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

Abstract

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

1. Introduction

It is well known that a scalar linear equation with delay ̇1𝑥(𝑡)+𝑒𝑥(𝑡−1)=0 (1.1) 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 𝑥̇(𝑡)+𝛼𝑥(𝑡−1)=0 (1.2) are oscillatory while, for 𝛼≤1/𝑒, there exists an eventually positive solution.

In [1] the third author considered the equation𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡−𝜏)=0, (1.3) 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)≤𝑎_𝑘(𝑡) if 𝑡→∞ where 𝑎_𝑘1(𝑡)∶=+𝜏𝑒𝜏8𝑒𝑡²+𝜏8𝑒(𝑡ln𝑡)²𝜏+⋯+8𝑒(𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡)². (1.4) 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 𝑎(𝑡)>𝑎_𝑘−2(𝑡)+𝜃𝜏8𝑒(𝑡ln𝑡ln₂𝑡⋯ln_𝑘−1𝑡)² (1.5) if 𝑡→∞. Then all solutions of (1.3) oscillate.

In this theorem for 𝑘≥1, ln_𝑘𝑡=ln(ln_𝑘−1𝑡), ln₀𝑡=𝑡, 𝑡>exp_𝑘−21 where exp_𝑘𝑡=exp(exp_𝑘−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 delay𝑥̇(𝑡)+𝑎(𝑡)𝑥(𝑡−𝜏(𝑡))=0, (1.6) 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]). Let 𝑡−𝜏(𝑡)≥𝑡₀−𝜏(𝑡₀) if 𝑡≥𝑡₀. Let an integer 𝑘≥0 exists such that 𝑎(𝑡)≤𝑎_𝑘𝜏(𝑡) for 𝑡→∞, where 𝑎_𝑘𝜏1(𝑡)∶=+𝑒𝜏(𝑡)𝜏(𝑡)8𝑒𝑡²+𝜏(𝑡)8𝑒(𝑡ln𝑡)²+⋯+𝜏(𝑡)8𝑒(𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡)². (1.7) If moreover ^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉≤1,when𝑡→∞,lim_𝑡→∞1𝜏(𝑡)⋅(𝑡ln𝑡ln₂𝑡⋯ln_𝑘𝑡)=0, (1.8)(1.9) then there exists an eventually positive solution 𝑥 of (1.6) for 𝑡→∞.

Theorem 1.3 (see [7]). Let one assume that 𝑡−𝜏(𝑡)≥𝑡₀−𝜏(𝑡₀) if 𝑡≥𝑡₀ and 1𝑎(𝑡)≤𝜏(𝑡)exp[−^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉] (1.10) 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 equation𝑥̇(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑥(ℎ_𝑖(𝑡))=0,𝑡≥𝑡₀, (2.1) subject to the following conditions:

(a1)𝑏_𝑖∶[𝑡₀,∞)→[0,∞), 𝑖=1,…,𝑚, are Lebesgue measurable functions essentially bounded in each finite interval [𝑡₀,𝑏] with 𝑏>𝑡₀.

(a2)ℎ_𝑖∶[𝑡₀,∞)→ℝ are Lebesgue measurable functions, ℎ_𝑖(𝑡)≤𝑡, 𝑡∈[𝑡₀,∞), and limsup_𝑡→∞ℎ_𝑖(𝑡)=+∞, 𝑖=1,…,𝑚.

Along with (2.1) we consider an initial value problem𝑥̇(𝑡)+^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑥(ℎ_𝑖(𝑡))=𝑓(𝑡),𝑡≥𝑡₀,𝑥(𝑡)=𝜑(𝑡),𝑡<𝑡₀,𝑥(𝑡₀)=𝑥₀. (2.2)(2.3) 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:𝑥̇(𝑡)+^𝑚_𝑖=1𝑐_𝑖(𝑡)𝑥(𝑔_𝑖(𝑡))=0. (2.4) Let (a1) holds with 𝑏_𝑖(𝑡)∶=𝑐_𝑖(𝑡) and (a2) holds with ℎ_𝑖(𝑡)∶=𝑔_𝑖(𝑡), 𝑖=1,2,…,𝑚.

Lemma 2.4 (see [9, 10]). Let (2.4) have a nonoscillatory solution. If 𝑏_𝑖(𝑡)≤𝑐_𝑖(𝑡),𝑔_𝑖(𝑡)≤ℎ_𝑖(𝑡),𝑡≥𝑡₀, (2.5) then (2.1) has a nonoscillatory solution as well.
Suppose that all solutions of (2.4) are oscillatory. If 𝑏_𝑖(𝑡)≥𝑐_𝑖(𝑡),𝑔_𝑖(𝑡)≥ℎ_𝑖(𝑡),𝑡≥𝑡₀, (2.6) then all solutions of (2.1) are oscillatory as well.

