and multiple pairs of time-varying delays

Attractivity analysis on a neoclassical growth system incorporating patch structure

and multiple pairs of time-varying delays

Qian Cao

College of Mathematics and Physics, Hunan University of Arts and Science, Changde 415000, Hunan, P. R. China

Received 27 June 2021, appeared 5 October 2021 Communicated by Leonid Berezansky

Abstract. In this paper, we focus on the global dynamics of a neoclassical growth sys- tem incorporating patch structure and multiple pairs of time-varying delays. Firstly, we prove the global existence, positiveness and boundedness of solutions for the ad- dressed system. Secondly, by employing some novel differential inequality analyses and the fluctuation lemma, both delay-independent and delay-dependent criteria are estab- lished to ensure that all solutions are convergent to the unique positive equilibrium point, which supplement and improve some existing results. Finally, some numerical examples are afforded to illustrate the effectiveness and feasibility of the theoretical findings.

Keywords: global attractivity, neoclassical growth system, patch structure, multiple pairs of time-varying delay.

2020 Mathematics Subject Classification: 34C25, 34D05, 34K13, 34K25.

1 Introduction

Under the assumptions that labor and capital are fully allocated and the output market is adjusted immediately, Day proposed a discrete-time neoclassical growth model in literature [5], which has unimodal feedback production function. As we all know, there is an inevitable time lag between the acquisition of information and the implementation of decisions, but the model proposed by Day ignores the influence of delays and cannot fully explain the actual economic situation. To revise this drawback and better characterize the long-term behavior of economics, Matsumoto and Szidarovszky [25] introduced the delayed neoclassical growth equation

x⁰(t) =−δx(t) +Px^γ(t−τ)e^{−σx(t−τ)}, (1.1) where x(t)labels the capital per labor at timet,δ is the sum of labor growth rate and capital depreciation rate multiplied by average saving rate, τdesignates the delay in the production

BEmail: caoqianj2019@126.com

function, γ denotes a proxy for measuring returns to scale of the production function, σ is regarded as a strength of a ‘negative influence’ produced by adding concentration of capital and is settled via a damaging degree of energy resources or natural environment. If γ = 1, the model (1.1) is the famous Nicholson’s blowflies model, whose dynamic behavior has been extensively studied in recent years [1,3,13,15–20,22,23,27,31,32,37]. However, for the case of γ 6= 1, there are relatively few studies devoted to model (1.1) and its extended models [4,7,24,26,33,34].

Recently, regarding that the identical production function usually contains different delays, L. Berezansky and E. Braverman put forward a dynamic model of the form in [2],

x⁰(t) =

∑

F_j(t,x(t−τ₁(t)), . . . ,x(t−τ_l(t)))−G(t,x(t)), t ≥t₀, (1.2) where l and m are positive integers, G describes the instantaneous mortality rate, and each F_j(j ∈ I := {1, 2,· · · ,m})is the feedback control relying on the values of the stable variable with distinctive delays τ₁(t),τ₂(t), . . . ,τ_l(t). Manifestly, (1.2) contains the modified delayed differential neoclassical growth model

x⁰(t) =β(t)

"

−δx(t) +

∑

P_jx^γ(t−g_j(t))e^{−σx(t−hj(t))}

#

, γ∈(0, 1), (1.3) which in the caseh_k ≡ g_k agrees with the traditional model [33].

In general, when each nonlinear function of the model contains only a small enough time delay, it will inherit some features of non time delay systems. For example, all the non- oscillatory solutions with respect to the unique positive equilibrium point are convergent.

Moreover, as long as the time delay is small enough, the global attractivity for the positive equilibrium point has been shown in [2,30]. And the existence, oscillation, persistence, peri- odicity and stability of positive solutions have been widely explored for the single time-delay system (1.3) and similar models withg_j(t)≡ h_j(t)[4,7,24,26,33,34]. However, when the same nonlinear function of the model incorporates two or more time delays, chaotic oscillation of the system will occur, which will increase the difficulty in the study of the dynamics of such systems. Therefore, this issue has attracted the attention of many scholars. More recently, Huang et al. [21] studied the attractivity for the scalar equation (1.3). Meanwhile, since the financial environment of some capitals is fragmented, and the natural separation of the space area is separate, the above scalar neoclassical growth model can be naturally generalized to the patch structure system [8,36], the scalar equation (1.3) can be normally extended to the following system incorporating patch structure and multiple pairs of time-varying delays:

x⁰_i(t) =β(t)

