On Stability for Impulsive Delay Differential Equations and Application to a Periodic Lasota-Wazewska Model

We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the context of recent literature.

(Communicated by the associate editor name) Abstract. We consider a class of scalar delay differential equations with impulses and satisfying an Yorke-type condition, for which some criteria for the global stability of the zero solution are established. Here, the usual requirements about the impulses are relaxed. The results can be applied to study the stability of other solutions, such as periodic solutions. As an illustration, a very general periodic Lasota-Wazewska model with impulses and multiple time-dependent delays is addressed, and the global attractivity of its positive periodic solution analysed. Our results are discussed within the context of recent literature.

1.
Introduction. The high impact of differential equations with impulses in terms of their application in population dynamics, disease modelling and other fields has lead to an increasing interest in the theory of impulsive systems. Recently, the theoretical analysis of existence and regularity of solutions to impulsive systems with delays, as well as the study of concrete impulsive models used in mathematical biology, have become an important area of research.
In this paper, we study a class of scalar impulsive differential equations with an instantaneous negative feedback term and (possibly unbounded) delays.
Without loss of generality, we suppose that t → t−τ (t) is non-decreasing; otherwise, whenever necessary, we can replace t → t − τ (t) by d(t) := inf s≥t (s − τ (s)), which is non-decreasing and satisfies the above conditions for t − τ (t). For t ≥ 0, denote by P C(t) = P C([−τ (t), 0]; R) the space of real functions that are piecewise continuous functions on [−τ (t), 0] and left continuous on (−τ (t), 0], with the norm φ t := sup We consider a scalar impulsive delay differential equation (DDE) of the form ∆(x(t k )) := x(t + k ) − x(t k ) = I k (x(t k )), k = 1, 2, . . . , (1.1) where: x ′ (t) is the left-hand derivative of x(t), (t k ) k is an increasing sequence of positive numbers with t k → ∞, a : [0, ∞) → [0, ∞) and I k : R → R are continuous functions; x t denotes the restriction of x(t) to the interval [t − τ (t), t], with x t ∈ P C(t) given by f (t, ϕ) is a functional defined for t ≥ 0 and ϕ ∈ P C(t) with some regularity discussed below. We also assume that f (t, 0) = 0 for t ≥ 0 and I k (0) = 0 for k ∈ N, thus x ≡ 0 is a solution of (1.1). The particular case of (1.1) with a(t) ≡ 0 reads as ∆(x(t k )) = I k (x(t k )), k = 1, 2, . . . , (1.2) and has been studied by many authors (see e.g. [2,18,20,21,22,23,24]). In the present study, the aim however is to take full advantage of the negative instantaneous feedback given by the term a(t)x(t) on the left-hand side of (1.1).
A rigorous abstract formulation of the existence of solutions problem for (1.1), or for more general impulsive DDEs, has been well established in the literature, and will be omitted here (see e.g., [1,7,8,10,18,19] and references therein for more details). We need however to set some properties for f , as well as clarify the space of initial conditions.
For a compact interval [α, β] ⊂ R, denote by B([α, β]; R) the space of bounded functions from [α, β] to R equipped with the supremum norm, and P C([α, β]; R) the subspace of B([α, β]; R) of functions that are piecewise continuous on [α, β] and left continuous on (α, β]. Now, define the space P C = P C((−∞, 0]; R) as the space of functions from (−∞, 0] to R for which the restriction to each compact interval [α, β] ⊂ (−∞, 0] is in the closure of P C([α, β]; R) in B([α, β]; R). A function ϕ ∈ P C is continuous everywhere except at most for an enumerable number of points s for which ϕ(s − ), ϕ(s + ) exist and ϕ(s − ) = ϕ(s). Denote by BP C the subspace of all bounded functions in P C, BP C = {ϕ ∈ P C : ϕ is bounded on (−∞, 0]}, with the supremum norm ϕ = sup s≤0 |ϕ(s)|. Clearly, each space P C(t) can be taken as a subspace of BP C. For equations without impulses, the subspaces of continuous functions C, BC and C(t) of P C, BP C and P C(t), respectively, will be considered. For . For the impulsive DDE (1.1), we consider initial conditions of the form In view of our purposes, we may suppose that the extension F : [0, ∞) × BP C → R of f is continuous or piecewise continuous (for simplicity, we abuse the language and refer to f as being continuous or piecewise continuous as well), but more general frameworks are allowed. For f piecewise continuous, under very general conditions (which are satisfied with the hypotheses we shall imposed in Section 2), from [1,10,17,19] it follows that the initial value problem (1.1)-(1.3) has a unique solution x(t) defined on [t 0 , ∞). This solution will be denoted by x(t, t 0 , ϕ).
