Citation: Songbai Guo, Wanbiao Ma. Remarks on a variant of Lyapunov-LaSalle theorem[J]. Mathematical Biosciences and Engineering, 2019, 16(2): 1056-1066. doi: 10.3934/mbe.2019050
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As we know, one of the most common ways to study the asymptotic stability for a system of delay differential equations (DDEs) is the Lyapunov functional method. For DDEs, the Lyapunov-LaSalle theorem (see [6,Theorem 5.3.1] or [11,Theorem 2.5.3]) is often used as a criterion for the asymptotic stability of an autonomous (possibly nonlinear) delay differential system. It can be applied to analyse the dynamics properties for lots of biomathematical models described by DDEs, for example, virus infection models (see, e.g., [2,3,10,14]), microorganism flocculation models (see, e.g., [4,5,18]), wastewater treatment models (see, e.g., [16]), etc.
In the Lyapunov-LaSalle theorem, a Lyapunov functional plays an important role. But how to construct an appropriate Lyapunov functional to investigate the asymptotic stability of DDEs, is still a very profound and challenging topic.
To state our purpose, we take the following microorganism flocculation model with time delay in [4] as example:
{˙x(t)=1−x(t)−h1x(t)y(t),˙y(t)=rx(t−τ)y(t−τ)−y(t)−h2y(t)z(t),˙z(t)=1−z(t)−h3y(t)z(t), | (1.1) |
where
G={ϕ=(ϕ1,ϕ2,ϕ3)T∈C+:=C([−τ,0],R3+) : ϕ1≤1, ϕ3≤1}. |
In model (1.1), there exists a forward bifurcation or backward bifurcation under some conditions [4]. Thus, it is difficult to use the research methods that some virus models used to study the dynamics of such model.
Clearly, (1.1) always has a microorganism-free equilibrium
L(ϕ)=ϕ2(0)+r∫0−τϕ1(θ)ϕ2(θ)dθ, ϕ∈G. | (1.2) |
The derivative of
˙L(ut)=(rx(t)−1−h2z(t))y(t)≤(r−1−h2z(t))y(t). | (1.3) |
Obviously, if
However, we can not get
lim inft→∞z(t)≥h1h1+rh3. | (1.4) |
If
˙V(ut)≤[r−1−h1h2ε(h1+rh3)]y(t)≤0, t≥T. |
Obviously, for all
In this paper, we will expand the view of constructing Lyapunov functionals, namely, we first give a new understanding of Lyapunov-LaSalle theorem (including its modified version [9,15,19]), and based on it establish some global stability criteria for an autonomous delay differential system.
Let
˙u(t)=g(ut), t≥0, | (2.1) |
where
˙L(ϕ)=˙L(ϕ)|(2.1)=lim sups→0+L(us(ϕ))−L(ϕ)s. |
Let
u(t)=u(t,ϕ):=(u1(t,ϕ),u2(t,ϕ),⋯,un(t,ϕ))T |
denote a solution of system (2.1) satisfying
U(t):=ut(⋅):X→X (which also satisfies U(t):¯X→¯X), |
and for
OT(ϕ):={ut(ϕ):t≥T}. |
Let
The following Definition 2.1 and Theorem 2.1 (see, e.g., [6,Theorem 5.3.1], [11,Theorem 2.5.3]) can be utilized in dynamics analysis of lots of biomathematical models in the form of system (2.1).
