# Asymptotic Behaviour of a Two-Dimensional Differential System with a Finite Number of Nonconstant Delays under the Conditions of Instability

Authors: Zdeněk Šmarda; Josef Rebenda
Journal: Abstract and Applied Analysis
 (2012)

---

## Abstract

The asymptotic behaviour of a real two-dimensional differential systemx′(t)=A(t)x(t)+∑k=1mBk(t)x(θk(t))+h(t,x(t),x(θ1(t)),…,x(θm(t))) with unbounded nonconstant delays t-θk(t)≥0 satisfying limt→∞θk(t)=∞ is studied under the assumption of instability. Here, A, Bk, and h are supposed to be matrix functions and a vector function. The conditions for the instable properties of solutions and the conditions for the existence of bounded solutions are given. The methods are based on the transformation of the considered real system to one equation with complex-valued coefficients. Asymptotic properties are studied by means of a Lyapunov-Krasovskii functional and the suitable Ważewski topological principle. The results generalize some previous ones, where the asymptotic properties for two-dimensional systems with one constant or nonconstant delay were studied.


---

## Body

## 1. Introduction

Consider the real two-dimensional system(1.1)x′(t)=A(t)x(t)+∑k=1mBk(t)x(θk(t))+h(t,x(t),x(θ1(t)),…,x(θm(t))),
where θk(t) are real functions, A(t)=(aij(t)), Bk(t)=(bijk(t))(i,j=1,2;k=1,…,m) are real square matrices, and h(t,x,y)=(h1(t,x,y1,…,ym),h2(t,x,y1,…,ym)) is a real vector function, x=(x1,x2), yk=(y1k,y2k). It is supposed that the functions θk, aij are locally absolutely continuous on [t0,∞), bijk are locally Lebesgue integrable on [t0,∞), and the function h satisfies Carathéodory conditions on [t0,∞)×ℝ2(m+1).There are a lot of papers dealing with the stability and asymptotic behaviour ofn-dimensional real vector equations with delay. Among others we should mention the recent results [1–13]. Since the plane has special topological properties different from those of n-dimensional space, where n≥3 or n=1, it is interesting to study the asymptotic behaviour of two-dimensional systems by using tools that are typical and effective for two-dimensional systems. The convenient tool is the combination of the method of complexification and the method of Lyapunov-Krasovskii functional. For the case of instability, it is useful to add to this combination the version of Ważewski topological principle formulated by Rybakowski in the papers [14, 15]. Using these techniques, we obtain new and easy applicable results on stability, asymptotic stability, instability, or boundedness of solutions of the system (1.1).The main idea of the investigation, the combination of the method of complexification and the method of Lyapunov-Krasovskii functional, was introduced for ordinary differential equations in the paper by Ráb and Kalas [16] in 1990. The principle was transferred to differential equations with delay by Kalas and Baráková [17] in 2002. The results in the case of instability were obtained for ODEs by Kalas and Osička [18] in 1994 and for delayed differential equations by Kalas [19] in 2005.We extend such type of results to differential equations with a finite number of nonconstant delays. We introduce the transformation of the considered real system to one equation with complex-valued coefficients. We present sufficient conditions for the instability of a solution and for the existence of a bounded solution. The applicability of the results is demonstrated with several examples.At the end of this introduction we append a brief overview of notation used in the paper and the transformation of the real system to one equation with complex-valued coefficients.ℝ is the set of all real numbers,ℝ+ the set of all positive real numbers,ℝ+0 the set of all nonnegative real numbers,ℝ- the set of all negative real numbers,ℝ-0 the set of all nonpositive real numbers,ℂ the set of all complex numbers,𝒞 the class of all continuous functions [-r,0]→ℂ,ACloc(I,M) the class of all locally absolutely continuous functions I→M,Lloc(I,M) the class of all locally Lebesgue integrable functions I→M,K(I×Ω,M) the class of all functions I×Ω→Msatisfying Carathéodory conditions on I×Ω,Rez the real part of z,Imz the imaginary part of z, andz̅ the complex conjugate of z.Introducing complex variablesz=x1+ix2, w1=y11+iy12,..., wm=ym1+iym2, we can rewrite the system (1.1) into an equivalent equation with complex-valued coefficients(1.2)z′(t)=a(t)z(t)+b(t)z̅(t)+∑k=1m[Ak(t)z(θk(t))+Bk(t)z̅(θk(t))]+g(t,z(t),z(θ1(t)),…,z(θm(t))),
where θk∈ACloc(J,ℝ) for k=1,…,m, Ak,Bk∈Lloc(J,ℂ), a,b∈ACloc(J,ℂ), g∈K(J×ℂm+1,ℂ), J=[t0,∞).The relations between the functions are as follows:(1.3)a(t)=12(a11(t)+a22(t))+i2(a21(t)-a12(t)),b(t)=12(a11(t)-a22(t))+i2(a21(t)+a12(t)),Ak(t)=12(b11k(t)+b22k(t))+i2(b21k(t)-b12k(t)),Bk(t)=12(b11k(t)-b22k(t))+i2(b21k(t)+b12k(t)),g(t,z,w1,…,wm)=h1(t,12(z+z̅),12i(z-z̅),12(w1+w̅1),…,12i(wm-w̅m))+ih2(t,12(z+z̅),12i(z-z̅),12(w1+w̅1),12i(w1-w̅1),…,12i(wm-w̅m)).
Conversely, putting(1.4)a11(t)=Re[a(t)+b(t)],a12(t)=Im⁡[b(t)-a(t)],a21(t)=Im⁡[a(t)+b(t)],a22(t)=Re[a(t)-b(t)],b11k(t)=Re[Ak(t)+Bk(t)],b12k(t)=Im⁡[Bk(t)-Ak(t)],b21k(t)=Im⁡[Ak(t)+Bk(t)],b22k(t)=Re[Ak(t)-Bk(t)],h1(t,x,y1,…,ym)=Reg(t,x1+ix2,y11+iy12,…,ym1+iym2),h2(t,x,y1,…,ym)=Im⁡g(t,x1+ix2,y11+iy12,…,ym1+iym2),
the equation (1.2) can be written in the real form (1.1) as well.