As an important example of a one-dimensional DDE appearing in mathematical biology, we refer to the well-known Lasota-Wazewska equation [15] to model the survival of read blood cells. Generalisations of this equation with periodic coefficients and multiple delays have received much attention from mathematicians and other researchers (see [3,6] and references therein). More recently ( [5,9,16]), impulses have been added to such models, as in where all the coefficients and delays are periodic functions with a common period ω > 0 and 0 < t 1 < t 2 < . . . with t k+p = t k + ω, k = 1, 2, . . . for some positive integer p. The investigation in this paper was partially inspired by the works of Tang [13], Yan [17] and Zhang [23]. Another strong motivation to study the stability of impulsive models of the form (1.1) was to apply criteria of stability for such models to the Lasota-Wazewska impulsive system (1.4), and obtain generalisations of Liu and Takeuchi's result in [9].
Often the entire space BPC is not a suitable set of initial conditions, and more restrictive sets should be considered. A set S ⊆ BP C is called an admissible set of initial conditions if ϕ ∈ S ⇒ x t (·, t 0 , ϕ) ∈ S, t ≥ t 0 . For models from mathematical biology as (1.4), clearly only positive solutions are meaningful, and therefore admissible. In this paper we establish sufficient conditions for the stability of the zero solution of the impulsive DDE (1.1). These results can be applied to study the stability of other solutions, such as periodic solutions. We recall here some stability definitions for an admissible set of initial conditions S ⊆ BP C. Definition 1.1. Let f (t, 0) = 0 for t ≥ 0 and I k (0) = 0, k ∈ N. We say that the solution x ≡ 0 of (1.1) is stable in S if for any ε > 0 and t 0 ≥ 0, there exists δ = δ(t 0 , ε) > 0 such that The solution x ≡ 0 of (1.1) is said to be asymptotically stable in S if it is stable and for any t 0 ≥ 0, there exists δ = δ(t 0 ) > 0 such that The solution x ≡ 0 of (1.1) is said to be globally attractive in S if all solutions of (1.1) with initial conditions in S tend to zero as t → ∞. Finally, the solution x ≡ 0 of (1.1) is globally asymptotically stable if it is stable and global attractive. If it is well understood which set S we are dealing with, we omit the reference to it.
The remainder of this paper is organized in two sections. Section 2 deals with the stability and global asymptotic stability of the zero solution of the impulsive DDE (1.1). First, a main set of assumptions for (1.1) is introduced, and a brief comparison with other hypotheses considered in the literature is presented. Sufficient conditions for the global attractivity of zero are established by treating separately non-oscillatory and oscillatory solutions of (1.1). In Section 3, the global asymptotic stability of a positive ω-periodic solution to (1.4) is studied by using the results in Section 2. The particular case of constant delays τ i (t) = m i ω for m i positive integers (1 ≤ i ≤ n), is further analysed; in this situation, better results are obtained when the impulses in (1.4) are given by linear functions I k (u) = b k u. A comparison of our criteria with recent results in the literature is also included.
2. Stability. In this section, we address the stability and global attractivity of the trivial solution of (1.1) in BP C, but another admissible set of initial conditions S ⊆ BP C can be chosen.
The main assumptions that will be imposed are taken from the ones described below. Hypothesis (H3) will be chosen in alternative to (H2). Occasionally, weaker versions of these assumptions will be considered. (H1) there exist positive sequences (a k ) and (b k ) such that (H3) (i) the sequence P n = n k=1 a k is convergent; (H4) there exist piecewise continuous functions λ 1 , where M t (ϕ) := max 0, sup ϕ(θ) is the Yorke's functional on P C(t); (H5) there exists T > 0 with T − τ (T ) ≥ 0 such that where the coefficients α i := α i (T ) are given by We observe that the hypotheses (H1) and (H4) imply I k (0) = 0 and f (t, 0) = 0 for k ∈ N, t ≥ 0, thus x = 0 is an equilibrium point of (1.1). In (H5) above, we make use of the standard convention that the product B(t) is equal to one if the number of factors is zero.
