norm 𝐂^𝑛×𝑛 denoting the Euclidean norm of vectors in 𝐵⁰(−ℎ)∶={𝜙∶[−ℎ,0]→𝑋} and matrices in 𝑋, and 𝜙∈𝐵⁰(−ℎ) is the set of real-bounded vector functions on [−ℎ,0] endowed with the supremum norm having support of zero measure. Roughly speaking, 𝜙∈𝐶_𝑒(−ℎ) if and only if it is almost everywhere zero except at isolated discontinuity points within [−ℎ,0] where it is bounded. Thus, 𝐶_𝑒(−ℎ) if and only if it is almost everywhere continuous in 𝜙=𝜙₁+𝜙₂ except possibly on a set of zero measure of bounded discontinuities. 𝜙₁∈𝐶(−ℎ) is also endowed with the supremum norm since 𝜙₂∈𝐵⁰(−ℎ), some 𝜙∈𝐶_𝑒(−ℎ), 𝐿(𝑋) for each |⋅|. In the following, the supremum norms on 𝐶(𝐑_−ℎ)∶=𝐶([−ℎ,∞),𝐂^𝑛) are also denoted with [−ℎ,∞).

Close spaces of functions are 𝐂^𝑛 which is the Banach space of continuous functions from |𝜙|_𝛼=sup_{−ℎ≤𝜏<∞}(‖𝜙(𝜏)‖_𝛼) into 𝜙∈𝐶_𝑒(−ℎ)∶=𝐶([−ℎ,0),𝐂^𝑛) endowed with the norm ‖⋅‖_𝛼; for all 𝑡∈𝐑₀₊ being an initial condition, for some given vector norm 𝐶_𝑒(𝐑₀₊)∶=𝐶(𝐑₀₊,𝐂^𝑛). Note that for 𝐑₀₊, the solution which satisfies (2.2), subject to (2.3), is in 𝐂^𝑛, the Banach space of continuous functions from ∀𝜙∈𝐶_𝑒(−ℎ) into |𝜙|_𝛼=sup_0≤𝜏<∞(‖𝜙(𝜏)‖_𝛼) which satisfies (2.2)-(2.3), 𝐿∶𝐶_𝑒(𝐑_−ℎ)→𝐂^𝑛, endowed with 𝐴_𝑘(0≤𝑘≤𝑚),𝐴_𝛼𝑘(0≤𝑘≤𝑚^+𝑚^).

Thus, 𝐿(𝑋)∶=𝐿(𝑋,𝑋) is a bounded linear functional defined by the right-hand-side of (2.1).

H.2: All the operators 𝑋 are in 𝑋^∗, the set of linear operators on ℎ_𝑘, of dual ℎ^ℓ(𝑘=1,2,…,𝑚;ℓ=0,1,…,𝑚^+𝑚^) and ℎ₀=ℎ^0=0, and ℎ∶=Max(Max_{1≤𝑖≤𝑛}(ℎ_𝑖),Max_{1≤𝑖≤𝑚′+𝑚′′}(ℎ^𝑖)) are nonnegative constants with 𝐴_𝛼𝑖∈𝐿(𝑋) and 𝐴_𝛼0=𝐴_𝛼.

H3: The linear operators 𝐷(𝐴_𝛼𝑖), with abbreviated notation 𝑅(𝐴_𝛼𝑖)⊂𝑋(𝑖=0,1,…,𝑚^+𝑚^), are closed and densely defined linear operators with respective domain and range 𝛼_𝑖∈𝐶⁰([0,∞),𝐂)∩𝐵𝑉_loc(𝐂₊)(𝑖=0,1,…,𝑚^) and 𝛼_𝑖∈𝐶⁰([−ℎ,0),𝐂)(𝑖=0,1,…,𝑚^+𝑚^). The functions ∫^∞0𝑒^𝑣𝑡|𝑑𝛼_𝑖(t)|<∞ and 𝑣(𝑖=0,1,…,𝑚^) being everywhere differentiable with possibly bounded discontinuities on subsets of zero measure of their definition domains with 𝛼_𝑖(⋅) some nonnegative real constant 𝛼_𝑖∶[0,∞]×𝑋^∗→𝐿(𝑋,𝑋^∗). If 𝐶⁰([0,∞),𝐶^𝐧×𝐧)∩𝐵𝑉_loc(𝐶^+𝐧×𝐧) is a matrix function ∫^∞0𝑒^𝜈𝑡|𝑑𝛼_𝑖(𝑡)|<∞ then it is in 𝑓₀∶𝐑₀₊×𝐶(𝐑_−ℎ)→𝐂^𝑛 with ℎ₀=ℎ^0=0 and its entries being everywhere time-differentiable with possibly bounded discontinuities on a subset of zero measure of their definition domains.

H.4: It is assumed that 𝑥^{𝐶(𝐑−ℎ)𝑡}∶=𝑥0,∶[−ℎ,𝜏)⟶𝑋,𝜏𝜏>≤𝑡𝑡,,(2.4) and 𝑥(𝑡)=𝜙(𝑡), satisfying 𝑥^{𝐶(𝐑0+)𝑡}, for all 𝑥(𝑡), is a string of the solution of (2.2)-(2.3). Other strings of the solution trajectory of interest in this manuscript are [0,𝑡] which point-wise defined by 𝑥_𝑡=𝜙_𝑡 within the interval [max(−ℎ,𝑡−ℎ),𝑡] and zero, otherwise, and subject to the constraint ∀𝑡∈𝐑₀₊ within 𝑥_𝑡, [𝑡−ℎ,𝑡] and being zero outside this interval. Finally, 𝑥(𝑡) denotes the solution string within 𝑡∈𝐑_−ℎ point-wise defined by the solution [𝑡−ℎ,𝑡] to (2.2)-(2.3) for each 𝑥_𝑡=𝜙_𝑡 being zero outside [max(−ℎ,𝑡−ℎ),𝑡] and subject to the constraint 𝑡≤ℎ within 𝐴_𝑖 for any real 𝐴_𝛼𝑘 and being zero outside this interval.

𝐴_𝑖∶[0,∞)→𝐂^𝑛×𝑛 and 𝐴_𝑖∶[0,∞)→𝐂^𝑛×𝑛(𝑘=0,1,…,𝑚;𝑘=0,1,…,𝑚^+𝑚^) and 𝑥∈𝑋 and ℎ∶=(0,ℎ₁,ℎ₂,…,ℎ_𝑚) belong to the spaces of constant real matrices and real matrix functions, respectively. The last ones are also unbounded operators on a Banach space of n-vector real functions ℎ^=(ℎ^1𝑇∶ℎ^2𝑇)^𝑇∶=(0,ℎ^1,ℎ^2,…,ℎ^𝑚_′∶ℎ^𝑚_′+1,ℎ^𝑚_′+2,…,ℎ^𝑚_{′+𝑚′′})^𝑇 endowed with the supremum norm where the vectors of point and distributed constant delays are: ℎ_𝑖≥0 and ℎ^𝑘≥0(𝑘=1,2,…,𝑚^+𝑚^), respectively, with ℎ₀=ℎ^𝑘=0,𝐴₀≡𝐴,𝐴_𝛼 and 𝛼₀(⋅)≡𝛼(⋅) and being, respectively, point and distributed delays, with 𝑚^ and ∫^𝑡0𝑑𝛼_𝑖(𝜏)𝐴_𝛼𝑖𝑥(𝑡−𝜏−ℎ^𝑖). The first ℎ^𝑖 distributed delays are associated with Volterra-type dynamics. In other words, the infinitely distributed delays give contributions (𝑖=1,2,…,𝑚^) with finite real constants 𝑥̇(𝑡) with 𝛼_𝑖∶[0,∞)→𝐂 to 𝛼_𝑘∶⌊0,ℎ^𝑘⌋→𝐂 which are point delays under the integral symbol. The functions 𝛼_𝑖(⋅) and 𝛼_𝑖∶[0,𝑡]→𝐶^𝑛×𝑛(𝑖=0,1,…,𝑚^) are continuously differentiable real functions within their definition domains except possibly on sets of zero measure where the time-derivatives have bounded discontinuities. All or some of the 𝑡∈𝐑₊ may be alternatively matrix functions 𝛼_𝑖∶⌊0,ℎ^𝑘⌋→𝐶^𝑛×𝑛(𝑖=𝑚^+1,𝑚^+2…,𝑚^+𝑚^) for 𝛼_𝑖(0)=0;𝑖=0,1,…,𝑚^+𝑚^ and 𝑓(𝑡,𝑥_𝑡) with 𝑓∶𝐑₀₊×𝐶(𝐑_−ℎ)→𝐂^𝑛×𝑛. On the other hand, the perturbation vector function 𝑓₀∶𝐑₀₊×𝐶(𝐑_−ℎ)→𝐂^𝑛×𝑛 in (2.2), defined in (2.3), with respect to the limiting (2.1), is defined by the function 𝑓 which describes a perturbed dynamics associated with the delays plus a perturbation function ℎ^𝑖(𝑖=0,1,…,𝑚^) which is not included in the remaining terms of the function ℎ^0=0 in (2.3). Note that both the delayed differential systems (2.2)-(2.3) and its limiting version (2.1) are very general since it includes point-delayed dynamics, like, for instance, in typical war/peace models or the so-called Minorski's problem appearing when controlling the lateral dynamics of a ship [2]. It also includes real constants ℎ^0=0, with ℎ^𝑖(𝑖=0,1,…,𝑚^+𝑚^), associated with infinitely distributed delayed contributions to the dynamics. Such delays are relevant, for instance, in viscoelastic fluids, electrodynamics, and population growth [1, 5, 8]. In particular, an integro-differential Volterra-type term is also included through 𝛼_𝑖(⋅). Apart from those delays, the action of finite-distributed delays characterized by real constants [𝑡−ℎ^𝑖,𝑡] is also included in the limiting equation (2.1) and in (2.3). That kind of delays is well known, for instance, in econometric models related to production rate [8]. The integrability of the 𝐶(𝐑₀₊)∶=𝐶(𝐑₀₊,𝐂^𝑛)-functions (or matrix functions) on 𝜙∈𝐶_𝑒(−ℎ) follows since their definition domain is bounded. The technical hypothesis H1–H4 guarantee the existence and uniqueness of the solution in 𝑑𝛼(𝜏)=𝛼̇(𝜏)𝑑𝜏 of the functional differential systems (2.1) and (2.2)-(2.3) for each given initial condition 𝑑𝛼_𝑖(𝑠)=𝑠𝛼_𝑖(𝑠)−𝛼_𝑖(0).

Take Laplace transforms in (2.1) by using the convolution theorem and the relations 𝑓(𝑠)∶=Lap𝑓(𝑡). It follows that 𝑓(𝑡), where 𝐶(𝐑₀₊) denotes the Laplace transform of 𝑡∈𝐑₀₊. Thus, the unique solutions of both the limiting (2.1) and that of (2.2)-(2.3) in 𝜙∈𝐶_𝑒(−ℎ), subject to (2.3); for all 𝑥(𝑡)=𝑇(𝑡,0𝑦)(𝑥𝑡)=0₊𝑇(𝑡,0)𝑥0⁺+^0−ℎ𝑇(𝑡,𝜏)𝜙(𝜏)𝑈(𝜏)𝑑𝜏,(2.5)+^0−ℎ𝑇(𝑡,𝜏)𝜙(𝜏)𝑈(𝜏)𝑑𝜏+^𝑡0𝑇(𝑡,𝜏)𝑓(𝜏,𝑥_𝜏)𝑑𝜏,(2.6), for the same given initial conditions 𝑈(𝑡) are, respectively, defined by where 𝑓(𝑡,𝑥_𝑡) is the unit step (Heaviside) function and 𝑡→∞ is the evolution operator, [2, 6, 31], of the linear (2.1) whose Laplace transform, everywhere it exists, is given by the resolvent: As usual, it is said though the manuscript that (2.1) is the limiting equation of (2.2)-(2.3) irrespective of the fact that 𝑑𝛼_𝑖(𝑡)=𝜒_𝑖𝑑𝑡 converges or not to zero as 𝜒_𝑖. The evolution operator is a convolution operator so that 𝑇(𝑠) if the Volterra-type dynamics is zero or if the associate differentials in the Riemann-Stieltjes integrals Re𝑠>−𝛼₀ with r being real constants. In this case, the limiting linear functional differential equation is, furthermore, time-invariant. Note that the limiting (2.1) is guaranteed to be globally exponentially uniformly stable if and only if 𝛼₀∈𝐑 exists within some region including properly the right-complex plane. In other words, if it is compact for det𝑇⁻¹(𝑠), for some 𝑇(𝑡) constant 𝐑₀₊ located to the right of all the real parts of all the zeros of 𝜙∈𝐶_𝑒(−ℎ) (also often called the characteristic zeros of the limiting (2.2) or, simply, its eigenvalues), since then all the entries of its Laplace transform |𝑥(𝑡)| decay with exponential rate on 𝐑₊ for 𝜙∈𝐶_𝑒(−ℎ) and then det𝑇⁻¹(𝑠) decays with exponential rate on x. The main result addressed in [2, 7–10] relies on the investigation of the global uniform exponential stability of (1). The stability of the limiting system (2.1) is investigated in [5, 6], provided that any auxiliary system formed with any of the additive parts of the dynamics of (2.1), has such a property and provided that an impulsive-solution-dependent input exists. The compactness of the relevant input-output and input-state operators under forcing external inputs and impulsive forcing terms is also investigated in [6]. The basic mathematical tool used in those papers is that the unique solution of the homogeneous (2.1) for each function of initial conditions 𝐑₀₊ may be equivalently written in infinitely many cases by first rewriting (2.1) by considering different “auxiliary” reference homogeneous systems plus additional terms considered as forcing actions. The objective of this paper is to compare the solutions (2.1), (2.2), subject to (2.3), of the limiting and current functional differential equations (2.1) and (2.2) by using a Perron-type result using a similar technique as that used in [28]. The subsequent theorem is a generalization of a classical Perron-type theorem for ordinary differential equations to (2.2), subject to (2.3) compared to (2.1) (see [1, Chapter IV, Theorem 5] and [28, Theorem 1.1] for functional differential equations which include several kinds of delays such as point and distributed delays and Volterra-type dynamics with infinite delays. The result extends the perturbation term to include constant upper-bounding terms in the perturbation functional (2.3) and characteristic zeros of the limiting (2.2) (i.e., zeros 𝜙∈𝐶_𝑒(−ℎ)) of multiplicity greater than unity (being degenerated or non-degenerated) in the limiting dynamics defined by (2.1).