## 2. Preliminaries

We consider (1.2) in the case when (2.1)liminf⁡t→∞(|Ima(t)|-|b(t)|)>0
and study the behavior of solutions of (1.2) under this assumption. This situation corresponds to the case when the equilibrium 0 of the autonomous homogeneous system (2.2)x′=Ax,
where A is supposed to be regular constant matrix, is a centre or a focus. See [16] for more details.Regarding (2.1) and since the delay functions θk satisfy limt→∞θk(t)=∞, there are numbers T1≥t0, T≥T1, and μ>0 such that
(2.3)|Ima(t)|>|b(t)|+μfort≥T1,t≥θk(t)≥T1fort≥T(k=1,…,m).Denote(2.4)γ̃(t)=Ima(t)+(Ima(t))2-|b(t)|2sgn⁡(Ima(t)),c̃(t)=-ib(t).Notice that the above-defined functionγ̃ need not be positive.Since|γ̃(t)|>|Ima(t)| and |c̃(t)|=|b(t)|, the inequality(2.5)|γ̃(t)|>|c̃(t)|+μ
is valid for t≥T1. It can be easily verified that γ̃,c̃∈ACloc([T1,∞),ℂ).For the rest of this section we will denote(2.6)ϑ̃(t)=Re(γ̃(t)γ̃′(t)-c̃̅(t)c̃′(t))-|γ̃(t)c̃′(t)-γ̃′(t)c̃(t)|γ̃2(t)-|c̃(t)|2.The instability and boundedness of solutions are studied subject to suitable subsets of the following assumptions.(i)
The numbersT1≥t0, T≥T1, and μ>0 are such that (2.3) holds.(ii)
There exist functionsϰ̃,κ̃k,ϱ:[T,∞)→ℝ such that(2.7)|γ̃(t)g(t,z,w1,…,wm)+c̃(t)g̅(t,z,w1,…,wm)|≤ϰ̃(t)|γ̃(t)z+c̃(t)z̅|+∑k=1mκ̃k(t)|γ̃(θk(t))wk+c̃(θk(t))w̅k|+ϱ(t)
for t≥T, z,wk∈ℂ(k=1,…,m), where ϱ is continuous on [T,∞).(iin)
There exist numbersRn≥0and functionsϰ̃n,κ̃nk:[T,∞)→ℝsuch that(2.8)|γ̃(t)g(t,z,w1,…,wm)+c̃(t)g(t,z,w1,…,wm)|≤ϰ̃n(t)|γ̃(t)z+c̃(t)z|+∑k=1mκ̃nk(t)|γ̃(θk(t))wk+c̃(θk(t))w̅k|
fort≥τn≥T,|z|+∑k=1m|wk|>Rn.(iii)
β̃∈ACloc([T,∞),ℝ-0)is a function satisfying
(2.9)θk′(t)β̃(t)≤-λ̃k(t)a.e.on⁡[T,∞),
where λ̃k is defined for t≥T by(2.10)λ̃k(t)=κ̃k(t)+(|Ak(t)|+|Bk(t)|)|γ̃(t)|+|c̃(t)||γ̃(θk(t))|-|c̃(θk(t))|.(iiin)
β̃n∈ACloc[T,∞),ℝ-0) is a function satisfying(2.11)θk′(t)β̃n(t)≤-λ̃nk(t)a.e.on⁡[τn,∞),
where λ̃nk is defined for t≥T by (2.12)λ̃nk(t)=κ̃nk(t)+(|Ak(t)|+|Bk(t)|)|γ̃(t)|+|c̃(t)||γ̃(θk(t))|-|c̃(θk(t))|.(ivn)
Λ̃n is a real locally Lebesgue integrable function satisfying the inequalities β̃n′(t)≥Λ̃n(t)β̃n(t), Θ̃n(t)≥Λ̃n(t) for almost all t∈[τn,∞), where Θ̃n is defined by(2.13)Θ̃n(t)=Rea(t)+ϑ̃(t)-ϰ̃n(t)+mβ̃n(t).Obviously, ifAk, Bk, κ̃k, and θk′ are locally absolutely continuous on [T,∞) and λ̃k(t)≥0, θk′(t)>0, the choice β̃(t)=-maxk=1,…,m[λ̃k(t)(θk′(t))-1] is admissible in (iii). Similarly, if Ak, Bk, κ̃nk, and θk′ are locally absolutely continuous on [T,∞) and λ̃nk(t)≥0, θk′(t)>0, the choice β̃n(t)=-maxk=1,…,m[λ̃nk(t)(θk′(t))-1] is admissible in (iiin).Denote(2.14)Θ̃(t)=Rea(t)+ϑ̃(t)-ϰ̃(t).
From assumption (i) it follows that (2.15)|ϑ̃|≤|Re(γ̃γ̃′-c̃̅c̃′)|+|γ̃c′-γ̃′c|γ̃2-|c̃|2≤(|γ̃′|+|c̃′|)(|γ̃|+|c̃|)γ̃2-|c̃|2=|γ̃′|+|c̃′||γ̃|-|c̃|≤1μ(|γ̃′|+|c̃′|);
therefore the function ϑ̃ is locally Lebesgue integrable on [T,∞), assuming that (i) holds true. If the relations β̃n∈ACloc([T,∞),ℝ-), ϰ̃n∈Lloc([T,∞),ℝ), and β̃n′(t)/β̃n(t)≤Θ̃n(t)  for almost all t≥τn together with conditions (i) and (iin) are fulfilled, then we can choose Λ̃n(t)=Θ̃n(t) for t∈[T,∞) in (ivn).