In the remainder of this paper, we shall use the notation many interesting models in the literature; in this case, instead of (H3)(i) it will be sufficient to impose (H2)(i). However, it is useful to consider the alternative hypothesis (H3)(ii), which in particular allows dealing with (1.2). The constraint on the impulses given by (H1) implies in particular that . We now compare our hypotheses with the ones in the literature, in particular in references [17,23], two major sources of inspiration for the analysis in this section. As far as we know, the hypotheses on the impulses, (H1) and either (H2)(i) or (H3)(i), are novel and strongly relax the usual requirements in the literature on stability for impulsive DDEs under Yorke-type conditions. In fact, typically either I k are supposed to be linear functions, or condition (2.1) is assumed with a k = 1 for all k ∈ N, i.e., which implies |x(t + k )| ≤ |x(t − k )|, hence forces the solutions to approach zero at each instant of impulse [2,4,17,20,21,23].
Zhang [23] treated only the case of system (1.2), and proved the global attractivity of its zero solution provided that the impulsive functions I k and f satisfy (2.5), (H3)(ii), (H4), and the additional generalised " 3 2 -type condition": In [17], Yan considered (1.1) with a set of more restrictive assumptions: again, the impulsive functions I k were subject to condition (2.5), the Yorke condition (H4) was imposed with λ 1 (t) = λ 2 (t) =: λ(t) and an extra condition to deal with non-oscillatory solutions was added; moreover, instead of hypothesis (H5) or (2.6), for the global attractivity of the trivial solution of (1.1) Yan imposed the restriction where B(t) is defined as in (H5). In the case λ 1 (t) = λ 2 (t) = λ(t), it is clear that there are however positive functions a(t) for which the condition σ < 3 2 is less restrictive than α 1 α 2 < 1. In this situation, it would be convenient to achieve stability results under hypotheses similar to or less restrictive than (2.7). This will be the subject of a forthcoming paper. Nonetheless we should emphasise that the main idea here was to take full advantage of the negative feedback term a(t)x(t), rather than working with a 3 2 -type condition. This kind of approach has also been taken for non-impulsive DDEs: we shall refer later in this section to the work of Tang [13] (see also [6,14] for some alternative criteria), where the DDE x t ) as in (1.1) and c a positive constant, was studied assuming (H2)(ii) and the following Yorke condition: We start our analysis with an auxiliary result from [17]. Let x(t) be a solution of (1.1) on [0, ∞) and define y(t) by To prove the global asymptotic stability of the trivial solution, we consider separately oscillatory and non-oscillatory solutions to (1.1). We recall that a solution x(t) is oscillatory if it is not eventually zero and has arbitrarily large zeros; otherwise, x(t) is non-oscillatory. First, we establish criteria about the asymptotic behaviour of all non-oscillatory solutions.
Then, all non-oscillatory solutions of (1.1) are bounded. If in addition (H2)(ii) holds, then all non-oscillatory solutions of (1.1) converge to zero as t → ∞.
Proof. From (H1), we have Take a solution x(t) of (1.1) and let y(t) be defined by (2.8). From (2.10), we have For a non-oscillatory solution x(t), assume that x(t) > 0 for t ≫ 0 (the situation is analogous if x(t) < 0 for t ≫ 0). Then y(t) > 0 for large t and, from (2.9) and (H4*), y ′ (t) ≤ y ′ (t)+a(t)y(t) ≤ 0 for t ≫ 0, t = t k . In particular, there are c 0 , w ≥ 0 such that y(t) ց c 0 and e A(t) y(t) ց w as t → ∞, where A(t) is as in (2.4). On the other hand, from (H2)(i) and (2.11) we have 0 < When (H2)(ii) is not satisfied, the convergence to zero of non-oscillatory solutions is obtained if (H3)(ii) is imposed and, rather than having P k simply bounded, P k is required to be convergent. Lemma 2.3. Assume (H1), (H4*), and either (H2) or (H3). Then, all nonoscillatory solutions of (1.1) converge to zero as t → ∞.
