# On the Critical Case in Oscillation for  Differential Equations with a Single Delay and with Several Delays

Authors: Jaromír Baštinec; Leonid Berezansky; Josef Diblík; Zdeněk Šmarda
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

New nonoscillation and oscillation criteria are derived for scalar delay differential equationsx˙(t)+a(t)x(h(t))=0,a(t)≥0,h(t)≤t,t≥t0, and x˙(t)+∑k=1mak(t)x(hk(t))=0,ak(t)≥0,hk(t)≤t,
and t≥t0, in the critical case including equations with several unbounded delays,
without the usual assumption that the parameters
a,h,ak, and hk of the equations are continuous functions. These conditions improve and extend some known oscillation results in the critical case for delay differential equations.


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## Body

## 1. Introduction

It is well known that a scalar linear equation with delay(1.1)ẋ(t)+1ex(t-1)=0
has a nonoscillatory solution as t→∞. This means that there exists an eventually positive solution. The coefficient 1/e is called critical with the following meaning: for any α>1/e, all solutions of the equation (1.2)ẋ(t)+αx(t-1)=0
are oscillatory while, for α≤1/e, there exists an eventually positive solution.In [1] the third author considered the equation(1.3)ẋ(t)+a(t)x(t-τ)=0,
where a:[t0,∞)→(0,∞), t0∈ℝ (throughout this paper we assume that t0≥0 is sufficiently large), is a continuous function and the delay τ>0 is a constant. For the critical case, he obtained the following result.Theorem 1.1.
(a)
Let an integer k≥0 exists such that a(t)≤ak(t) if t→∞ where
(1.4)ak(t):=1eτ+τ8et2+τ8e(tlnt)2+⋯+τ8e(tlntln2t⋯lnkt)2.
Then there exists an eventually positive solution x of (1.3).
(b)
Let an integer k≥2 and θ>1, θ∈ℝ, exist such that
(1.5)a(t)>ak-2(t)+θτ8e(tlntln2t⋯lnk-1t)2
if t→∞. Then all solutions of (1.3) oscillate.In this theorem fork≥1, lnkt=ln(lnk-1t), ln0t=t, t>expk-21 where expkt=exp(expk-1t), exp0t=t, and exp-1t=0.Further results on the critical case for (1.3) can be found in [2–6]. Theorem 1.1 was generalized in [7] for the following equation with a variable delay(1.6)ẋ(t)+a(t)x(t-τ(t))=0,
where a:[t0,∞)→(0,∞), t0∈ℝ, and τ:[t0,∞)→(0,∞), t0∈ℝ, are continuous functions.The main results of this paper include the following.Theorem 1.2 (see [7]).
Lett-τ(t)≥t0-τ(t0) if t≥t0. Let an integer k≥0 exists such that a(t)≤akτ(t) for t→∞, where
(1.7)akτ(t):=1eτ(t)+τ(t)8et2+τ(t)8e(tlnt)2+⋯+τ(t)8e(tlntln2t⋯lnkt)2.
If moreover
(1.8)∫t-τ(t)t1τ(ξ)dξ≤1,whent→∞,(1.9)limt→∞τ(t)·(1tlntln2t⋯lnkt)=0,
then there exists an eventually positive solution x of (1.6) for t→∞.Theorem 1.3 (see [7]).
Let one assume thatt-τ(t)≥t0-τ(t0) if t≥t0 and
(1.10)a(t)≤1τ(t)exp[-∫t-τ(t)t1τ(ξ)dξ]
as t→∞. Then there exists an eventually positive solution x of (1.6).In this paper we obtain new nonoscillation and oscillation sufficient conditions for (1.6) in the critical case, independent of Theorems 1.1–1.3. We also obtain nonoscillation and oscillation conditions for equations with several delays, including equations with unbounded delays. To the best of our knowledge, we are the first to investigate the critical case of such equations.