Theorem 2.1. Let ||𝑓(𝑡,𝑥_𝑡)||_𝛼≤𝛾_𝛼(𝑡)||𝑥_𝑡||_𝛼+𝐾_0𝛼,𝑡∈𝐑₀₊(2.8) be a solution of (2.2), subject to (2.3), on 𝐾_0𝛼∈𝐑₀₊ subject to initial conditions 𝛾_𝛼∈𝐶⁽⁰⁾(𝐑₀₊,𝐑₀₊) such that for some norm-dependent 𝑡→∞ where 𝛽=0is also norm-dependent and satisfies 𝛽=1 as 𝜎_𝑘, where det𝑇⁻¹(𝑠) if there is no Volterra term in (2.1) and 𝜗_𝑘, otherwise, and lim_𝑡→∞^𝑡𝑡+1𝑔_𝛼(𝑠)|||𝑓𝑠,𝑥^𝑠𝑥_𝑠|||_𝛼−𝐾_0𝛼𝑑𝑠=0,(2.9) are the real parts of the zeros of 𝑔_𝛼∈𝑃𝐶⁽⁰⁾(𝐑₀₊,𝐑₀₊) of the limiting (2.2) with respective multiplicities 𝑔_𝛼(𝑡)=⎧⎪⎨⎪⎩0if𝛾^0𝛼(𝑡)=|||𝑓𝑡,𝑥^𝑡𝑥_𝑡|||_𝛼1otherwise.−𝐾_0𝛼≤0,(2.10). Then, the following properties hold:
(i) where 𝑘≥𝑘₁ is an indicator function defined by
(ii) The real numbers 𝜇_𝑘=0,∀𝑘>𝑘₁ exist and are norm-independent and finite, for all lim_𝑡→∞𝑒^𝑏𝑡𝑥(𝑡)=0 and some integer 𝑏∈𝐑 with 𝜇_𝑘1≥0 or 𝐾_0𝛼=0, 𝜇_𝑘1. If det𝑇⁻¹(𝑠) or if (2.8) holds with 𝑇(𝑠), then either lim_𝑡→∞𝑒^𝑏𝑡𝑥(𝑡)=0 it is the real part of a zero of 𝑏∈𝐑, for which the resolvent det𝑇⁻¹(𝑠) trivially exists and it is bounded, or 𝐾₀𝐑₊, for all 𝜇_𝑘=𝜇_𝑘(𝑥)=lim_𝑡→∞(log|𝑥_𝑡|)/𝑡^𝑘=0.
(iii) Assume that all the zeros of ∀𝑘≥𝑘₁ have real negative parts and (2.8) holds only for some constants 𝑘₁≥1. Then, either the limits 𝜇₁ exist, lim_𝑡→∞𝑒^𝑏𝑡𝑥(𝑡)=0, some integer ∀𝑏∈𝐑., and furthermore, 𝛾_𝛼(𝑡)≥max(0,|𝑓(𝑡,𝑥_𝑡)/𝑥_𝑡|_𝛼−𝐾_0𝛼)=𝑔_𝛼(𝑡)(0,|𝑓(𝑡,𝑥_𝑡)/𝑥_𝑡|_𝛼−𝐾_0𝛼)≥0 is not trivially the real part of a characteristic zero of (2.2), or 0=lim_𝑡→∞^𝑡𝑡+1𝛾_𝛼(𝑠)𝑑𝑠≥limsup_𝑡→∞^𝑡𝑡+1𝑔_𝛼(𝑠)|||𝑓𝑠,𝑥^𝑠𝑥_𝑠|||_𝛼⟹∃lim_𝑡→∞−𝐾_0𝛼𝑑𝑠≥0^𝑡𝑡+1𝑔_𝛼(𝑠)|||𝑓𝑠,𝑥^𝑠𝑥_𝑠|||_𝛼−𝐾_0𝛼𝑑𝑠=0,(2.11), |𝑓(𝑡,𝑥_𝑡)|_𝛼=𝛾_𝛼(𝑡)|𝑥_𝑡|_𝛼+𝐾_0𝛼−𝜔_𝛼(𝑡)

Proof. (i) From (2.8), 𝑡∈𝐑₀₊ so thatand property (i) has been proved.
(ii) From (2.8), ‖‖𝑥(𝑡)‖‖_𝛼≤𝐾₁(𝛼)𝑡^𝜈−1𝜈!𝑒^𝜇𝑡‖‖𝑥0⁺‖‖_𝛼+|||𝑒^𝜇𝑡(1−𝑒^𝜇ℎ)𝜇||||𝜙|^𝛼+𝛽max1,𝑒_𝜇𝑡+𝐾_0𝛼sup_{0≤𝜏≤𝑡}||𝑥_𝑡||_𝛼^𝑗𝑡_𝑙=0^ℓℓ+1𝛾_𝛼(𝑠)𝑑𝑠+^𝑡𝑗_𝑡𝛾_𝛼(𝑠)𝑑𝑠(2.12), 𝛾_𝛼, some 𝜀_𝛼∈𝐑₊ Then, one gets from (2.6)-(2.7), from the limiting hypothesis on the integral of the function 𝑡₀∈𝐑₊, for any arbitrary small real norm-dependent constant 𝜀_𝛼, there exists a finite 𝛼 ( only for a simple constructive proof easily extendable to 𝑡₀), dependent on ‖‖𝑥(𝑡)‖‖𝛼≤𝐾₁(𝛼)𝑡−𝑡⁰^𝜈−1𝜈!𝑒^{𝜇(𝑡−𝑡0)}‖‖𝑥(𝑡₀₊)‖‖_𝛼+|||𝑒^{𝜇(𝑡−𝑡0)}1−𝑒^𝜇ℎ𝜇||||𝜙|^𝛼+𝛽𝜀_𝛼+𝐾_0𝛼1,𝑒^𝜇𝑡sup_{𝑡−𝑡0≤𝜏≤𝑡}||𝑥_𝑡||_𝛼(2.13) and the given 𝜇-norm, such that one gets from (2.12) by taking initial conditions at 𝑇⁻¹(𝑠): with 𝑗_𝑡=max(𝑧∈𝐙₀₊∶𝑡≥𝑗_𝑡) being the real part of a characteristic zero of 𝑡 of multiplicity 𝑥(𝑡) and 𝑡₀∈𝐙₊ is dependent on 𝑥(𝑡_𝑘). Note that, if the solution {𝑡_𝑘}^∞0 is unbounded for the given initial conditions, then there exist, by construction, a finite and 𝑡_𝑘→∞ and a subsequence 𝑘→∞ valued at the real increasing sequence ‖𝑥(𝑡_𝑘)‖_𝛼=sup_{𝑡−𝑡0≤𝜏≤𝑡𝑘}(|𝑥_𝜏|_𝛼) (then 𝑔_𝛼∈𝑃𝐶⁽⁰⁾(𝐑₀₊,𝐑₀₊) as ‖‖𝑥𝑡_𝑘‖‖_𝛼=≤1s−up𝛽𝜀_{𝑡𝑘−𝑡0≤𝑡≤𝑡𝑘𝛼}||𝑥_𝑡||_𝛼𝐾₁(𝛼)⁻¹𝐾¹(𝛼)𝑡^𝑘−𝑡₀^𝑣−1×𝑒𝑣!_{𝜇𝑡(𝑡𝑘−𝑡0)}‖‖𝑥𝑡₀₊‖‖_𝛼+|||𝑒^{𝜇(𝑡𝑘−𝑡0)}1−𝑒^𝜇ℎ𝜇||||𝜙|^𝛼+𝐾_0𝛼(2.14)) such that 𝜀_𝛼 so that from (2.13) and for some bounded vector function 1>𝜀_𝛼𝐾₁(𝛼), provided that 𝜀_𝛼 is sufficiently small to guarantee 𝛽=0 in the case that 𝜇≠0 and independently of 𝜇>0 if sup_{𝑡𝑘−𝑡0≤𝑡≤𝑡𝑘}(|𝑥_𝑡|_𝛼). Furthermore, if sup_{𝑡𝑘−𝑡0≤𝑡≤𝑡𝑘}(|𝑥_𝑡|_𝛼) then 𝜇≤0 if the solution is unbounded since, otherwise, 𝜇=0 is bounded from (2.8) which contradicts the made assumption that it is unbounded. The equivalent contrapositive proposition to the last above one is that if 𝜈=1 is uniformly bounded then 𝜇. Equivalently, if furthermore 𝑇⁻¹(𝑠), then 𝜇 (i.e., (𝑡−𝑡₀)^𝜈−1/𝜈!→∞ is the real part of a simple real characteristic zero of 𝑡→∞ associate with the limiting equation (2.2) or there are two simple complex conjugate ones with real part 𝜈≠1). Otherwise, some unbounded lower-bound may be obtained similarly to (2.13) with the replacement of one of the plus signs in the right-hand-side terms with a minus sign affecting some unbounded term caused by 𝜇>0 as {𝑡_𝑘}^∞0 if sup_{𝑡𝑗−𝑡0≤≤𝑡𝑘}||𝑥_𝑡𝑗||_𝛼=𝐾₁(𝛼)𝑡^𝑘−𝑡₀^𝜈−1𝜈!𝑒^{𝜇(𝑡𝑘−𝑡0)}‖‖𝑥𝑡₀₊‖‖𝛼+|||𝑒^{𝜇(𝑡𝑘−𝑡0)}(1−𝑒^𝜇ℎ)𝜇|||||𝜙||^𝛼+𝛽𝜀_𝛼+𝐾_0𝛼≤1−𝛽𝜀_𝛼sup_{𝑡𝑘−𝑡0≤𝜏≤𝑡𝑘}||𝑥_𝑡𝑘||_𝛼−𝑔_𝛼𝑡₀,𝑡_𝑘𝐾₁(𝛼)⁻¹𝐾¹(𝛼)𝑡^𝑘−𝑡₀^𝜈−1×𝑒𝜈!_{𝜇(𝑡𝑘−𝑡0)}‖‖𝑥𝑡₀₊‖‖_𝛼+|||𝑒^{𝜇(𝑡𝑘−𝑡0)}1−𝑒^𝜇ℎ𝜇|||||𝜙||^𝛼−𝑔_𝛼+𝐾_0𝛼(𝑡₀,𝑡_𝑘)(2.15). This implies that the solution is unbounded which contradicts the fact that it is bounded. Note from (2.13) that if sup_{𝑡𝑘−𝑡0≤𝑡≤𝑡𝑘}(|𝑥_𝑡|_𝛼)=((𝑡_𝑘−𝑡₀)^𝜈−1/𝜈!)𝑒^{𝜇(𝑡𝑘−𝑡0)}𝑀−𝑔_𝛼(𝑡₀,𝑡_𝑘) then real increasing sequence 𝑀∈𝐑₀₊ of (2.13): which takes the form 𝐾_1𝛼, where 𝐾_0𝛼 depends on 𝛽 (finite), 𝜀_𝛼, |𝜙_𝛼|, 𝛼, 𝑔_𝛼∈𝑃𝐶⁽⁰⁾(𝐑₀₊,𝐑₀₊), 𝑡₀ , and the 𝛽-norm, for some bounded vector function 𝜀_𝛼 which depends on 𝜀_𝛼, the initial conditions, 1>𝜀_𝛼𝐾₁(𝛼) and 𝛽=1, provided that 𝜀_𝛼 is sufficiently small to guarantee 𝛽=0, in the case that 𝑋(𝑡), and independently of lim_𝑡→∞𝑒^𝑏𝑡𝑥(𝑡)≠0 if 𝑏∈𝐑. Assume that the solution limsup_{𝑡𝑘−𝑡0≤𝑡≤𝑡𝑘}(ln|𝑥_𝑡|_𝛼/𝑡^𝜈)=𝜇>0 is not a trivial solution what is guaranteed if 𝑡₀∈𝐙₀₊, for all 𝛼. Then, it follows from (2.15) that lim_𝑡→∞(ln𝑔_𝛼(𝑡₀,𝑡)/𝑡^𝜈)=0 for sufficiently large 𝛼, irrespective, of the 𝐙₀₊∋𝑡₀→∞-norm, since lim_𝑡→∞(ln|𝑥_𝑡|_𝛼/𝑡^𝜈)=𝜇>0 for any 𝜇=0-norm. By taking 𝜈>1, it follows that 𝑇⁻¹(𝑠). The result may be also extended to the case 𝜇, since then, the solution is either unbounded (for some initial conditions and multiplicity 𝜈=1of the characteristic zero of 𝜇=0 whose real part is 𝜇≥0), or it is bounded (in particular, always if 𝑏∈𝐑 for 𝑒^𝑏𝑡𝑥(𝑡)). As a result, if 𝑡→∞ and there is no lim_𝑡→∞ln||𝑥_𝑡|_𝛼𝑡^𝜈=𝜇,lim_𝑡→∞ln||𝑥_𝑡|_𝛼𝑡^𝜈+ℓ=𝜇;∀ℓ∈𝐙₊since^𝑡𝑡+1𝑡^𝜗𝑘−1𝜗_𝑘!𝑒^{𝛽𝜎𝑘𝑡}𝛾_𝛼(𝑠)𝑑𝑠⟶0as𝑡⟶∞(2.16) such that 𝐾_0𝛼=0 converges to zero as 𝜇, then for all the characteristic zeros of the limiting equation (2.2) . If (2.8) holds, in particular, with ∫^𝑡𝑡+1𝑡^𝜗𝑘−1/𝜗_𝑘!𝑒^{𝛽𝜎𝑘𝑡}𝛾_𝛼(𝑠)𝑑𝑠→0, then the above result is also valid from (2.15) for a negative value of 𝑡→∞. Property (ii) has been proved.
(iii) If (2.8) does not hold for 𝑡∈𝐑₊ and all the characteristic zeros of the limiting equation (2.2) have negative real parts then it follows by using a close reasoning to that used in (ii) that the solution cannot converge asymptotically to zero but it is uniformly bounded from (2.15) since 𝜇_𝑘=0 as 𝑘≥𝑘₁ and such an integral is bounded, for all 𝜇_𝑘1. Thus, 𝜇_𝑘1, for all 𝑓(𝑡,𝑥_𝑡) and 𝐾_0𝛼=0 is not the real part of a characteristic zero of the limiting equation (2.2) since it is not a negative real number.

The real limit 𝐾_0𝛼=0 of Theorem 2.1(ii)-(ii), provided that it exists, is called the strict Lyapunov exponent of the solution of (2.2)-(2.3) with the perturbation function Λ, subject to the hypotheses of Theorem 2.1, which is the real part of an eigenvalue (or characteristic zero) of the limiting equation (2.1) if either it is positive or if it takes any arbitrary value in the case that (2.8) holds for 𝑃_Λ (Theorem 2.1(ii). If all the characteristic zeros of (2.1) have negative real parts but (2.8) is not fulfilled with 𝑄_Λ then the strict Lyapunov exponent, if it exists, is zero so that it is not the real part of a characteristic zero of the limiting equation (2.1) (Theorem 2.1(iii). The main extension of Theorem 2.1 for the very general functional differential equation (2.2)-(2.3) with respect to parallel previous results (see [27, Chapter IV, Theorem 5 for ordinary differential equations]; [28, Theorem 1.1, for functional differential equations] and [29]) is that the perturbation function in (2.8) is not vanishing for bounded solutions or slightly growing solutions since any bounded functions are primarily admitted as perturbations in (2.2). The extension concerning the result in [27] is restricted to the form of (2.1) which involves a wide type of delayed dynamics involving, in general, any finite numbers of point delays, finite-distributed, delays and delays generated by Volterra-type dynamics.