Lemma 2.5 (see [9, 10]). Let exist 𝑡₀ such that ^𝑡_{min𝑖=1,…,𝑚{max{𝑡0,ℎ𝑖(𝑡)}}}^𝑚_𝑗=1𝑏_𝑗1(𝑠)𝑑𝑠≤𝑒,𝑡≥𝑡₀. (2.7) 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 𝑢(𝑡)≥^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑒^{∫𝑡ℎ𝑖(𝑡)𝑢(𝑠)𝑑𝑠},𝑡≥𝑡₀,𝑢(𝑡)=0,𝑡<𝑡₀. (2.8)

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 Theorem 1.1 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).

Theorems 1.2 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 integer 𝑘≥0, 𝛿>0, and 𝑡→∞ we define𝐴_𝑘1(𝑡)∶=+𝛿𝑒𝛿𝜏(𝑡)8𝑒𝜏(𝑡)𝑠²+𝛿8𝑒𝜏(𝑡)(𝑠ln𝑠)²𝛿+⋯+8𝑒𝜏(𝑡)(𝑠ln𝑠ln₂𝑠⋯ln_𝑘𝑠)², (3.1) where𝑠=𝑝(𝑡)∶=^𝑡_𝑡01𝜏(𝜉)𝑑𝜉. (3.2)