## 2. Preliminaries

We consider a scalar delay differential equation(2.1)ẋ(t)+∑i=1mbi(t)x(hi(t))=0,t≥t0,
subject to the following conditions:(a1)bi:[t0,∞)→[0,∞), i=1,…,m, are Lebesgue measurable functions essentially bounded in each finite interval [t0,b] with b>t0.(a2)hi:[t0,∞)→ℝ are Lebesgue measurable functions, hi(t)≤t, t∈[t0,∞), and lim supt→∞hi(t)=+∞, i=1,…,m.Along with (2.1) we consider an initial value problem(2.2)ẋ(t)+∑i=1mbi(t)x(hi(t))=f(t),t≥t0,(2.3)x(t)=φ(t),t<t0,x(t0)=x0.
We also assume that the following hypothesis holds:(a3)f:[t0,∞)→ℝ is a Lebesgue measurable function essentially bounded in each finite interval [t0,b] with b>t0, and φ:(-∞,t0)→ℝ is a Borel measurable bounded function.Definition 2.1.
A function absolutely continuous on each interval[t0,b] with b>t0, x:ℝ→ℝ, is called  a solution of problem (2.2), (2.3) if it satisfies (2.2) for almost all t∈[t0,∞) and equalities (2.3) for t≤t0.Lemma 2.2 (see [8]).
Let(a1)–(a3) hold. Then there exists exactly one solution of problem (2.2), (2.3).Definition 2.3.
One will say that (2.1) has a nonoscillatory solution if, for some problem (2.2), (2.3) with f(t)≡0, t≥t0, there exists an eventually positive solution. Otherwise, all solutions of (2.1) oscillate.To formulate a comparison result, consider the following equation:(2.4)ẋ(t)+∑i=1mci(t)x(gi(t))=0.
Let (a1) holds with bi(t):=ci(t) and (a2) holds with hi(t):=gi(t), i=1,2,…,m.Lemma 2.4 (see [9, 10]).
Let (2.4) have a nonoscillatory solution. If
(2.5)bi(t)≤ci(t),gi(t)≤hi(t),t≥t0,
then (2.1) has a nonoscillatory solution as well.
Suppose that all solutions of (2.4) are oscillatory. If
(2.6)bi(t)≥ci(t),gi(t)≥hi(t),t≥t0,
then all solutions of (2.1) are oscillatory as well.Lemma 2.5 (see [9, 10]).
Let existt0 such that
(2.7)∫mini=1,…,m{max{t0,hi(t)}}t‍∑j=1mbj(s)ds≤1e,t≥t0.
Then there exists a positive solution of (2.1) for t≥t0.Lemma 2.6 (see [9, 10]).
A nonoscillatory solution of (2.1) exists if and only if, for some t0, there exists a nonnegative locally integrable function u(t)≥0,  t∈ℝ, such that
(2.8)u(t)≥∑i=1mbi(t)e∫hi(t)tu(s)ds,t≥t0,u(t)=0,t<t0.


