Asymptotic Comparison of the Solutions of Linear Time-Delay Systems with Point and Distributed Lags with Those of Their Limiting Equations
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.
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 , 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, . 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 . In , 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  is devoted to investigate the existence and the global asymptotic stability of a periodic -mild solution for the -periodic logistic system with time-varying generating operators and -0-periodic impulsive perturbations on Banach spaces. In , a close problem is solved based on a generalized Gronwall's lemma. In , 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 .
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].
The following sets are used through the manuscript:where , and are the sets of real, complex and integer numbers, respectively, The complex imaginary unity is .
A finite subset of consecutive positive integers starting with 1 is denoted by . The set will be used to define the solution of functional differential equations on including its initial condition in .
is the set of m-vector real functions of class and definition domain and is the set of m-vector real functions in whose th derivative is piecewise continuous. Similar sets of functions are defined when the ranges are complex as and .
For the delayed system, is the inverse Laplace transform of the resolvent mapping , which is holomorphic where it exists, with 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 of definition domain by denoting any vector or induced matrix norm, that is, for the -norm, for the -norm, and so forth. Similar notations are used for the corresponding matrix-induced norms. In particular, stands for (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 for vectors but not for matrices. In the case of vectors, the Euclidean norm coincides with its -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 as that is, acts on the dynamics of from to for finite and as .
Dom(H) is the definition domain of the operator H and is the spectrum (i.e., the set of distinct eigenvalues) of the square matrix . The matrix measure of the norm-dependent complex-valued matrix is defined by .
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 as under the following technical hypothesis.
H.1: The initial conditions of both differential equations (2.1) and (2.2) are real n-vector functions where , with ; that is, the set of continuous mappings from into the Banach space with norm denoting the Euclidean norm of vectors in and matrices in , and is the set of real-bounded vector functions on endowed with the supremum norm having support of zero measure. Roughly speaking, if and only if it is almost everywhere zero except at isolated discontinuity points within where it is bounded. Thus, if and only if it is almost everywhere continuous in except possibly on a set of zero measure of bounded discontinuities. is also endowed with the supremum norm since , some , for each . In the following, the supremum norms on are also denoted with .
Close spaces of functions are which is the Banach space of continuous functions from into endowed with the norm ; for all being an initial condition, for some given vector norm . Note that for , the solution which satisfies (2.2), subject to (2.3), is in , the Banach space of continuous functions from into which satisfies (2.2)-(2.3), , endowed with .
Thus, is a bounded linear functional defined by the right-hand-side of (2.1).
H.2: All the operators are in , the set of linear operators on , of dual and , and are nonnegative constants with and .
H3: The linear operators , with abbreviated notation , are closed and densely defined linear operators with respective domain and range and . The functions and being everywhere differentiable with possibly bounded discontinuities on subsets of zero measure of their definition domains with some nonnegative real constant . If is a matrix function then it is in with 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 and , satisfying , for all , is a string of the solution of (2.2)-(2.3). Other strings of the solution trajectory of interest in this manuscript are which point-wise defined by within the interval and zero, otherwise, and subject to the constraint within , and being zero outside this interval. Finally, denotes the solution string within point-wise defined by the solution to (2.2)-(2.3) for each being zero outside and subject to the constraint within for any real and being zero outside this interval.
and and and 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 endowed with the supremum norm where the vectors of point and distributed constant delays are: and , respectively, with and and being, respectively, point and distributed delays, with and . The first distributed delays are associated with Volterra-type dynamics. In other words, the infinitely distributed delays give contributions with finite real constants with to which are point delays under the integral symbol. The functions and 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 may be alternatively matrix functions for and with . On the other hand, the perturbation vector function in (2.2), defined in (2.3), with respect to the limiting (2.1), is defined by the function which describes a perturbed dynamics associated with the delays plus a perturbation function which is not included in the remaining terms of the function 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 . It also includes real constants , with , 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 . Apart from those delays, the action of finite-distributed delays characterized by real constants 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 . The integrability of the -functions (or matrix functions) on follows since their definition domain is bounded. The technical hypothesis H1–H4 guarantee the existence and uniqueness of the solution in of the functional differential systems (2.1) and (2.2)-(2.3) for each given initial condition .
Take Laplace transforms in (2.1) by using the convolution theorem and the relations . It follows that , where denotes the Laplace transform of . Thus, the unique solutions of both the limiting (2.1) and that of (2.2)-(2.3) in , subject to (2.3); for all , for the same given initial conditions are, respectively, defined by where is the unit step (Heaviside) function and 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 converges or not to zero as . The evolution operator is a convolution operator so that if the Volterra-type dynamics is zero or if the associate differentials in the Riemann-Stieltjes integrals with 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 exists within some region including properly the right-complex plane. In other words, if it is compact for , for some constant located to the right of all the real parts of all the zeros of (also often called the characteristic zeros of the limiting (2.2) or, simply, its eigenvalues), since then all the entries of its Laplace transform decay with exponential rate on for and then decays with exponential rate on . 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 . The basic mathematical tool used in those papers is that the unique solution of the homogeneous (2.1) for each function of initial conditions 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 . 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 ) of multiplicity greater than unity (being degenerated or non-degenerated) in the limiting dynamics defined by (2.1).