"

−δ^¯_ix_i(t) +

∑

a_ijx_j(t) +

∑

P_ijx^γ_i(t−g_ij(t))e^{−σijxi(t−hij(t))}

#

, γ∈(0, 1), (1.4) where i ∈ Q := {1, 2, . . . ,n}, x_i stands for the amount of the capital per labor in the patch i, a_ij designates the dispersal coefficient of the capital from patch j to patch i, m accounts for the number of population reproductive types, P_ijx^γ_i(t−g_ij(t))e^{−σijxi(t−hij(t))} describes the time-dependent reproduction function which is related to the incubation delayh_ij(t)and the maturation delayg_ij(t), andx^γ_i e^−σijxi acquires the maximum reproduce rate atx_i(t) = _σ^γ

ij. For more detailed biological significance, one can directly refer to [8,21,36] and their references quoted therein.

Hereafter, by changing the variables

δ¯_i =δ_i−a_ii with a_ii <0, (1.4) can be rewritten as

x⁰_i(t) = β(t)

"

−δ_ix_i(t) +

∑

a_ijx_j(t) +

∑

P_ijx^γ_i(t−g_ij(t))e^{−σijxi(t−hij(t))}

#

, γ∈(0, 1), i∈Q.

(1.5) It should be pointed out that, the dynamic characteristics of neoclassical growth model in- corporating patch structure and multiple pairs of time-varying delays have not been fully studied. To the best of our knowledge, we have only found that the author of [36] established the attractivity results of the system (1.5) when g_ij(t) ≡ h_ij(t) (i ∈ Q, j∈ I). However, there is no research on the dynamic behavior of the model (1.5) with g_ij(t)6=h_ij(t) (i∈ Q, j∈ I).

According to the above discussions, our goal is to establish the global attractivity condi- tions of the unique positive equilibrium point for the system (1.5) under g_ij(t) 6= h_ij(t)(i ∈ Q, j ∈ I). Briefly speaking, the contributions of this article can be summarized as below. 1) The boundedness and persistence on the solutions of system (1.5) are established by exploit- ing some novel differential inequality analyses; 2) Under certain assumptions, with the aid of the fluctuation lemma, some sufficient criteria ensuring the global attractivity of system (1.5) are obtained for the first time, which improve and generalize all recent works reported in [21,36]; 3) Numerical simulations involving comparison discussions are afforded to reveal the obtained theoretical results.

The remaining of this work is arranged as follows. In Section 2, some necessary lemmas and assumptions are listed. In Section 3, the global attractivity of the unique positive equi- librium point for the addressed system is demonstrated. To evidence our theoretical results, some numerical experiments are carried out in Section 4. Conclusions are given in Section 5.

2 Preliminary results

Throughout this manuscript, N⁺ labels the set of all positive integers and Rⁿ (R¹ = R) designates the n-dimensional real vectors set. For a bounded real function u, let u⁺ = sup_ϑ∈Ru(ϑ), u⁻=inf_ϑ∈Ru(ϑ).

With the biological applications in mind, we assume thatδ_i > 0, P_ij > 0, σ_ij > 0, β⁻ >0 and

r_i =max (

max

1≤j≤m

sup

t∈_R

g_ij(t), max

1≤j≤m

sup

t∈_R

h_ij(t) )

, r = max

1≤i≤n{r_i}.

Likewise, g_ij, h_ij, β : R −→ (0,+∞) (i ∈ Q, j ∈ I)are bounded and continuous functions, A= (a_ij)_n×nis an irreducible and cooperative matrix witha_ij ≥0(i6= j), and

∑

a_ij =−a_ii, for alli∈ Q. (2.1)

In addition, suppose that there exists a positive constantN^∗ such that

−δ_i(N^∗)^1−γ+

∑

P_ije^−σijN∗ =0, for alli∈Q, (2.2)

which implies that(N^∗,N^∗, . . . ,N^∗)is a positive equilibrium point of system (1.5).

DenoteC= _∏ⁿ_i=1C([−r_i, 0],R)be a Banach space involving the supremum normk · k, and C+=_∏ⁿ_i=1C([−r_i, 0],[0, +∞)). Also, we set x_t(t₀,ϕ)(x(t;t₀,ϕ))for an admissible solution of (1.5) obeying the initial conditions:

x_t0 = ϕ, ϕ∈C+ and ϕ_i(0)>0, i∈Q, (2.3) and[t₀,η(ϕ))be the maximal right-interval of existence.