Definition 2.1. We call
(ⅰ)
(ⅱ)
Theorem 2.1 (Lyapunov-LaSalle theorem [11]). Let
In Theorem 2.1, a Lyapunov functional
X={ϕ=(ϕ1,ϕ2,⋯,ϕn)T∈C:ϕi(0)>0}, | (2.2) |
which can ensure
However, we will assume that
Corollary 2.1. Let the solution
Proof. It is clear that if
Remark 2.1. It is not difficult to find that in the modified Lyapunov-LaSalle theorem (see, e.g., [9,15,19]), if
Remark 2.2. In fact, we can see that a bounded
From Corollary 2.1, we may consider the global properties of system (2.1) on the larger space than
Let
Theorem 3.1. Suppose that the following conditions hold:
(ⅰ) Let
˙L(φ)≤−w(φ)b(φ), | (3.1) |
where
(ⅱ) There exist
k1≤φ≤k2, w(φ)≥(w01,w02,⋯,w0k)≡w0=w0(k1,k2)≫0, |
and
Then
Proof. To obtain
lim inft→∞w(ut(ϕ)):=(lim inft→∞w1(ut(ϕ)),lim inft→∞w2(ut(ϕ)),⋯,lim inft→∞wk(ut(ϕ)))=(limm→∞f1(t1m),limm→∞f2(t2m),⋯,limm→∞fk(tkm)). |
For each sequence
lim inft→∞wi(ut(ϕ))=limm→∞wi(utim(ϕ))=wi(ϕi). |
By the condition (ⅱ),
˙L(φ)≤−w(φ)b(φ)≤−w0b(φ)2≤0. |
Hence,
Next, we show that
˙L(ut(ψ))≤−w(ut(ψ))b(ut(ψ)), ∀t≥0. |
By (ⅱ),
Remark 3.1. By
Next, we will give an illustration for Theorem 3.1. Now, we reconsider the global stability for the infection-free equilibrium
{˙x(t)=s−dx(t)−cx(t)y(t)−βx(t)v(t),˙y(t)=e−μτβx(t−τ)v(t−τ)−py(t),˙v(t)=ky(t)−uv(t), | (3.2) |
where
In [1], we know
G={ϕ∈C([−τ,0],R3+):ϕ1≤x0}⊂C+:=C([−τ,0],R3+). |
Indeed, by Theorem 3.1, we can extend the result of [1] to the larger set
Corollary 3.1. If
Proof. It is not difficult to obtain
L(ϕ)=ϕ1(0)−x0−x0lnϕ1(0)x0+a1ϕ2(0)+a1e−μτ∫0−τβϕ1(θ)ϕ3(θ)dθ+a2ϕ3(0), | (3.3) |
where
a1=2(kβx0+ucx0)pu−e−μτkβx0,a2=2(pβx0+e−μτcβx20)pu−e−μτkβx0. |
Let
w(φ)≡(dφ1(0),a1p−a2k−cx0,a2u−a1e−μτβφ1(0)−βx0)≥(dx0,a1p−a2k−cx0,a2u−a1e−μτβx0−βx0)=(dx0,cx0,βx0)≡w0≫0, |
where
The derivative of
˙L1(ut)=d(x0−x(t))(1−x0x(t))+x0(cy(t)+βv(t))−x(t)(cy(t)+βv(t))+a1e−μτβx(t)v(t)−a1py(t)+a2ky(t)−a2uv(t)≤−dx(t)(x0−x(t))2−(a1p−a2k−cx0)y(t)−(a2u−a1e−μτβx(t)−βx0)v(t)=−w(ut)b(ut). |
Therefore, it follows from Theorem 3.1 that
In [3,Theorem 3.1], the infection-free equilibrium
Theorem 3.2. In the condition (ii) of Theorem 3.1, if the condition that
Proof. In the foundation of the similar argument as in the proof of Theorem 3.1, we have that
˙L(ut(ψ))≤−w0b(ut(ψ))≤0. |
Hence,
Next, by using Theorem 3.2, we will give the global stability of the equilibrium
˙L(ut)≤−w(ut)b(ut), | (3.4) |
where
w(ut)=1+h2zt(0)−r=1+h2z(t)−r,b(ut)=yt(0)=y(t). |
Let
p(t)=rh1xt(−τ)+yt(0)=rh1x(t−τ)+y(t), t≥τ. |
Then we have
lim inft→∞x(t)≥1r+1, lim inft→∞z(t)≥h1h1+rh3. | (3.5) |
Thus, for any
(1/(r+1),0,h1/(h1+rh3))T≤φ≤(1,r/h1,1)T,w(φ)=1+h2φ3(0)−r≥1+h1h2/(h1+rh3)−r≡w0>0, |
and
Thus, we only need to obtain the solutions of a system are bounded and then may establish the upper- and lower-bound estimates of
Corollary 3.2. Let
a(φ(0))≤L(φ), ˙L(φ)≤−w0b(φ), 0≪wT0∈Rk, | (3.6) |
where
Proof. Since
a(u(t,ϕ))≤L(ut(ϕ))≤L(uT(ϕ)), t∈[T,εϕ), |
and the fact that
Corollary 3.3. Assume that
a(|φ(0)−E|)≤L(φ), ˙L(φ)≤−w0b(φ), 0≪wT0∈Rk, | (3.7) |
where
Proof. It follows from Corollary 3.2 that the boundedness of
ut(ϕ)∈B(ut(E),ε)=B(E,ε), |
where
a(|u(t,ϕ)−E|)≤L(ut(ϕ))≤L(uT(ϕ))<a(ε), |
which yields
Lemma 3.1. ([13,Lemma 1.4.2]) For any infinite positive definite function
By Lemma 3.1, we have the following remark.