Theorem 2.1. Let be a solution of (2.2), subject to (2.3), on subject to initial conditions such that
for some norm-dependent where is also norm-dependent and satisfies as , where if there is no Volterra term in (2.1) and , otherwise, and are the real parts of the zeros of of the limiting (2.2) with respective multiplicities . Then, the following properties hold:
(i) where is an indicator function defined by
(ii) The real numbers exist and are norm-independent and finite, for all and some integer with or , . If or if (2.8) holds with , then either it is the real part of a zero of , for which the resolvent trivially exists and it is bounded, or , for all .
(iii) Assume that all the zeros of have real negative parts and (2.8) holds only for some constants . Then, either the limits exist, , some integer , and furthermore, is not trivially the real part of a characteristic zero of (2.2), or ,
Proof. (i) From (2.8), so thatand property (i) has been proved.
(ii) From (2.8), , , some Then, one gets from (2.6)-(2.7), from the limiting hypothesis on the integral of the function , for any arbitrary small real norm-dependent constant , there exists a finite ( only for a simple constructive proof easily extendable to ), dependent on and the given -norm, such that one gets from (2.12) by taking initial conditions at : with being the real part of a characteristic zero of of multiplicity and is dependent on . Note that, if the solution is unbounded for the given initial conditions, then there exist, by construction, a finite and and a subsequence valued at the real increasing sequence (then as ) such that so that from (2.13) and for some bounded vector function , provided that is sufficiently small to guarantee in the case that and independently of if . Furthermore, if then if the solution is unbounded since, otherwise, 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 is uniformly bounded then . Equivalently, if furthermore , then (i.e., is the real part of a simple real characteristic zero of associate with the limiting equation (2.2) or there are two simple complex conjugate ones with real part ). 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 as if . This implies that the solution is unbounded which contradicts the fact that it is bounded. Note from (2.13) that if then real increasing sequence of (2.13): which takes the form , where depends on (finite), , , , , , and the -norm, for some bounded vector function which depends on , the initial conditions, and , provided that is sufficiently small to guarantee , in the case that , and independently of if . Assume that the solution is not a trivial solution what is guaranteed if , for all . Then, it follows from (2.15) that for sufficiently large , irrespective, of the -norm, since for any -norm. By taking , it follows that . The result may be also extended to the case , since then, the solution is either unbounded (for some initial conditions and multiplicity of the characteristic zero of whose real part is ), or it is bounded (in particular, always if for ). As a result, if and there is no such that converges to zero as , then for all the characteristic zeros of the limiting equation (2.2) . If (2.8) holds, in particular, with , then the above result is also valid from (2.15) for a negative value of . Property (ii) has been proved.
(iii) If (2.8) does not hold for 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 as and such an integral is bounded, for all . Thus, , for all and 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 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 , 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 (Theorem 2.1(ii). If all the characteristic zeros of (2.1) have negative real parts but (2.8) is not fulfilled with 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 ) 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  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 is a finite set of eigenvalues of (2.1), then and denote the generalized eigenspace associated with and the corresponding complementary subspace of , respectively. The phase space is decomposed by into the direct sum . The projections of the solution of (2.2), subject to (2.3), for any initial condition , onto the above subspaces are denoted by and , respectively, . Note that, although the initial conditions of (2.2)-(2.3) are in general in , the corresponding unique solution of (2.2), subject to (2.3), for are in . The whole solutions in which includes any given initial condition then satisfying , , and the differential equation (2.2), subject to (2.3), for are in where is the complementary subspace of in . The projections of the solution onto those subspaces are and , respectively, . The following technical result follows.
Proof. The first relation follows from , and the superposition principle for linear systems building the solution for by projecting the function of initial conditions into the complementary subspaces and in subject to the constraint , for all . The second relation follows from again from the superposition principle with , for all . The third relation follows from and the superposition principle applied to the solution at .
The intuitive meaning of Lemma 2.2 is that for , is decomposed uniquely as a sum of a function in and another one in its complementary in , even for initial conditions in , rather than in the more restrictive set . However, the complementary set of in replaces , since for any given . Note that , is untrue except for .
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 . Consider generalized eigenspaces , and for and, also, generalized eigenspaces , and for , where the spectral sets , and each generating the two corresponding eigenspaces, are defined by,
Then, the following properties hold,
(i) (i.1) if ,(i.2) if and, furthermore, (2.8) holds with ,(i.3) if all the eigenvalues of (2.1) have negative real parts and, furthermore, (2.8) does not hold with ,(i.4) if any of the following conditions hold: (1) No eigenvalue of (2.1) is in , (2) No eigenvalue of (2.1) is in and, furthermore, (2.8) does not hold with , (3) (2.8) holds with .