Theorem 3.2. Let for 𝑡₀ sufficiently large and 𝑡≥𝑡₀: 𝜏(𝑡)>0 a.e. 1/𝜏(𝑡) be a locally integrable function, lim_𝑡→∞(𝑡−𝜏(𝑡))=∞,^∞_𝑡01𝜏(𝜉)𝑑𝜉=∞, (3.3) and let there exists 𝑡₁>𝑡₀ such that 𝑡−𝜏(𝑡)≥𝑡₀, 𝑡≥𝑡₁.
(a) If there exists a 𝛿∈(0,∞) such that ^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉≤𝛿,𝑡≥𝑡₁ (3.4) and, for a fixed integer, 𝑘≥0,𝑎(𝑡)≤𝐴_𝑘(𝑡),𝑡≥𝑡₁, (3.5) then there exists an eventually positive solution of (1.6).
(b) If there exists a 𝛿∈(0,∞) such that ^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉≥𝛿,𝑡≥𝑡₁, (3.6) and, for a fixed integer 𝑘≥2 and 𝜃>1, 𝜃∈ℝ, 𝑎(𝑡)>𝐴_𝑘−2(𝑡)+𝜃𝛿8𝑒𝜏(𝑡)(𝑠ln𝑠ln₂𝑠⋯ln_𝑘−1𝑠)² (3.7) 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 ℎ(𝑡)∶=𝑡−𝜏(𝑡). (3.8) Since, by (3.2), 𝑠>0, 𝑝 is a strictly increasing function and, hence, there exists an inverse function 𝑡=𝑝⁻¹(𝑠). Denote 𝑦(𝑠)∶=𝑥(𝑡)=𝑥(𝑝⁻¹(𝑠)),𝑙(𝑠)∶=𝑝(ℎ(𝑡))=𝑝(ℎ(𝑝⁻¹(𝑠))). (3.9) Since ℎ(𝑡)≤𝑡, we have 𝑙(𝑠)≤𝑠 (by (3.9) and (3.2)). From (3.9) we also have 𝑙(𝑠)=_𝑡^ℎ(𝑡)₀1𝜏(𝜉)𝑑𝜉. (3.10) Substituting 𝑥(𝑡)=𝑦(𝑠) in (1.6), we have 𝑥̇(𝑡)=𝑦(̇𝑠)/𝜏(𝑡) and (using (3.9)) 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)). (3.11) Hence, (1.6) takes the form 𝑦(̇𝑠)+𝜏(𝑡)𝑎(𝑡)𝑦(𝑙(𝑠))=0, (3.12) where 𝜏(𝑡)𝑎(𝑡)=𝜏(𝑝⁻¹(𝑠))𝑎(𝑝⁻¹(𝑠)). Equality 𝑦(𝑠)=𝑥(𝑡) (3.13) 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)) 𝑠−𝑙(𝑠)=^𝑡_𝑡01𝜏(𝑠)𝑑𝑠−_𝑡^ℎ(𝑡)₀1𝜏(𝑠)𝑑𝑠=^𝑡_ℎ(𝑡)1𝜏(𝑠)𝑑𝑠=^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝑠)𝑑𝑠≤𝛿. (3.14) Hence 𝑙(𝑠)≥𝑠−𝛿. (3.15) Consider an equation 𝑦(̇𝑠)+𝑎^∗_𝑘(𝑠)𝑦(𝑠−𝛿)=0, (3.16) where 𝑎^∗_𝑘(𝑠) is defined similar to 𝑎_𝑘(𝑡) by (1.4), where 𝜏 is replaced by 𝛿 and 𝑡 by 𝑠, that is, 𝑎^∗_𝑘1(𝑠)∶=+𝛿𝑒𝛿8𝑒𝑠²+𝛿8𝑒(𝑠ln𝑠)²𝛿+⋯+8𝑒(𝑠ln𝑠ln₂𝑠⋯ln_𝑘𝑠)². (3.17) By Theorem 1.1, (3.16) has a positive solution. Equation (3.12) is of type (2.1) with 𝑚=1,𝑏₁(𝑠)=𝜏(𝑝⁻¹(𝑠))𝑎(𝑝⁻¹(𝑠)),ℎ₁(𝑠)=𝑙(𝑠). (3.18) Now we use comparison of Lemma 2.4 where (2.4) is replaced by (3.16), that is, 𝑚=1,𝑐₁(𝑠)=𝑎^∗_𝑘(𝑠),𝑔₁(𝑠)=𝑠−𝛿. (3.19) Since, by (3.5) and (3.1), 𝑏₁(𝑠)=𝜏(𝑝⁻¹(𝑠))𝑎(𝑝⁻¹(𝑠))≤𝜏(𝑝⁻¹(𝑠))𝐴_𝑘(𝑝⁻¹(𝑠))=𝑎^∗_𝑘(𝑠)=𝑐₁(𝑠) (3.20) and, by (3.15), 𝑔₁(𝑠)=𝑠−𝛿≤𝑙(𝑠)=ℎ₁(𝑠), (3.21) 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 Theorem 3.2 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 ̇𝑡𝑥(𝑡)+𝑎(𝑡)𝑥(2)=0,𝑡≥𝑡₀=1, (3.22) where 𝑎∶[1,∞)→(0,∞). Here 𝑡𝜏(𝑡)=2,𝑠=𝑝(𝑡)=^𝑡_𝑡01𝜏(𝜉)𝑑𝜉=^𝑡₁2𝜉𝑑𝜉=2ln𝑡. (3.23) We set 𝛿∶=^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉=^𝑡_𝑡/22𝜉𝑑𝜉=2ln2. (3.24) In accordance with Theorem 3.2 (case (a) where 𝑘=0), (3.22) has a nonoscillatory solution if 𝑎(𝑡)≤𝐴₀1(𝑡)=+𝛿𝑒𝛿𝜏(𝑡)8𝑒𝜏(𝑡)𝑠²=1+(𝑒ln2)𝑡ln28𝑒𝑡(ln𝑡)². (3.25) Since (by Theorem 3.2, case (b) with 𝑘=2) all solutions of (3.22) oscillate if 𝑎(𝑡)>𝐴₀(𝑡)+𝜃𝛿8𝑒𝜏(𝑡)(𝑠ln𝑠)²=1+(𝑒ln2)𝑡ln28𝑒𝑡(ln𝑡)²+𝜃ln28𝑒𝑡(ln𝑡)²(ln(2ln𝑡))², (3.26) we conclude that the value 𝑎(𝑡)=𝑎^∗1(𝑡)∶=+(𝑒ln2)𝑡ln28𝑒𝑡(ln𝑡)² (3.27) is a critical value for the nonoscillation of (3.22).
The above statement is corroborated by Lemma 2.5 since, for 𝑚=1, ℎ₁(𝑡)=𝑡−𝜏(𝑡)=𝑡/2, 𝑏₁(𝑡)=𝑎^∗(𝑡), and 𝑡∈[1,2], ^𝑡_{max{𝑡0,ℎ1(𝑡)}}𝑏₁(𝑠)𝑑𝑠=^𝑡_𝑡0𝑎^∗(𝑠)𝑑𝑠=^𝑡₁𝑎^∗(𝑠)𝑑𝑠=ln𝑡≤1𝑒ln2𝑒 (3.28) and for 𝑡≥2^𝑡_{max{𝑡0,ℎ1(𝑡)}}𝑏₁(𝑠)𝑑𝑠=^𝑡_ℎ1(𝑡)𝑎^∗(𝑠)𝑑𝑠=^𝑡_𝑡/2𝑎^∗=(𝑠)𝑑𝑠_𝑡𝑡/211(𝑒ln2)𝑠𝑑𝑠=𝑒, (3.29) and (2.7) turns into an equality for 𝑡≥2.