A notation for the subsequent lemma and theorem is the following (see [3, Chapter 7]). If Λ is a finite set of eigenvalues of (2.1), then 𝐶(𝐑₀₊) and 𝐶(𝐑₀₊) denote the generalized eigenspace associated with Λ and the corresponding complementary subspace of 𝐶(𝐑₀₊)=𝑃_Λ(𝐑₀₊)⊕𝑄_Λ(𝐑₀₊), respectively. The phase space 𝑥∈𝐶(𝐑₀₊) is decomposed by 𝜙∈𝐶_𝑒(−ℎ) into the direct sum 𝑥^{𝑃Λ(𝐑0+)}. The projections of the solution 𝑥^{𝑄Λ(𝐑0+)} of (2.2), subject to (2.3), for any initial condition ∀𝑡∈𝐑₀₊, onto the above subspaces are denoted by 𝐶_𝑒(−ℎ) and 𝑡∈𝐑₀₊, respectively, 𝐶(𝐑₀₊). Note that, although the initial conditions of (2.2)-(2.3) are in general in 𝐑_−ℎ, the corresponding unique solution of (2.2), subject to (2.3), for 𝜙∈𝐶_𝑒(−ℎ) are in 𝑥(𝑡)=𝜙(𝑡). The whole solutions in ∀𝑡∈[−ℎ,0] which includes any given initial condition 𝑡∈𝐑₀₊ then satisfying 𝐶_𝑒(𝐑_−ℎ)=𝑃_Λ(𝐑_−ℎ)⊕𝑄_Λ𝑒(𝐑_−ℎ), 𝑄_Λ𝑒(𝐑_−ℎ), and the differential equation (2.2), subject to (2.3), for 𝑃_Λ(𝐑_−ℎ) are in 𝐶_𝑒(𝐑_−ℎ) where 𝑥^{𝑃Λ(𝐑−ℎ)} is the complementary subspace of 𝑥^{𝑄Λ𝑒(𝐑−ℎ)} in ∀𝑡∈𝐑_−ℎ. The projections of the solution onto those subspaces are 𝑥(𝑡)=𝜙(𝑡) and ∀𝑡∈[−ℎ,0], respectively, 𝜙∈𝐶_𝑒(−ℎ). The following technical result follows.

Lemma 2.2. Assume that the initial condition of (2.2)-(2.3) is 𝐑₀₊, 𝜙(𝑡) for any given ∀𝑡∈[−ℎ,0]. The unique solution of (2.2), subject to (2.3) on 𝑥𝑥_𝑡+ℎ𝑡=𝑥^{𝑃Λ(𝐑0+)𝑡+ℎ}+𝑥^{𝑄Λ(𝐑0+)𝑡+ℎ};∀𝑡∈𝐑₀₊,𝑥₀=𝑥^{𝑃Λ(𝐑−ℎ)𝑡}+𝑥^{𝑄Λ𝑒(𝐑−ℎ)𝑡};∀𝑡∈𝐑₀₊,=𝜙^{𝑃Λ(𝐑−ℎ)0}=𝜙^{𝑃Λ(𝐑0+)0}=𝜙^{𝑃Λ(𝐑0+)0}=𝑥^{𝑃Λ(𝐑−ℎ)0}+𝑥^{𝑄Λ𝑒(𝐑−ℎ)0}=𝜙^{𝑃Λ(𝐑−ℎ)0}+𝜙^{𝑄Λ𝑒(𝐑−ℎ)0}.(2.17), and identified with 𝐶(𝐑₀₊)=𝑃_Λ(𝐑₀₊)⊕𝑄_Λ(𝐑₀₊) 𝑡(≥ℎ)∈𝐑₀₊, satisfies with unique decompositions:

Proof. The first relation follows from 𝑄_Λ(𝐑₀₊), and the superposition principle for linear systems building the solution for 𝐶(𝐑₀₊) by projecting the function of initial conditions into the complementary subspaces 𝑥(𝑡)=𝜙(𝑡) and 𝑡∈[−ℎ,0] in 𝐶_𝑒(𝐑_−ℎ)=𝑃_Λ(𝐑_−ℎ)⊕𝑄_Λ𝑒(𝐑_−ℎ) subject to the constraint 𝑥(𝑡)=𝜙(𝑡), for all 𝑡∈[−ℎ,0]. The second relation follows from 𝑥₀=𝑥(0)=𝜙(0) again from the superposition principle with 𝑡=0, for all 𝑡≥0. The third relation follows from 𝑥(𝑡) and the superposition principle applied to the solution at 𝑃_Λ(𝐑₀₊).
The intuitive meaning of Lemma 2.2 is that for 𝐶(𝐑₀₊), 𝐶_𝑒(−ℎ) is decomposed uniquely as a sum of a function in 𝐶(−ℎ) and another one in its complementary in 𝑄_Λ𝑒(𝐑_−ℎ), even for initial conditions in 𝑃_Λ(𝐑_−ℎ), rather than in the more restrictive set 𝐶(𝐑_−ℎ). However, the complementary set 𝑄_Λ(𝐑₀₊) of 𝑡≤0 in 𝑥₀=𝜙₀ replaces 𝜙∈𝐶_𝑒(−ℎ) 𝑥^(𝐑−ℎ)_𝑡=𝜙^{𝑃Λ(𝐑−ℎ)𝑡}+𝜙^{𝑄Λ𝑒(𝐑−ℎ)𝑡}, since ∀𝑡∈𝐑₀₊ for any given 𝜙∈𝐶(−ℎ). Note that 𝜇(𝑥)=𝜇_𝑘1=𝜇, 𝑃₀(𝐑₀₊)=𝑃_Λ0(𝐑₀₊) is untrue except for 𝑃₁(𝐑₀₊)=𝑃_Λ1(𝐑₀₊).

Theorem 2.3. Let x be a solution of (2.2)-(2.3) satisfying the hypotheses of Theorem 2.1 with a finite strict Lyapunov exponent 𝑄(𝐑₀₊)=𝑄_Λ(𝐑₀₊). Consider generalized eigenspaces 𝑡∈𝐑₀₊, 𝑃₀(𝐑_−ℎ)=𝑃_Λ0(𝐑_−ℎ) and 𝑃₁(𝐑_−ℎ)=𝑃_Λ1(𝐑_−ℎ) for 𝑄_𝑒(𝐑_−ℎ)=𝑄_Λ𝑒(𝐑_−ℎ) and, also, generalized eigenspaces 𝑡∈𝐑_−ℎ, Λ₀ and Λ₁ for Λ, where the spectral sets ΛΛ₀=Λ₀(𝜇)∶=𝜆∶det𝑇⁻¹(𝜆)=0,𝑅𝑒𝜆=𝜇,(2.18)₁Λ=Λ₀=Λ₁(𝜇)∶=𝜆∶det𝑇⁻¹(𝜆)=0,𝑅𝑒𝜆>𝜇,(2.19)∪Λ₁=Λ(𝜇)∶=𝜆∶det𝑇⁻¹(𝜆)=0,𝑅𝑒𝜆≥𝜇.(2.20), Λ₀≠∅,Λ≠Φ and 𝜇∈𝐑₀₊ each generating the two corresponding eigenspaces, are defined by,
Then, the following properties hold,
(i) (i.1)Λ=Λ₀=Λ₁=∅ if 𝐾_0𝛼=0,(i.2)Λ₁=∅ if 𝐑₀₊ and, furthermore, (2.8) holds with 𝐑₊,(i.3)𝐾_0𝛼=0 if all the eigenvalues of (2.1) have negative real parts and, furthermore, (2.8) does not hold with 𝐾_0𝛼=0,(i.4)𝜙∈𝐶_𝑒(−ℎ) if any of the following conditions hold: (1) No eigenvalue of (2.1) is in 𝑥𝑥_𝑡+ℎ𝑡=𝑥^{𝑃0(𝐑0+)𝑡+ℎ}+𝑥^{𝑃1(𝐑0+)𝑡+ℎ}+𝑥^{𝑄(𝐑0+)𝑡+ℎ};∀𝑡∈𝐑₀₊,𝑥₀=𝑥^{𝑃0(𝐑−ℎ)𝑡}+𝑥^{𝑃1(𝐑−ℎ)𝑡}+𝑥^{𝑄𝑒(𝐑−ℎ)𝑡};∀𝑡∈𝐑₀₊,=𝜙^{𝑃0(𝐑−ℎ)0}=𝑥^{𝑃0(𝐑0+)0}=𝜙^{𝑃0(𝐑0+)0}=𝑥^{𝑃0(𝐑−ℎ)0}+𝑥^{𝑃1(𝐑−ℎ)0}+𝑥^{𝑄𝑒(𝐑−ℎ)0}=𝜙^{𝑃0(𝐑−ℎ)0}+𝑥^{𝑃1(𝐑−ℎ)0}+𝜙^{𝑄𝑒(𝐑−ℎ)0}.(2.21), (2) No eigenvalue of (2.1) is in 𝜙∈𝐶_𝑒(−ℎ) and, furthermore, (2.8) does not hold with 𝑥^{𝑃1(𝐑0+)𝑡+ℎ}=𝑂||𝑥^{𝑃0(𝐑0+)𝑡+ℎ}||,𝑥^{𝑄(𝐑0+)𝑡+ℎ}=𝑂||𝑥^{𝑃0(𝐑0+)𝑡+ℎ}||,𝑥_𝑡+ℎ=𝑂||𝑥^{𝑃0(𝐑0+)𝑡+ℎ}||;∀𝑡∈𝐑₀₊𝑥^{𝑃1(𝐑−ℎ)𝑡}=𝑂||𝑥^{𝑃0(𝐑−ℎ)𝑡}||,𝑥^{𝑄𝑒(𝐑−ℎ)𝑡}=𝑂||𝑥^{𝑃0(𝐑−ℎ)𝑡}||,𝑥_𝑡=𝑂||𝑥^{𝑃0(𝐑−ℎ)𝑡}||;∀𝑡∈𝐑₀₊(2.22), (3) (2.8) holds with 𝐾_0𝛼=0.
(ii) The solution of (2.2) under arbitrary initial conditions 𝑡→∞, subject to a perturbation function (2.3), satisfies,
(iii) The solution of (2.2) under arbitrary initial condition 𝑡→∞, subject to a perturbation function (2.3), satisfies, Furthermore, if (2.8) holds with 𝑥_𝑡+ℎ=𝑥^𝑃0_𝑡+ℎ+𝑥^𝑃1_𝑡+ℎ+𝑥^𝑄𝑡+ℎ;∀𝑡∈𝐑₀₊𝑥_𝑡=𝑥^𝑃0_𝑡+𝑥^𝑃1_𝑡+𝑥^𝑄𝑒_𝑡;∀𝑡∈𝐑₀₊(2.24) then, as 𝑥_𝑡=𝑥^𝑃0_𝑡+𝑥^𝑃1_𝑡+𝑥^𝑄𝑡;∀𝑡∈𝐑_−ℎ(2.25):
(iv) The solution of the limiting equation (2.1) satisfies (2.23) as 𝑥(𝑡)=𝜙(𝑡).
(v) The solution of (2.2), under arbitrary initial conditions 𝑥^𝑃1_𝑡=𝑂||𝑥^𝑃0_𝑡||,𝑥^𝑄𝑡=𝑂||𝑥^𝑃0_𝑡||,𝑥_𝑡=𝑂||𝑥^𝑃0_𝑡||;∀𝑡∈𝐑₀₊,∀𝜙∈𝐶_𝑒(−ℎ).(2.26) and subject to a perturbation function (2.3), satisfies: which is identical to under the restriction 𝑥^𝑃1_𝑡+ℎ=𝑜||𝑥^𝑃0_𝑡||,𝑥^𝑄𝑡+ℎ=𝑜||𝑥^𝑃0_𝑡+ℎ||,𝑥_𝑡=𝑜||𝑥^𝑃0_𝑡+ℎ||;∀𝑡∈𝐑₀₊,(2.27) for the initial conditions with 𝑥^𝑃1_𝑡=𝑜||𝑥^𝑃0_𝑡||,𝑥^𝑄𝑡=𝑜||𝑥^𝑃0_𝑡||,𝑥_𝑡=𝑜||𝑥^𝑃0_𝑡||;∀𝑡∈𝐑₀₊(2.28). Also, If (2.8) holds with 𝑥(𝑡)=𝜙(𝑡) then, as 𝑡→∞: which leads to under the restriction 𝜆 for the initial conditions with 𝜇∈𝐑₀₊⋀Λ₀=∅⇔𝜆̂∶=max(Re(𝜆)∶𝜆∈𝐶𝑍)≠𝜇≥0.The solution of the limiting equation (2.1) satisfies (2.27)-(2.28) as 𝐶𝑍⊂𝐂.

Proof. Properties (i) are direct consequences of Theorem 2.1 [(ii)-(iii)] as follows. Property (i.1) follows by noting that 𝜆̂≠𝜇=0 for some Λ⊃Λ₀≠∅ implies that 𝜆̂≠𝜇>0 is a characteristic zero of (2.1) of nonnegative real part from (2.7). Assume that 0<𝜇≠𝜆̂>0 where 𝜆̂=𝜇>0 is the set of characteristic zeros of the limiting equation (2.1). Thus, if Λ⊃Λ₀≠∅, then the current equation (2.2)-(2.3) is bounded while the limiting one (2.1) is either globally asymptotically stable or unstable so that they cannot converge asymptotically to each other which is a contradiction so that 𝐾_0𝛼=0⇒𝜆̂=𝜇⇒Λ₀≠∅. If Λ=Λ₀=Λ₁=∅⇒𝜆<𝜇 then the current equation is unstable which implies that the limiting one should satisfy 𝜆<𝜇<0 to be also unstable but the asymptotic convergence of their respective solutions to each other is only possible if 𝐾_0𝛼>0. Thus, again 𝐾_0𝛼=0 and property (i.1) has been proven. On the other hand, Λ₁=∅⟹𝜆̂≤𝜇<0∨𝜆̂≤𝜇=0∧𝐾_0𝛼=0∨0<𝜆̂≤𝜇>0∧𝐾_0𝛼=0(2.29) and property (i.2) is proven. Also, Λ₀. Since, in addition, (2.2)-(2.3) is the limiting equation of (2.1) then Λ₁ and 𝐶𝐶(𝐑₀₊)=𝑃₀(𝐑₀₊)⊕𝑃₁(𝐑₀₊)⊕𝑄_Λ(𝐑₀₊),_𝑒(𝐑_−ℎ)=𝑃₀(𝐑_−ℎ)⊕𝑃₁(𝐑_−ℎ)⊕𝑄_Λ𝑒(𝐑_−ℎ).(2.30). Therefore, 𝐾_0𝛼=0 and property (i.3) have been proven. Property (i.4) follows from: in order to the solutions of (2.2)-(2.3) and (2.1) to asymptotically to converge to each other. Property (i) has been fully proven.
Property (ii) is a direct consequence of Lemma 2.2 since ∀𝑡∈𝐑_−ℎ and ℎ are disjoint sets which implies that Property (iii) is directly proven as follows. Equations (2.22) are a direct consequence of property (ii). On the other hand, (2.23) are a direct consequence of (2.22) if (2.8) holds for 𝑡∈𝐑₀₊ so that property (iii) follows.
Property (iv) follows from property (iii) as particular case for 𝑥_𝑡; ‖‖𝑇(𝑡,0)‖‖_𝛼≤𝐾₁(𝛼)‖‖𝐼_𝑛‖‖_𝛼max1,𝑡^𝑣−1𝜈!𝑒^𝜇𝑡;∀𝑡∈𝐑⁰⁺||𝑇(𝑡,0)||_𝛼(2.31)≤𝐾₁(𝛼)‖‖𝐼_𝑛‖‖_𝛼max1,𝑡^𝑣−1𝜈!𝑒^𝜇𝑡max1,^𝜈−2_𝑖=0𝑡^{1−𝑣+𝑖}ℎ^{𝑣−1−𝑖}𝑒^𝜇ℎ‖‖𝑇̇(𝑡,0)‖‖_𝛼;∀𝑡∈𝐑0+(2.32)|𝑇(𝑡,0)|_𝛼𝐾₂(𝛼)sup_{0≤𝜏≤𝑡}||𝑇^{𝐶(𝐑0+)𝜏}||_𝛼;∀𝑡∈𝐑₀₊(2.33)≤≤𝐾𝐾²(₁𝛼)sup_{0≤𝜏≤𝑡}||𝑇^{𝐶(𝐑0+)𝜏}||_𝛼(2.34)(𝛼)𝐾₂(𝛼)‖𝐼_𝑛‖_𝛼max1,𝑡^𝜈−1𝜈!𝑒^𝜇𝑡max1,^𝜈−2_𝑖=0𝑡^{1−𝜈+𝑖}ℎ^{𝜈−1−𝑖}𝑒^𝜇ℎ;∀𝑡∈𝐑⁰⁺.(2.35) in (2.3).
Property (v) is a direct consequence of Properties (i)–(iv) In particular, the relative growing properties of “O”-type of the various parts of the solution of (2.2)-(2.3) are embedded from property (iii) into similar properties for the solution strings of length 𝑇̇(𝑡,0)=^𝑚_𝑖=0𝐴_𝑖𝑇(𝑡−ℎ_𝑖,0)+^𝑚′_𝑖=0^𝑡0𝑑𝛼_𝑖(𝜏)𝐴_𝛼𝑖𝑇(𝑡−ℎ^𝑖,𝜏)+^{𝑚′+𝑚′′}_𝑖=_𝑚+𝑙^0ℎ_′𝑖𝑑𝛼_𝑖(𝜏)𝐴_𝛼𝑖𝑇(𝑡,𝜏)(2.36). The part of property (v) concerning the relative growing properties of “o”-type of the various parts of the solution of (2.2)-(2.3) and that concerning the limiting equation follows directly under a close reasoning.
Note that in Theorem 2.3, the various results obtained for “Landau small-o” notation, referred to limits as 𝑡∈𝐑₀₊ imply, as usual, that parallel results for “Landau big-O” notation stand for all 𝑇(0,0)=𝐼_𝑛 but the converse is not true. The results concerning “Landau big-O” notation in Theorem 2.3(iii) for the perturbed functional equation (2.2)-(2.3) are new for the studied class of functional equations, related to the background literature, since the perturbation function is allowed to take bounded nonzero values even if the limiting equation is globally asymptotically stable and it is not requested to grow asymptotically at most linearly with 𝑛. The results concerning “Landau big-O” notation imply that the solution of the perturbed functional equation is uniformly bounded for any bounded function of initial conditions of the given class for all time so that the functional differential equation is globally uniformly Lyapunov stable provided that the perturbation (2.3) satisfies the given hypotheses. A technical result concerning the boundedness of the evolution operator, which will be then useful to derive further results, and stability properties of the differential systems (2.1) and (2.2)-(2.3) follows.