Proof. As above and without loss of generality, suppose that x(t) is an eventually positive solution of (1.1), and let y(t) be defined by (2.8). From the proof of Lemma For t ≫ 0, 0 < x(t) ≤ k:0≤t k <t a k y(t), and by (H3)(i) We now show that c + = c − = 0.
(2.10), we obtain as well, and from (2.13) we get w = −∞, which is a contradiction. Thus c = 0, and the proof is complete.
Remark 2.1. It is easy to verify from the above proof that the assumption (H3)(ii) is not needed at all if (H3)(i) holds with lim n n k=1 a k = 0.
The goal now is to show that (H1), (H4), (H5) are sufficient conditions for the trivial equilibrium to be a global attractor of the oscillatory solutions of (1.1). A first auxiliary result is crucial to establish upper and lower bounds for oscillatory solutions, and was inspired in arguments of [23].
Let y(t) be a solution of (2.9) on [0, ∞) and t 0 ≥ T such that y(t 0 ) = 0. Then, for any η > 0, the following conditions hold: Proof. We shall prove (i), the proof of (ii) being similar. If the assertion (i) is false, there exists T 0 > t 0 such that either y(T 0 ) > α 2 η or y(T 0 ) < −η. We consider these two situations separately.
Reasoning as above, we deduce that there is ξ 0 ∈ [t 0 , T 0 ) ∩ [T 0 − τ (T 0 ), T 0 ) such that y(ξ 0 ) = 0 and y(t) < 0 for ξ 0 < t < T 0 . Since Integrating over [ξ 0 , T 0 ], and using the inequality α 1 α 2 ≤ 1, we get We are now in the position to prove the main result in this section. As y(t) is continuous, there exists a sequence s n ր ∞ with s n ≥ T 0 such that y(s n ) > 0, y(s n ) are local maxima, and y(s n ) → u as n → ∞. We may assume that y(s) < y(s n ) for s n − s > 0 small. As in the proof of Lemma 2.4, by the Yorke condition (H4) we deduce that for each n ∈ N there exists ξ n ∈ [s n − τ (s n ), s n ) such that y(ξ n ) = 0 and y(s) > 0 for s ∈ (ξ n , s n ]. By (2.16), we have y(s) > −v ε for s ∈ [ξ n − τ (ξ n ), s n ]. Arguing as in the proof of Case 1 of Lemma 2.4(i), we conclude that y(s n ) ≤ α 2 v ε . Letting n → ∞ and ε → 0 + , we obtain Under the constraint α 1 α 2 < 1, (2.19) is possible only if u = 0 and v = 0, thus y(t) → 0 as t → ∞.
Even for the situation without impulses (2.20), when a(t) ≡ 0 but ∞ 0 a(t) dt < ∞, the additional requirement (H3)(ii) must be imposed together with (H4), otherwise zero need not attract the eventually monotone solutions, as shown by the next counter-example.
As a particular case, a criterion given by Tang [13] and stated below is obtained by considering the DDE without impulses (2.20) and taking c 1 = c 2 = 1 in Corollary 2.2.
where M t (ϕ) is as in (2.2). Then the zero solution of (2.20) is globally asymptotically stable.

A periodic Lasota-Wazewska model with impulses.
In this section, we study a periodic Lasota-Wazewska model with impulses (see e.g. [5,9,16]): where 0 < t 1 < t 2 < · · · < t k < . . . , t k → ∞, and (f 0 ) the functions a(t), b i (t), β i (t), τ i (t) are continuous, positive and ω-periodic, 1 ≤ i ≤ n, t ∈ R, for some constant ω > 0; (i 0 ) the functions I k : [0, ∞) → R are continuous with u + I k (u) > 0 for u > 0, k ∈ N; moreover, there is a positive integer p such that Special attention will be given to the particular case of (3.1) with ω-periodic constant delays τ i (t) = m i ω for m i positive integers, i = 1, . . . , n. For some related models, see also [4,11,12]. Without loss of generality, we may suppose that there are exactly p instants of impulses on the interval [0, ω], t 1 , t 2 , . . . , t p . With minimal changes, we can also consider a more general framework, with a(t), b i (t), β i (t), τ i (t) piecewise continuous functions. Due to the biological applications, we only consider positive solutions of (3.1), corresponding to initial conditions is easy to see that if φ ∈ C + 0 , then N (t, 0, φ) is defined and positive for t ≥ 0. Some criteria for the existence of an ω-periodic solution to (3.1) have been established. Namely, the following result is a consequence of Theorem 2.3 in Li et al. [5]: where I ∞ = lim sup Let us however mention that the assumption I k (u) ≥ 0 for all u ≥ 0, k ∈ N in [5] is quite strong, since it requires that the impulses are always positive. In view of the biological meaning of the model, the natural constraint is only that I k (u) + u > 0 for u > 0. On the other hand, as we shall see, Liu and Takeuchi [9] considered the particular case of (3.1) with ω-periodic constant delays τ i (t) = m i ω for m i positive integers, i = 1, . . . , n, and linear impulses I k (u) = b k u with constants b k > −1 (see system (3.14) addressed later in this section), for which the existence of a positive ω-periodic solution was proven under a very general condition.