## 3. Results

Theorem 3.1.
Let assumptions (i), (ii0), (iii0), and (iv0) be fulfilled for some  τ0≥T. Suppose there exist  t1≥τ0 and  ν∈(-∞,∞) such that
(3.1)inf⁡t≥t1[∫t1tΛ̃0(s)ds-ln⁡(|γ̃(t)|+|c̃(t)|)]≥ν.
If z(t) is any solution of (1.2) satisfying
(3.2)min⁡θ(t1)≤s≤t1|z(s)|>R0,Δ(t1)>R0e-ν,
where
(3.3)θ(t)=min⁡k=1,…,mθk(t),Δ(t)=(|γ̃(t)|-|c̃(t)|)|z(t)|+β̃0(t)max⁡θ(t)≤s≤t|z(s)|∑k=1m∫θk(t)t(|γ̃(s)|+|c̃(s)|)ds,
then
(3.4)|z(t)|≥Δ(t1)|γ̃(t)|+|c̃(t)|exp⁡[∫t1tΛ̃0(s)ds]
for all t≥t1, for which z(t) is defined.In the proof we use the following Lemma.Lemma 3.2.
Leta1,  a2,  b1,  b2∈ℂ,  |a2|>|b2|. Then,
(3.5)Rea1z+b1z̅a2z+b2z̅≥Re(a1a̅2-b1b̅2)-|a1b2-a2b1||a2|2-|b2|2
for z∈ℂ, z≠0.The proof is analogous to that of Lemma 1 in [20, page 101] or to the proof of Lemma in [16, page 131].Proof of Theorem3.1.
Letz(t)  be any solution of (1.2) satisfying (3.2). Consider the Lyapunov functional
(3.6)V(t)=U(t)+β̃0(t)∑k=1m∫θk(t)tU(s)ds,
where
(3.7)U(t)=|γ̃(t)z(t)+c̃(t)z̅(t)|.
For brevity we shall denote wk(t)=z(θk(t)) and we shall write the function of variable t simply without indicating the variable t, for example, γ̃ instead of γ̃(t).
In view of (3.6), we have
(3.8)V′=U′+β0′̃∑k=1m∫θk(t)tU(s)ds+mβ̃0|γ̃z+c̃z̅|-∑k=1mθk′β̃0|γ̃(θk(t))wk+c̃(θk(t))w̅k|
for almost all t≥t1 for which z(t) is defined and U′(t) exists. Put 𝒦={t≥t1:z(t)exists,|z(t)|>R0}. Clearly U(t)≠0 for t∈𝒦. The derivative U′(t) exists for almost all t∈𝒦.
Sincez(t) is a solution of (1.2), we obtain
(3.9)UU′=Re[(γ̃z̅+c̃̅z)(γ̃′z+γ̃z′+c̃′z̅+c̃z̅′)]=Re{(γ̃z̅+c̃̅z)[γ̃′z+c̃′z̅+γ̃(az+bz̅+∑k=1m(Akwk+Bkw̅k)+g)+c̃(a̅z̅+b̅z)+∑k=1m(A̅kw̅k+B̅kwk+g̅)]}=Re{(∑k=1m(Akwk+Bkw̅k)+g)(γ̃z̅+c̃̅z)[(∑k=1m(Akwk+Bkw̅k)+g)γ̃′z+c̃′z̅+(γ̃a+c̃b̅)z+(γ̃b+c̃a̅)z̅+γ̃(∑k=1m(Akwk+Bkw̅k)+g)+c̃(∑k=1m(A̅kw̅k+B̅kwk)+g̅)]}
for almost all t∈𝒦. Taking into account
(3.10)(γ̃a+c̃b̅)c̃=(γ̃b+c̃a̅)γ̃,
we get
(3.11)UU′≥Re{(γ̃z̅+c̃̅z)(γ̃a+c̃b̅)(z+c̃γ̃z̅)}+Re{(γ̃z̅+c̃̅z)[γ̃∑k=1m(Akwk+Bkw̅k)+c̃∑k=1m(A̅kw̅k+B̅kwk)]}+Re{(γ̃z̅+c̃̅z)(γ̃g+c̃g̅)}+Re{(γ̃z̅+c̃̅z)(γ′̃z+c̃′z̅)}≥U2Re(a+c̃γ̃b̅)-U(|γ̃|+|c̃|)(∑k=1m|Akwk+Bkw̅k|)-U|γ̃g+c̃g̅|+U2Reγ̃′z+c̃′z̅γ̃z+c̃z̅.
By the use of Lemma3.2, we get
(3.12)Reγ̃′z+c̃′z̅γ̃z+c̃z̅≥ϑ̃.
The last inequality together with (2.12), taken for n=0, assumption (ii0), and the relation
(3.13)Re(a+c̃γ̃b̅)=Rea
yields
(3.14)UU′≥U2(Rea+ϑ̃-ϰ̃0)-U∑k=1m(κ̃0k|γ̃(θk)wk+c̃(θk)w̅k|)-U(|γ̃|+|c̃|)(∑k=1m|Ak||wk|+|Bk||w̅k||γ̃(θk)|-|c̃(θk)|(|γ̃(θk)|-|c̃(θk)|))≥U2(Rea+ϑ̃-ϰ̃0)-U{∑k=1m[κ̃0k+(|Ak|+|Bk|)|γ̃|+|c̃||γ̃(θk)|-|c̃(θk)|]|γ̃(θk)wk+c̃(θk)w̅k|}≥U2(Rea+ϑ̃-ϰ̃0)-U∑k=1mλ̃0k|γ̃(θk)wk+c̃(θk)w̅k|
for almost all t∈𝒦.
Consequently,(3.15)U′≥U(Rea+ϑ̃-ϰ̃0)-∑k=1mλ̃0k|γ̃(θk)wk+c̃(θk)w̅k|