Now, we present two lemmas to reveal the positiveness and boundedness of (1.5).

Lemma 2.1. x(t) =x(t;t₀,ϕ)has positiveness and boundedness on[t₀, +_∞).

Proof. By Theorem 5.2.1 in [28], we have thatx_t(t₀,ϕ)∈C+for allt∈[t₀,η(ϕ)). This, together with (1.5) and (2.3), follows that

x_i(t) = ϕ_i(0)e⁻

R_t

t0(δi−aii)β(s)ds

+e⁻

R_t

t0(δi−aii)β(s)dsZ _t

t0

β(s)

×

"

∑

a_ijx_j(s) +

∑

P_ijx^γ_i (s−g_ij(s))e^{−σijxi(s−hij(s))}

# e

Rs

t0(δ_i−a_ii)β(v)dv

ds

>0 for allt ∈[t₀,η(ϕ))andi∈Q. (2.4) Fort> t₀, leti₀ ∈QandT_i0 ∈ [t₀−r_i0, t]such that

x_i0(T_i0) = max

t0−ri0≤s≤tx_i0(s) =max

i∈Q

t0−maxri≤s≤tx_i(s)

. WhenT_i0 ∈[t₀−r_i0, t₀], it is easily seen that

kxs(t₀,ϕ)k ≤x_i0(T_i0) =kϕk for alls ∈[t₀, t]. (2.5) IfT_i0 ∈ (t0, t], (1.5), (2.1) and (2.4) lead to

0≤ x⁰_i0(T_i0)

= β(T_i0)

"

−δ_i0x_i0(T_i0) +

∑

a_i0jx_j(T_i0) +

∑

P_i0jx^γ_i

0(T_i0−g_i0j(T_i0))e^{−σi0jxi0(Ti0−hi0j(Ti0))}

#

≤ β(T_i0)

"

−δ_i0x_i0(T_i0) +

∑

a_i0jx_i0(T_i0) +

∑

P_i0jx_i^γ

0(T_i0)e^{−σi0jxi0(Ti0−hi0j(Ti0))}

#

≤ β(T_i0)x^γ_i

0(T_i0)

"

−δ_i0x¹_i^−γ

0 (T_i0) +

∑

P_i0j

# , which yields

kxs(t₀,ϕ)k ≤x_i0(T_i0)≤max

i∈Q

∑^mj=1P_ij δ_i

!₁₋¹_γ

for all s∈(t₀, t]. (2.6) From (2.5) and (2.6), we obtain thatx(t)has boundedness on[t₀, η(ϕ)), and

kxt(t₀,ϕ)k ≤x_i0(T_i0)≤max

i∈Q

∑^mj=1P_ij δ_i

!₁₋¹_γ

+kϕk=:X^ϕ for all t ∈[t₀, η(ϕ)). (2.7) This, together with Theorem 2.3.1 in [9], follows η(ϕ) = +∞, and finishes the evidence of Lemma2.1.

Lemma 2.2. lim inf_t→+_∞x_i(t)>0for all i∈Q.

Proof. To obtain a contradiction, we suppose thatl=min

i∈Q lim inf

t→+_∞ x_i(t) =0. Let m(t) =max

ξ :ξ ≤t

there is ˆi∈Qsatisfying x_iˆ(ξ) =min

i∈Q

t0min≤s≤tx_i(s)

.

Then, limt→+_∞m(t) = +∞. Likewise, for a strictly monotone increasing infinite sequence {t_p}_p≥1, there are ˆi∈Qand a subsequence{t_pk}_k≥1 ⊆ {t_p}_p≥1 agreeing with

xˆi(m(tp_k)) = min

t0≤s≤t_pkxˆi(s) =min

i∈Q

t0min≤s≤t_pkx_i(s)

and lim

k→+_∞xˆi(m(tp_k)) =0. (2.8) Owing to (1.5), (2.1), (2.7) and (2.8), we derive

0≥ x_i⁰_ˆ(m(t_pk))

≥ β(m(tp_k))

"

−δˆixˆi(m(tp_k)) +xˆi(m(tp_k))

∑

aijˆ

+

∑

Pijˆx^γ_ˆ

i (m(tp_k)−gˆij(m(tp_k)))e^{−σˆijxˆi(m(tpk)−hˆij(m(tpk)))}

#

≥ β(m(tp_k))

"

−δˆixˆi(m(tp_k)) +

∑

Pˆijx^γ_ˆ

i (m(tp_k))e^−σijˆXϕ

#

for all m(tp_k)>t0, and

δiˆ≥

∑

Pijˆ

1 x¹_ˆ^−γ

i (m(tp_k))^e

−σijˆX^ϕ

, for allm(tp_k)>t₀. (2.9) By taking limits, (2.8) and (2.9) give usδ_iˆ≥+∞, which yields a contradiction and finishes the proof.