## 3. Differential Equation with a Single Delay

Equation (1.6) is a special case of (2.1) for m=1, b1(t)=a(t), and h1(t)=t-τ(t).Our first result is a simple consequence of Theorem1.1 and Lemma 2.4. Theorem 1.1 was obtained under the assumption that a(t) and τ(t) are continuous functions. But the proof of this theorem remains valid even for more general conditions (a1)-(a2).Theorem 3.1.
(A)
Let τ>0,  0≤τ(t)≤τ, for t→∞,  and let condition (a) of Theorem 1.1 holds. Then (1.6) has a nonoscillatory solution.
(B)
Let τ(t)≥τ>0 for t→∞, and let condition (b) of Theorem 1.1 holds. Then all solutions of (1.6) oscillate.Proof.
(A) We set h1(t):=t-τ(t), g1(t):=t-τ. Obviously h1(t)≥g1(t) for t→∞. By Theorem 1.1, (1.3) has a nonoscillatory solution. By Lemma 2.4 (with m=1, b1(t)=c1(t)=a(t)), (1.6) also has a nonoscillatory solution.
(B) The proof of this part is much the same (using Theorem 1.1 and Lemma 2.4) as the proof of part (A).Theorems1.2 and 1.3 can be applied to equations with one unbounded delay. Here, we want to give some new nonoscillation and oscillation conditions for equations with one delay, also including equations with unbounded delays. We remove some conditions of Theorems 1.2 and 1.3, in particular conditions (1.8) and (1.9). Moreover, the delay function τ(t) used in Theorems 1.2 and 1.3 as a coefficient appears in our conditions in both integral and nonintegral expressions.For every integerk≥0, δ>0, and t→∞ we define(3.1)Ak(t):=1eδτ(t)+δ8eτ(t)s2+δ8eτ(t)(slns)2+⋯+δ8eτ(t)(slnsln2s⋯lnks)2,
where(3.2)s=p(t):=∫t0t1τ(ξ)dξ.Theorem 3.2.
Let fort0 sufficiently large and t≥t0:  τ(t)>0 a.e. 1/τ(t) be a locally integrable function,
(3.3)limt→∞(t-τ(t))=∞,∫t0∞1τ(ξ)dξ=∞,
and let there exists t1>t0 such that t-τ(t)≥t0, t≥t1.
(a)
If there exists a δ∈(0,∞) such that
(3.4)∫t-τ(t)t1τ(ξ)dξ≤δ,t≥t1
and, for a fixed integer, k≥0,(3.5)a(t)≤Ak(t),t≥t1,
then there exists an eventually positive solution of (1.6).
(b)
If there exists a δ∈(0,∞) such that
(3.6)∫t-τ(t)t1τ(ξ)dξ≥δ,t≥t1,
and, for a fixed integer k≥2 and θ>1, θ∈ℝ,
(3.7)a(t)>Ak-2(t)+θδ8eτ(t)(slnsln2s⋯lnk-1s)2
if t≥t1, then all solutions of (1.6) oscillate.Proof.
(a) For the proof we will use a transformation applied to delay equations for the first time in [11]. Consider (1.6) for t≥t1. Denote
(3.8)h(t):=t-τ(t).
Since, by (3.2), s>0, p is a strictly increasing function and, hence, there exists an inverse function t=p-1(s). Denote
(3.9)y(s):=x(t)=x(p-1(s)),l(s):=p(h(t))=p(h(p-1(s))).
Since h(t)≤t, we have l(s)≤s (by (3.9) and (3.2)). From (3.9) we also have
(3.10)l(s)=∫t0h(t)1τ(ξ)dξ.
Substituting x(t)=y(s) in (1.6), we have ẋ(t)=ẏ(s)/τ(t) and (using (3.9))
(3.11)x(t-τ(t))=x(h(t))=x(h(p-1(s)))=x(p-1[p(h(p-1(s)))])=x(p-1(l(s)))=y(l(s)).
Hence, (1.6) takes the form
(3.12)ẏ(s)+τ(t)a(t)y(l(s))=0,