for almost all t∈𝒦. Inequality (3.15) together with relation (3.8) gives
(3.16)V′≥U(Rea+ϑ̃-ϰ̃0+mβ̃0)-∑k=1m(λ̃0k+θk′β̃0)|γ̃(θk)wk+c̃(θk)w̅k|+β̃0′∑k=1m∫θk(t)t|γ̃(s)z(s)+c̃(s)z̅(s)|ds.
Using (2.11) and (2.13) for n=0, we obtain
(3.17)V′(t)≥U(t)Θ̃0(t)+β̃0′(t)∑k=1m∫θk(t)tU(s)ds.
Hence, in view of (iv0),
(3.18)V′(t)-Λ̃0(t)V(t)≥0
for almost all t∈𝒦.
Multiplying (3.18) by exp[-∫t1tΛ̃0(s)ds] and integrating over [t1,t], we get
(3.19)V(t)exp⁡[-∫t1tΛ̃0(s)ds]-V(t1)≥0
on any interval [t1,ω), where the solution z(t) exists and satisfies the inequality |z(t)|>R0. Now, with respect to (3.6), (3.7), and β̃0≤0, we have
(3.20)(|γ̃(t)|+|c̃(t)|)|z(t)|≥V(t)≥V(t1)exp⁡[∫t1tΛ̃0(s)ds]≥Δ(t1)exp⁡[∫t1tΛ̃0(s)ds].
If (3.2) is fulfilled, there is R>R0 such that Δ(t1)>Re-ν. By virtue of (3.1), and (3.2), we can easily see that
(3.21)|z(t)|≥Δ(t1)|γ̃(t)|+|c̃(t)|exp⁡[∫t1tΛ̃0(s)ds]≥Re-νeν=R
for all t≥t1, for which z(t) is defined.To obtain results on the existence of bounded solutions, we shall suppose that (1.2) satisfies the uniqueness property of solutions. Moreover, we suppose that the delays are bounded, that is, that the functions θk satisfy the condition (3.22)t-r≤θk(t)≤tfort≥t0+r,
where r>0 is a constant. Our assumptions imply the existence of numbers T1=t0+r, T≥T1, and μ>0 such that((5′))|Im⁡a(t)|>|b(t)|+μfort≥T1,t≥θk(t)≥t-rfort≥T(k=1,…,m).In view of this, we replace (2.3) in assumption (i) with ((5′)). All other assumptions we keep in validity.In the proof of the following theorem we shall utilize Ważewski topological principle for retarded functional differential equations of Carathéodory type. Details of this theory can be found in the paper of Rybakowski [15].Theorem 3.3.
Let conditions (i), (ii), and (iii) be fulfilled, and letΛ̃,  θk′(k=1,…,m) be continuous functions such that the inequality  Λ̃(t)≤Θ̃(t) holds a. e. on  [T,∞), where  Θ̃ is defined by (2.14). Suppose that  ξ:[T-r,∞)→ℝ is a continuous function such that
(3.23)Λ̃(t)+β̃(t)∑k=1mθk′(t)exp⁡[-∫θk(t)tξ(s)ds]-ξ(t)>ϱ(t)C-1exp⁡[-∫Ttξ(s)ds]
for t∈[T,∞] and some constant C>0. Then, there exist t2>T and a solution z0(t) of (1.2) satisfying
(3.24)|z0(t)|≤C|γ̃(t)|-|c̃(t)|exp⁡[∫Ttξ(s)ds]
for t≥t2.Proof.
Write (1.2) in the form
((2′))z′=F(t,zt),
where F:J×𝒞→ℂ is defined by
(3.25)F(t,ψ)=a(t)ψ(0)+b(t)ψ̅(0)+∑k=1m[Ak(t)ψ(θk(t)-t)+Bk(t)ψ̅(θk(t)-t)]+g(t,ψ(0),ψ(θ1(t)-t),…,ψ(θm(t)-t))
and zt is the element of 𝒞 defined by the relation zt(θ̃)=z(t+θ̃), θ̃∈[-r,0]. Let τ>T. Put
(3.26)Ũ(t,z,z̅)=|γ̃(t)z+c̃(t)z̅|-φ(t),φ(t)=Cexp⁡[∫Ttξ(s)ds],Ω0={(t,z)∈(τ,∞)×C:Ũ(t,z,z̅)<0},ΩŨ={(t,z)∈(τ,∞)×C:Ũ(t,z,z̅)=0}.
