Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order

Abstract

Some new sufficient conditions for oscillation of all solutions of the first-order linear neutral delay differential equations are obtained. Our new results improve many well-known results in the literature. Some examples are inserted to illustrate our results.

1. Introduction

A neutral delay differential equation (NDDE) is a differential equation in which the highest-order derivative of the unknown function is evaluated both at the present state at time 𝑡 and at the past state at time 𝑡−𝑘 for some positive constant 𝑘.

In the last two decades, there has been an increasing interest in obtaining sufficient conditions for the oscillation and/or nonoscillation of solutions of neutral delay differential equations. Particularly, we mention the papers by Ladas and Sficas [1], Chuanxi and Ladas [2], Ruan [3], Elabbasy and Saker [4], Kulenović et al. [5], and Karpuz and Öcalan [6] who investigated NDDEs with variable coefficients. To a large extent, this is due to its theoretical interest as well as to its importance in applications. It suffices to note that NDDEs appear in the study of networks containing lossless transmission lines (as in high-speed computers where the lossless transmission lines are used to interconnect switching circuits) in population dynamics and also in many applications in epidemics and infection diseases. We refer reader to [1–18] for relevant studies on this subject.

In this paper, we consider the linear first-order NDDE of the type 𝑥(𝑡)−𝑝𝑥(𝑡−𝜏)^′+𝑞₁𝑥(𝑡)+𝑞₂(𝑡)𝑥(𝑡−𝜎)=0;𝑡≥𝑡₀, (1) where  𝑝,𝑞₁,𝜏,𝜎∈(0,∞) and 𝑞₂(𝑡)∈𝐶[[𝑡₀,∞),ℝ]. When  𝑞₁≡0 and 𝑞₂(𝑡)=𝑞,𝑞  is a constant, Jaroš [9] established some new oscillation conditions for all solutions of (1), and his technique was based on the study of the characteristic equation 𝜆−𝜆𝑝𝑒^−𝜆𝜏+𝑞𝑒^−𝜆𝜎=0. (2)

Zhang [19], Ladas and Sficas [1], Grammatikopoulos et al. [10], and Yu et al. [8] considered (1) when 𝑞₁≡0, and they obtained some sufficient conditions for oscillation of (1). The purpose of this work is to present some new sufficient conditions under which all solutions of (1) are oscillatory. In order to achieve this object, we are first concerned with NDDE (1) with constant coefficients (when 𝑞₂(𝑡)≡𝑞₂,𝑞₂  is a constant). That is, 𝑥(𝑡)−𝑝𝑥(𝑡−𝜏)^′+𝑞₁𝑥(𝑡)+𝑞₂𝑥(𝑡−𝜎)=0,𝑡≥0. (3)

Some illustrating examples are given. In some sense, the established results extend and improve some previous investigations such as [1, 8–10, 19].

As usual, a solution of (1) is said to be oscillatory if it has arbitrarily large zeros and nonoscillatory if it is eventually positive or eventually negative. A function  𝑥(𝑡)   is called eventually positive (or negative) if there exists  𝑡₀  such that  𝑥(𝑡)>0   (or  𝑥(𝑡)<0) for all𝑡≥𝑡₀. Equation (1) is called oscillatory if all its solutions are oscillatory; otherwise, it is called nonoscillatory.

2. Main Results

In this section, we give some new sufficient conditions for the oscillation of all solutions of (1) and (3). This is done by using the following well-known lemmas which are from [11, 12].

Lemma 1. Consider the NDDE 𝑥(𝑡)+𝑝𝑥(𝑡−𝜏)^′+^𝑛_𝑖=1𝑞_𝑖𝑥𝑡−𝜎_𝑖=0,𝑡≥𝑡₀, (4) where  𝜏≥0,𝑞_𝑖>0,   and 𝜎_𝑖≥0𝑓𝑜𝑟𝑎𝑙𝑙𝑖=1,2,…,𝑛.
Let 𝑥(t)   be a positive solution of (4). Set 𝑧(𝑡)=𝑥(𝑡)−𝑝𝑥(𝑡−𝜏). (5)
If 𝑝≥−1, then 𝑧(𝑡) is a positive and decreasing solution of (4); that is, 𝑧^′(𝑡)+𝑝𝑧^′(𝑡−𝜏)+^𝑛_𝑖=1𝑞_𝑖𝑧𝑡−𝜎_𝑖=0,𝑡≥𝑡₀. (6)

Lemma 2. Let 𝑝 and 𝜏  be positive constants. Let 𝑥(𝑡)   be an eventually positive solution of the delay differential inequality 𝑥^′(𝑡)+𝑝𝑥(𝑡−𝜏)≤0. (7)
Then for 𝑡 sufficiently large, 𝑥(𝑡−𝜏)≤𝐵𝑥(𝑡), (8) where 4𝐵=𝑝𝜏². (9)

Our main results can now be given as follows.

Theorem 3. Consider NDDE (3). Assume that (i)𝑞₂∈[0,∞),𝜎≥𝜏, 0<𝑝𝑒^𝑞1𝜏<1 and(ii)𝜏(𝑞₁𝑝𝑒^𝑞1𝜏+𝑞₂𝑒^𝑞1𝜎)>(1−𝑝𝑒^𝑞1𝜏𝑚)²/𝑚,
where 𝑚 is the unique real root of the equation 1−𝑝𝑒^𝑞1𝜏1𝑚=ln𝑚,1≤𝑚≤𝑝^{𝑒1𝜏−𝑞}. (10)
Then all solutions of (3) are oscillatory.