where τ(t)a(t)=τ(p-1(s))a(p-1(s)). Equality
(3.13)y(s)=x(t)
implies that the oscillation properties of (1.6) and (3.12) are equivalent. We have (by (3.2), (3.10), (3.8), and (3.4))
(3.14)s-l(s)=∫t0t1τ(s)ds-∫t0h(t)1τ(s)ds=∫h(t)t1τ(s)ds=∫t-τ(t)t1τ(s)ds≤δ.
Hence
(3.15)l(s)≥s-δ.
Consider an equation
(3.16)ẏ(s)+ak*(s)y(s-δ)=0,
where ak*(s) is defined similar to ak(t) by (1.4), where τ is replaced by δ and t by s, that is,
(3.17)ak*(s):=1eδ+δ8es2+δ8e(slns)2+⋯+δ8e(slnsln2s⋯lnks)2.
By Theorem 1.1, (3.16) has a positive solution. Equation (3.12) is of type (2.1) with
(3.18)m=1,b1(s)=τ(p-1(s))a(p-1(s)),h1(s)=l(s).
Now we use comparison of Lemma 2.4 where (2.4) is replaced by (3.16), that is,
(3.19)m=1,c1(s)=ak*(s),g1(s)=s-δ.
Since, by (3.5) and (3.1),
(3.20)b1(s)=τ(p-1(s))a(p-1(s))≤τ(p-1(s))Ak(p-1(s))=ak*(s)=c1(s)
and, by (3.15),
(3.21)g1(s)=s-δ≤l(s)=h1(s),
equation (3.12) and, due to (3.13), equation (1.6) also has a positive (i.e., nonoscillatory) solution. Part (b) can be proved in much the same way.Now we want to compare Theorem3.2 and Theorems 1.2 and 1.3 for equations with unbounded delays. Note that Theorem 1.1 is not valid for such equations and Theorem 1.2 contains additional restrictions (1.8), (1.9). Theorem 1.3 is not explicitly valid for the critical case.Example 3.3.
Let (1.6) be of the form
(3.22)ẋ(t)+a(t)x(t2)=0,t≥t0=1,
where a:[1,∞)→(0,∞). Here
(3.23)τ(t)=t2,s=p(t)=∫t0t1τ(ξ)dξ=∫1t2ξdξ=2lnt.
We set
(3.24)δ:=∫t-τ(t)t1τ(ξ)dξ=∫t/2t2ξdξ=2ln2.
In accordance with Theorem 3.2 (case (a) where k=0), (3.22) has a nonoscillatory solution if
(3.25)a(t)≤A0(t)=1eδτ(t)+δ8eτ(t)s2=1(eln2)t+ln28et(lnt)2.
Since (by Theorem 3.2, case (b) with k=2) all solutions of (3.22) oscillate if
(3.26)a(t)>A0(t)+θδ8eτ(t)(slns)2=1(eln2)t+ln28et(lnt)2+θln28et(lnt)2(ln(2lnt))2,
we conclude that the value
(3.27)a(t)=a*(t):=1(eln2)t+ln28et(lnt)2
is a critical value for the nonoscillation of (3.22).
The above statement is corroborated by Lemma2.5 since, for m=1, h1(t)=t-τ(t)=t/2, b1(t)=a*(t), and t∈[1,2],
(3.28)∫max{t0,h1(t)}tb1(s)ds=∫t0ta*(s)ds=∫1ta*(s)ds=lnteln2≤1e
and for t≥2(3.29)∫max{t0,h1(t)}tb1(s)ds=∫h1(t)ta*(s)ds=∫t/2ta*(s)ds=∫t/2t1(eln2)sds=1e,
and (2.7) turns into an equality for t≥2.To apply Theorem1.2, we verify condition (1.8). But, unfortunately, for (3.22) we have (3.30)∫t-τ(t)t1τ(ξ)dξ=∫t/2t2ξdξ=2ln2≐1.386>1.
Thus, this theorem is not applicable to (3.22).To compare Theorem1.3 with Theorem 3.2 we set a(t):=a*(t) (where a* is defined by (3.27)) in (3.22). By Theorem 3.2, (3.22) has a nonoscillatory solution. By Theorem 1.3, (3.22) has a nonoscillatory solution if (we set τ(t):=t/2 in (1.10)) (3.31)a(t)≤1τ(t)exp[-∫t-τ(t)t1τ(ξ)dξ]=2texp[-∫t/2t2ξdξ]=12t.
But in our case (3.32)a(t)=a*(t)=1(eln2)t+ln28et(lnt)2≥11.885t>12t,
and Theorem 1.3 fails for this equation.