It can be easily verified thatΩ0 is a polyfacial set generated by the functions Û(t)=τ-t, Ũ(t,z,z̅) (see Rybakowski [15, page 134]). It holds that ΩŨ⊂∂Ω0. As (|γ̃(t)|+|c̃(t)|)|z(t)|≥|γ̃(t)z+c̃(t)z̅|, we have
(3.27)|z|≥φ(t)|γ̃(t)|+|c̃(t)|=C|γ̃(t)|+|c̃(t)|exp⁡[∫Ttξ(s)ds]>0
for (t,z)∈ΩŨ. It holds that
(3.28)D+Û(t)=∂∂t(τ-t)=-1<0.
Let (t*,ζ)∈ΩŨ and ϕ∈𝒞 be such that ϕ(0)=ζ and (t*+θ̃,ϕ(θ̃))∈Ω0 for all θ̃∈[-r,0). If (t,ψ)∈(τ,∞)×𝒞, then
(3.29)D+Ũ(t,ψ(0),ψ̅(0)):=limsup⁡h→0+(1h)[Ũ(t+h,ψ(0)+hF(t,ψ),ψ̅(0)+hF̅(t,ψ))-Ũ(t,ψ(0),ψ̅(0))]=∂Ũ(t,ψ(0),ψ̅(0))∂t+∂Ũ(t,ψ(0),ψ̅(0))∂zF(t,ψ)+∂Ũ(t,ψ(0),ψ̅(0))∂z̅F̅(t,ψ).
Therefore,
(3.30)D+Ũ(t,ψ(0),ψ̅(0))=|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|Reγ̃′(t)ψ(0)+c̃′(t)ψ̅(0)γ̃(t)ψ(0)+c̃(t)ψ̅(0)-φ′(t)+12|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|-1×{[γ̃(t)(γ̃(t)ψ̅(0)+c̃̅(t)ψ(0))+(γ̃(t)ψ(0)+c̃(t)ψ̅(0))c̃̅(t)]F(t,ψ)+[c̃(t)(γ̃(t)ψ̅(0)+c̃̅(t)ψ(0))+γ̃(t)(γ̃(t)ψ(0)+c̃(t)ψ̅(0))]F̅(t,ψ)}
provided that the derivatives γ̃′(t), c̃′(t) exist and that ψ(0)≠0. Thus,
(3.31)D+Ũ(t,ψ(0),ψ̅(0))=|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|Reγ′̃(t)ψ(0)+c′̃(t)ψ̅(0)γ̃(t)ψ(0)+c̃(t)ψ̅(0)-φ′(t)+|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|-1Re{γ̃(t)(γ̃(t)ψ̅(0)+c̃̅(t)ψ(0))F(t,ψ)+c̃(t)(γ̃(t)ψ̅(0)+c̃̅(t)ψ(0))F̅(t,ψ)}=|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|Reγ′̃(t)ψ(0)+c′̃(t)ψ̅(0)γ̃(t)ψ(0)+c̃(t)ψ̅(0)-φ′(t)+|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|-1Re{(γ̃(t)ψ̅(0)+c̃̅(t)ψ(0))×(γ̃(t)F(t,ψ)+c̃(t)F̅(t,ψ))}.
Using (3.10), (3.13), and (ii), similarly to the proof of Theorem 3.1, we obtain
(3.32)D+Ũ(t,ψ(0),ψ̅(0))≥|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|Rea(t)-∑k=1m|Ak(t)ψ(θk(t)-t)+Bk(t)ψ̅(θk(t)-t)|(|γ̃(t)|+|c̃(t)|)-ϰ̃(t)|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|-∑k=1mκ̃k(t)|γ̃(θk(t))ψ(θk(t)-t)+c̃(θk(t))ψ̅(θk(t)-t)|+ϑ̃(t)|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|-ϱ(t)-φ′(t)
and consequently, with respect to (iii),
(3.33)D+Ũ(t,ψ(0),ψ̅(0))≥(Rea(t)+ϑ̃(t)-ϰ̃(t))|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|-∑k=1mλ̃k(t)|γ̃(θk(t))ψ(θk(t)-t)+c̃(θk(t))ψ̅(θk(t)-t)|-ϱ(t)-φ′(t)≥Θ̃(t)|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|+β̃(t)∑k=1mθk′(t)|γ̃(θk(t))ψ(θk(t)-t)+c̃(θk(t))ψ̅(θk(t)-t)|-ϱ(t)-φ′(t)≥Λ̃(t)|γ̃(t)ψ(0)+c̃(t)ψ̅(0)|+β̃(t)∑k=1mθk′(t)|γ̃(θk(t))ψ(θk(t)-t)+c̃(θk(t))ψ̅(θk(t)-t)|-ϱ(t)-φ′(t)
for almost all t∈(τ,∞) and for ψ∈𝒞 sufficiently close to ϕ. Replacing t and ψ by t* and ϕ, respectively, in the last expression, we get