(ii) The solution of (2.2) under arbitrary initial conditions , subject to a perturbation function (2.3), satisfies,
(iii) The solution of (2.2) under arbitrary initial condition , subject to a perturbation function (2.3), satisfies, Furthermore, if (2.8) holds with then, as :
(iv) The solution of the limiting equation (2.1) satisfies (2.23) as .
(v) The solution of (2.2), under arbitrary initial conditions and subject to a perturbation function (2.3), satisfies: which is identical to under the restriction for the initial conditions with . Also, If (2.8) holds with then, as : which leads to under the restriction for the initial conditions with .The solution of the limiting equation (2.1) satisfies (2.27)-(2.28) as .
Proof. Properties (i) are direct consequences of Theorem 2.1 [(ii)-(iii)] as follows. Property (i.1) follows by noting that for some implies that is a characteristic zero of (2.1) of nonnegative real part from (2.7). Assume that where is the set of characteristic zeros of the limiting equation (2.1). Thus, if , 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 . If then the current equation is unstable which implies that the limiting one should satisfy to be also unstable but the asymptotic convergence of their respective solutions to each other is only possible if . Thus, again and property (i.1) has been proven. On the other hand, and property (i.2) is proven. Also, . Since, in addition, (2.2)-(2.3) is the limiting equation of (2.1) then and . Therefore, 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 and 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 so that property (iii) follows.
Property (iv) follows from property (iii) as particular case for ; 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 . 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 imply, as usual, that parallel results for “Landau big-O” notation stand for all 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 . 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 subject to initial conditions (i.e., the th identity matrix) and , . Thus, it satisfies also the unforced (2.2) (i.e., for ). This leads directly to (2.31). However, (2.32) follows by using the Newton binomial to expand and the fact that the maximum of the real exponential function within the real interval is reached at the boundary. Equation (2.33) follows by the inspection of (2.36) for some norm-dependent 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 , the evolution operator as a function of time is of exponential order whose norm time-function satisfies: which converges exponentially to zero as if the strict Lyapunov exponent 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 for any admissible function of initial conditions. The differential equation (2.2), subject to (2.3) is globally uniformly Lyapunov stable if and its solution satisfies: for large and converges exponentially to zero (i.e., it is globally uniformly exponentially Lyapunov stable) if and the perturbation function has an upper-bounding function with .
(ii) It follows directly from (2.12).
The evolution operator explicits the solutions of the limiting equation (2.5) and the perturbed one (2.6) for each function of initial conditions. Then, let be the solution semigroup of the linear autonomous equation (2.1), which is unique for for each and whose infinitesimal generator is satisfying , . Thus, the string of the solution of the limiting functional differential equation (2.1) within 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, is a -dimensional complex row vector. The phase space for (2.42) on is . Corresponding spaces of functions taking into account the more general spaces for initial conditions are , and . Let be a finite set of eigenvalues of (2.1) and let be a basis for the generalized eigenspace , [27, 28]. Then, there exists a square -matrix , with , such that the subsequent relations hold: The relations (2.43) yield via direct computation property (i) of the subsequent result since commutes with . Property (ii) is a direct consequence of (2.36) subject to and for .
Proposition 2.5. The two following properties hold.
(i)The following relations hold, for all : (ii)The evolution operator of the solution of (2.1) is uniquely given by with and for .
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 can be extended to by , where is of dimension compatible with the order of . Let be the complementary eigenspace to . Now, use appropriate notations for the corresponding subspaces on and their extensions to to consider more general initial conditions (on ) for (2.1) and (2.2)-(2.3) than bounded continuous functions in a Banach space leading to the uniquely defined decompositions and . Then, given a function of initial conditions the decomposition is unique. Also, the unique solution of (2.1) and that of (2.2), subject to (2.3), are uniquely decomposable in as via the direct sum of subspaces . The solution iincluding initial conditions defined by for is uniquely decomposable in as via the direct sum of subspaces .
3. Asymptotic Behavior and Asymptotic Comparison
The string solution (2.6) of (2.2)-(2.3) for , point-wise defined by , , any given , and may be expressed equivalently via the solution semigroup of the limiting equation (2.1) as where , defined bywith is the unique solution of the limiting equation (2.1), and the kernel of , is defined by where is the fundamental matrix of (2.1) with initial values and , . The following technical result holds.
Lemma 3.1. The following relations hold: where
Also, the following relations hold for , for all , being sufficiently small: for some irrespective of the multiplicity of the eigenvalue of the limiting equation (2.1) whose real part is .
Proof. Equations (3.5)–(3.7) hold for any from Theorem 2.3, the definition of the set 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 for and for and . 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 , the above relations hold also for any