A new generalization of Halanay type inequality and its applications

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R E S E A R C H

Open Access

A new generalization of Halanay-type

inequality and its applications

Haiyang Wen

1

_{, Shi Shu}

1

_{and Liping Wen}

1*

*_{Correspondence:}

[email protected] 1_{School of Mathematics and} Computational Science, Xiangtan University, Xiangtan, P.R. China

Abstract

In this paper, in order to study the dissipativity of nonlinear neutral functional differential equations, a generalization of the Halanay inequality is given. We apply this generalized Halanay inequality to an analysis of the dissipativity of two classes of nonlinear neutral delay integro-differential equations and the sufficient conditions are presented to ensure these systems are dissipative.

Keywords: Neutral delay integro-diﬀerential equations; Dynamical systems; Halanay inequality; Functional diﬀerential equations

1 Introduction

In 1966, in order to discuss the stability of the zero solution of the delay diﬀerential equa-tion

u(t) =Au(t) +Bu(t–τ), τ > 0,

Halanay introduced the following lemma (see [1] p. 378).

Lemma 1.1(Basic Halanay inequality) Assume thatτ≥0and v(t)is a positive function deﬁned on[t0–τ, +∞),with derivative v(t)on[t0, +∞).If

v(t)≤–αv(t) +β sup

t–τ≤s≤t

v(s), t≥t0,

whereα>β> 0,then there existγ > 0and k> 0such that

v(t)≤kexp–γ(t–t0)

, t≥t0.

The Halanay inequality became a powerful tool in the stability theory of delay differ-ential equations, therefore many authors improved or generalized it to more general type and used it for investigating the stability and dissipativity of various functional differential equations. We refer the reader to the papers, for instance, of Baker and Tang [2], Agarwal, Kim and Sen [3,4], Baker [5], Liz and Trofimchuk [6], Tian [7], Wen, Yu and Wang [8,9], Liu et al. [10], Wang [11], Hien et al. [12], and Gan [13].

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On the other hand, many interesting problems in physics and engineering are modeled by dissipative dynamical systems. These systems are characterized by the property of pos-sessing a bounded absorbing set, which all trajectories enter in a finite time and thereafter remain inside of. In the study of dissipative systems, this asymptotic behavior of the system is of interest and important (see [14]). In 1994, Humphries and Stuart [15] first studied the analytical and numerical dissipativity of initial value problems (IVPs) in ODEs. Hereafter, a number of results on the analytical and numerical dissipativity with respect to various types of differential equations are presented (such as found in [16–21]).

In this paper, we first present a more general Halanay-type inequality in Sect.2. Then, in Sect.3, we use this inequality to discuss the analytical dissipativity of two classes of nonlinear neutral delay integro-differential equations (NDIDEs) and some sufficient con-ditions which ensure the systems to be dissipative are given. Finally, the paper ends with a conclusion.

2 The generalized Halanay inequality

For simplicity of presentation, we denotef[t1,t2]_:=_sup

t1≤ξ≤t2f(ξ) andf

[t1,+∞)_:=_sup

ξ≥t1f(ξ)

for a bounded functionf.

Theorem 2.1 Assume thatτ≥0and u(t),w(t)are non-negative functions deﬁned on[t0–

τ, +∞),with derivative u(t)on[t0, +∞).If ⎧

⎨ ⎩

u(t)≤R1(t) +A(t)u(t) +B(t)u[t–τ,t]+C(t)w(t) +D(t)w[t–τ,t],

w(t)≤R2(t) +F(t)u(t) +G(t)u[t–τ,t]+H(t)w[t–τ,t],

(2.1)

for t≥t0and

w[t0–τ,t0]≤ r2

1 –H0

+F0+G0 1 –H0

u[t0–τ,t0]_, _(2.2)

and there exists a constantσ> 0such that

A(t) +B(t) +C(t) +D(t)F0+G0 1 –H0 ≤

–σ, ∀t≥t0. (2.3)