To apply Theorem 1.2, we verify condition (1.8). But, unfortunately, for (3.22) we have ^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉=^𝑡_𝑡/22𝜉𝑑𝜉=2ln2≐1.386>1. (3.30) Thus, this theorem is not applicable to (3.22).

To compare Theorem 1.3 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)) 1𝑎(𝑡)≤𝜏(𝑡)exp[−^𝑡_{𝑡−𝜏(𝑡)}12𝜏(𝜉)𝑑𝜉]=𝑡exp[−^𝑡_𝑡/22𝜉1𝑑𝜉]=.2𝑡 (3.31) But in our case 𝑎(𝑡)=𝑎^∗1(𝑡)=+(𝑒ln2)𝑡ln28𝑒𝑡(ln𝑡)²≥1>11.885𝑡,2𝑡 (3.32) and Theorem 1.3 fails for this equation.

4. Differential Equation with Several Delays

We start with the following question: for what functions 𝑏(𝑡)≥0 and delay 𝜎>0 the equatioṅ1𝑥(𝑡)+𝑏(𝑡)𝑥(𝑡−𝜎)+𝑒𝜏𝑥(𝑡−𝜏)=0 (4.1) can have a nonoscillatory solution? It is easy to see that 𝑏 should be vanishing.

Theorem 4.1. Let liminf_𝑡→∞𝑏(𝑡)=𝑏>0,𝜏>0, and 𝜎≥0. Then all solutions of (4.1) are oscillatory.