For system (3.1) with (f 0 ), (i 0 ) fulfilled, we now impose the following additional hypotheses: (i 1 ) there exist constants a 1 , . . . , a p and b 1 , . . . , b p , with b k > −1, and such that x − y ≤ a k , x, y ≥ 0, x = y, k = 1, . . . , p; Assume now that there exists a positive ω-periodic solution N * (t), and effect the change of variables where f (t, ϕ) =  Now, we insert this transformed system into the framework of the previous sections: (3.2) has the form (1.1), where the function f (t, x t ) may have jump discontinuities at the points t such that t − τ i (t) = t k for some 1 ≤ i ≤ n, k ∈ N.
Proof. Write c i (t), f i (t, ϕ) as in (3.3). Take t ≥ 0, i ∈ {1, . . . , n} and ϕ ∈ S i (t). We have On the other hand, by Lagrange's intermediate value theorem, . The next criterion follows as an immediate consequence of Theorem 2.3.
We now proceed with a deeper analysis of the case of (3.1) with time independent delays multiple of the period, where m i are positive integers, i = 1, . . . , n. In this situation, (3.2) becomes whereĨ k (u) = I k N * (t k ) + u − I k N * (t k ) are as in (3.3).
In what follows we establish upper estimates for α 1 , α 2 defined by (2.25) which, although not as sharp as the ones in Theorem 3.1, are easier to handle. To simplify the exposition, assume I k (0) = 0 for k ∈ N -which is natural from a biological point of view -, so that I k (N * (t k )) ≥ b k N * (t k ); otherwise, I k (N * (t k )) ≥ I k (0) + b k N * (t k ), 1 ≤ k ≤ p, and straightforward changes should be introduced in the computations below. Theorem 3.3. Consider (3.1) with τ i (t) ≡ m i ω (m i ∈ N) and denotem = max 1≤i≤n m i . Assume (f 0 ), (i 0 )-(i 2 ) with I k (0) = 0 for k ∈ N, and that there is a positive ω-periodic solution N * (t) of system (3.1). If Bm βN * (e βN * − 1) Proof. Condition (i 2 ) implies B ≥ 1 for B defined above, hence for t ≥ 0 and For the sake of simplicity, in what follows we suppose that the coefficients b k in (i 1 ) satisfy b k ∈ (−1, 0] (1 ≤ k ≤ p); otherwise we may replace b k by min{0, b k }. Now, choose λ 1,i as in (3.4) and λ 2,i as in (3.6). For (3.7) we obtain Since N * (t) is an ω-periodic solution of (3.1), for t > 0, t = t k , (3.9) with N * (t) having possible jumps at the points t k . From (3.9), for t > 0 we derive (3.11) The estimates (3.10) and (3.11) yield In a similar way, we obtain Clearly, condition σ 1 σ 2 < 1 is equivalent to (3.8).
As a by-product, we obtain some results for DDEs without impulses by setting b k = a k = 0 for 1 ≤ k ≤ p in the above theorems. (3.12) where ω > 0, m i ∈ N and the coefficient functions satisfy (f 0 ). Letm = max 1≤i≤n m i . Then, there is a positive ω-periodic solution N * (t), which is a global attractor of all positive solutions if one of the following conditions holds: (i) α 1 α 2 < 1 for 1 − e −m ω 0 a(u) du < 1. Proof. Here, we use Lemma 3.3 and Corollary 2.1 directly. The Yorke condition (2.2) is satisfied with λ 1 (t) = b(t)e −N * (t) and λ 2 (t) = b(t). For α 1 (t), α 2 (t) given by (2.21) we obtain t+s t a(u) du ds.