Then,for t≥t0,we have ⎧

⎨ ⎩

u(t)≤γ_σ∗+φe–μ∗(t–t0)_,

w(t)≤ γ2

1–H0+

F0+G0

1–H0

γ∗ σ +

F0+G0eμ∗τ

1–H0eμ∗τ φe

–μ∗(t–t0)_,

(2.4)

where R1(t),R2(t), –A(t),B(t),C(t),D(t),F(t),G(t),H(t)are non-negative,continuous and

bounded functions deﬁned on[t0, +∞); ⎧

⎨ ⎩

F0=F[t0,+∞), G0=G[t0,+∞), C0=C[t0,+∞), D0=D[t0,+∞),

H0=H[t0,+∞), γ1=R[1t0,+∞), γ2=R[2t0,+∞), φ=u[t0–τ,t0],

and0 <H(t)≤H0< 1,γ∗=γ1+C_1–0+_HD₀0γ2.The constantμ∗> 0is deﬁned as

μ∗=inf

t≥t0

μ(t) :μ(t) +A(t) +B(t)eμ(t)τ₊_C₍_t_{) +}_D₍_t₎_eμ(t)τF0+G0e μ(t)τ 1 –H0eμ(t)τ

= 0

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Specially,if R1(t) =R2(t)≡0, (2.4)degenerates into following form: ⎧

⎨ ⎩

u(t)≤φe–μ∗(t–t0)_,

w(t)≤F0+G0eμ∗τ

1–H0eμ∗τ φe

–μ∗(t–t0)_, t≥t0. (2.5)

Proof First, ifτ = 0, from the second formula of (2.1), we have

w(t)≤ R2(t) 1 –H(t)+

F(t) +G(t)

1 –H(t) u(t). (2.6)

Substituting (2.6) into the ﬁrst formula of (2.1) shows that (2.1) degenerates into a diﬀer-ential inequality

u(t)≤R(t) +A(t)u(t), t≥t0,

where

⎧ ⎨ ⎩

R(t) =R1(t) + (C(t) +D(t))_1–R2_H(t₍)_t₎,

A(t) =A(t) +B(t) + (C(t) +D(t))F_1–(t)+_HG₍_t(₎t). Noting the condition (2.3), it is can be proved that

u(t)≤γ

∗

σ

1 –exp

t

t0

A(s)ds

+u(t0)exp

t

t0

A(s)ds

.

The combination of this formula and (2.6) shows that (2.4) holds with

μ∗=inf

t≥t0

–A(t) –B(t) –C(t) +D(t)F(t) +G(t) 1 –H0

.

It is obvious thatμ∗> 0 under the assumption (2.3).

In the following we assume thatτ > 0. For any givent∈[t0, +∞), we deﬁne function

E(μ) on [0,1_τln_H1 0) by

E(μ) :=μ+A(t) +B(t)eμτ+C(t) +D(t)eμτF0+G0e μτ

1 –H0eμτ

. (2.7)

From (2.7) we can see that

E(0) < 0, lim

μ→1_τln_H1 0–0

E(μ) = +∞, E(μ) > 0.

Therefore, there exists a uniqueμ∈(0,1_τ ln_H1

0) such that

μ+A(t) +B(t)eμτ₊_C₍_t_{) +}_D₍_t₎_eμτF0+G0e μτ

1 –H0eμτ

= 0, (2.8)

which deﬁnes an implicit functionμ(t) fort≥t0. It is obvious thatμ∗≥0. Now we prove

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In fact, if this is not true. Let H0 satisfying 0 <H0<H0 < 1 and let 0 <ε1<min{σ₂,

–1_τ lnH0,1τln( σ

2Q+ 1)}, where

Q=B0+

C0H0(G0+F0H0) +D0H0(F0+G0) +D0G0(1 –H0)

(H0–H0)(1 –H0)

,

andB0=B[t0,+∞).

Then there would existt∗≥t0such thatμˆ:=μ(t∗) <ε1and

ˆ

μ+At∗+Bt∗eμτˆ +Ct∗+Dt∗eμτˆ F0+G0e

ˆ

μτ

1 –H0eμτˆ

= 0. (2.9)

Substituting (2.3) into (2.9) gives

0 =μˆ+At∗+Bt∗eμτˆ ₊_C_t∗₊_D_t∗_eμτˆ F0+G0eˆ μτ

1 –H0eμτˆ

≤ε1–σ+

B0+

C0(F0H0+G0) +D0(F0+G0) +D0G0eε1τ(1 –H0))

(1 –H0eε1τ)(1 –H0)

eε1τ _{– 1}

≤ε1–σ+Q

eε1τ _{– 1}_<_ε

1–σ+

σ 2 < 0,

which is a contradiction.

In order to verify (2.4), we ﬁrst show that, for anyε> 0,

⎧ ⎨ ⎩

u(t) <γ_σ∗+ε+φe–μ∗(t–t0)_,

w(t) < γ2

1–H0+

F0+G0

1–H0 (

γ∗ σ +ε) +

F0+G0eμτ

1–H0eμτφe

–μ∗(t–t0)_,

t≥t0. (2.10)

In fact, whent=t0, (2.10) is evident by using (2.2).