(3.34)Λ̃(t*)|γ̃(t*)ϕ(0)+c̃(t*)ϕ̅(0)|+β̃(t*)∑k=1mθk′(t*)|γ̃(θk(t*))ϕ(θk(t*)-t*)+c̃(θk(t*))ϕ̅(θk(t*)-t*)|-ϱ(t*)-φ′(t*)≥Λ̃(t*)|γ̃(t*)ζ+c̃(t*)ζ̅|+β̃(t*)∑k=1mθk′(t*)φ(θk(t*))-ϱ(t*)-φ′(t*)≥Λ̃(t*)φ(t*)+β̃(t*)∑k=1mθk′(t*)φ(θk(t*))-ϱ(t*)-φ′(t*)=Λ̃(t*)Cexp⁡[∫Tt*ξ(s)ds]+β̃(t*)∑k=1mθk′(t*)Cexp⁡[∫Tθk(t*)ξ(s)ds]-ϱ(t*)-Cξ(t*)exp⁡[∫Tt*ξ(s)ds]={Λ̃(t*)+β̃(t*)∑k=1mθk′(t*)exp⁡[-∫θk(t*)t*ξ(s)ds]-ξ(t*)}Cexp⁡[∫Tt*ξ(s)ds]-ϱ(t*)>0.
Therefore, in view of the continuity, D+Ũ(t,ψ(0),ψ̅(0))>0 holds for ψ sufficiently close to ϕ and almost all t sufficiently close to t*. Hence, Ω0 is a regular polyfacial set with respect to ((2′)).
ChooseZ={(t2,z)∈Ω0∪ΩŨ}, where t2>τ+r is fixed. It can be easily verified that Z∩ΩŨ is a retract of ΩŨ, but Z∩ΩŨ is not a retract of Z. Let η∈𝒞 be such that η(0)=1 and 0≤η(θ)<1 for θ∈[-r,0). Define the mapping p:Z→𝒞 for (t2,z)∈Z by the relation
(3.35)p(t2,z)(θ)=φ(t2+θ)η(θ)(γ̃2(t2+θ)-|c̃(t2+θ)|2)φ(t2)[(γ̃(t2)γ̃(t2+θ)-c̃̅(t2)c̃(t2+θ))z+(γ̃(t2+θ)c̃(t2)-γ̃(t2)c̃(t2+θ))z̅]
The mapping p is continuous, and it holds that
(3.36)p(t2,z)(0)=zfor(t2,z)∈Z,p(t2,0)(θ)=0forθ∈[-r,0].
Since
(3.37)γ̃(t2+θ)p(t2,z)(θ)+c̃(t2+θ)p(t2,z)(θ)¯=φ(t2+θ)η(θ)φ(t2)(γ̃(t2)z+c̃(t2)z̅),
we have
(3.38)|γ̃(t2)z+c̃(t2)z̅|<φ(t2),(3.39)|γ̃(t2+θ)p(t2,z)(θ)+c̃(t2+θ)p(t2,z)(θ)¯|<φ(t2+θ)
for (t2,z)∈Z∩Ω0 and θ∈[-r,0]. Clearly, inequality (3.39) holds also for (t2,z)∈Z∩ΩŨ and θ∈[-r,0).
Using a topological principle for retarded functional differential equations (see Rybakowski [15, Theorem 2.1]), we see that there is a solution z0(t) of (1.2) such that (t,z0(t))∈Ω0 for all t≥t2 for which the solution z0(t) exists. Obviously, z0(t) exists for all t≥t2 and
(3.40)(|γ̃(t)|-|c̃(t)|)|z0(t)|≤|γ̃(t)z0(t)+c̃(t)z̅0(t)|≤φ(t)fort≥t2.
Hence
(3.41)|z0(t)|≤φ(t)|γ̃(t)|-|c̃(t)|fort≥t2.Theorem 3.4.
Suppose that hypotheses (i), (ii), (iin), (iii), (iiin), and (ivn) are fulfilled for  τn≥T and  n∈ℕ, where  Rn>0,  infn∈ℕRn=0. Let  Λ̃,  θk′ be continuous functions satisfying the inequality  Λ̃(t)≤Θ̃(t) a. e. on  [T,∞), where  Θ̃ is defined by (2.14). Assume that  ξ:[T-r,∞)→ℝ is a continuous function such that
(3.42)Λ̃(t)+β̃(t)∑k=1mθk′(t)exp⁡[-∫θk(t)tξ(s)ds]-ξ(t)>ρ(t)C-1exp⁡(-∫Ttξ(s)ds)
for t∈[T,∞) and some constant C>0. Suppose that
(3.43)limsup⁡t→∞[∫Tt(Λ̃n(s)-ξ(s))ds+ln⁡|γ̃(t)|-|c̃(t)||γ̃(t)|+|c̃(t)|]=∞,(3.44)lim⁡t→∞[β̃n(t)max⁡θ(t)≤s≤texp⁡[∫Tsξ(σ)dσ]|γ̃(s)|-|c̃(s)|∑k=1m∫θk(t)t(|γ̃(s)|+|c̃(s)|)ds]=0,(3.45)inf⁡τn≤s≤t<∞[∫stΛ̃n(σ)dσ-ln⁡(|γ̃(t)|+|c̃(t)|)]≥ν
for n∈ℕ, where θ(t)=mink=1,…,mθk(t) and ν∈(-∞,∞). Then, there exists a solution z0(t) of (1.2) such that
(3.46)lim⁡t→∞min⁡θ(t)≤s≤t|z0(s)|=0.Proof.
By the use of Theorem3.3 we observe that there is a t2≥T and a solution z0(t) of (1.2) with property
(3.47)|z0(t)|≤C|γ̃(t)|-|c̃(t)|exp⁡[∫Ttξ(s)ds]