Remark 3.2. Corollary 3.2 is easily extended to (3.12), however our interest here is to compare the statement in Corollary 3.2 with [3]. Using an iterative technique, Graef et al. [3]. showed that the ω-periodic positive solution N * (t) of (3.13) is globally attractive if With the notations of the above proof, clearly α 1 < σ. On the other hand, observe that σ = mω 0 a(s)N * (s) ds; whether α 2 ≤ σ or not depends on the relative sizes of the functions a(t), b(t) and the delay mω. Therefore, the criteria in Corollary 3.2 and in [3] are not comparable without further information on the coefficient functions and delay.
We now study the particular case of (3.1) with τ i (t) ≡ m i ω (m i positive integers) and the impulses given by I k (u) = b k u: (3.14) As before, we assume that a(t), b i (t), β i (t) are as in (f 0 ), and that (i 0 ) holds, i.e., b k > −1 for k ∈ N, 0 < t 1 < t 2 < · · · < t p < ω and where p is some positive integer. In this setting, assumption (i 2 ) translates as Under these assumptions, the existence of a positive ω-periodic solution N * (t) of (3.14) follows from [9]. The next theorem recovers the criterion for its global asymptotic stability established in [9].
Theorem 3.4. Consider system (3.14) with b k > −1 for all k, and assume (f 0 ), (3.15) and (3.16). For N * (t) a positive ω-periodic solution, whose existence is given in [9], N * (t) is globally attractive if Proof. After translating the ω-periodic solution N * (t) to the origin, we obtain system (3.7) withĨ k (x(t k )) = b k x(t k ), k = 1, 2, . . . . We now effect the change of variables in (2.8), which in this situation reads as and obtain an equivalent DDE with piecewise coefficients but no impulses, given by Reasoning as in Lemma 3.3, we deduce that for (3.19) the Yorke condition (2.2) holds with λ 1 (t) = λ 2 (t) = n i=1b i (t)β i (t). Next, we observe that and apply Corollary 2.2 to (3.19) to obtain the result.
Remark 3.3. Yan [16] considered model (3.14) with the following additional constraint: the function t → Θ(t) := 0≤t k <t is ω-periodic. As pointed out by Liu and Takeuchi [9], the condition of Θ(t) being ω-periodic is too restrictive: it implies (3.15) and that Under these assumptions, Yan [16] gave additional sufficient conditions for the existence and global attractivity of a unique ω-periodic solution of (3.14). Nevertheless, Liu and Takeuchi remarked that Yan's proof was not complete. In turn, they proved themselves the existence of a positive ω-periodic solution N * (t) to (3.14) if which is always true if (3.16) is satisfied, and showed that N * (t) is globally attractive if, in addition to (3.16), the above condition (3.17) was imposed. The technique in [9] is quite different from the approach in [16]: the latter adapts the method in [3], whereas the main idea in [9] is to assume a Yorke-type condition of the form (2.24), and use [13]. Although this scenario is a particular situation of the general situation treated in our Corollary 2.2, the work in [9] was a strong motivation for the investigation carried out in this section.
Remark 3.4. For the situation of (3.1) with τ i (t) ≡ m i ω and impulsive functions I k satisfying (i 1 ) with b k < a k for some k ∈ {1, . . . , p}, the change of variables (3.20) is not suitable for our purposes, since it transforms (3.1) into (3.21) together with the impulsive conditions ∆y(t k ) =Î k (y(t k )), k ∈ N, whereÎ k satisfy Therefore (H2)(i) is never satisfied, and the results in Section 2 cannot be invoked.
Following the approach in Theorem 3.3, we now get estimates for α 1 , α 2 which are easier to verify, although they are not as refined as in (3.22).  (1 + b k ) β i (t), implies the global attractivity of the positive ω-periodic solution N * (t) of (3.14) (see also Remark 3.3, for a comment in [9]). In any case, the results in Theorems 3.5 and 3.6 are not easily comparable with the claim in [16].