Proof. Consider first the equation ̇1𝑥(𝑡)+𝑏𝑥(𝑡−𝜎)+𝑒𝜏𝑥(𝑡−𝜏)=0. (4.2) Suppose that (4.2) has a nonoscillatory solution. We set 𝑏𝑚=2,₁(𝑡)=𝑐₁(𝑡)∶=𝑏,𝑏₂(𝑡)=𝑐₂1(𝑡)∶=,𝑔𝑒𝜏₁(𝑡)∶=𝑡−𝜎,ℎ₂(𝑡)=𝑔₂ℎ(𝑡)∶=𝑡−𝜏,₁(𝑡)∶=𝑡. (4.3) Since 𝑔₁(𝑡)≤ℎ₁(𝑡), by Lemma 2.4, the equation ̇1𝑥(𝑡)+𝑏𝑥(𝑡)+𝑒𝜏𝑥(𝑡−𝜏)=0 (4.4) has a nonoscillatory solution. After the substitution 𝑥(𝑡)=𝑒^−𝑏𝑡𝑦(𝑡), (4.4) takes a form ̇𝑒𝑦(𝑡)+^𝑏𝜏𝑒𝜏𝑦(𝑡−𝜏)=0. (4.5) Since 𝑒^𝑏𝜏>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 large 𝑡₀, we have 𝑏(𝑡)≥𝑏, 𝑡≥𝑡₀. We set 𝑏𝑚=2,₁(𝑡)∶=𝑏(𝑡)≥𝑐₁𝑏(𝑡)∶=𝑏,₂(𝑡)=𝑐₂1(𝑡)∶=,ℎ𝑒𝜏₁(𝑡)=𝑔₁(𝑡)∶=𝑡−𝜎,ℎ₂(𝑡)=𝑔₂(𝑡)∶=𝑡−𝜏. (4.6) 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 integer 𝑘≥0 and 𝜏>0 exist such that, for all sufficiently large 𝑡, inequalities 𝑡−ℎ_𝑖(𝑡)≤𝜏,𝑖=1,2,…,𝑚,_𝑚𝑖=1𝑏_𝑖(𝑡)≤𝑎_𝑘(𝑡), (4.7)(4.8) where 𝑎_𝑘 is defined by (1.4), are valid. Then there exists an eventually positive solution 𝑥 of (2.1).
(b) Let an integer 𝑘≥2, 𝜏>0, and 𝜃>1 exist such that, for all sufficiently large 𝑡, inequalities 𝑡−ℎ_𝑖(𝑡)≥𝜏,𝑖=1,2,…,𝑚,_𝑚𝑖=1𝑏_𝑖(𝑡)≥𝑎_𝑘−2(𝑡)+𝜃𝜏8𝑒(𝑡ln𝑡ln₂𝑡⋯ln_𝑘−1𝑡)², (4.9)(4.10) where 𝑎_𝑘−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 Theorem 1.1, the equation 𝑥̇(𝑡)+𝑎_𝑘(𝑡)𝑥(𝑡−𝜏)=0 (4.11) has a nonoscillatory solution. By Lemma 2.6 (with 𝑚=1, 𝑏₁(𝑡)=𝑎_𝑘(𝑡), and ℎ₁(𝑡)=𝑡−𝜏), there exist a 𝑡₀ and a locally integrable function 𝑢∶ℝ→[0,∞) such that 𝑢(𝑡)≥𝑎_𝑘(𝑡)𝑒^{∫𝑡𝑡−𝜏𝑢(𝑠)𝑑𝑠},𝑡≥𝑡₀,𝑢(𝑡)=0,𝑡<𝑡₀. (4.12) We have ^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑒^{∫𝑡ℎ𝑖(𝑡)𝑢(𝑠)𝑑𝑠}≤(^𝑚_𝑖=1𝑏_𝑖(𝑡))𝑒^{∫𝑡𝑡−𝜏𝑢(𝑠)𝑑𝑠}≤𝑎_𝑘(𝑡)𝑒^{∫𝑡𝑡−𝜏𝑢(𝑠)𝑑𝑠}. (4.13) Hence 𝑢(𝑡)≥^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑒^{∫𝑡ℎ𝑖(𝑡)𝑢(𝑠)𝑑𝑠},𝑡≥𝑡₀,𝑢(𝑡)=0,𝑡<𝑡₀. (4.14) 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 𝑢(𝑡)≥^𝑚_𝑖=1𝑏_𝑖(𝑡)𝑒^{∫𝑡ℎ𝑖(𝑡)𝑢(𝑠)𝑑𝑠}≥(𝑎_𝑘−2(𝑡)+𝜃𝜏8𝑒(𝑡ln𝑡ln₂𝑡⋯ln_𝑘−1𝑡)²)𝑒^{∫𝑡𝑡−𝜏𝑢(𝑠)𝑑𝑠},𝑡≥𝑡₀,𝑢(𝑡)=0,𝑡<𝑡₀. (4.15) Hence (using Lemma 2.6 again), an equation 𝑥̇(𝑡)+(𝑎_𝑘−2(𝑡)+𝜃𝜏8𝑒(𝑡ln𝑡ln₂𝑡⋯ln_𝑘−1𝑡)²)𝑥(𝑡−𝜏)=0 (4.16) 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) ̇1𝑥(𝑡)+8𝑒𝑡²1𝑥(𝑡−𝜎)+𝑒𝑥(𝑡−1)=0 (4.17) 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. Let 𝑡₀ be sufficiently large, for 𝑡≥𝑡₀, 𝜏(𝑡)>0 a.e., 1/𝜏(𝑡) let locally integrable function, lim_𝑡→∞(𝑡−𝜏(𝑡))=∞,^∞_𝑡01𝜏(𝜉)𝑑𝜉=∞,lim_𝑡→∞ℎ_𝑖(𝑡)=+∞,𝑡−ℎ_𝑖(𝑡)≤𝜏(𝑡),𝑖=1,2,…,𝑚, (4.18)(4.19) and let there exists 𝑡₁>𝑡₀ such that 𝑡−𝜏(𝑡)≥𝑡₀ if 𝑡≥𝑡₁.
(a) If there exists a 𝜎∈(0,∞) such that ^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉≤𝜎,𝑡≥𝑡₁, (4.20) and, for a fixed integer 𝑘≥0, ^𝑚_𝑖=1𝑏_𝑖(𝑡)≤𝐴_𝑘(𝑡),𝑡≥𝑡₁, (4.21) where 𝐴_𝑘(𝑡) is defined by (3.1), (3.2), then there exists an eventually positive solution of (2.1).
(b) If there exists a 𝜎∈(0,∞) such that ^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉≥𝜎,𝑡≥𝑡₁, (4.22) and, for a fixed integer 𝑘≥2 and 𝜃>1, 𝜃∈ℝ, ^𝑚_𝑖=1𝑏_𝑖(𝑡)≥𝐴_𝑘−2(𝑡)+𝜃𝜎8𝑒𝜏(𝑡)(𝑠ln𝑠ln₂𝑠⋯ln_𝑘−1𝑠)²,𝑡≥𝑡₁, (4.23) where 𝐴_𝑘−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 Theorem 3.2, the equation 𝑥̇(𝑡)+(^𝑚_𝑖=1𝑏_𝑖(𝑡))𝑥(𝑡−𝜏(𝑡))=0 (4.24) has a nonoscillatory solution. Equation (4.24) is of a form of (2.4) with 𝑐_𝑖(𝑡)=𝑏_𝑖(𝑡), 𝑔_𝑖(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 𝑏_𝑖(𝑡)≤𝑐_𝑖(𝑡) and, due to (4.19), 𝑔_𝑖(𝑡)≤ℎ_𝑖(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): ̇𝛼𝑥(𝑡)+𝑡𝑡𝑒ln3𝑥(3)+ln38𝑒𝑡(ln𝑡)²𝑡𝑥(2)=0,𝑡≥𝑡₀=1. (4.25) First let 0<𝛼≤1. We set 𝑚=2, 𝑡₁∶=3, 𝜏(𝑡)=∶2𝑡/3, and 𝜎∶=^𝑡_{𝑡−𝜏(𝑡)}1𝜏(𝜉)𝑑𝜉=^𝑡_{𝑡−(2𝑡)/3}32𝜉𝑑𝜉=^𝑡_𝑡/3332𝜉𝑑𝜉=2ln3. (4.26) Moreover, we put ℎ₁𝑡(𝑡)=3,ℎ₂𝑡(𝑡)=2,𝑏₁𝛼(𝑡)=𝑡𝑒ln3,𝑏₂(𝑡)=ln38𝑒𝑡(ln𝑡)²,𝑡−ℎ_𝑘2(𝑡)≤𝜏(𝑡)=3𝑡,𝑘=1,2,𝑠=^𝑡_𝑡01𝜏(𝜉)𝑑𝜉=^𝑡₁332𝜉𝑑𝜉=2ln𝑡. (4.27) By (3.1), 𝐴₀1(𝑡)=+𝜎𝑒𝜎𝜏(𝑡)8𝑒𝜏(𝑡)𝑠²=1+𝑡𝑒ln3ln38𝑒𝑡(ln𝑡)²≥𝑏₁(𝑡)+𝑏₂(𝑡). (4.28) All conditions of Theorem 4.4 part (a) hold, hence (4.25) has a nonoscillatory solution.