If we suppose (2.10) is not true fort>t0, then there would exist someε0> 0 andς>t0

such that whent<ς

⎧ ⎨ ⎩

u(t) <γ_σ∗+ε0+φe–μ

∗₍_t_–_t 0)_,

w(t) < γ2

1–H0+

G0+F0

1–H0 (

γ∗ σ +ε0) +

F0+G0eμ∗τ

1–H0eμ∗τ φe

–μ∗(t–t0)_,

(2.11)

while whent=ς, at least one of the following two equalities is true:

u(ς) = γ

∗

σ +ε0+φe

–μ∗(ς–t0) _(2.12)

and

w(ς) = γ2 1 –H0

+G0+F0 1 –H0

γ∗ σ +ε0

+F0+G0e μ∗τ

1 –H0eμ∗τ

φe–μ∗(ς–t0)_. _(2.13)

However, from the second formula of (2.1), whenς–τ≥t0, we have

w(ς)≤R2(ς) +F(ς)u(ς) +G(ς) sup

ς–τ≤ξ≤ς

u(ξ) +H(ς) sup

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<R2(ς) +F(ς)

γ∗

σ +ε0+φe

–μ∗(ς–t0)

+G(ς)

γ∗

σ +ε0+φe

–μ∗(ς–τ–t0)

+H(ς)

γ2

1 –H0

+F0+G0 1 –H0

γ∗ σ +ε0

+F0+G0e μ∗τ

1 –H0eμ∗τ

φe–μ∗(ς–τ–t0)

=R2(ς) +H(ς)

γ2

1 –H0

+

F(ς) +G(ς) +H(ς)F0+G0 1 –H0

γ σ +ε0

+

F(ς) +G(ς)eμ∗τ₊F0+G0e μ∗τ

1 –H0eμ∗τ

H0eμ

∗_τ

φe–μ∗(ς–t0)

≤ γ2

1 –H0

+F0+G0 1 –H0

γ∗ σ +ε0

+F0+G0e μ∗τ

1 –H0eμ∗τ

φe–μ∗(ς–t0)_, _(2.14)

and, whenς–τ <t0, we have

w(ς)≤R2(ς) +F(ς)u(ς) +G(ς)max

sup

t0–τ≤ξ≤t0

u(ξ), sup

t0≤ξ≤ς u(ξ)

+H(ς)max

sup

t0–τ≤ξ≤t0

w(ξ), sup

t0≤ξ≤ς w(ξ)

<R2(ς) +F(ς)

γ∗

σ +ε0+φe

–μ∗(ς–t0)

+G(ς)

γ∗

σ +ε0+φ

+H(ς)

γ2

1 –H0

+F0+G0 1 –H0

γ∗ σ +ε0

+F0+G0e μ∗τ

1 –H0eμ∗τ

φ

≤ γ2

1 –H0

+F0+G0 1 –H0

γ∗ σ +ε0

+F0+G0e μ∗τ

1 –H0eμ∗τ

φe–μ∗(ς–t0)_. _(2.15)

Hence (2.14) and (2.15) show that (2.13) is not true. Therefore we need only consider the case that (2.12) holds and we shall obtain a contradiction. Set

v(t) =γ

∗

σ +ε0+φe

–μ∗(t–t0)_, _z₍_t_{) =}_v₍_t_{) –}_u₍_t_).

Thenz(t) > 0 fort<ςandz(ς) = 0 andz(ς)≤0. Hence from the ﬁrst formula of (2.1) we have

z(ς) =v(ς) –u(ς)

≥–φμ∗e–μ∗(ς–t0)

–R1(ς) +A(ς)u(ς) +B(ς)u[ς–τ,ς]+C(ς)w(ς) +D(ς)w[ς–τ,ς]

. (2.16)

Ifς–τ≥t0, it follows from (2.11), (2.12), (2.16) and the deﬁnition ofγ∗that

z(ς)≥–γ∗–

γ∗ σ +ε0

A(ς) +B(ς) +C(ς) +D(ς)F0+G0 1 –H0

–φe–μ∗(ς–t0)

×

μ∗+A(ς) +B(ς)eμ∗τ+C(ς) +D(ς)eμ∗τF0+G0e μ∗τ

1 –H0eμ∗τ

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From the deﬁnition of the functionμ(t), we have

μ(ς) +A(ς) +B(ς)eμ(ς)τ+C(ς) +D(ς)eμ(ς)τF0+G0e μ(ς)τ

1 –H0eμ(ς)τ

= 0.