Lemma 2.3. Lemma 2.2 indicates that(0, 0, . . . , 0)is unstable.

3 Global attractivity analysis

First, we present a delay-independent criterion to assure the attractivity for nonoscillatory solutions of system (1.5).

Proposition 3.1. If

mini∈Q lim inf

t→+_∞ x_i(t)≥ N^∗ (or max

i∈Q

lim sup

t→+_∞

x_i(t)≤N^∗), thenlim sup_t→+∞x_i(t) =N^∗(or lim inf_t→+_∞x_i(t) =N^∗)for all i∈Q.

Proof. We just need to deal with the case that mini∈Q lim inf

t→+_∞ x_i(t)≥ N^∗,

since the situation is entirely analogous for the case that max_i∈Qlim sup_t→+∞x_i(t)≤ N^∗.

Sety_i(t) =x_i(t)−N^∗(i∈Q), it is evident that lim sup

t→+_∞

y_i(t)≥0 for alli∈ Q. (3.1)

Let i^∗ ∈ Qbe such an index as lim sup_t→+∞y_i^∗(t) = max_i∈Qlim sup_t→+∞y_i(t). We state that

lim sup

t→+_∞

y_i^∗(t) =0.

Otherwise, lim sup_t→+∞y_i^∗(t)>0. Owing to the fluctuation lemma [29, Lemma A.1.], it is an easy matter to find a sequence{t_k}_k≥1obeying

k→+lim_∞t_k = +∞, lim

k→+_∞y_i^∗(t_k) =lim sup

t→+_∞

y_i^∗(t), lim

k→+_∞y⁰_i∗(t_k) =0. (3.2) Due to (1.5) and (2.1), we gain

y⁰_i∗(t_k) =β(t_k)

"

−δ_i^∗x_i^∗(t_k) +

∑

a_i^∗_jy_j(t_k) +

∑

P_i^∗_jx_i^γ∗(t_k−g_i^∗_j(t_k))e^{−σi∗jxi∗(tk−hi∗j(tk))}

#

. (3.3)

Because β(t), x_i^∗(t−g_i^∗_j(t)) and x_i^∗(t−h_i^∗_j(t)) are bounded on [t₀, +∞), we can select a subsequence of {t_k} (for convenience of exposition, we still label by {t_k}) satisfying that lim_k→+∞β(t_k), lim_k→+∞y_l(t_k), lim_k→+∞x_i^∗(t_k −g_i^∗_j(t_k)) and lim_k→+∞x_i^∗(t_k −h_i^∗_j(t_k)) exist for alll∈Q\{i^∗}andj∈ I. Moreover, 0<β⁻≤lim_k→+_∞β(t_k), and

N^∗ ≤ lim

k→+_∞x_i^∗(t_k−h_i^∗_j(t_k)), lim

k→+_∞x_i^∗(t_k−g_i^∗_j(t_k))≤ N^∗+ lim

k→+_∞y_i^∗(t_k). (3.4) With the help of (3.4), we regard two cases as follow.

Case 1. If lim_k→+_∞x_i^∗(t_k−h_i^∗_j(t_k)) = N^∗ for all j∈ I, by taking limits, (2.1), (2.2), (3.2), and (3.3) reveal that

0= lim

k→+_∞y⁰_i^∗(t_k)

≤ lim

k→+_∞β(t_k)

"

−δ_i^∗ lim sup

t→+_∞

y_i^∗(t) +N^∗

!

+lim sup

t→+_∞

y_i^∗(t)

∑

a_i^∗_j

+

∑

P_i^∗_j lim sup

t→+_∞

y_i^∗(t) +N^∗

!γ

e^{−σi∗jN∗}

#

≤ lim

k→+_∞β(t_k) lim sup

t→+_∞

y_i^∗(t) +N^∗

!γ

−δ_i^∗ lim sup

t→+_∞

y_i^∗(t) +N^∗

!1−γ

+

∑

P_i^∗_je^{−σi∗jN∗}





< lim

k→+_∞β(t_k) lim sup

t→+_∞

y_i^∗(t) +N^∗

!γ"

−δ_i^∗(N^∗)^1−γ+

∑

P_i^∗_je^{−σi∗jN∗}

#

=0,

which leads to a contradiction, and suggests that lim sup_t→+∞y_i^∗(t) =0.