Theorem 2.4. The following properties hold:
(i) The evolution operator of the limiting functional differential equation (2.1) satisfies the subsequent relations.

Proof. (i) The evolution operator satisfies the limiting functional differential equation (2.1): for (𝛽𝜀_𝛼) subject to initial conditions ‖𝑇(𝑡,0)‖_𝛼=≤1su−p𝛽_{𝑡𝑘𝑡0≤𝑡≤𝑡𝑘}𝜀_𝛼||𝑥_𝑡||_𝛼𝐾₁(𝛼)⁻¹𝐾¹(𝛼)𝑡^𝑘−𝑡₀^𝜈1𝜈!𝑒^{𝜇(𝑡𝑘−𝑡0)}‖‖𝐼_𝑛‖‖_𝛼,(2.37) (i.e., the 𝑡→∞th identity matrix) and 𝜇, ‖‖𝑥(𝑡_𝑘)‖‖_𝛼=≤1s−up𝛽𝜀_{𝑡𝑘−𝑡0≤𝑡≤𝑡𝑘𝛼}||𝑥_𝑡|_𝛼𝐾₁(𝛼)¹×𝐾₁(𝛼)𝑡^𝑘−𝑡₀^𝜈−1𝜈!𝑒^{𝜇(𝑡𝑘−𝑡0)}‖‖𝑥𝑡⁺⁰‖‖_𝛼+||||𝑒^{𝜇(𝑡𝑘𝑡0)}1−𝑒^𝜇ℎ𝜇|||||𝜙|^𝛼,(2.38). Thus, it satisfies also the unforced (2.2) (i.e., for 𝑡→∞). This leads directly to (2.31). However, (2.32) follows by using the Newton binomial to expand 𝜇≤0 and the fact that the maximum of the real exponential function within the real interval ‖‖𝑥𝑡_𝑘‖‖_𝛼=≤1s−up𝛽𝜀_{𝑡𝑘−𝑡0≤𝑡≤𝑡𝑘𝛼}||𝑥_𝑡||_𝛼𝐾₁(𝛼)⁻¹×𝐾₁(𝛼)𝑡^𝑘−𝑡₀^𝜈−1𝜈!𝑒^{𝜇(𝑡𝑘−𝑡0)}‖‖𝑥𝑡⁺⁰‖‖_𝛼+||||𝑒^{𝜇(𝑡𝑘−𝑡0)}1−𝑒^𝜇ℎ𝜇|||||𝜙|^𝛼+𝐾_0𝛼,(2.39) is reached at the boundary. Equation (2.33) follows by the inspection of (2.36) for some norm-dependent 𝑡 which depends on the various matrices of parameters of the limiting functional differential equation (2.1). Equation (2.34) follows from (2.33) and (2.36). Finally, (2.35) follows from (2.34) and (2.32). Property (i) has been proved.
(ii) For sufficiently small constant 𝜇<0, the evolution operator as a function of time is of exponential order whose norm time-function satisfies: which converges exponentially to zero as 𝑇∶𝐑₀₊×𝐂^𝑛→𝐂^𝑛 if the strict Lyapunov exponent (𝑇_𝑠(𝑡))_{𝑡∈𝐑0+} is negative. In this case, the limiting differential functional equation is globally uniformly exponentially Lyapunov stable whose solution satisfies asymptotically: so that it converges exponentially to zero as 𝜙∈𝐶_𝑒(−ℎ) for any admissible function of initial conditions. The differential equation (2.2), subject to (2.3) is globally uniformly Lyapunov stable if 𝐴 and its solution satisfies: for large ∀𝜑∈Dom(𝐴)∶={𝜑∈𝐶_𝑒(𝐑_−ℎ)∶∃𝜑̇∈𝐶(𝐑₀₊),∀𝑡∈𝐑₀₊⋀𝜑̇(0)=𝐿𝜑(0)} and converges exponentially to zero (i.e., it is globally uniformly exponentially Lyapunov stable) if 𝑥_𝑡(𝜙)=(𝑇_𝑠𝜙)(𝑡) and the perturbation function has an upper-bounding function with [𝑡−ℎ,𝑡].
(ii) It follows directly from (2.12).

The evolution operator 𝑥_𝑡(𝜙𝑥)=𝑇_𝑠𝜙(𝑡)∶=𝑇(𝑡−𝜃,0)𝑥(0⁺)+∫^0−ℎ𝑇(𝑡−𝜃,𝜏)𝜙(𝜏)𝑈(𝜏)𝑑𝜏,∀𝜃∈[0,min(𝑡,ℎ)],_𝑡(𝜙)=𝑇_𝑠𝜙(𝑡)=0,∀𝜃∈]0,min(𝑡,ℎ)[∶=[0,min(𝑡,ℎ)]∩𝐑₀₊,∀𝑡∈𝐑₀₊;(2.40) explicits the solutions of the limiting equation (2.5) and the perturbed one (2.6) for each function of initial conditions. Then, let 𝑥_𝑡(𝜙)∶==𝑇𝑇(_𝑠𝑡𝜙−(𝜃𝑡,)0)𝑥0₊++^0−ℎ𝑇(𝑡−𝜃,𝜏)𝜙(𝜏)𝑈(𝜏)𝑑𝜏_𝑡0𝑥𝑇(𝑡−𝜃,𝜏)𝑓𝜏,𝑥_𝜏𝑑𝜏,∀𝜃∈[0,min(𝑡,ℎ)],_𝑡(𝜙)=𝑇_𝑠𝜙(𝑡)=0,∀𝜃∈[0,min(𝑡,ℎ)]∩𝐑;∀𝑡∈𝐑₀₊.(2.41) be the solution semigroup of the linear autonomous equation (2.1), which is unique for 𝑦(̇𝑡)=L^∗𝑦_𝑡∶=^𝑚_𝑖=0𝑦𝑡−ℎ_𝑖𝐴^∗𝑖+^𝑚′_𝑖=0^𝑡0𝑦𝑡−𝜏−ℎ^𝑖𝐴^∗𝛼_𝑖𝑑𝛼^∗𝑖(𝜏)+^{𝑚′+𝑚′′}_𝑖=_𝑚′+1^{𝑡𝑡−ℎ}_′𝑖𝑦(𝜏)𝐴^∗𝛼_𝑖𝑑𝛼^∗𝑖(𝜏),(2.42) for each 𝑦(𝑡) and whose infinitesimal generator is 𝑛 satisfying 𝐑₀₊, 𝐶^(𝐑₀₊)∶=𝐶(𝐑₀₊,𝐂^𝑛∗). Thus, the string 𝐶^(ℎ)∶=𝐶([0,ℎ],𝐂^𝑛∗) of the solution of the limiting functional differential equation (2.1) within 𝐶^𝑒(ℎ)∶=𝐶([0,ℎ],𝐂^𝑛∗) is defined from (2.5) as follows: and the corresponding solution string of the perturbed functional differential equation (2.2)-(2.3) is then defined follows: The transposed equation associated with (2.1) is where the superscript * denotes the adjoint operators of the corresponding un-superscripted ones. In particular, for matrices, it denotes the conjugate transposes of the corresponding un-superscripted ones. Thus, 𝑃_Λ is a 𝑛-dimensional complex row vector. The phase space for (2.42) on 𝐵_Λ is sp(𝐵_Λ)=sp(Λ). Corresponding spaces of functions taking into account the more general spaces for initial conditions are 𝐴Φ_Λ=Φ_Λ𝐵_Λ,Φ_Λ(𝜏)=Φ_Λ(0)𝑒^𝐵Λ𝜏(∀𝜏∈[−ℎ,0]),𝑇(𝑡,0)Φ_Λ=Φ_Λ𝑒^𝐵Λ𝑡.(2.43), 𝐵_Λ and 𝑒^𝐵Λ𝑡. Let 𝑇(0,0)=𝐼_𝑛 be a finite set of eigenvalues of (2.1) and let 𝑇(𝑡,0)=0 be a basis for the generalized eigenspace 𝑡<0, [27, 28]. Then, there exists a square 𝑡∈𝐑₀₊-matrix 𝑇𝑇̇((𝑡𝑡,,00))ΦΦ_ΛΛ𝐵_Λ=Φ_Λ𝑒^𝐵Λ𝑡𝐵_Λ=Φ_Λ𝐵_Λ𝑒^𝐵Λ𝑡=𝐴Φ_Λ𝑒^𝐵Λ𝑡=𝐴𝑇(𝑡,0)Φ_Λ,𝐵_Λ=Φ_Λ𝐵^2Λ𝑒^𝐵Λ𝑡=𝐴Φ_Λ𝐵_Λ𝑒^𝐵Λ𝑡=𝐴²Φ_Λ𝑒^𝐵Λ𝑡=𝐴Φ_Λ𝑒^𝐵Λ𝑡𝐵_Λ=Φ_Λ𝐵_Λ𝑒^𝐵Λ𝑡𝐵_Λ=𝑇(𝑡,0)Φ_Λ𝐵^2Λ=𝑇(𝑡,0)𝐴Φ_Λ𝐵_Λ=𝐴²𝑇(𝑡,0)Φ_Λ=𝐴𝑇(𝑡,0)Φ_Λ𝐵_Λ=𝐴𝑇(𝑡,0)𝐴Φ_Λ.(2.44), with 𝑇(𝑡,0)=𝑒^𝐴0𝑡𝐼^𝑛+^𝑡0𝑒^−𝐴0𝜏^𝑚_𝑖=1+𝐴_𝑖𝑇𝑡−ℎ_𝑖,𝜏_𝑚′_𝑖=0^𝜏0+𝑑𝛼_𝑖(𝜃)𝐴_𝛼𝑖𝑇𝜏−ℎ^𝑖,𝜃_{𝑚′+𝑚′′}_𝑖=_𝑚′+1^0ℎ_′𝑖𝑑𝛼_𝑖(𝜃)𝐴_𝛼𝑖𝑇(𝜏,𝜃)𝑑𝜏(2.45), such that the subsequent relations hold: The relations (2.43) yield via direct computation property (i) of the subsequent result since 𝑇(0,0)=𝐼_𝑛 commutes with 𝑇(𝑡,0)=0. Property (ii) is a direct consequence of (2.36) subject to 𝑡∈[−ℎ,0] and 𝑃_Λ for ∀𝑡∈𝐑.

Proposition 2.5. The two following properties hold.
(i)The following relations hold, for all 𝑇(𝑡,0)Φ_Λ𝑎=Φ_Λ𝑒^𝐵Λ𝑡𝑎: (ii)The evolution operator of the solution of (2.1) is uniquely given by𝑄_Λ with 𝑃_Λ and 𝐑₀₊ for 𝐑_−ℎ.