for t≥t2. Suppose that (3.46) is not satisfied. Then, there is ε0>0 such that
(3.48)limsup⁡t→∞min⁡θ(t)≤s≤t|z0(s)|>ε0.
Choose N∈ℕ such that
(3.49)max⁡{RN,2μRNe-ν}<ε0.
It holds that
(3.50)min⁡θ(τ)≤s≤τ|z0(s)|>max⁡{RN,2μRNe-ν}
for some τ>max{T,τN,t2}. In view of (3.44), we can suppose that
(3.51)|β̃N(τ)|Cmax⁡θ(τ)≤s≤τexp⁡[∫Tsξ(σ)dσ]|γ̃(s)|-|c̃(s)|∑k=1m∫θk(τ)τ(|γ̃(s)|+|c̃(s)|)ds<12RNe-ν.
Therefore, taking into account (2.5), (3.47), (3.50), and (3.51) and the nonpositiveness of βN, we have
(3.52)(|γ̃(τ)|-|c̃(τ)|)|z0(τ)|+β̃N(τ)max⁡θ(τ)≤s≤τ|z0(s)|∑k=1m∫θk(τ)τ(|γ̃(s)|+|c̃(s)|)ds≥(|γ̃(τ)|-|c̃(τ)|)|z0(τ)|+β̃N(τ)Cmax⁡θ(τ)≤s≤τexp⁡[∫Tsξ(σ)dσ]|γ̃(s)|-|c̃(s)|∑k=1m∫θk(τ)τ(|γ̃(s)|+|c̃(s)|)ds≥μ2μRNe-ν-12RNe-ν>RNe-ν.
Moreover, (3.45) implies that
(3.53)inf⁡τ≤t<∞[∫τtΛ̃N(s)ds-ln⁡(|γ̃(t)|+|c̃(t)|)]≥ν>-∞.
By Theorem 3.1, we obtain an estimation
(3.54)|z0(t)|≥Ψ(τ)|γ̃(t)|+|c̃(t)|exp⁡[∫τtΛ̃N(s)ds]
for all t≥τ, Ψ being defined by
(3.55)Ψ(τ)=(|γ̃(τ)|-|c̃(τ)|)|z0(τ)|+β̃N(τ)max⁡θ(τ)≤s≤τ|z0(s)|∑k=1m∫θk(τ)τ(|γ̃(s)|+|c̃(s)|)ds.
Relation (3.47) together with (3.54) yields
(3.56)Ψ(τ)|γ̃(t)|+|c̃(t)|exp⁡[∫τtΛ̃N(s)ds]≤C|γ̃(t)|-|c̃(t)|exp⁡[∫Ttξ(s)ds],
that is
(3.57)∫Tt[Λ̃N(s)-ξ(s)]ds+ln⁡|γ̃(t)|-|c̃(t)||γ̃(t)|+|c̃(t)|≤∫TτΛ̃N(s)ds-ln⁡[C-1Ψ(τ)]
for t≥τ. However, the last inequality contradicts (3.43) and Theorem 3.4 is proved.From Theorem3.1 we easily obtain several corollaries.Corollary 3.5.
Let the assumptions of Theorem3.1 be fulfilled with  R0>0. If
(3.58)liminf⁡t→∞[∫t1tΛ̃0(s)ds-ln⁡(|γ̃(t)|+|c̃(t)|)]=ς>ν,
then for any ε, 0<ε<R0eς-ν, there is t2≥t1 such that
(3.59)|z(t)|>ε
for all t≥t2, for which z(t) is defined.Proof.
Without loss of generality we can assume thatε>R0. Choose χ, 0<χ<1 such that R0<ε<χR0eς-ν. In view of (3.58), there is t2≥t1 such that
(3.60)∫t1tΛ̃0(s)ds-ln⁡(|γ̃(t)|+|c̃(t)|)>ς+ln⁡χ
for t≥t2. Hence,
(3.61)∫t1tΛ̃0(s)ds-ln⁡(|γ̃(t)|+|c̃(t)|)>ν+ln⁡εR0
for t≥t2. Estimation (3.4) together with (3.2) now yields
(3.62)|z(t)|>R0e-νeνεR0=ε
for all t≥t2, for which z(t) is defined.Corollary 3.6.
Let the assumptions of Theorem3.1 be fulfilled with  R0>0. If
(3.63)lim⁡t→∞[∫t1tΛ̃0(s)ds-ln⁡(|γ̃(t)|+|c̃(t)|)]=∞,
then for any ε>0 there exists t2≥t1 such that (3.59) holds for all t≥t2, for which z(t) is defined.The efficiency of Theorem3.1 and Corollary 3.6 is demonstrated in the following example.Example 3.7.
Consider (1.2) where a(t)≡4+3i, b(t)≡2, Ak(t)≡0, Bk(t)≡0 for k=1,…,m, θk(t)=t+(1/2k)(coskt-1), g(t,z,w1,…,wm)=3z+∑k=1m(1/2m)e-twk.
Obviously,t-(1/k)≤θk(t)≤t and 1/2≤θk′(t)≤(3/2). Suppose that t0=1 and T≥2. Then, γ̃≡3+5, c̃≡-2i. Further,