Similarly, one can show (by Theorem 4.4 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 delays𝑥̇(𝑡)+𝑏₁(𝑡)𝑥(ℎ₁(𝑡))+𝑏₂(𝑡)𝑥(ℎ₂(𝑡))=0,𝑡≥𝑡₀, (5.1) where 𝑏_𝑖∶[𝑡₀,∞)→[0,∞), 𝑖=1,2,ℎ₁(𝑡)=𝑡−𝜏,ℎ₂(𝑡)=𝑡−𝜎, (5.2) and 𝜏, 𝜎 are positive constants. Let liminf_𝑡→∞𝑏₁(𝑡)=𝑝,liminf_𝑡→∞𝑏₂(𝑡)=𝑞. (5.3) In accordance with [12] we say that (5.1) is in a critical state if there exists 𝜆₀≥0 such that 𝜆₀=𝑝𝑒^𝜆0𝜏+𝑞𝑒^𝜆0𝜎 (5.4) and, for any 𝜆>0, 𝜆≠𝜆₀, we have 𝜆<𝑝𝑒^𝜆𝜏+𝑞𝑒^𝜆𝜎. (5.5)

Theorem 5.1 (see [12]). Let (5.1) be in a critical case, 𝑝>0, 𝑞>0, and liminf_𝑡→∞[𝑏₁(𝑡+𝜏)−𝑝]𝑡=𝛼,liminf_𝑡→∞[𝑏₂(𝑡+𝜎)−𝑞]𝑡=𝛽, (5.6) where −∞<𝛼,𝛽≤+∞. If 𝛼𝑒^𝜆0𝜏+𝛽𝑒^𝜆0𝜎>0, then all solutions of (5.1) oscillate.

Theorem 5.2 (see [12]). Let liminf_𝑡→∞𝑏₁1(𝑡)=𝑒𝜏,liminf_𝑡→∞𝑏₂(𝑡)=0,liminf_𝑡→∞[𝑏₁1(𝑡+𝜏)−𝑒𝜏]𝑡=𝛼,liminf_𝑡→∞𝑡𝑏₂(𝑡+𝜎)=𝛽, (5.7) where −∞<𝛼≤+∞, 𝛽>0. If 𝑒𝛼+𝑒^𝜎/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. Let 𝑡−ℎ₁(𝑡)≤𝜏 and 𝑡−ℎ₂(𝑡)≤𝜎 for 𝑡≥𝑡₀, limsup_𝑡→∞[𝑏₁(𝑡)−𝑝]𝑡=𝛼,limsup_𝑡→∞[𝑏₂(𝑡)−𝑞]𝑡=𝛽, (5.8) where −∞<𝛼, 𝛽<+∞, and let there exists a 𝜆₀>0 such that 𝜆₀≥𝑝𝑒^𝜆0𝜏+𝑞𝑒^𝜆0𝜎,𝛼𝑒^𝜆0𝜏+𝛽𝑒^𝜆0𝜎<0. (5.9)(5.10) Then (5.1) has a nonoscillatory solution.