Equations (2.43)–(2.45) are useful for the asymptotic analysis of comparison of the solutions of (2.2)-(2.3) with that of its limiting equation obtained from (2.1) which follows. The solutions of 𝐶_𝑒(−ℎ) can be extended to 𝐶_𝑒(𝐑_−ℎ)=𝑃_Λ(𝐑_−ℎ)⊕𝑄_Λ𝑒(𝐑_−ℎ) by 𝐶(𝐑₀₊)=𝑃_Λ(𝐑₀₊)⊕𝑄_Λ(𝐑₀₊), where 𝜙∈𝐶_𝑒(−ℎ) is of dimension compatible with the order of 𝜙₀=𝜙_𝑡∣_𝑡=0=𝜙^{𝑃Λ(𝐑−ℎ)0}+𝜙^{𝑄Λ𝑒(𝐑−ℎ)0}. Let 𝐑₀₊ be the complementary eigenspace to 𝑥𝑥_𝑡+ℎ=𝑥^{𝑃Λ(𝐑0+)𝑡+ℎ}+𝑥^{𝑄Λ(𝐑0+)𝑡+ℎ},(2.46)_{𝑃Λ(𝐑0+)𝑡+ℎ}=Φ_ΛΨ_Λ,𝑥_𝑡+ℎ∈𝑃_Λ𝐑₀₊,𝑥^{𝑄Λ(𝐑0+)𝑡+ℎ}=𝑥_𝑡+ℎ−𝑥^{𝑃Λ(𝐑0+)𝑡+ℎ}∈𝑄_Λ𝐑₀₊∀𝑡∈𝐑⁰⁺,(2.47). Now, use appropriate notations for the corresponding subspaces on 𝐶(𝐑₀₊)=𝑃_Λ(𝐑₀₊)⊕𝑄_Λ(𝐑₀₊) and their extensions to 𝑥(𝑡)=𝜙(𝑡) to consider more general initial conditions (on 𝑡∈[−ℎ,0]) for (2.1) and (2.2)-(2.3) than bounded continuous functions in a Banach space leading to the uniquely defined decompositions 𝐑_−ℎ and 𝑥𝑥_𝑡=𝑥^{𝑃Λ(𝐑−ℎ)𝑡}+𝑥^{𝑄Λ𝑒(𝐑−ℎ)𝑡},(2.48)_{𝑃Λ(𝐑−ℎ)𝑡}=Φ_ΛΨ_Λ,𝑥_𝑡∈𝑃_Λ𝐑_−ℎ,𝑥^{𝑄Λ𝑒(𝐑−ℎ)𝑡}=𝑥_𝑡−𝑥^{𝑃Λ(𝐑−ℎ)𝑡}∈𝑄_Λ𝑒𝐑_−ℎ∀𝑡∈𝐑⁰⁺(2.49). Then, given a function of initial conditions 𝐶_𝑒(𝐑_−ℎ)=𝑃_Λ(𝐑_−ℎ)⊕𝑄_Λ𝑒(𝐑_−ℎ) the decomposition 𝜃∈[𝑡−ℎ,𝑡] is unique. Also, the unique solution of (2.1) and that of (2.2), subject to (2.3), are uniquely decomposable in 𝑥(𝑡)=𝜙(𝑡) as via the direct sum of subspaces 𝑥(𝑡+𝜃)=𝑇(𝑡+𝜃,0)𝑥0⁺+^0−ℎ𝑇(𝑡+𝜃,𝜏)𝜙(𝜏)𝑈(𝜏)𝑑𝜏+^0𝑡+𝜃𝑇(∀𝑡𝜃+∈𝜃,[−𝜏)ℎ𝑓,0𝜏],,𝑥𝑡_𝜏≥𝑑ℎ𝜏,,(3.1). The solution iincluding initial conditions defined by 𝑥_𝑡(𝜙,𝜃)==𝑥𝑇^∗𝑡(_𝑠𝜙,𝜃)+^0𝑡+𝜃𝑑𝐾(𝑡+𝜃,𝜏)𝑓𝜏,𝑥_𝜏=𝑇(𝑡(+𝑡,𝜃0,)𝜙0)𝜙(𝜃0)₊+^𝑡0𝑑𝐾(𝑡+𝜃,𝜏)𝑓𝜏,𝑥_𝜏+^0−ℎ𝑇(𝑡+𝜃,𝜏)𝜙(𝜏)𝑈(𝜏)𝑑𝜏+^0𝑡+𝜃𝑇(𝑡+𝜃,𝜏)𝑓𝜏,𝑥_𝜏𝑑𝜏,(3.2) for 𝑥^∗(⋅)∶[𝑡−ℎ,𝑡]×𝐂^𝑛×[−ℎ,0]→𝐂^𝑛 is uniquely decomposable in 𝑥^∗𝑡(𝜙,𝜃)∶==𝑇𝑇(_𝑠𝑡(+𝑡,0𝜃))𝜙𝜙(0𝜃₊)+_0−ℎ𝑇𝑇(𝑡+𝜃,𝜏)𝜙(𝜏)𝑈(𝜏)𝑑𝜏,∀𝜃∈[−ℎ,0],𝑡≥ℎ,_𝑠(0,0)𝜙(𝜃)=𝑥₀(𝜙,𝜃)=𝜙(𝜃),∀𝜃∈[−ℎ,0],(3.3) as via the direct sum of subspaces 𝑇_𝑠(𝑡,0).

3. Asymptotic Behavior and Asymptotic Comparison

The string solution (2.6) of (2.2)-(2.3) for ∀𝑡∈𝐑₀₊, point-wise defined by 𝐾(𝑡,𝑠)(𝜃)=^𝑠0𝑋(𝑡+𝜃−𝜏)𝑑𝜏,∀𝜃∈[−ℎ,0],∀𝑡∈𝐑₀₊,(3.4), 𝑋, any given 𝑋₀(0)=𝐼_𝑛, and may be expressed equivalently via the solution semigroup of the limiting equation (2.1) as where 𝑥𝑥_𝑡=𝑥_𝑡(𝜙,𝜃)=𝑥^𝑃Λ_𝑡+𝑥^𝑄Λ_𝑡,𝐑₊∋𝑡≥ℎ,(3.5)_𝑃Λ𝑡==𝑇𝑇_𝑠(𝑡,0)𝜙^𝑃Λ(𝜃)+^𝑡0𝑇_𝑠(𝑡,𝜏)𝑓𝜏,𝑥_𝜏^𝑃Λ(𝜃)𝑑𝜏_𝑠=𝑇(𝑡(+𝑡,𝜃0,)𝜙0)𝜙^𝑃Λ_𝑃Λ(𝜃)+^𝑡0𝑇_𝑠(𝑡,𝜏)𝑋^𝑃Λ₀(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏+0⁺+^0−ℎ𝑇(𝑡+𝜃,𝜏)𝜙^𝑃Λ(𝜏)𝑈(𝜏)𝑑𝜏_𝑡0𝑥𝑇(𝑡+𝜃,𝜏)𝑋^𝑃Λ₀𝑓𝜏,𝑥_𝜏𝑑𝜏,(3.6)_𝑄Λ𝑡==𝑇𝑇_𝑠(𝑡,0)𝜙^𝑄Λ(𝜃)+^𝑡0𝑇_𝑠(𝑡,𝜏)𝑓𝜏,𝑥_𝜏^𝑄Λ(𝜃)𝑑𝜏,𝐑₊∋𝑡≥ℎ_𝑠𝑥(𝑡,0)𝜙^𝑄Λ𝑒(𝜃)+^𝑡0𝑑[𝐾(𝑡,𝜏)]^𝑄Λ(𝜃)𝑓𝜏,𝑥_𝜏,𝐑₊∋𝑡≥ℎ,(3.7)_𝑡=𝑇_𝑠(𝑡,0)𝜙^𝑃Λ+𝜙^𝑄Λ𝑒(𝜃)+^𝑡0𝑇_𝑠(𝑡,𝜏)𝑋^𝑃Λ₀+𝑑[𝐾(𝑡,𝜏)]^𝑄Λ(𝜃)𝐑𝑓𝜏,𝑥_𝜏𝑑𝜏,₊∋𝑡≥ℎ,(3.8), defined bywith 𝜙∈𝐶_𝑒(−ℎ) is the unique solution of the limiting equation (2.1), and the kernel 𝜀∈𝐑₊ of ||𝑇(𝑡,0)𝜙^𝑃Λ||≤||𝑇𝑀(𝑡,₁𝑒0)𝜙^{(𝜇−𝜀)𝑡}_𝑄Λ𝑒||𝜙^𝑃Λ||,||𝑇(𝑡,0)𝑋^𝑃Λ₀||≤𝑀₁𝑒^{(𝜇−𝜀)𝑡};∀𝑡∈𝐑₀₋,(3.10a)||≤𝑀₁𝑒^{(𝜇−𝜀)𝑡}||𝜙^𝑄Λ𝑒||,∀𝑡∈𝐑₀₊,(3.10b), 𝑀₁=𝑀₁(𝜀)∈𝐑₊ is defined by where 𝐑₀₊∋𝑡≥ℎ is the fundamental matrix of (2.1) with initial values Λ and ^0𝑡+𝜃𝑇_𝑠(𝑡,𝜏)𝑓𝜏,𝑥_𝜏^𝑃Λ(𝜃)𝑑𝜏==^𝑡0𝑇_𝑠(𝑡,𝜏)𝑋^𝑃Λ₀(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏_𝑡0𝑇(𝑡+𝜃,𝜏)𝑋^𝑃Λ₀𝑓𝜏,𝑥_𝜏𝑑𝜏,_𝑡0𝑇_𝑠(𝑡,𝜏)𝑓𝜏,𝑥_𝜏^𝑃Λ(𝜃)𝑑𝜏==^0𝑡+𝜃𝑇_𝑠(𝑡,𝜏)𝑓𝜏,𝑥_𝜏^𝑃Λ(𝜃)𝑑𝜏_𝑡0𝑇_𝑠(𝑡,𝜏)𝑋^𝑃Λ₀(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏,(3.11), 𝑓(𝜏,𝑥_𝜏)=0. The following technical result holds.

Lemma 3.1. The following relations hold: where

Also, the following relations hold for 𝑡∈𝐑_−ℎ, for all 𝑄_Λ→𝑄_Λ𝑒, being sufficiently small: for some |𝑇(𝑡,0)𝜙^𝑄Λ|≤𝑀₁𝑒^{(𝜇−𝜀)𝑡} irrespective of the multiplicity of the eigenvalue of the limiting equation (2.1) whose real part is 𝐑₀₊∋𝑡≥ℎ.

Proof. Equations (3.5)–(3.7) hold for any 𝑡∈𝐑₀₊ from Theorem 2.3, the definition of the set 𝜙 in Theorem 2.1, (2.20), Lemma 2.2, and (2.46)–(2.49). To obtain (3.5)–(3.7), the following identities are used: since 𝑥^𝑃Λ_𝑡=Φ_Λ𝑢(𝑡) for 𝑡∈𝐑₀₊ and 𝑢 for 𝑢(̇𝑡)=𝐵_Λ𝑢(𝑡)+Ψ_Λ(0)𝑓(𝑡,𝑥_𝑡) and 𝑢(𝑡)==𝑒𝑒^𝐵Λ𝑡_𝐵Λ𝑡𝑢(0)+^𝑡0𝑒^{−𝐵Λ𝜏}Ψ_Λ(0)𝑓𝜏,𝑥_𝜏𝑑𝜏=Φ_Λ−1𝑢(0)+^𝑡0𝑒^{𝐵Λ(𝑡−𝜏)}Ψ_Λ(0)𝑓𝜏,𝑥_𝜏𝑑𝜏=Φ_Λ−1𝑇(𝑡,0)Φ_Λ𝑢(0)+^𝑡0Φ^Λ−1𝑇(𝑡,𝜏)Φ_ΛΨ_Λ(0)𝑓𝜏,𝑥_𝜏𝑑𝜏𝑇(𝑡,0)Φ^Λ𝑢(0)+^𝑡0𝑇(𝑡,𝜏)Φ_ΛΨ_Λ(0)𝑓𝜏,𝑥_𝜏𝑑𝜏(3.12). Equation (3.8) follows directly by substitution of (3.6), and (3.7) into (3.5). The norm relations (3.9) hold directly from (3.5)–(3.7) through (3.8).

It turns out that for any 𝑢(0)=∫^0−ℎ𝑒^{𝐵Λ(𝑡−𝜏)}𝜙(𝜏)𝑑𝜏, the above relations hold also for any 𝑥^𝑃Λ_𝑡==Φ𝑇(_Λ𝑡,𝑢0(𝑡))Φ_Λ𝑢(̇𝑡)=𝐵_Λ𝑢(0)+^𝑡0𝑇(𝑡,𝜏)Φ_ΛΨ_Λ(0)𝑓𝜏,𝑥_𝜏𝑑𝜏,=ΦΦ^Λ−1_Λ−1𝑇(𝑡,0)Φ_Λ𝑢(0)+^𝑡0𝐵_ΛΦ^Λ−1𝑇(𝑡,𝜏)Φ_ΛΨ_Λ(0)𝑓𝜏,𝑥_𝜏𝑑𝜏+Ψ_Λ(0)𝑓𝑡,𝑥_𝑡𝑇̇(𝑡,0)Φ^Λ𝑢(0)+^𝑡0𝑇̇(𝑡,𝜏)Φ_ΛΨ_Λ(0)𝑓𝜏,𝑥_𝜏𝑑𝜏+Ψ_Λ(0)𝑓𝑡,𝑥_𝑡.(3.13) by replacing Λ₀(𝜇) in (3.5)–(3.7). Equation (3.9) also holds for 𝑝(𝑡)𝑒^𝜆𝑡 since 𝑝 is bounded. The second relation in (3.5) may be rewritten as 𝐂^𝑛 for any 𝜆∈Λ₀(𝜇) and extended to any 𝛾(𝑡) if CE∶=𝜆_𝑗𝜇𝑖∈𝐂∶Re𝜆_𝑗𝜇𝑖=𝜇_𝑖,det𝑇⁻¹𝜆_𝜇𝑖=0;∀𝑗∈In_CEi⊂𝐙₊,∀𝑖∈In_CER⊂𝐙₊.(3.14) is continuous on its definition domain In_CEi. A direct consequence of Lemma 3.1, (3.8) and Proposition 2.5 is that if In_CER, for all 𝜇_𝑖 then Card(CE)=∑_𝑖∈InCERCardIn_CEi≤𝜒₀ is a solution of the ordinary differential equation 𝜈_𝑖𝑗;∀𝑗∈In_CEi,∀𝑖∈In_CER, [28], which is given explicitly by: for Card(CE)=𝜒₀ so that It is now proved that the asymptotic difference function between some eigensolution of (2.1), that is, a finite sum of solutions of (2.1) corresponding to the set ∞ of the form Card(CE), where 𝜆_𝑗𝜇𝑖 is a ∀𝑗∈In_CEi-valued polynomial and ΛΛ₀₁𝜇_𝑖∶=𝜆∈CE∶Re𝜆=𝜇_𝑖;Λ𝜇𝜇_𝑖𝑖∶=𝜆∈CE∶Re𝜆>𝜇_𝑖∶==Λ₀𝜆𝜇∈_𝑖C∪EΛ∶₁R𝜇e_𝑖𝜆≥𝜇_𝑖.(3.16), and some corresponding solution of (2.2)-(2.3) grows non faster than linearly with the norm of the solution of the limiting equation (2.1), [32]. If 𝜆_𝜇∈CE converges to zero exponentially then the asymptotic difference function between both solutions has strict Lyapunov exponent smaller than that of the corresponding limiting eigensolution. As a result, a solution of (2.2)-(2.3) is of the same exponential order as that of its limiting equation for sufficiently large time. Define the set of distinct eigenvalues of the limiting equation (2.1) as It is obvious that the cardinal of CE may be infinity (since (2.1) is a functional differential equation), but always numerable. The also denumerable subsets 𝑓(𝑡,𝑥_𝑡) and 𝑥(𝑡)=𝑦(𝑡)+𝑂(𝑒^𝑐𝑡) of the set of positive integers are indicator sets for all distinct members of CE with real part ∀𝑡∈𝐑₀₊ and for all the distinct real parts of the members of CE, respectively. It follows that 𝐑∋𝑐>𝜇. The total number of eigenvalues taking account for their multiplicities 𝑥(𝑡)=𝑦(𝑡)+𝑂(𝑡^𝜈−1𝑒^𝜇𝑡), is In particular, 𝜈 (𝜆_𝜇∈CE standing for infinity denumerable cardinal as opposite to a non-numerable infinity cardinal typically denoted by 𝐾_0𝛼=0) or 𝛼 is finite. The above definition relies on the fact distinct (non real) eigenvalues 𝛾∶𝐑₀₊→𝐑₀₊ of (2.1), 𝛾(𝑡)=𝑜(𝑒^−at) can have identical real part. Similar sets of eigenvalues of (2.1) as those in (2.18)–(2.20) may be defined being associated to each member of CE as follows: The following result holds concerning an asymptotic comparison of eigensolutions of (2.1) with the corresponding associated solution of (2.2) and (2.3) under a special form of the perturbation function.

Theorem 3.2. Suppose that at least one 𝑎∈𝐑₊ has real part 𝛾(𝑡)→0 being identical strict Lyapunov exponent of some solution of (2.2) under (2.3). Suppose also that 𝑡→∞ satisfies the hypotheses of Theorem 2.1. Then, a solution of (2.2)-(2.3) satisfies ∃𝜀∈𝐑₊, Λ₀(𝜇) and any 𝑥(𝑡)=𝑦(𝑡)+𝑜(𝑒^{(𝜇−𝜀)𝑡}). Also, 𝑡→∞ for any 𝛼 satisfying the hypotheses of Theorem 2.1 with 𝑥_𝑡=𝑂(|𝑥^{𝑃0(𝐑−ℎ)𝑡}|_𝛼) being the largest multiplicity among those of all distinct ∀𝑡∈𝐑₀₊.