(3.64)|γ̃(t)g(t,z,w1,…,wm)+c̃(t)g̅(t,z,w1,…,wm)|≤3|γ̃(t)z+c̃(t)z̅|+∑k=1m12me-t|γ̃(θk(t))wk+c̃(θk(t))w̅k|.Takingϰ̃0(t)≡3, κ̃0k(t)=(1/2m)e-t, τ0=T, R0=0, ϑ̃(t)≡0, β̃0(t)=-(1/m)e-t, Λ̃0(t)=Θ̃0(t)=1-e-t (>0) in Theorem 3.1, we have(3.65)θk′(t)β̃0(t)≤-λ̃0k(t),β̃0′(t)≥Θ̃0(t)β̃0(t)
for t∈[T,∞) and Theorem 3.1 and Corollary 3.6 are applicable to the considered equation.As a corollary of Theorem3.3 we obtain sufficient conditions for the existence of a bounded solution of (1.2) or the existence of a solution z0(t) of (1.2) satisfying limt→∞z0(t)=0.Corollary 3.8.
Let the assumptions of Theorem3.3 be satisfied. If
(3.66)limsup⁡t→∞[1|γ̃(t)|-|c̃(t)|exp⁡(∫Ttξ(s)ds)]<∞,
then there is a bounded solution z0(t) of (1.2). If
(3.67)lim⁡t→∞[1|γ̃(t)|-|c̃(t)|exp⁡(∫Ttξ(s)ds)]=0,
then there is a solution z0(t) of (1.2) such that
(3.68)lim⁡t→∞z0(t)=0.The next example shows how Theorem3.3 and Corollary 3.8 (namely the first part) can be used.Example 3.9.
Consider (1.2) where a(t)≡4+3i, b(t)≡i, Ak(t)≡0, Bk(t)≡0, θk(t)=t-e-kt for k=1,…,m, g(t,z,w1,…,wm)=(1/2)z+∑k=1m(1/4m)wk+e-t.
Obviouslyt-1≤θk(t)≤t and θk′(t)=1+ke-kt≥1>0 for t≥0. Suppose that t0=1 and T≥2. Then,
(3.69)γ̃(t)=Im⁡a(t)+(Im⁡a(t))2-|b(t)|2sgn⁡(Im⁡a(t))≡3+22,c̃(t)=-ib(t)≡1.
Further,
(3.70)|γ̃(t)g(t,z,w1,…,wm)+c̃(t)g̅(t,z,w1,…,wm)|≤|γ̃|+|c̃||γ̃|-|c̃|12|γ̃(t)z+c̃(t)z̅|+|γ̃|+|c̃||γ̃|-|c̃|∑k=1m[14m|γ̃(θk(t))wk+c̃(θk(t))w̅k|]+e-t=22|γ̃(t)z+c̃(t)z̅|+2∑k=1m[14m|γ̃(θk(t))wk+c̃(θk(t))w̅k|]+e-t.
If we takeϰ̃(t)≡2/2, κ̃k(t)=2/4m, ϑ̃(t)≡0, β̃(t)=-2/4m, Λ̃(t)=Θ̃(t)=4-(2/2) in Theorem 3.3, we observe that
(3.71)θk′(t)β̃(t)=-(1+ke-kt)24m≤-24m=-λ̃k(t)
for t∈[T,∞). Then, for ξ≡0 and C=1, we have
(3.72)Λ̃(t)+β̃(t)∑k=1mθk′(t)exp⁡[-∫θk(t)tξ(s)ds]-ξ(t)=4-22-24m∑k=1m(1+ke-kt)≥4-22-24m⋅2m=4-2>1>e-t=ρ(t)C-1exp⁡(-∫Ttξ(s)ds)
for t∈[T,∞) and
(3.73)limsup⁡t→∞[1|γ̃(t)|-|c̃(t)|exp⁡(∫Ttξ(s)ds)]=12+22<∞,
and hence the assertions of Theorem 3.3 and the first part of Corollary 3.8 hold true.


## 4. Summary

We investigated the problem of instability and asymptotic behaviour of real two-dimensional differential system with a finite number of nonconstant delays. We focused on the case corresponding to the situation when the equilibrium 0 of the autonomous system (2.2) is a focus or a centre and it is unstable. We obtained several criteria for instability properties of the solutions as well as conditions for the existence of bounded solutions. We used the methods of complexification, the method of Lyapunov-Krasovskii functional and a Ważewski topological principle for retarded functional differential equations of Carathéodory type. At the end we supplied several corollaries and explanatory examples.



---

*Source: 952601-2012-08-08.xml*