Proof. There exist 𝑡₁>𝑡₀+max(𝜏,𝜎) and 𝜖>0 such that, owing to (5.8) and (5.10), 𝑏₁(𝑡)≤𝑝+𝛼+𝜖𝑡,𝑏₂(𝑡)≤𝑞+𝛽+𝜖𝑡,(𝛼+𝜖)𝑒^𝜆0𝜏+(𝛽+𝜖)𝑒^𝜆0𝜎≤0 (5.11) if 𝑡≥𝑡₁. By Lemma 2.4, with 𝑚=2, 𝑐₁(𝑡)∶=𝑝+𝛼+𝜖𝑡,𝑐₂(𝑡)∶=𝑞+𝛽+𝜖𝑡,𝑔₁𝑔(𝑡)∶=𝑡−𝜏,₂(𝑡)∶=𝑡−𝜎 (5.12) in (2.4), the existence of a nonoscillatory solution of the equation 𝑥̇(𝑡)+(𝑝+𝛼+𝜖𝑡)𝑥(𝑡−𝜏)+(𝑞+𝛽+𝜖𝑡)𝑥(𝑡−𝜎)=0 (5.13) implies the existence of a nonoscillatory solution of (5.1).
By Lemma 2.6, for the existence of a positive solution of (5.13), it is sufficient to find a nonnegative solution of the inequality 𝑢(𝑡)≥(𝑝+𝛼+𝜖𝑡)𝑒^{∫𝑡𝑡−𝜏𝑢(𝑠)𝑑𝑠}+(𝑞+𝛽+𝜖𝑡)𝑒^{∫𝑡𝑡−𝜎𝑢(𝑠)𝑑𝑠}, (5.14) where 𝑡≥𝑡₁. Substituting 𝑢(𝑡) for 𝜆₀ in inequality (5.14), we have 𝜆₀≥𝑝𝑒^𝜆0𝜏+𝑞𝑒^𝜆0𝜎+(𝛼+𝜖)𝑒^𝜆0𝜏+(𝛽+𝜖)𝑒^𝜆0𝜎𝑡. (5.15) 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. Let 𝑡−ℎ₁(𝑡)≤𝜏, 𝑡−ℎ₂(𝑡)≤𝜎 for 𝑡≥𝑡₀, limsup_𝑡→∞[𝑏₁1(𝑡)−𝑒𝜏]𝑡=𝛼,limsup_𝑡→∞𝑡𝑏₂(𝑡)=𝛽, (5.16) where 𝛼,𝛽∈ℝ. If 𝑒𝛼+𝑒^𝜎/𝜏𝛽<0, (5.17) then (5.1) has a nonoscillatory solution.