Case 2. If for some j∈ I, N^∗ < lim_k→+∞x_i^∗(t_k−h_i^∗_j(t_k)), it follows from (2.1), (2.2), (3.2) and (3.3) that

0= lim

k→+_∞y⁰_i∗(t_k)

< lim

k→+_∞β(t_k)

"

−δ_i^∗ lim

k→+_∞x_i^∗(t_k) +

∑

a_i^∗_j lim

k→+_∞y_j(t_k) +

∑

P_i^∗_j

k→+lim_∞x^γ_i∗(t_k−g_i^∗_j(t_k))

e^{−σi∗jN∗}

#

< lim

k→+_∞β(t_k) lim sup

k→+_∞

y_i^∗(t) +N^∗

!γ"

−δ_i^∗(N^∗)^1−γ+

∑

P_i^∗_je^{−σi∗jN∗}

#

=0,

which is also a contradiction and proves the above statement. This finishes the proof of Proposition3.1.

Corollary 3.2. If for any i ∈ Q,x_i(t)is eventually nonoscillatory about N^∗, i.e., there is T^∗ obeying that

x_i(t)≥ N^∗(or x_i(t)≤N^∗) for all t ≥T^∗ and i∈ Q.

Thenlimt→+_∞x_i(t) =N^∗for all i ∈Q.

Remark 3.3. Corollary 3.2 shows that a delay-independent criterion has been established to guarantee that all non-oscillatory solutions of the system (1.5) are convergent to its unique positive equilibrium point.

Remark 3.4. It is obvious that all conclusions in Theorem 3.1, Theorem 3.2 of [21] and the results of Theorem 3.1 in [36] are special ones of Proposition3.1.

Theorem 3.5. Letσ=max_i∈Qmax_j∈Iσ_ij, suppose that, for all i∈ Q, δ_iσN^∗(e^{(δi−aii)β+r}−1)

δ_i−a_ii ≤1, (3.5)

and

0<σN^∗δ_i 1−e^{−r(δi−aii)β+}

δ_i[1−e(1−e^{−r(δi−aii)β+})]−a_iie^{−r(δi−aii)β+} ≤1, (3.6) hold. Thenlim_t→+_∞x_i(t) =N^∗for all i ∈Q.

Proof. Let

z_i(t) =σ(x_i(t)−N^∗), i∈Q, we have from (1.5) that

z⁰_i(t) +σδ_iβ(t)N^∗+δ_iβ(t)z_i(t)

=β(t)

∑

a_ijz_j(t) +σβ(t)

∑

P_ij

z_i(t−g_ij(t)) σ +N^∗

γ

e⁻

σij zi(t−_hij(t))

σ −σ_ijN^∗, (3.7)

and

z_i(t)e

Rt

t0(δi−aii)β(v)dv0

=

"

∑

a_ijβ(t)z_j(t) +σβ(t)

∑

P_ij

z_i(t−g_ij(t)) σ +N^∗

γ

×e⁻

σij zi(t−_hij(t))

σ −σ_ijN^∗−σβ(t)δ_iN^∗

e

R_t

t0(δi−aii)β(v)dv

, t≥t₀, i∈ Q. (3.8) To finish the verification, we shall reveal that

mini∈Q lim inf

t→+_∞ z_i(t) =max

i∈Q

lim sup

t→+_∞

z_i(t) =0.

In view of Corollary 3.2, we only need to treat the case that for each T^∗ > t₀, there are t^∗,t^∗∗∈(T^∗, +∞)such that

mini∈Q z_i(t^∗)<0 and max

i∈Q z_i(t^∗∗)>0. (3.9) Set

µ=lim sup

t→+_∞

z_i1(t) =max

i∈Q

lim sup

t→+_∞

z_i(t), λ=lim inf

t→+_∞ z_i2(t) =min

i∈Q lim inf

t→+_∞ z_i(t). (3.10) Owing to (3.9), we gain

λ≤0≤µ.

Now, it suffices to evidence thatλ=µ=0. Contrarily, eitherµ>0 orλ<0 is valid.