Remark 3.2. If there exists an infinite positive definite function
Corollary 3.4. In Corollary 3.2, if the condition
For a dissipative system (2.1), we will give the upper- and lower-bound estimates of
Lemma 3.2. Let
Proof. For any
Theorem 3.3. Suppose that there exist
\begin{equation} k_{1}\leq\liminf\limits_{t\rightarrow\infty}u_{t}(\phi)(\theta)\leq \limsup\limits_{t\rightarrow\infty}u_{t}(\phi)(\theta)\leq k_{2}, ~\forall \phi\in X, ~\forall\theta\in\lbrack-\tau, 0], \label{kk}% \end{equation} | (3.8) |
where
\begin{align*} \liminf\limits_{t\rightarrow\infty}u_{t}(\phi)(\theta) & : = \left( \liminf\limits_{t\rightarrow\infty}u_{1t}(\phi)\left( \theta\right) , \cdots, \liminf\limits_{t\rightarrow\infty}u_{nt}(\phi)\left( \theta\right) \right) ^{T}, \\ \limsup\limits_{t\rightarrow\infty}u_{t}(\phi)(\theta) & : = \left( \limsup\limits_{t\rightarrow\infty}u_{1t}(\phi)\left( \theta\right) , \cdots, \limsup\limits_{t\rightarrow\infty}u_{nt}(\phi)\left( \theta\right) \right) ^{T}. \end{align*} |
Then
Proof. Clearly,
|\dot{u}(t, \varphi)|\leq M_{1}, ~\forall t\geq0, ~\forall\varphi\in M. |
It follows from the invariance of
In this paper, we first give a variant of Theorem 2.1, see Corollary 2.1. In fact, the modified version of Lyapunov-LaSalle theorem (see, e.g., [9,15,19]) is to expand the condition (ⅰ) of Definition 2.1, while Corollary 2.1 is mainly to expand the condition (ⅱ) of Definition 2.1. More specifically, we assume that
As a result, the criteria for the global attractivity of equilibria of system (2.1) are given in Theorem 3.1 and Theorem 3.2, respectively. As direct consequences, we also give the corresponding particular cases of Theorem 3.1 and Theorem 3.2, see Corollaries 3.2, 3.3 and 3.4, respectively. The developed theory can be utilized in many models (see, e.g., [2,3,9,10,14]). The compactness and the upper- and lower-bound estimates of
This work was supported in part by the General Program of Science and Technology Development Project of Beijing Municipal Education Commission (No. KM201910016001), the Fundamental Research Funds for Beijing Universities (Nos. X18006, X18080 and X18017), the National Natural Science Foundation of China (Nos. 11871093 and 11471034). The authors would like to thank Prof. Xiao-Qiang Zhao for his valuable suggestions.
The authors declare there is no conflict of interest in this paper.
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