## 4. Differential Equation with Several Delays

We start with the following question: for what functionsb(t)≥0 and delay σ>0 the equation(4.1)ẋ(t)+b(t)x(t-σ)+1eτx(t-τ)=0
can have a nonoscillatory solution? It is easy to see that b should be vanishing.Theorem 4.1.
Letlim inft→∞b(t)=b>0,τ>0, and σ≥0. Then all solutions of (4.1) are oscillatory.Proof.
Consider first the equation(4.2)ẋ(t)+bx(t-σ)+1eτx(t-τ)=0.
Suppose that (4.2) has a nonoscillatory solution. We set
(4.3)m=2,b1(t)=c1(t):=b,b2(t)=c2(t):=1eτ,g1(t):=t-σ,h2(t)=g2(t):=t-τ,h1(t):=t.
Since g1(t)≤h1(t), by Lemma 2.4, the equation
(4.4)ẋ(t)+bx(t)+1eτx(t-τ)=0
has a nonoscillatory solution. After the substitution x(t)=e-bty(t), (4.4) takes a form
(4.5)ẏ(t)+ebτeτy(t-τ)=0.
Since ebτ>1, all solutions of (4.5) are oscillatory by Theorem 1.1 (b). This is a contradiction. Hence, all solutions of (4.2) are oscillatory.
For sufficiently larget0, we have b(t)≥b, t≥t0. We set
(4.6)m=2,b1(t):=b(t)≥c1(t):=b,b2(t)=c2(t):=1eτ,h1(t)=g1(t):=t-σ,h2(t)=g2(t):=t-τ.
Now, Lemma 2.4 implies the statement of the theorem.We consider general equation (2.1) with delays subject to restrictions (a1), (a2).Theorem 4.2.
(a) Let an integerk≥0 and τ>0 exist such that, for all sufficiently large t,  inequalities
(4.7)t-hi(t)≤τ,i=1,2,…,m,(4.8)∑i=1mbi(t)≤ak(t),
where ak is defined by (1.4), are valid. Then there exists an eventually positive solution x of (2.1).
(b) Let an integerk≥2, τ>0, and θ>1 exist such that, for all sufficiently large t, inequalities
(4.9)t-hi(t)≥τ,i=1,2,…,m,(4.10)∑i=1mbi(t)≥ak-2(t)+θτ8e(tlntln2t⋯lnk-1t)2,
where ak-2 is defined by (1.4), are valid. Then all solutions of (2.1) oscillate.Proof.
Let the assumptions of case (a) be valid. Then, by Theorem1.1, the equation
(4.11)ẋ(t)+ak(t)x(t-τ)=0
has a nonoscillatory solution. By Lemma 2.6 (with m=1, b1(t)=ak(t), and h1(t)=t-τ), there exist a t0 and a locally integrable function u:ℝ→[0,∞) such that
(4.12)u(t)≥ak(t)e∫t-τtu(s)ds,t≥t0,u(t)=0,t<t0.
We have
(4.13)∑i=1mbi(t)e∫hi(t)tu(s)ds≤(∑i=1mbi(t))e∫t-τtu(s)ds≤ak(t)e∫t-τtu(s)ds.
Hence
(4.14)u(t)≥∑i=1mbi(t)e∫hi(t)tu(s)ds,t≥t0,u(t)=0,t<t0.
Now using Lemma 2.6 again, we conclude that there exists an eventually positive solution x of (2.1).
Let the assumptions of case (b) be valid. Suppose, on the contrary, that (2.1) has a nonoscillatory solution. Using calculations similar to those of the previous part of the proof, one can deduce that (by Lemma 2.6) there exist a t0 and a locally integrable function u(t)≥0 such that