Therefore, it is easy to see that

μ∗+A(ς) +B(ς)eμ∗τ+C(ς) +D(ς)eμ∗τF0+G0e μ∗τ

1 –H0eμ∗τ

=μ∗–μ(ς) +B(ς)eμ∗τ –eμ(ς)τ+C(ς)

F0+G0eμ

∗_τ

1 –H0eμ∗τ

–F0+G0e μ(ς)τ

1 –H0eμ(ς)τ

+D(ς)

F0+G0eμ

∗_τ

1 –H0eμ∗τ

eμ∗τ–F0+G0e μ(ς)τ

1 –H0eμ(ς)τ

eμ(ς)τ

≤0,

which substituting into (2.17) and noting the condition (2.3), gives

w(ς) =v(ς) –u(ς)≥σ ε0> 0. (2.18)

Ifς–τ<t0, it follows from (2.16) that

z(ς)≥–φμ∗e–μ∗(ς–t0)_–_R

1(ς) –A(ς)u(ς) –B(ς)max

φ,u[t0,ς]

–C(ς)w(ς) –D(ς)maxw[t0–τ,t0]_,_w[t0,ς]_.

Thus we also can get (2.18) by simple derivation. This is in contradiction with our result w(ς)≤0. Therefore the inequality (2.10) must hold for any given ε> 0. Sinceε> 0 is arbitrary, we letε→0 and obtain (2.4), which completes the proof of Theorem2.1.

Remark2.2 IfR1(t) =R2(t) =C(t) =F(t)≡0, we can obtain expression (2.5). Particularly,

if we further assume thatC(t) =F(t)≡0, then (2.5) degenerates into a conclusion which is present in [11].

3 Dissipativity of two classes of nonlinear neutral functional differential equations

In this section, we consider several simple applications of Theorem2.1to the study of dissipativity for two classes of nonlinear neutral functional diﬀerential equations.

LetXbe a real or complex, ﬁnite-dimensional or inﬁnite-dimensional Hilbert space with the inner product·,·and the corresponding norm · .

3.1 Dissipativity of nonlinear neutral delay integro-differential equations (NNDIDEs)

Consider the IVPs in NNDIDEs as follows:

⎧ ⎨ ⎩

y(t) =f(t,y(t),y(t–τ),y(t–τ),_tt_–_τg(t,ξ,y(ξ))dξ), t≥t0,

y(t) =φ(t), y(t) =φ(t), t0–τ≤t≤t0,

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where τ are positive constant, the functions f : [t0, +∞)×X×X×X×X→X, g :

[t0, +∞)×[t0–τ, +∞)×X→X,φ: [t0–τ,t0]→Xare assumed to be continuous

func-tions and for anyt≥t0,y,u,v,w∈X,f andgsatisfy the conditions:

⎧ ⎪ ⎪ ⎨ ⎪ ⎪ ⎩

2Ref(t,y,u,v,w),y ≤αy2₊_β_f₍_t_{, 0,}_u_,_v_,_w₎2_,

f(t,y,u,v,w)2≤γ1+Lyy2+ωf(t, 0,u,v,w)2,

f(t, 0,u,v,w)2_≤_γ

2+Luu2+Lvv2+Lww2,

(3.2)

and

g(t,ξ,u)≤λu, t–τ ≤ξ≤t,t≥t0, (3.3)

where –α,β,γ1,γ2,ω,λ,Ly,Lu,Lv,Lware all non-negative real constants.

Theorem 3.1 Let problem(3.1)satisfy(3.2)and(3.3)with Lvω< 1,and initial value

func-tionφ(t)satisfy

max

t0–τ≤t≤t0

φ(t)2≤γ1+ωγ2 1 –ωLv

+Ly+ω(Lu+λ

2_τ2_L

w)

1 –ωLv

max

t0–τ≤t≤t0

φ(t)2.