Proof. There exist 𝑡₁>𝑡₀+max(𝜏,𝜎) and 𝜖>0 such that, owing to (5.16) and (5.17), 𝑏₁1(𝑡)≤+𝑒𝜏𝛼+𝜖𝑡,𝑏₂(𝑡)≤𝛽+𝜖𝑡,𝑒(𝛼+𝜖)+𝑒^𝜎/𝜏(𝛽+𝜖)≤0 (5.18) if 𝑡≥𝑡₁. By Lemma 2.4, with 𝑚=2, 𝑐₁1(𝑡)=+𝑒𝜏𝛼+𝜖𝑡,𝑐₁(𝑡)=𝛽+𝜖𝑡,𝑔₁𝑔(𝑡)=𝑡−𝜏,₂(𝑡)=𝑡−𝜎 (5.19) in (2.4), the existence of a nonoscillatory solution of the equation ̇1𝑥(𝑡)+(+𝑒𝜏𝛼+𝜖𝑡)𝑥(𝑡−𝜏)+(𝛽+𝜖𝑡)𝑥(𝑡−𝜎)=0 (5.20) 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 1𝑢(𝑡)≥(+𝑒𝜏𝛼+𝜖𝑡)𝑒^{∫𝑡𝑡−𝜏𝑢(𝑠)𝑑𝑠}+(𝛽+𝜖𝑡)𝑒^{∫𝑡𝑡−𝜎𝑢(𝑠)𝑑𝑠}, (5.21) where 𝑡≥𝑡₁. Put 𝑢(𝑡)=1/𝜏, 𝑡≥𝑡₁ in inequality (5.21). We have 1𝜏≥1𝑒𝜏𝑒+𝑒(𝛼+𝜖)+𝑒^𝜎/𝜏(𝛽+𝜖)𝑡. (5.22) 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 𝑏₁1(𝑡)=+𝑒𝜏|sin𝑡|−2𝑡,𝑏₂(𝑡)=|cos𝑡|+1𝑡 (5.23) and with ℎ₁(𝑡), ℎ₂(𝑡) defined by (5.2), that is, ̇1𝑥(𝑡)+(+𝑒𝜏|sin𝑡|−2𝑡)𝑥(𝑡−𝜏)+|cos𝑡|+1𝑡𝑥(𝑡−𝜎)=0, (5.24) where 𝑡≥𝑡₀. Since liminf_𝑡→∞𝑏₁1(𝑡)=𝑒𝜏,liminf_𝑡→∞𝑏₂(𝑡)=0,liminf_𝑡→∞[𝑏₁1(𝑡+𝜏)−𝑒𝜏]𝑡=−2,liminf_𝑡→∞𝑡𝑏₂(𝑡+𝜎)=1, (5.25) then, by Theorem 5.2 (with 𝛼=−2 and 𝛽=1), all solutions of (5.24) oscillate if 𝜎𝜏>1+ln2. (5.26) Since limsup_𝑡→∞[𝑏₁1(𝑡)−𝑒𝜏]𝑡=−1,limsup_𝑡→∞𝑡𝑏₂(𝑡)=2, (5.27) by Theorem 5.4 (with 𝛼=−1 and 𝛽=2), (5.24) has a nonoscillatory solution if 𝜎𝜏<1−ln2. (5.28)

5.1. Generalization for Equations with Several Delays

It is easy to generalize Theorem 5.3 for a general equation (2.1) with several delays. Denote 𝑝_𝑖∶=liminf_𝑡→∞𝑏_𝑖(𝑡), (5.29) 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. Let 𝜏_𝑖, 𝑖=1,2,…,𝑚, be positive constants such that 𝑡−ℎ_𝑖(𝑡)≤𝜏_𝑖 (5.30) for 𝑡≥𝑡₀, limsup_𝑡→∞[𝑏_𝑖(𝑡)−𝑝_𝑖]𝑡=𝛼_𝑖, (5.31) where 𝛼_𝑖∈ℝ, and let there exists a 𝜆₀>0 such that 𝜆₀≥^𝑚_𝑖=1𝑝_𝑖𝑒^{𝜆0𝜏𝑖},_𝑚𝑖=1𝛼_𝑖𝑒^{𝜆0𝜏𝑖}<0. (5.32) Then (2.1) has a nonoscillatory solution.

The following statement generalizes Theorem 5.4. We will formulate this result for (2.4).

Theorem 5.7. Let 𝐼⊂{1,…,𝑚} be a set of indices such that 𝑔_𝑘ℎ(𝑡)≤¹ℎ(𝑡),if𝑘∈𝐼,₂(𝑡),if𝑘∉𝐼, (5.33) where ℎ_𝑖∶[𝑡₀,∞)→ℝ and ℎ_𝑖(𝑡)≤𝑡. Let 𝑏₁(𝑡)∶=_𝑘∈𝐼𝑐_𝑘(𝑡),𝑏₂(𝑡)∶=_𝑘∉𝐼𝑐_𝑘(𝑡). (5.34) If, for functions 𝑏_𝑖, ℎ_𝑖, 𝑖=1,2, all assumptions of Theorem 5.4 are true, then (2.4) has a nonoscillatory solution.

The proof of Theorem 5.7 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].

Acknowledgments

J. Baštinec was supported by the Grant 201/10/1032 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30529 and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. L. Berezansky was partially supported by grant 25/5 “Systematic support of international academic staff at Faculty of Electrical Engineering and Communication, Brno University of Technology” (Ministry of Education, Youth and Sports of the Czech Republic) and by the Grant 201/10/1032 of the Czech Grant Agency (Prague). J. Diblík was supported by the Grant 201/08/0469 of the Czech Grant Agency (Prague), by the Council of Czech Government Grant MSM 00216 30519 and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology. Z. Šmarda was supported by the Council of Czech Government Grant MSM 00216 30503 and MSM 00216 30529, and by the Grant FEKT-S-10-3 of Faculty of Electrical Engineering and Communication, Brno University of Technology.