We only deal with the case thatµ>0 occurs. (λ<0 can be treated similarly.)

If λ = 0, i.e., λ = min_i∈Qlim inf_t→+_∞z_i(t) = 0. By Proposition 3.1, one can see that µ=lim sup_t→+∞z_i1(t) =0.

When µ > 0 andλ < 0, on account of the fluctuation lemma [29, Lemma A.1.], one can take two strictly monotone increasing infinite sequences{l_q}_q≥1,{s_q}_q≥1satisfying that

z_i1(lq)>0, lq →+∞, z_i1(lq)→µ, z⁰_i1(lq)→0 asq→+∞, (3.11) and

z_i2(s_q)<0, s_q→+∞, z_i2(s_q)→λ, z⁰_i2(s_q)→0 asq→+∞. (3.12) Note that a bounded sequence has a convergent subsequence, we can presume that for allj∈ I,

q→+lim_∞β(l_q) = β^∗, lim

q→+_∞z_i1(l_q−g_i

1j(l_q)) =z^j_i

1, lim

q→+_∞z_i(l_q) =z^l_i (i∈ Q\ {i₁}), (3.13) and

q→+lim_∞β(sq) =β^∗∗, lim

q→+_∞z_i2(sq−g_i

2j(sq)) =z^j_i

2, lim

q→+_∞z_i(sq) =z^s_i (i∈Q\ {i₂}). (3.14) To obtain a contradiction, we divide our proof into three steps.

First, we assert that there exists H₁ > 0 obeying that, for any q ≥ H₁, there is Lq ∈ [l_q−r_i1,l_q)agreeing with

z_i1(L_q) =0, and z_i1(t)>0, for all t∈(L_q,l_q). (3.15)

If not, there exists a subsequence of {l_q}(do not relabel) such that

z_i1(t)>0, for allt ∈[l_q−r_i1,l_q), q=1, 2, . . . (3.16) Subsequently,

0≤ lim

q→+_∞z_i1(l_q−g_i1j(l_q))≤ µ for all j∈ I, (3.17) and

z⁰_i1(lq) =β(lq)

∑

a_i1jz_j(lq) +σβ(lq)

∑

P_i1j

z_i1(l_q−g_i1j(l_q))

σ +N^∗

γ

e⁻

σi1j zi1(lq−_hi 1j(lq)) σ −σ_i1jN^∗

−σδ_i1β(lq)N^∗−δ_i1β(lq)z_i1(lq)

< β(lq)

∑

a_i1jz_j(lq) +σβ(lq)

∑

P_i1j

z_i1(l_q−g_i1j(l_q))

σ +N^∗

γ

e^−σi1jN∗

−σδ_i1β(l_q)N^∗−δ_i1β(l_q)z_i1(l_q). (3.18) By taking limit, (3.11), (3.13), (3.17) and (3.18) lead to

0≤a_i1i1β^∗ lim

q→+_∞z_i1(l_q) +β^∗

∑

a_i1j lim

q→+_∞z_j(l_q) +σβ^∗

∑

P_i1j





q→+lim_∞z_i1(lq−g_i1j(lq))

σ +N^∗





γ

e^−σi1jN∗−σδ_i1β^∗N^∗−δ_i1β^∗ lim

q→+_∞z_i1(l_q)

≤σβ^∗

∑

P_i1j





q→+lim_∞z_i1(lq−g_i1j(lq))

σ +N^∗





γ

e^−σi1jN∗−σβ^∗δ_i1

N^∗+ ^µ σ

≤σβ^∗

N^∗+ ^µ σ

γ

"

∑

P_i1je^−σi1jN∗−δ_i1

N^∗+ ^µ σ

1−γ

#

<0,

which is a contradiction and validates the above assertion.

Similarly, from (3.12) and (3.14), one can find H₁^∗ > 0 such that for any q ≥ H₁^∗, there is S_q∈[s_q−r_i2,s_q)such that

z_i2(Sq) =0, and z_i2(t)<0, for all t∈(Sq,sq). (3.19) Secondly, we show

e^−µ−1≤ λ≤0≤ µ≤e^−λ−1. (3.20) For any 0 < ε < σ(N^∗+ ^λ_σ) = σlim inft→+_∞x_i2(t), (3.10) suggests that one can select a positive integerq^∗ >H₁+H₁^∗ satisfying

λ−ε <z_i(t)<µ+ε for allt >min{l_q^∗,s_q^∗} −2r and i∈Q. (3.21)