Let y(t)be the solution of(3.1).Assume that there exists a constantσ> 0such that

α+β(Lu+LvLy+λ

2_τ2_L

w)

1 –Lvω

≤–σ. (3.4)

Then

(1)for any t≥t0we have ⎧

⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎩

y(t)2_≤β(γ2+Lvγ1)

(1–Lvω)σ +φ0e

–μ∗(t–t0)_,

y(t)2_≤γ1+ωγ2

1–ωLv +

Ly+ω(Lu+λ2τ2Lw)

1–ωLv

β(γ2+Lvγ1) (1–Lvω)σ +Ly+ω(Lu+λ2τ2Lw)eμ∗τ

1–ωLveμ∗τ φ0e

–μ∗(t–t0)_,

(3.5)

whereφ0=maxt0–τ≤t≤t0φ(t)

2_,_μ∗_{> 0}_{is given as follows}_:

μ∗= inf

t≥t0

μ(t) :μ(t) +α+βLu+λ2τ2Lw

eμ(t)τ

+Ly+ω(Lu+Lwλ

2_τ2₎_eμ(t)τ 1 –ωLveμ(t)τ

βLveμ(t)τ = 0

. (3.6)

(2)the system is dissipative,for anyε> 0the open ball

B=B

0,

β(γ2+Lvγ1)

(1 –Lvω)σ

+ε

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Proof Let

⎧ ⎨ ⎩

u(t) =y(t)2_,

w(t) =y(t)2_, t≥t0–τ (3.7)

and

z(t) =

t t–τ

gt,ξ,y(ξ)dξ, t≥t0.

Then whent≥t0, from (3.2) we have

u(t) = d dt

y(t),y(t)

= 2Rey(t),ft,y(t),y(t–τ),y(t–τ),z(t)

≤αu(t) +βft, 0,y(t–τ),y(t–τ),z(t)2

≤αu(t) +βγ2+Luu(t–τ) +Lvw(t–τ) +Lwz(t)

2

. (3.8)

Noting (3.3) one obtains

z(t)≤λ

t t–τ

y(ξ)dξ

≤λτ max

t–τ≤ξ≤t

y(ξ),

which gives

z(t)2≤λ2τ2 max

t–τ≤ξ≤tu(ξ). (3.9)

Substituting (3.9) into (3.8), we have

u(t)≤βγ2+αu(t) +β

Lu+Lwλ2τ2

u[t–τ,t]+βLvw(t–τ). (3.10)

On the other hand, from the second formula of (3.2) and (3.9) we have

w(t) =ft,y(t),y(t–τ),y(t–τ),z(t)2

≤γ1+Lyu(t) +ω

γ2+Luu(t–τ) +Lvw(t–τ) +Lwz(t)

2

≤γ1+ωγ2+Lyu(t) +ωLvw(t–τ) +ω

Lu+Lwλ2τ2

u[t–τ,t]_. _(3.11)

Therefore, combining of (3.10) and (3.11), fort≥t0we have ⎧

⎨ ⎩

u(t)≤βγ2+αu(t) +β(Lu+Lwλ2τ2)u[t–τ,t]+βLvw[t–τ,t],

w(t)≤γ1+ωγ2+Lyu(t) +ω(Lu+Lwλ2τ2)u[t–τ,t]+ωLvw[t–τ,t].

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Let

⎧ ⎨ ⎩

R1=βγ2, A=α, B=β(Lu+Lwλ2τ2), C= 0, D=βLv,

R2=γ1+ωγ2, F=Ly, G=ω(Lu+Lwλ2τ2), H=ωLv.

From Theorem2.1we can obtain (3.5) immediately. This completes the proof of

Theo-rem3.1.

3.2 Dissipativity of nonlinear neutral Volterra integro-differential equations (NNVIDEs)

Consider the IVPs in NNVIDEs as follows:

⎧ ⎨ ⎩

y(t) =f(t,y(t),y(t–τ),_tt_–_τK(t,s,y(s),y(s))ds), t≥t0,

y(t) =φ(t), y(t) =φ(t), t∈[t0–τ,t0],

(3.13)

whereτ> 0 is constant,φis a continuous function, and the functionsf : [t0, +∞)×X×

X×X→XandK: [t0, +∞)×[t0–τ, +∞)×X×X→Xsatisfy the conditions for any

t≥t0,y,u,v∈X: ⎧

⎪ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎪ ⎩

2Ref(t,y,u,v),y ≤γ +αy2+β1f(t, 0,u,v)2,

f(t,y,u,v)2_≤_L

yy2+β2f(t, 0,u,v)2,

f(t, 0,u,v)2≤Luu2+Lvv2,

K(t,s,u,v) ≤μu+Lkv, (t,s)∈D,

(3.14)

whereD={(t,s) :t∈[t0, +∞),s∈[t–τ,t]},γ,β1,β2,μ,Ly,Lu,Lvare non-negative real

con-stants andα≤0.