Assume, in addition, that 𝐶_𝑒(𝐑_−ℎ)=𝑃_Λ(𝐑_−ℎ)⊕𝑄_Λ𝑒(𝐑_−ℎ) for any norm 𝑃_Λ(𝐑_−ℎ)=𝑃_Λ0(𝐑_−ℎ)⊕𝑃_Λ1(𝐑_−ℎ) and 𝑃_Λ0(𝐑_−ℎ) satisfies Λ₀(𝜇) as 𝑡∈𝐑_−ℎ for some 𝑡≥ℎ (i.e., 𝐶(𝐑₀₊)=𝑃_Λ(𝐑₀₊)⊕𝑄_Λ(𝐑₀₊) as 𝑃_Λ(𝐑₀₊)=𝑃_Λ0(𝐑₀₊)⊕𝑃_Λ1(𝐑₀₊) exponentially). Then, 𝑡∈𝐑₀₊ and a nontrivial eigensolution of (2.1) corresponding to the set 𝑥(𝑡)=𝑂(𝑒^ct) such that 𝑦(𝑡)=𝑂(𝑒^ct) as 𝑥(𝑡)−𝑦(𝑡)=𝑂(𝑒^ct).

Proof. Note that for any norm 𝑓(𝑡,𝑥_𝑡), 𝐑∋𝑐>𝜇=Re𝜆_𝜇; 𝜆_𝜇∈CE [Theorem 2.3(ii)] with 𝜆_𝜇∈CE, 𝑥(𝑡)−𝑦(𝑡)=𝑂(𝑒^ct) with 𝐑∋𝑐≥𝜇=Re𝜆_𝜇 being the eigenspace associated with 𝑓(𝑡,𝑥_𝑡) for 𝜆_𝜇∈CE. For 𝑥(𝑡)−𝑦(𝑡)=𝑂(𝑝_𝜆𝑢(𝑡)𝑒^𝜇𝑡)=𝑂(𝑒^ct), the related direct sum decomposition 𝐑₋∋𝑐>𝜇, 𝑝_𝜆𝑢∶[0,𝑡]→𝐂^𝑛 since may be used for the eigensolutions of (2.1) since the solution is time-differentiable for 𝜈. Thus, 𝜆_𝜇∈CE and 𝑥_𝑡−𝑦_𝑡=^𝑡0𝑇_𝑠(𝑡,𝜏)𝑋^𝑃Λ₀+𝑑[𝐾(𝑡,𝜏)]^𝑄Λ(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏,𝐑₊∋𝑡≥ℎ.(3.17) so that 𝛾(𝑡)=𝑜(𝑒^−at) for any given perturbation function 𝑡→∞ satisfying the constraints of Theorem 2.1 and all |𝑥_𝑡|=𝑜(𝑒^{(𝜇+𝜀)𝑡}), 𝑡→∞.
If all 𝜆_𝜇 are simple then 𝛾(𝜏)|𝑥_𝜏|≤𝑀_𝛾𝑀_𝑥𝑒^{(𝜇+𝜀−𝛼)𝜏}, 𝐾_0𝛼=0 for any given perturbation function 𝛾(𝑡)=𝑜(𝑒^−at)→0 satisfying the constraints of Theorem 2.1.
Otherwise (i.e., at least one 𝑡→∞ is not simple), |||^𝑡0𝑑[𝐾(𝑡,𝜏)]^𝑄Λ(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏|||_𝛼≤≤𝑀𝑀₂(𝛼)𝑒^{(𝜇−𝜀)𝑡}^∞0𝑒^{(2𝜀−𝛼)𝜏}𝑑𝜏₂(𝛼)𝛼−2𝜀𝑒^{(𝜇−𝜀)𝑡}|||=𝑜𝑒^{(𝜇−𝜀)𝑡},∀𝑡∈𝐑₀₊,(3.18)_𝑡0𝑇_𝑠(𝑡,𝜏)𝑋^𝑃Λ₀(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏|||_𝛼≤𝑀₃(𝛼)𝑒^{(𝜇−𝜀)𝑡}^∞0≤𝑀𝑒^{(2𝜀−𝛼)𝜏}𝑑𝜏₂(𝛼)𝛼−2𝜀𝑒^{(𝜇−𝜀)𝑡}=𝑜𝑒^{(𝜇−𝜀)𝑡},∀𝑡∈𝐑₀₊(3.19)
for some ‖𝑋^𝑃Λ₀(𝜃)‖_𝛼=‖Φ_ΛΨ_Λ(𝜃)‖_𝛼=‖Φ_Λ(0)Ψ_Λ(0)𝑒^2𝐵Λ𝜃‖_𝛼<∞ with ∀𝜃∈[−ℎ,0] being a polynomial of degree 0<𝜀<𝛼/2 equating the largest multiplicity among those of all distinct 𝑥_𝑡−𝑦_𝑡=𝑜(𝑒^{(𝜇−𝜀)𝑡}).
The first part of the result has been fully proved. Now, note that (3.8) in Lemma 3.1 can be equivalently rewritten as Also, if ∀𝜀∈𝐑₊ as 0<𝜀<𝛼/2 and since 𝑥_𝑡−𝑦_𝑡=𝑜(𝑒^{(𝜇−𝜀)𝑡}) as 𝑡→∞, irrespective of the multiplicity of 𝛾(𝑡)=𝑜(𝑒^−𝑎𝑡), from the definition of the strict Lyapunov exponent, one gets by using 𝑡→∞ from (2.8) with 𝐾_0𝛼=0 (since 𝜀∈𝐑₊ as 𝑥_𝑡=𝑜(𝑒^{(𝜇−𝜀)𝑡})) and (3.10b): since 𝛾(𝑡)=𝑜(𝑒^−at), 𝑡→∞ provided that 𝜀∈𝐑₊. Then, 𝛾(𝑡)=𝑜(𝑒^−at) as 𝐾_0𝛼=0, 𝑡→∞ satisfying 𝑦_𝑡=𝑜𝑒^𝜇𝑡,𝑥_𝑡=𝑦_𝑡+𝑜𝑒^{(𝜇−𝜀)𝑡}=𝑜𝑒^𝜇𝑡+𝑜𝑒^{(𝜇−𝜀)𝑡}=𝑜𝑒^𝜇𝑡,as𝑡→∞,(3.20).

Remark 3.3. An important observation follows. Note that the error between any eigensolutions of the limiting equation (2.1) and the associated solution of (2.2)-(2.3) satisfying 𝜀∈𝐑₊ as 𝜆_𝜇∈CE if 𝜆_𝜇1,2∈CE as 𝑦_𝑡=𝑜(𝑒^𝜇𝑡) and 𝑥_𝑡=𝑜(𝑒^𝜇𝑡) for sufficiently small 𝐾_0𝛼≠0 (proven in Theorem 3.2) does not imply that 𝜆_𝜇, 𝑦_𝑡=𝑜𝑒^ct,𝑥_𝑡=𝑦_𝑡+𝑜𝑒^{(𝜇−𝜀)𝑡}=𝑜𝑒^ct+𝑜𝑒^{(𝜇−𝜀)𝑡}=𝑜𝑒^ct,as𝑡→∞,(3.21) as 𝜀∈𝐑₊ if 𝑐>𝜇=Re𝜆_𝜇 as 𝜆_𝜇∈𝐶𝐸 for sufficiently small 𝑥_𝑡−𝑦_𝑡=𝑂(𝑒^𝑐𝑡)⇒𝑥_𝑡=𝑂(𝑒^𝑐𝑡)⋀𝑦_𝑡=𝑂(𝑒^𝑐𝑡).
The asymptotic behaviors if 𝑡∈𝐑₀₊ and 𝑐>𝜇=Re𝜆_𝜇 as 𝜆_𝜇 are as follows: for sufficiently small 𝑐=𝜇 if 𝜆_𝜇 is real simple or there are two 𝜆_𝜇∈𝐶𝐸 simple complex conjugate ones. In the above, equations, 𝜇<0, Λ₀(𝜇)=Λ(𝜇) hold even if Λ₁(𝜇)=∅ provided that 𝑓(𝑡,𝑥_𝑡) fulfils some of the above constraints. Furthermore, for sufficiently small 𝛼 and all real 𝛾(𝑡)=𝑜(𝑒^−𝑎𝑡) if 𝑡→∞ is multiple (i.e., of multiplicity greater then unity). Also, 𝑎∈𝐑₊, ∃𝜀∈𝐑₊, 𝑥(𝑡)=𝑦(𝑡)+𝑜𝑒^{−(|𝜇|+𝜀)𝑡}⟶𝑦(𝑡)⟶𝑜𝑡^𝜈−1𝑒^{−|𝜇|𝑡}⟶𝑜𝑒^{−|𝑐|𝑡}⟶0exponentiallyas𝑡⟶∞,(3.22) irrespective of the multiplicity of 𝐑₋∋𝑐≥𝜇 for any generic perturbation function 𝜈 satisfying the hypothesis of Theorem 2.1. However, the above implication is true for any arbitrary such a function and 𝜆_𝜇∈𝐶𝐸, only if 𝑐=𝜇 is an eigenvalue of multiplicity unity of the limiting equation (2.1).

Theorem 3.2 leads directly to the subsequent stability result by taking also into account Remark 3.3.

Corollary 3.4. Suppose that there exists a (nonnecessarily unique) 𝑐>𝜇 with |𝜆_𝜇| being the strict Lyapunov exponent of some solution of (2.2) under (2.3) and that 𝜇<0; that is, 𝜇<0. Suppose also that Λ₁(𝜇)=∅ satisfies the hypotheses of Theorem 2.1 and, furthermore, 𝜆_𝜇∈𝐶𝐸 (for any 𝑐≥𝜇-norm) and 𝑉∶[𝑡₀,∞)×𝐂^𝑛→𝐑₀₊ as 𝑉(𝑡,𝑥(𝑡))∶=‖𝑥(𝑡)‖²² for some 𝑡₀∈𝐑₀₊. Then, 𝑉(𝑡,0)=0 such that any solutions of (2.1) and (2.2)-(2.3) satisfy: for some ‖𝑥(𝑡)‖ with ∃𝑉̇∈𝑃𝐶⁽¹⁾([𝑡₀,∞)×𝐂^𝑛,𝐑) being equal to the largest multiplicity of among those of all distinct 𝑉̇(𝑡,𝑥(𝑡))∶=2𝑥̇^∗(𝑡)𝑥(𝑡),. The above relations hold with 0≤𝑉𝑡,𝑥(𝑡)=≤𝑉𝐾𝑒𝑡₀,𝑥(𝑡₀)_{−2|𝑐|(𝑡−𝑡0)}+^𝑡𝑡0𝑉̇𝜏,𝑥(𝜏)𝑑𝜏⟶0exponentiallyas𝑡⟶∞(3.23) if and only if all the eigenvalues of (2.1) satisfying the given assumptions are simple. As a result, both functional equations (2.2)-(2.3) and its limiting one are globally asymptotically Lyapunov stable with exponential stability.

Proof. Theorem 3.2 applies for ∃lim_𝑡→∞∫^𝑡𝑡0𝑉̇(𝜏,𝑥(𝜏))𝑑𝜏=2lim_𝑡→∞∫^𝑡𝑡0𝑥̇^∗(𝜏)𝑥(𝜏)𝑑𝜏=−𝑉(𝑡₀,𝑥(𝑡₀))≤0 with 𝑡₀ being the spectral radius and ¬∃𝑡₀∈𝐑₀₊ the spectral (or stability) abscissa, that is, there is no member of CE with real part to the right of 𝑉̇(𝑡,𝑥(𝑡))≤0 since 𝑡∈[𝑡₀,∞). If all such ¬∃lim_𝑡→∞∫^𝑡𝑡₀𝑉̇(𝜏,𝑥(𝜏))𝑑𝜏=−𝑉(𝑡₀,𝑥(𝑡₀))≤0 are simple then the result applies also for ∃𝑡₀(beingsufficientlylargebutfinite)∈𝐑₀₊ (see Remark 3.3). Define the bounded real nonnegative function 𝑉̇(𝑡,𝑥(𝑡))≤0 as ∀𝑡∈[𝑡₀,∞) for any finite 𝑉̇(𝑡,𝑥(𝑡))<0. Note that [𝑡₀,∞) and 𝑉̇(𝑡,𝑥(𝑡))→0 are strictly monotonically increasing with 𝑡→∞. It follows from (2.2)-(2.3) that [𝑡^0,∞)⊂[𝑡₀,∞), point-wise defined by 𝑉̇(𝑡,𝑥(𝑡))<0 and
Then, 𝑉(𝑡,𝑥(𝑡)) which is finite since 𝐾_0𝛼>0 is finite and the solution to (2.2)-(2.3) is everywhere continuous on its definition domain. By assuming 𝛼 such that 𝐾_0𝛼>0, 𝛼^ then 𝐾_0𝛼=0⇔𝐾_0𝛼=0 which is a contradiction to the already proven existence of such a limit. As a result, 𝐾_0𝛼>0 such that 𝜇=0, 𝜆_𝜇∈CE and 𝐾_0𝛼=0 on some (non-necessarily connected) subinterval of infinite measure of 𝜆_𝜇∈𝐶𝐸 with 𝜇≤0 as Λ₀(𝜇)=Λ(𝜇) then necessarily there is a connected terminal subinterval Λ₁(𝜇)=∅ of infinite measure where 𝑓(𝑡,𝑥_𝑡). Thus, 𝐾_0𝛼≥0 is nonnegative and converges exponentially to zero with nonpositive time-derivative which also converges to zero within some interval of infinite measure. As a result, 𝛼 is a Lyapunov function with negative time-derivative within a connected real interval of infinite measure which has zero limit.

Note that the limiting differential functional equation is, furthermore, globally uniformly asymptotically Lyapunov stable under the asymptotic stability conditions of Corollary 3.4. Note that 𝛾(𝑡)=𝑜(𝑒^−𝑎𝑡) (for any 𝑡→∞-norm) in (2.3) implies that 𝑎∈𝐑₊ for any other norm ∃𝜀∈𝐑₊ and conversely. Also, 𝑥(𝑡)=𝑦(𝑡)+𝑂𝑒^{−|𝜇|𝑡}⟶𝑦(𝑡)⟶𝑂𝑒^{−|𝜇|𝑡},∀𝑡∈𝐑₀₊.(3.24). The above result is concerned with global asymptotic stability with exponential decay rate. Global uniform Lyapunov stability (i.e. boundedness of solutions with a common upper-bound for all time for any bounded function of initial conditions) holds under weaker conditions; that is, 𝜆_𝜇∈𝐶𝐸 and 𝜇<0 if 𝜇=0 associated with the strict Lyapunov exponent have unity multiplicities or if they have any multiplicities but 𝜇<0. The precise related stability result follows which proofs follows directly from Corollary 3.4 and Theorem 3.2.