(4.15)u(t)≥∑i=1mbi(t)e∫hi(t)tu(s)ds≥(ak-2(t)+θτ8e(tlntln2t⋯lnk-1t)2)e∫t-τtu(s)ds,t≥t0,u(t)=0,t<t0.
Hence (using Lemma 2.6 again), an equation
(4.16)ẋ(t)+(ak-2(t)+θτ8e(tlntln2t⋯lnk-1t)2)x(t-τ)=0
should have a nonoscillatory solution. Due to θ>1 being arbitrary, we easily get a contradiction to statement (b) of Theorem 1.1.Example 4.3.
We show that equation of type (2.1)
(4.17)ẋ(t)+18et2x(t-σ)+1ex(t-1)=0
has a nonoscillatory solution for any positive σ≤1. Indeed, set m=2, τ=1, h1(t)=t-σ, h2(t)=t-τ=t-1, b1(t)=1/(8et2), and b2(t)=1/e. Then (4.8), where k=0, holds and Part (a) of Theorem 4.2 is valid.Now we consider (2.1) with unbounded delays.Theorem 4.4.
Lett0 be sufficiently large, for t≥t0,  τ(t)>0 a.e., 1/τ(t) let locally integrable function,
(4.18)limt→∞(t-τ(t))=∞,∫t0∞1τ(ξ)dξ=∞,(4.19)limt→∞hi(t)=+∞,t-hi(t)≤τ(t),i=1,2,…,m,
and let there exists t1>t0 such that t-τ(t)≥t0 if t≥t1.
(a) If there exists aσ∈(0,∞) such that
(4.20)∫t-τ(t)t1τ(ξ)dξ≤σ,t≥t1,
and, for a fixed integer k≥0,
(4.21)∑i=1mbi(t)≤Ak(t),t≥t1,
where Ak(t) is defined by (3.1), (3.2), then there exists an eventually positive solution of (2.1).
(b) If there exists aσ∈(0,∞) such that
(4.22)∫t-τ(t)t1τ(ξ)dξ≥σ,t≥t1,
and, for a fixed integer k≥2 and θ>1, θ∈ℝ,
(4.23)∑i=1mbi(t)≥Ak-2(t)+θσ8eτ(t)(slnsln2s⋯lnk-1s)2,t≥t1,
where Ak-2(t) is defined by (3.1), (3.2), then all solutions of (2.1) oscillate.Proof.
Let the assumptions of case (a) be valid. By Theorem3.2, the equation
(4.24)ẋ(t)+(∑i=1mbi(t))x(t-τ(t))=0
has a nonoscillatory solution. Equation (4.24) is of a form of (2.4) with ci(t)=bi(t), gi(t)=t-τ(t), i=1,2,…,m. This means that we see (4.24) as an equation with m delayed terms.
Compare (4.24) with (2.1). We have bi(t)≤ci(t) and, due to (4.19), gi(t)≤hi(t), i=1,2,…,m. By Lemma 2.4, (2.1) has a nonoscillatory solution.
The proof of part (b) can be carried out in a way similar to that of the proof of part (a) and, therefore,  it is omitted.Example 4.5.
Consider the equation of the type of (2.1):
(4.25)ẋ(t)+αteln3x(t3)+ln38et(lnt)2x(t2)=0,t≥t0=1.
First let 0<α≤1. We set m=2, t1:=3, τ(t)=:2t/3, and
(4.26)σ:=∫t-τ(t)t1τ(ξ)dξ=∫t-(2t)/3t32ξdξ=∫t/3t32ξdξ=32ln3.
Moreover, we put
(4.27)h1(t)=t3,h2(t)=t2,b1(t)=αteln3,b2(t)=ln38et(lnt)2,t-hk(t)≤τ(t)=23t,k=1,2,s=∫t0t1τ(ξ)dξ=∫1t32ξdξ=32lnt.
By (3.1),
(4.28)A0(t)=1eστ(t)+σ8eτ(t)s2=1teln3+ln38et(lnt)2≥b1(t)+b2(t).
All conditions of Theorem 4.4 part (a) hold, hence (4.25) has a nonoscillatory solution.Similarly, one can show (by Theorem4.4 part (b)) that, for α>1, all solutions of (4.25) are oscillatory.