Theorem 3.2 Assume that (3.13)satisﬁes(3.14)with 2β2τ2LvL2k< 1, and initial value

functionφ(t)satisﬁes

max

t0–τ≤t≤t0

φ(t)2≤Ly+β2(Lu+ 2τ

2_μ2_L

v)

1 – 2β2τ2LvL2k

max

t0–τ≤t≤t0

φ(t)2.

Assume there exists a constantσ> 0such that

α+β1

Lu+ 2τ2Lv(L2kLy+μ2)

1 – 2β2τ2LvL2_k ≤

–σ. (3.15)

Let y(t)be the solution of(3.13).Then (1)for any t≥t0we have

⎧ ⎨ ⎩

y(t)2_≤γ σ +φ0e–

μ∗(t–t0)_,

y(t)2≤Ly+β2(Lu+2τ2μ2Lv)

1–2β2τ2_L_v_L2 k

γ σ +

Ly+β2(Lu+2τ2μ2Lv)eμ∗τ

1–2β2τ2LvL2_keμ∗τ φ0e

–μ∗(t–t0)_,

whereφ0=supt0–τ≤ξ≤t0φ(ξ)

2_,_μ∗_{> 0}_{is deﬁned as}

μ∗=inf

t≥t0

μ(t) :μ(t) +α+Lu+ 2τ

2_L

v(L2kLy+μ2)

1 – 2β2τ2LvL2keμ(t)τ

β1eμ(t)τ = 0

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(2)the system is dissipative,for anyε> 0the open ball

B= B

0,

γ σ +ε

is an absorbing set.

Proof Let

⎧ ⎨ ⎩

u(t) =y(t)2_,

w(t) =y(t)2_, t≥t0–τ

and

z(t) =

t t–τ

Kt,s,y(s),y(s)ds), t≥t0.

From (3.14) we can obtain

z(t)2≤τ2

μ max

t–τ≤ξ≤ty(ξ)+Lkt–maxτ≤ξ≤ty

_(ξ₎2

≤2τ2μ2u[t–τ,t]+L2_kw[t–τ,t]

and

u(t) = 2Reft,y(t),y(t–τ),z(t),y(t)

≤γ +αu(t) +β1f

t, 0,y(t–τ),z(t)2

≤γ +αu(t) +β1

Luu(t–τ) +Lvz(t)

2

≤γ +αu(t) +β1

Lu+ 2μ2τ2Lv

u[t–τ,t]+ 2β1τ2LvL2kw[t–

τ,t] _(3.16)

and

w(t) =ft,y(t),y(t–τ),z(t)2

≤Lyy(t)2+β2f

t, 0,y(t–τ),z(t)2

≤Lyy(t)

2

+β2

Luu(t–τ) +Lvz(t)

2

≤Lyu(t) +β2

Lu+ 2Lvτ2μ2

u[t–τ,t]+ 2β2τ2LvL2kw[t–

τ,t]_. _(3.17)

It can be summarized from (3.16) and (3.17) that

⎧ ⎨ ⎩

u(t)≤γ+αu(t) +β1(Lu+ 2τ2μ2Lv)u[t–τ,t]+ 2β1τ2LvL2kw[t–τ,t],

w(t)≤Lyu(t) +β2(Lu+ 2τ2μ2Lv)u[t–τ,t]+ 2β2τ2LvL2kw[t–

τ,t]_. (3.18)

We denote

γ1=γ, A=α, B=β1

Lu+ 2τ2μ2Lv

(11)

γ2= 0, F=Ly, G=β2

Lu+ 2τ2μ2Lv

, H= 2β2τ2LvL2k.

Then from Theorem2.1we can complete the proof of Theorem3.2.

Remark3.3 From a numerical point of view, it is important to study the potential of nu-merical methods in preserving the qualitative behavior of the analytical solutions. There-fore, the results of Theorem 3.1and Theorem3.2presented in this paper, provide the theoretical foundation for analyzing the dissipativity of the numerical methods when they are applied to the underlying systems.

Acknowledgements

This work is supported by Hunan Key Laboratory for Computation and Simulation in Science and Engineering.

Funding

This work is supported by NSF of China (No. 11371302, 11571291).

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Three authors contributed equally to writing of this paper. Three authors read and approved the ﬁnal manuscript.

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional aﬃliations.

Received: 9 August 2018 Accepted: 29 October 2018

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