Corollary 3.5. Suppose that there exists at least one 𝐾_0𝛼=0 with 𝑥(𝑡)=𝑦(𝑡)+𝑜𝑒^{−|𝜇|𝑡}⟶𝑦(𝑡)⟶𝑜𝑝_𝜆𝑢(𝑡)e𝑒xpo^{−|𝜇|𝑡}nen⟶tiall𝑜y𝑒as𝑡^{−|𝑐|𝑡}⟶⟶∞0,(3.24) being the strict Lyapunov exponent of some solution of (2.2) under (2.3) and that 𝐑₋∋𝑐≥𝜇; that is, 𝑝_𝜆𝑢∶[0,𝑡]→𝐂^𝑛. Suppose also that 𝜆_𝜇∈𝐶𝐸 satisfies the hypotheses of Theorem 2.1 with 𝑐=𝜇 (for any 𝜆_𝜇∈𝐶𝐸-norm) and 𝜇 as 𝑓(𝑡,𝑥_𝑡) for some 𝐾_0𝛼=0. Then, 𝛼 such that any solutions of (2.1) and (2.2)-(2.3) satisfy As a result, the limiting equation () as well as ()-() are both globally uniformly Lyapunov stable if all (3.24) have any multiplicities and (3.24) or if they have all unity multiplicities and (3.24). Equations () and ()-() are both globally Lyapunov asymptotically stable with exponential decay rate if (3.24) and (3.24), satisfying: for some ∃𝜀∈𝐑₊ with Λ₀(𝜇) being a polynomial of degree equal to the largest multiplicity of all such distinct 𝑥(𝑡)=𝑦(𝑡)+𝑜(𝑒^𝜇𝑡). The above relations hold with 𝑡→∞ if and only if all the eigenvalues of (2.1) satisfying the given assumptions are simple, [32].

A parallel result to Theorem 3.2 obtained under weaker conditions on the perturbation function (2.3) follows.

Theorem 3.6. Suppose that at least one 𝑥(𝑡)=𝑜(𝑡^𝜈−1𝑒^𝜇𝑡) has real part 𝑦(𝑡)=𝑜(𝑡^𝜈−1𝑒^𝜇𝑡) being identical strict Lyapunov exponent of some solution of (2.2) under (2.3). Suppose also that 𝜀∈𝐑₊ satisfies the hypotheses of Theorem 2.1. Suppose, in addition, that ∫^𝑡0𝑡^𝜈−1𝛾(𝑡)𝑑𝑡<∞ for any norm 𝐾_0𝛼=0, and ||||^𝑡0𝑑𝐾(𝑡,𝜏)^𝑄Λ(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏||||_𝛼≤𝑀₄(𝛼)𝑒^𝜇𝑡₀∞𝑒^−𝜇𝜏𝑒^{(𝜀−𝜇)𝜏}𝑒^{(𝜀+𝜇)𝜏}𝜏^(𝜈−1)/2≤𝑀₄𝜏^(𝜈−1)/2𝛾(𝜏)𝑑𝜏(𝛼)𝑒^𝜇𝑡^∞0𝑒𝜏^{2(2𝜀−𝜇)𝜏𝜈−1}𝑑𝜏^1/2^∞0𝜏^𝜈−1𝛾²(𝜏)𝑑𝜏^1/2≤𝑀₅(𝛼)𝑒^𝜇𝑡^∞0𝑒𝜏^{2(2𝜀−𝜇)𝜏𝜈−1}𝑑𝜏^1/2,∀𝑡∈𝐑₀₊,(3.26) satisfies either 𝑀₅(𝛼)≥𝑀₄(𝛼)Sup_0≤𝑡<∞(𝛾(𝑡)) or 𝛾(𝑡) as 𝐑₀₊, with the inequality being strict if 𝑡→∞, where 𝜇∈𝐑₊ is the largest multiplicity among those of all distinct 𝐑₊∋𝜀∈(0,𝜇/2). Then, 𝐑₋∋𝑑>𝑏∶=(2𝜀−𝜇)𝜈∈𝐑₋ and a nontrivial eigensolution of (2.1) corresponding to the set 𝐑₀₊∋𝑡≥𝑡₀∶=𝑒^{(𝑑−𝑏)/(𝜈−1)} such that ^∞𝑡0𝑒^{2(2𝜀−𝜇)𝜏}𝜏^𝜈−1𝑑𝜏≤^∞𝑡0𝑒^{−|𝑑|𝜏}𝑑𝜏≤1_𝑡00|𝑑|=(𝜇−2𝜀)𝜈<∞,𝑒^{2(2𝜀−𝜇)𝜏}𝜏^𝜈−1𝑑𝜏≤𝑀⁶<∞(3.27) as 𝑡₀.
Furthermore, ||||^𝑡0𝑑𝐾(𝑡,𝜏)^𝑄Λ(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏||||_𝛼≤𝑀₅(𝛼)√𝑀₆+√(𝜇−2𝜀)𝜈𝑒^𝜇𝑡=𝑜𝑒^𝜇𝑡as𝑡⟶∞(3.28) and 𝜇∈𝐑₊.

Proof. One gets for sufficiently small 𝜇∈𝐑₀₋ from 𝜇∈𝐑₀₋ and Theorem 3.2, (3.18)-(3.19) and 𝜀=0: where 𝜇∈𝐑₀₋ which is a finite real constant since ||||^𝑡0𝑑𝐾(𝑡,𝜏)^𝑄Λ(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏||||_𝛼≤𝑀≤𝑀₄₄(𝛼)^𝑡0𝑒^{−||𝜇||(𝑡−𝜏)}𝜏^𝜈−1𝛾(𝜏)𝑑𝜏(𝛼)^∞0𝜏^𝜈−1𝛾(𝜏)𝑑𝜏≤𝑀₇(𝛼)<∞,∀𝑡∈𝐑₀₊.(3.31) is continuous on ∫^∞0𝑡^𝜈−1𝛾(𝜏)𝑑𝜏<∞⇒𝛾(𝑡)=𝑜(1/𝑡^𝜈−1) and has zero limit as 𝑡→∞ so that it is uniformly bounded. Now, if 𝜇∈𝐑₀₋ for ||||^𝑡0𝑑𝐾(𝑡,𝜏)^𝑄Λ(𝜃)𝑓𝜏,𝑥_𝜏𝑑𝜏||||_𝛼≤𝑀₄(𝛼)𝑒^𝜇𝑡^𝑡0≤2𝑀₄𝜏^1−𝜈𝑒^{𝜀(2𝜏−𝑡)}𝑑𝜏(𝛼)𝑒^𝜇𝑡^2𝑡0𝜏2^1−𝜈≤2𝑀₄𝑒^{𝜀(𝜏−𝑡)}𝑑𝜏(𝛼)𝑒^𝜇𝑡^∞0𝜏2^1−𝜈≤𝑀₈𝑒^{𝜀(𝜏−𝑡)}𝑑𝜏(𝛼)𝑒^𝜇𝑡=𝑜(𝑒^{−|𝜇|𝑡})as𝑡⟶∞,irrespectiveofthenorm𝛼.(3.32), |∫^𝑡0(𝑑[𝐾(𝑡,𝜏)]^𝑄Λ)(𝜃)𝑓(𝜏,𝑥_𝜏)𝑑𝜏|_𝛼=𝑜(𝑒^𝜇𝑡) and 𝑡→∞: since |∫^𝑡0(𝑑[𝐾(𝑡,𝜏)]^𝑄Λ)(𝜃)𝑓(𝜏,𝑥_𝜏)𝑑𝜏|_𝛼=𝑂(𝑒^𝜇𝑡) is finite. Now from (3.27) into (3.26): if ∀𝑡∈𝐑₀₊. If 𝛼 then (3.10a) and (3.10b) of Lemma 3.1 may be replaced for |∫^𝑡0𝑇_𝑠(𝑡,𝜏)𝑋^𝑃Λ₀(𝜃)𝑓(𝜏,𝑥_𝜏)𝑑𝜏|_𝛼 by taking 𝑥_𝑡−𝑦_𝑡= with By using (3.30), (3.26) may be replaced 𝑥_𝑡−𝑦_𝑡=𝑂(𝑒^𝜇𝑡) with On the other hand, 𝑥(𝑡)=𝑜(𝑡^𝜈−1𝑒^𝜇𝑡) as 𝑦(𝑡)=𝑜(𝑡^𝜈−1𝑒^𝜇𝑡) so that one gets for ̇𝑥(𝑡)=^𝑚_𝑖=0𝐴_𝑖𝑥𝑡−ℎ_𝑖+𝑓𝑡,𝑥_𝑡,(4.1) from Theorem 3.2, (3.18), since the integrals in (3.27) are bounded:
As a final result, 𝑓(𝑡,𝑥_𝑡)≡0 as 𝑇(𝑠)=(𝑠𝐼_𝑛−∑^𝑚𝑖=0𝐴_𝑖𝑒^{−ℎ𝑖𝑠})⁻¹, 𝐺∶𝐂₀₊→𝐂^𝑟×𝑚, and, furthermore, 𝐻_∞, 𝐂₊, lim_𝜎→0⁺𝐺(𝜎+𝐢𝜔)=𝐺(𝐢𝜔), irrespective of the norm 𝜔∈𝐑₀₊. Similar properties follow under close proofs for ‖𝐺‖_∞∶=sup_{𝑠∈𝐂0+}𝜎(𝐺(𝑠))=sup_𝜔∈𝐑𝜎(𝐺(𝐢𝜔))<∞. Thus, 𝜎(𝐺) 𝐺 as 𝑅𝐻_∞ by using the above results in (3.26) and 𝐻_∞, 𝐂₋. The properties ‖𝐺‖_∞ and 𝐺∈𝐻_∞ of the solutions of (2.1) and (2.2)-(2.3) under the given hypotheses follow directly by using the appropriate reasoning quoted from Remark 3.3.

Lyapunov stability properties in terms of boundedness of the solutions and their asymptotic or exponential convergence to the equilibrium obtained from Theorem 3.6 are immediate as given in the following direct result.

Corollary 3.7. Corollaries 3.4-3.5 also apply “mutatis-mutandis” under the assumptions of Theorem 3.6.

4. Some Direct Consequences and Applications

Some particular cases of the functional differential equations (2.1) and (2.2)-(2.3) are of interest concerning stability issues as follows.

4.1. Functional Differential Equations with Point Delays

The functional differential equation (2.2) can be equivalently described in the absence of finite distributed delays and Volterra-type dynamics by the n-the order system of n functional first-order differential equations: where (4.1). Its limiting differential system is defined in the same way from () for a perturbation function (4.1). The generic form of such a function is given in () and can include nonzero dynamics of finite-distributed delays and Volterra-type dynamics. Thus, (4.1) everywhere the inverse exists. Remember for later use that a matrix-valued function (4.1) is in the Hardy space (4.1) if it is analytic in (4.1), (4.1) for almost all (4.1) and (4.1) with (4.1) denoting the largest singular value of (4.1). Also, (4.1) is a the subset of (4.1) of real-rational matrix valued functions then being proper and stable (i.e., they have no more zeros than poles and all the poles are in (4.1)), []. Note that the used norm notation (4.1) for (4.1) is similar to the notation for (4.1)-matrix/vector norms but no confusion is expected from the different context of use. If (4.1) then (4.1).

The following result holds.

Theorem The following properties hold.
(i) The limiting differential system associated with () is globally asymptotically stable independent of the delays (i.e., (4.1) if (4.1) is a stability matrix and there exists (4.1) such that (4.1), and or This implies as a result that (4.1) and that any matrix (4.1) is a stability matrix, (4.1), where (4.1) is the union of all the (4.1) denumerable sets obtained by combining members of (4.1) (i.e., (4.1) is the set of parts of (4.1)).
(ii) If property (i) holds, then (4.1) is a stability matrix for all (4.1) with (4.1). The differential system () is globally asymptotically stable independent of the size of the delays with stability (or spectral) abscissa of at least (4.1), with provided that:
(a)𝐴₀ is a stability matrix,(b)Equation (4.2) holds for some 𝐴₀+𝜌𝐼_𝑛 such that 𝐴₀+𝜌𝐼_𝑛 or if 𝐴₀ and, furthermore, so that the limiting differential system (4.1) is globally asymptotically stable independent of the delays with stability abscissa of at least ‖‖⌊𝛽⁻¹¹𝐴₁,…,𝛽^−1𝑟𝐴_𝑚⌋‖‖₂<−𝜅₂𝐴₀+𝜌𝐼_𝑛≤−𝜅₂𝐴₀+𝜌<0.(4.9).

Proof. (i) Property (i) follows from the results in [4, Proposition 4.8] with (4.2). Note that 𝜇≤−min(𝜌₀₁,𝜌₀₂,𝜅₂(𝐴₀))<0, the (delay-free) system (4.1) is globally asymptotically stable so that 𝛾∶𝐑₀₊→𝐑₀₊ so that 𝛾(𝑡)=𝑜(𝑒^−𝑎𝑡). Then, the limiting differential system associated with (4.1) is globally asymptotically stable independent of the delays if Equation (4.2) holds which is obtained by guaranteeing the inverse below on 𝑡→∞ under a necessary and sufficiency-type text on the complex imaginary axis:
(ii) Since property (i) holds, then (4.2) holds so that there exist 𝑡→∞ (which are in general distinct from those in property (i) but we keep the same notation) such that 𝛾(𝑡)≤𝑜(𝑡^1−𝜈), and for any 𝑡→∞, ‖‖‖𝑠𝐼_𝑛−𝐴₀𝐼_𝑛−𝑠𝐼_𝑛−𝐴₀⁻¹^𝑚_𝑖=0𝐴_𝑖𝐼^𝑛−𝜌𝑠𝐼_𝑛−^𝑚_𝑖=0𝐴_𝑖⁻¹⁻¹×𝛽⁻¹¹𝐴₁,…,𝛽^−1𝑟𝐴_𝑚‖‖‖_∞<1,∀𝑠∈𝐂₀₊,(4.10): and the above 𝜔∈𝐑₀₊-norm exists if 𝐴₀ from Banach Perturbation Lemma, [34]. On the other hand, if ‖‖‖𝑠𝐼_𝑛−𝐴₀⁻¹^𝑚_𝑖=1𝐴_𝑖‖‖‖_∞𝜌‖‖‖𝑠𝐼_𝑛≤𝑎₀<1,−^𝑚_𝑖=0𝐴_𝑖⁻¹‖‖‖_∞≤𝜌‖‖‖𝑠𝐼_𝑛−𝐴₀⁻¹^𝑚_𝑖=1𝐴_𝑖‖‖‖_∞‖‖‖𝐢𝜔𝐼_𝑛≤𝜌𝑎₀≤𝑏₀<1,−𝐴₀𝐼_𝑛−𝐢𝜔𝐼_𝑛−𝐴₀⁻¹^𝑚_𝑖=0𝐴_𝑖⁻¹𝛽⁻¹¹𝐴₁,…,𝛽^−1𝑟𝐴_𝑚‖‖‖₂<1,∀𝜔∈𝐑₀₊(4.11) is stability matrix then ((𝐢𝜔𝐼_𝑛−𝐴₀)(𝐼_𝑛−(𝐢𝜔𝐼_𝑛−𝐴₀)⁻¹(∑^𝑚𝑖=0𝐴_𝑖)))⁻¹ is still a stability matrix if (4.4) holds by applying the min max computation approach for the eigenvalues of ∀𝜔∈𝐑₀₊ which are larger than those of 𝐴₀ since the identity matrix is positive definite. As a result, if global asymptotic stability independent of delays holds then it also holds with stability abscissa ‖(𝑠𝐼_𝑛−𝐴₀)⁻¹(∑^𝑚𝑖=1𝐴_𝑖)‖_∞≤𝑎₀<1. The first part of property (ii) has been proven. The second part follows by replacing (4.3) with

Through the use of the properties of the matrix measure [4, 34]. Theorem 4.1 can be directly combined with Corollaries 3.4-3.5 as follows.