## 5. Differential Equation with Two Delays

In [12] the authors consider a differential equation with two delays(5.1)ẋ(t)+b1(t)x(h1(t))+b2(t)x(h2(t))=0,t≥t0,
where bi:[t0,∞)→[0,∞), i=1,2,(5.2)h1(t)=t-τ,h2(t)=t-σ,
and τ, σ are positive constants. Let (5.3)lim inft→∞b1(t)=p,lim inft→∞b2(t)=q.
In accordance with [12] we say that (5.1) is in a critical state if there exists λ0≥0 such that (5.4)λ0=peλ0τ+qeλ0σ
and, for any λ>0, λ≠λ0, we have (5.5)λ<peλτ+qeλσ.Theorem 5.1 (see [12]).
Let (5.1) be in a critical case, p>0,  q>0, and
(5.6)lim inft→∞[b1(t+τ)-p]t=α,lim inft→∞[b2(t+σ)-q]t=β,
where -∞<α,β≤+∞. If αeλ0τ+βeλ0σ>0, then all solutions of (5.1) oscillate.Theorem 5.2 (see [12]).
Let(5.7)lim inft→∞b1(t)=1eτ,lim inft→∞b2(t)=0,lim inft→∞[b1(t+τ)-1eτ]t=α,lim inft→∞tb2(t+σ)=β,
where -∞<α≤+∞, β>0. If eα+eσ/τβ>0, then all solutions of (5.1) oscillate.The aim of the following theorems is to obtain nonoscillation conditions for (5.1) in the above-mentioned critical case. This will complete the oscillation results given by Theorems 5.1 and 5.2. Note that, in Theorems 5.3 and 5.4 below, delays h1 and h2 being not defined by (5.2) are arbitrary and subject only to the restrictions indicated.Theorem 5.3.
Lett-h1(t)≤τ and t-h2(t)≤σ for t≥t0,
(5.8)lim supt→∞[b1(t)-p]t=α,lim supt→∞[b2(t)-q]t=β,
where -∞<α, β<+∞, and let there exists a λ0>0 such that
(5.9)λ0≥peλ0τ+qeλ0σ,(5.10)αeλ0τ+βeλ0σ<0.
Then (5.1) has a nonoscillatory solution.Proof.
There existt1>t0+max(τ,σ) and ϵ>0 such that, owing to (5.8) and (5.10),
(5.11)b1(t)≤p+α+ϵt,b2(t)≤q+β+ϵt,(α+ϵ)eλ0τ+(β+ϵ)eλ0σ≤0
if t≥t1. By Lemma 2.4, with m=2,
(5.12)c1(t):=p+α+ϵt,c2(t):=q+β+ϵt,g1(t):=t-τ,g2(t):=t-σ
in (2.4), the existence of a nonoscillatory solution of the equation
(5.13)ẋ(t)+(p+α+ϵt)x(t-τ)+(q+β+ϵt)x(t-σ)=0
implies the existence of a nonoscillatory solution of (5.1).
By Lemma2.6, for the existence of a positive solution of (5.13), it is sufficient to find a nonnegative solution of the inequality
(5.14)u(t)≥(p+α+ϵt)e∫t-τtu(s)ds+(q+β+ϵt)e∫t-σtu(s)ds,
where t≥t1. Substituting u(t) for λ0 in inequality (5.14), we have
(5.15)λ0≥peλ0τ+qeλ0σ+(α+ϵ)eλ0τ+(β+ϵ)eλ0σt.
Due to (5.9) and (5.11), we conclude that the last inequality holds, and, consequently, u(t)=λ0 is a solution of inequality (5.14). By Lemma 2.6, (5.13) has a nonoscillatory solution. Hence, (5.1) has a nonoscillatory solution, too.Theorem 5.4.
Lett-h1(t)≤τ,  t-h2(t)≤σ for t≥t0,
(5.16)lim supt→∞[b1(t)-1eτ]t=α,lim supt→∞tb2(t)=β,
where α,β∈ℝ. If
(5.17)eα+eσ/τβ<0,
then (5.1) has a nonoscillatory solution.Proof.
There existt1>t0+max(τ,σ) and ϵ>0 such that, owing to (5.16) and (5.17),
(5.18)b1(t)≤1eτ+α+ϵt,b2(t)≤β+ϵt,e(α+ϵ)+eσ/τ(β+ϵ)≤0
if t≥t1. By Lemma 2.4, with m=2,
(5.19)c1(t)=1eτ+α+ϵt,c1(t)=β+ϵt,g1(t)=t-τ,g2(t)=t-σ
in (2.4), the existence of a nonoscillatory solution of the equation
(5.20)ẋ(t)+(1eτ+α+ϵt)x(t-τ)+(β+ϵt)x(t-σ)=0
implies the existence of a nonoscillatory solution of (5.1). By Lemma 2.6, it is sufficient to find a nonnegative solution of the inequality
(5.21)u(t)≥(1eτ+α+ϵt)e∫t-τtu(s)ds+(β+ϵt)e∫t-σtu(s)ds,
where t≥t1. Put u(t)=1/τ, t≥t1 in inequality (5.21). We have
(5.22)1τ≥1eτe+e(α+ϵ)+eσ/τ(β+ϵ)t.
Due to (5.18), we conclude that the last inequality holds, and, consequently, u(t)=1/τ is a solution of inequality (5.21). Let u(t)=0 for t<t1. By Lemma 2.6, (5.20) has a nonoscillatory solution. Hence, (5.1) also has a nonoscillatory solution.Example 5.5.
Consider (5.1) with
(5.23)b1(t)=1eτ+|sint|-2t,b2(t)=|cost|+1t
and with h1(t), h2(t) defined by (5.2), that is,
(5.24)ẋ(t)+(1eτ+|sint|-2t)x(t-τ)+|cost|+1tx(t-σ)=0,
where t≥t0. Since
(5.25)lim inft→∞b1(t)=1eτ,lim inft→∞b2(t)=0,lim inft→∞[b1(t+τ)-1eτ]t=-2,lim inft→∞tb2(t+σ)=1,
then, by Theorem 5.2 (with α=-2 and β=1), all solutions of (5.24) oscillate if
(5.26)στ>1+ln2.
Since
(5.27)lim supt→∞[b1(t)-1eτ]t=-1,lim supt→∞tb2(t)=2,
by Theorem 5.4 (with α=-1 and β=2), (5.24) has a nonoscillatory solution if
(5.28)στ<1-ln2.


### 5.1. Generalization for Equations with Several Delays

It is easy to generalize Theorem5.3 for a general equation (2.1) with several delays. Denote (5.29)pi:=lim inft→∞bi(t),
where i=1,2,…,m. We omit the proof of this generalization since it is similar to that of Theorem 5.3.Theorem 5.6.
Letτi,  i=1,2,…,m,  be positive constants such that
(5.30)t-hi(t)≤τi
for t≥t0,
(5.31)lim supt→∞[bi(t)-pi]t=αi,
where αi∈ℝ, and let there exists a λ0>0 such that
(5.32)λ0≥∑i=1mpieλ0τi,∑i=1mαieλ0τi<0.
Then (2.1) has a nonoscillatory solution.The following statement generalizes Theorem5.4. We will formulate this result for (2.4).Theorem 5.7.
LetI⊂{1,…,m} be a set of indices such that
(5.33)gk(t)≤{h1(t),ifk∈I,h2(t),ifk∉I,
where hi:[t0,∞)→ℝ and hi(t)≤t. Let
(5.34)b1(t):=∑k∈Ick(t),b2(t):=∑k∉Ick(t).
If, for functions bi, hi, i=1,2, all assumptions of Theorem 5.4 are true, then (2.4) has a nonoscillatory solution.The proof of Theorem5.7 is omitted as it can be done easily using Lemma 2.4 and Theorem 5.4.


## 5.1. Generalization for Equations with Several Delays

It is easy to generalize Theorem5.3 for a general equation (2.1) with several delays. Denote (5.29)pi:=lim inft→∞bi(t),
where i=1,2,…,m. We omit the proof of this generalization since it is similar to that of Theorem 5.3.Theorem 5.6.
Letτi,  i=1,2,…,m,  be positive constants such that
(5.30)t-hi(t)≤τi
for t≥t0,
(5.31)lim supt→∞[bi(t)-pi]t=αi,
where αi∈ℝ, and let there exists a λ0>0 such that
(5.32)λ0≥∑i=1mpieλ0τi,∑i=1mαieλ0τi<0.
Then (2.1) has a nonoscillatory solution.The following statement generalizes Theorem5.4. We will formulate this result for (2.4).Theorem 5.7.
LetI⊂{1,…,m} be a set of indices such that
(5.33)gk(t)≤{h1(t),ifk∈I,h2(t),ifk∉I,
where hi:[t0,∞)→ℝ and hi(t)≤t. Let
(5.34)b1(t):=∑k∈Ick(t),b2(t):=∑k∉Ick(t).
If, for functions bi, hi, i=1,2, all assumptions of Theorem 5.4 are true, then (2.4) has a nonoscillatory solution.The proof of Theorem5.7 is omitted as it can be done easily using Lemma 2.4 and Theorem 5.4.


## 6. Concluding Remarks

In conclusion we note that there exist numerous results on nonoscillation for various classes of delay differential equations in a noncritical case. We refer, for example, to monographs [6, 9, 13, 14], recent papers [15–22], and references therein. Some of the books and papers mentioned discuss the critical case from various points of view different from our approach, and we mentioned them above. In the paper we investigate the critical case for delayed differential equations. It will be interesting as a motivation for further investigation along these lines to consider cases, critical is a sense to other classes of equations, in particular, for integrodifferential equations, differential equations with distributed delay, or differential equations of a neutral type. Finally, for nonoscillation results for difference equations we refer to [23–28].



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*Source: 417869-2010-09-23.xml*