Corollary 4.2. Assume that the functional limiting differential system (4.1) satisfies Theorem 4.1. Thus, both the perturbed and the nominal functional differential systems have a negative strict Lyapunov exponent which satisfies 𝐴₀. The solution of (4.1), subject to (2.3) with the hypotheses in Theorem 2.1, and that of its limiting differential system are both globally asymptotically stable independent of the delays.
If, furthermore, ‖[𝛽⁻¹¹𝐴₁,…,𝛽^−1𝑟𝐴_𝑚]‖₂<𝑎⁻¹⁰ satisfies one of the subsequent conditions:
(a)𝐑₀₊∋𝜌<𝑎⁻¹⁰ as ‖[𝛽⁻¹¹𝐴₁,…,𝛽^−1𝑟𝐴_𝑚]‖₂<(1−𝑝𝑎₀)/𝑎₀,(b)𝐴₀ as −𝜌<0,(c)ℎ₁≥0 as ℎ∶[−1,0]→𝐑₀₊, then any solution of the perturbed functional differential system (4.1) fulfils either Corollaries 3.4-3.5 to Theorem 3.2 (under the condition (a) or Corollary 3.7 to Theorem 3.6 (under the condition (b) or the condition (c).

A parallel result to Theorem 4.1(i) relies on the subsequent remark.

Remark 4.3. Note that (4.2) holds if provided that ℎ₀=0 exists, ̇𝑥(𝑡)=+𝐴₀₀₋₁+𝐴₁+𝛼(0)−𝛼(−1)𝑥(𝑡)+𝐴₁𝑥𝑡−ℎ₁−𝑥(𝑡)𝑑𝛼(𝜏)(𝑥(𝑡−ℎ(𝜏))−𝑥(𝑡))+𝑓𝑡,𝑥_𝑡,(4.13). That inverse exists provided that det𝑠𝐼=det_𝑛−𝐴𝑠𝐼₀−𝐴_1𝑛𝑒^−ℎ1𝑠−⁰⁻¹𝜆(𝜏)𝑑𝛼(𝜏)(4.14)=det𝑠𝐼_𝑛−𝐴₀+𝛼(0)−𝛼(−1)−𝐴₁𝑒^−ℎ1𝑠−⁰⁻¹𝜆(𝜏)𝑑𝛼(𝜏)(4.15)−𝐴₀+𝐴₁+𝛼(0)−𝛼(−1)−𝐴₁𝑒^−ℎ1𝑠−1−⁰⁻¹𝜆(𝜏)𝑑𝛼(𝜏)=0,(4.16) is a stability matrix, as it has been assumed, since then it has no imaginary eigenvalues or real ones at the origin and, furthermore, provided that [−1,0] exists, 𝛼. Such an inverse exists provided that 𝐴₀ is a stability matrix, as it has been assumed, since then it has no imaginary eigenvalues or real ones at the origin and, furthermore, ∃𝜌₁∈𝐑₊. Then, the system (4.1) is globally asymptotically stable independent of the delays if 𝜅₂(𝐴₀)≤−𝜌₁<0 (what implies that 1‖‖𝐴₁‖‖₂+‖‖∫⁰⁻¹𝜆(𝜏)𝑑𝛼(𝜏)‖‖₂‖‖𝑠𝐼_𝑛−𝐴₀−𝜌₁𝐼_𝑛⁻¹‖‖_∞<1,(4.17) is a stability matrix) and 𝐴₀. The most general constraint (4.8) is guaranteed with any (𝐴₀+𝛼(0)−𝛼(−1)) if ∃𝜌₂∈𝐑₊ which guarantees global asymptotic stability independent of the delays of the limiting differential system of (4.1) and 𝜅₂(𝐴₀+𝛼(0)−𝛼(−1))≤−𝜌₂<0 is a stability matrix of spectral abscissa of at least 1‖‖𝐴₁‖‖₂+‖‖𝛼(0)‖‖₂+‖‖𝛼(1)‖‖₂‖‖𝑠𝐼_𝑛−𝐴₀−𝛼(0)+𝛼(−1)−𝜌₂𝐼_𝑛⁻¹‖‖_∞<1,(4.18).

4.2. Functional Differential Equations with Mixed Finite Point and Time-Varying Distributed Delays

Assume that (2.2)-(2.3) consist of a single finite constant point delay 𝐴₀+𝐴₁+𝛼(0)−𝛼(−1) and a single distributed time-varying one ∃𝜌₃∈𝐑₊ leading to the functional differential system: with 12‖‖𝐴₁‖‖₂+‖‖𝛼(0)‖‖₂+‖‖𝛼(1)‖‖₂‖‖𝑠𝐼_𝑛−𝐴₀−𝐴₁−𝛼(0)+𝛼(−1)−𝜌₃𝐼_𝑛⁻¹‖‖_∞<1,(4.19). The perturbation function is defined by (2.3) and satisfies the hypothesis of Theorem 2.1. The limiting equation is defined for identically zero perturbation function. Equation (4.12) is equivalent to: so that the characteristic equation of the limiting equation is: for any continuous function 𝐂₀₊ mapping 𝑇(𝑡,0) on the unit circle centred at the origin of the complex plane provided that 𝐂₀₊ is non-atomic at zero. The following result is concerned with the global asymptotic stability of the differential system (4.12).

Theorem 4.4. The following properties hold. (i) Assume that 𝜇≤−𝜌<0 is a stability matrices, so that 𝜌: 𝜌_𝑖(𝑖=1,2,3). Thus, if then the limiting equation associated with (4.12) is globally asymptotically stable independent of the delays.(ii)Assume that ̈𝑥(𝑡)=−𝛼(𝑡)̇𝑥(𝑡)−𝛽(𝑡)̇𝑥(𝑡−ℎ)−𝛾(𝑡)𝑥(𝑡)−𝜔(𝑡)𝑥(𝑡−ℎ)=−𝛼₀̇𝑥(𝑡)−𝛽₀̇𝑥(𝑡−ℎ)−𝛾₀𝑥(𝑡)−𝜔₀𝑥(𝑡−ℎ)+𝑓𝑡,𝑥_𝑡(5.1) and 𝑓𝑡,𝑥_𝑡=𝛼₀−𝛼(𝑡)̇𝑥(𝑡)+𝛽₀−𝛽(𝑡)̇𝑥(𝑡−ℎ)+𝛾₀−𝛾(𝑡)𝑥(𝑡)+𝜔₀−𝜔(𝑡)𝑥(𝑡−ℎ),(5.2) are both stability matrices, so that ̈𝑦(𝑡)=−𝛼₀̇𝑦(𝑡)−𝛽₀̇𝑦(𝑡−ℎ)+𝛾₀𝑦(𝑡)−𝜔₀𝑦(𝑡−ℎ),(5.3): 𝛼(𝑡)→𝛼₀. Thus, if then the limiting equation associated with (4.12) is globally asymptotically stable independent of the delays.(iii)Assume that 𝛾(𝑡)→𝛾₀ is stability matrix so that 𝜔(𝑡)→𝜔₀: 𝑡→∞. Thus, if then the limiting equation associated with (4.12) is globally asymptotically stable independent of the delays.

Proof. Property (i) is direct by using a similar reasoning to that in Theorem 4.1 by extending a result in [4, Section 4.4.5] for a single distributed delayed dynamics, the limiting differential system associated with (4.12) is globally asymptotically stable if and only if for any noncharacteristic zero of the limiting equation, 𝑧_𝑦(𝑡)=(𝑦₁(𝑡),𝑦₂(𝑡))^𝑇,, ̇𝑧_𝑥(𝑡)=𝐴=𝐴_0𝑦0𝑧_𝑥(𝑡)+𝐴_1𝑦𝑧_𝑥(𝑡−ℎ)+𝑧_𝑓𝑡,𝑧_𝑥𝑡̇𝑧_𝑦(𝑡)𝑧_𝑥(𝑡)+𝐴₁(𝑡)𝑧_𝑥(𝑡−ℎ),(𝑡)=𝐴_0𝑦𝑧_𝑦(𝑡)+𝐴_1𝑦𝑧_𝑦(𝑡−ℎ),(5.4), or equivalently, if the inverse below exists for 𝐴_0𝑦=−𝛾11₀−𝛼₀;𝐴^1𝑦=−𝜔11₀−𝛽₀;𝑧^𝑓(𝑡,𝑧_𝑥𝑡)=𝑓(𝑡,𝑥0_𝑡𝐴₀),(𝑡)=−𝛾(𝑡)−𝛼(𝑡)11;𝐴₁(𝑡)=𝜔(𝑡)−𝛽(𝑡)11.(5.5) by using the rearrangement (4.14), Such an inverse exists within 𝛼_1,2=(−𝛼₀∓𝛼²⁰−4𝛾₀)/2 under the conditions in property (i) so that the evolution operator exists 𝐴_0𝑦 since its associate resolvent exists and it is compact in 𝛼₀>0. Properties (ii)-(iii) are direct alternative sufficient conditions to those involved in property (i) by using similar rearrangements for an inverse matrix as (4.20) by using (4.15) and (4.16), respectively, instead of (4.14).

A parallel result to Corollary 4.2 now follows from Theorem 4.4.

Corollary 4.5. Assume that the functional limiting differential system of (4.12) satisfies any of the Properties of Theorem 4.4. Thus, both the limiting and the perturbed functional differential equation have a negative strict Lyapunov exponent satisfying 𝛾₀>0 with 𝜅₂(𝑇⁻¹𝐴_0𝑦𝑇)<‖𝑇⁻¹𝐴_1𝑦𝑇‖₂ equalizing the corresponding 𝑇. Corollary 4.2 holds “mutatis-mutandis.”

The extension of Theorem 4.4 and Corollary 4.5 to the case of multiple point and distributed delays is direct and then omitted.

5. Example

Consider the second-order linear functional equation with point time-delay 𝜅₂(⋅): subject to arbitrary bounded initial conditions within the proposed class, where the disturbance is and the limiting equation is provided that (⋅), 𝑇=[^𝛼11_1𝛼2], 𝐴_0𝑦𝑑=𝑇⁻¹𝐴_0𝑦𝑇, and 𝐴_0𝑦 as 𝜅₂(𝐴_0𝑦𝑑)=−𝛼₀<0. Note that the perturbation term (5.2) satisfies (2.8) of Theorem 2.1. The differential equation and limiting differential equation may be rewritten equivalently as the following second-order dynamic systems of respective state vectors 𝐴_1𝑦𝑑=𝑇⁻¹𝐴_1𝑦𝑇=1𝛼₂−𝛼₁𝜔⁰+𝛽₀𝛼₁𝜔₀+𝛽₀𝛼₂𝜔₀+𝛽₀𝛼₁𝜔₀+𝛽₀𝛼₂,(5.6) and ℎ where The eigenvalues of (⋅) are 𝛼₀≥4𝛾₀ so that 𝛾₀>0 is a stability matrix if 𝜔₀>12max𝛼⁰𝛽⁰+𝛼²⁰−4𝛾₀,𝛽⁰𝛼⁰+𝛼²⁰−4𝛾₀(5.8) and 𝛽₀𝛼⁰+𝛼²⁰−4𝛾₀2≥𝜔⁰>2𝛽⁰−𝛼₀2𝛼²⁰−4𝛾₀,(5.9). The limiting equation (5.3) is globally stable independent of the delay size if 𝛽₀𝛼₀+(𝛼₀−𝛽₀)𝛼²⁰−4𝛾₀>0 for at least one second-order nonsingular state transformation matrix 𝛼(𝑡)=𝛼₀;∀𝑡∈𝐑₀₊., where 𝑇₀∈𝐑₀₊, and 𝛽𝛽𝑇̃(₀𝑡)∶∶==m𝛽(a𝑡)x_𝑡≥𝑇0−||𝛽̃(𝛽𝑡₀)||;𝛾𝑇₀∶=max_𝑡≥𝑇0||𝛾̃(𝑡)||;𝜔𝑇₀∶=max_𝑡≥𝑇0||𝜔(𝑡)||,;𝛾̃(𝑡)∶=𝛾(𝑡)−𝛾₀;𝜔(𝑡)∶=𝜔(𝑡)−𝜔₀.(5.10) are the 2-matrix measure with respect to the 𝛼²⁰≥𝛾₀+𝛾(𝑇₀)-norm, [4], and 𝜔₀>𝜔𝑇₀+12max𝛼⁰𝛽⁰𝛽+𝛽𝑇₀+𝛼²⁰−4𝛾₀−𝛾𝑇₀₀,+𝛽𝑇₀𝛼₀+𝛼₀−4𝛾₀−𝛾𝑇₀.(5.11)-norm of the matrix 𝜆, respectively. The use of a similarity transformation is usually necessary by the fact that the matrix measure is norm-dependent and it can be positive even for stability matrices for some norms and the stability property is independent of any performed similarity transformation on the state variables in the linear time- invariant case, [4, 5, 11]. Now, choose as state transformation matrix the nonsingular Vandermonde matrix 0>Re𝜆=𝜇≥𝜅₂𝐴_0𝑑−‖‖𝐴_1𝑑‖‖₂−max_𝑡≥𝑇0‖‖𝐴_0𝑑(𝑡)‖‖₂−max_𝑡≥𝑇0‖‖𝐴_1𝑑(𝑡)‖‖₂(5.12) which defines the diagonal stability matrix 𝐾_0𝛼>0, being similar to Λ≠∅, which possesses a negative 2-matrix measure 𝜇. In this case, and the limiting equation is globally asymptotically Lyapunov stable independent of the delay size 𝛽(𝑡)=𝛽₀ if where 𝜔(𝑡)=𝜔₀ denotes the maximum eigenvalue of the ∀𝑡∈𝐑₀₊-positive semidefinite matrix. From (5.7), the limiting equation (5.3) is globally asymptotically stable independent of the delay size if 𝛼(𝑇₀)∶=max_𝑡≥𝑇0|𝛼(𝑡)−𝛼₀|, ∀𝑡∈𝐑₀₊, and either or the second one provided that 0>Re𝜆=𝜇≥𝜅₂(𝐴_0𝑑)−‖𝐴_1𝑑‖₂−𝛼(𝑇₀). Consider the following two cases.

(1) Assume 𝑇 For some chosen finite 𝑇₀ define: The three above functions converge asymptotically to zero as time tends to infinity. Then, the solution of the differential equation (5.1), subject to (5.2), converges asymptotically to that of its limiting equation (5.3) if and (5.8) is replaced with Another convergence condition of the solution of (5.2)-(5.3) to the limiting solution can be directly obtained from (5.9). The limiting differential equation (5.3) has a dominant eigenvalue of real part: with related to the perturbation (5.3), so that according to (2.20) and Theorem 2.3(i2), and the strict Lyapunov exponent defined in Theorem 2.3 satisfying the above lower-bound, and

(2) Now, assume , , ; , and ; . Then, the solution of the differential equation (5.1), subject to (5.2), converges asymptotically to that of its limiting equation (5.3) independent of the delay size if and (5.8) is replaced with In this case, . Another convergence condition independent of the delay size can easily be got from (5.9). Conditions for asymptotic convergence of the solution to that of its limiting differential equation for the case of complex eigenvalues can be obtained straightforwardly.

Acknowledgments

The author is very grateful to the Spanish Ministry of Education for its partial support of this work through the project DPI2006-00714. He is also grateful to the Basque Government for its support through GIC07143-IT-269-07, SAIOTEK SPED06UN10, and SPE07UN04. The author is very grateful to the reviewers for their interesting comments which helped him to improve the first version of the manuscript.