Proof. Assume, for the sake of a contradiction, that (3) has a nonoscillatory solution  𝑥(𝑡). Without loss of generality, assume that  𝑥(𝑡)>0∀𝑡≥𝑡₀>0. Let 𝑥(𝑡)=𝑒^−𝑞1𝑡𝑦(𝑡). (11) So that 𝑦(𝑡) is also a positive solution of (3).
That is, 𝑦(𝑡)−𝑝₁𝑦(𝑡−𝜏)^′+²_𝑖=1𝑎_𝑖𝑦𝑡−𝜏_𝑖=0, (12) where 𝑝₁=𝑝𝑒^𝑞1𝜏,𝑎₁=𝑞₁𝑝𝑒^𝑞1𝜏,𝑎₂=𝑞₂𝑒^𝑞1𝜎,𝜏₁=𝜏,𝜏₂=𝜎. (13)
Set for 𝑡≥𝑡₀+2𝜏𝑧(𝑡)=𝑦(𝑡)−𝑝₁𝑦(),𝑤(𝑡)=𝑧()𝑡−𝜏𝑡−𝜏𝑧(𝑡). (14)
Thus it follows from Lemma 1 that 𝑧(𝑡)   is a positive and decreasing solution of 𝑧^′(𝑡)−𝑝₁𝑧^′(𝑡−𝜏)+²_𝑖=1𝑎_𝑖𝑧𝑡−𝜏_𝑖=0, (15) and in particular (as 𝜎>𝜏 implies that 𝑡−𝜏_𝑖≤𝑡−𝜏,𝑖=1,2.), it follows that 𝑧^′(𝑡)−𝑝₁𝑧^′(𝑡−𝜏)+𝑎₁+𝑎₂𝑧(𝑡−𝜏)≤0. (16)
But we have 𝑧^′(𝑡−𝜏)<0. (17)
This implies that 𝑧^′(𝑡)+𝑎₁+𝑎₂𝑧(𝑡−𝜏)≤0. (18)
Applying Lemma 2 with (18) we get 𝑧()(𝑡)4𝑡−𝜏≤𝐵𝑧;𝐵=𝜏²𝑎₁+𝑎₂². (19)
Then  𝑤(𝑡)  is bounded.
Dividing (16) by  𝑧(𝑡)>0 and integrating from  𝑡−𝜏  to  𝑡, we get ln𝑤(𝑡)≥𝑎₁+𝑎₂^𝑡_𝑡−𝜏𝑤(𝑠)𝑑𝑠−𝑝₁^𝑡_𝑡−𝜏𝑤(𝑠)𝑑((𝑑𝑠ln𝑧𝑠−𝜏))𝑑𝑠. (20)
Let 𝑚=lim_𝑡→∞inf𝑤(𝑡).
Then, it follows from (20) that for 𝜀>0 and sufficiently small, ln(𝑚+𝜀)≥𝑎₁+𝑎₂(𝑚−𝜀)𝜏+𝑝₁(𝑚−𝜀)ln(𝑚−𝜀). (21)
As 𝜀 is arbitrary, so we have 𝑎₁+𝑎₂𝜏≤1−𝑝₁𝑚ln𝑚𝑚. (22)
Let 𝐹(𝑚)=1−𝑝₁𝑚ln𝑚𝑚,1≤𝑚≤𝐵. (23)
Then 𝐹^′(𝑚)=1−𝑝₁𝑚−ln𝑚𝑚²,1≤𝑚≤𝐵. (24)
Let 𝑚 be the unique real root of the equation 1−𝑝₁1𝑚=ln𝑚,𝑚∈1,𝑝₁. (25)
Then max_𝑚≥1𝐹(𝑚)=𝐹(𝑚)=1−𝑝₁𝑚²𝑚. (26)
Hence 𝑞₁𝑝𝑒^𝑞1𝜏+𝑞₂𝑒^𝑞1𝜎𝜏≤1−𝑝𝑒^𝑞1𝜏𝑚²𝑚. (27)
This contradicts condition (ii) and then completes the proof.

Example 4. Consider the NDDE 𝑥(𝑡)−19𝑥𝜋𝑡−2^′+19𝑥(𝑡)+𝑥𝑡−5𝜋2=0,𝑡≥0. (28)
We note that 1𝑝=9,𝑞₁=19,𝑞₂𝜋=1,𝜏=2,𝜎=5𝜋2. (29)
Then we have (i)  0<𝑝𝑒^𝑞1𝜏=(1/9)𝑒^𝜋/18<1and𝜎≥𝜏,  (ii)  𝜏𝑞₁𝑝𝑒^𝑞1𝜏+𝑞₂𝑒^𝑞1𝜎=𝜋21𝑒81^𝜋/18+19𝑒^5𝜋/18>1−𝑝𝑒^𝑞1𝜏𝑚²𝑚=1−(1/9)𝑒^𝜋/1822, (30)where  𝑚=2  is the unique real root of the equation 11−9𝑒^𝜋/18𝑚=ln𝑚,𝑚∈1,9𝑒^−𝜋/18. (31) Then all the hypotheses of Theorem 3 are satisfied, and therefore every solution of (28) oscillates. (Indeed  𝑥(𝑡)=sin𝑡  is such a solution.)

Theorem 5. Consider the NDDE (1). Assume that (iii)  0<𝑝𝑒^𝑞1𝜏<1,𝜎=𝜏,  and 𝑞₂(𝑡)∈𝐶[[𝑡₀,∞),(0,∞)]  is periodic with period  𝜏, (iv)  lim_𝑡→∞inf∫^𝑡_𝑡−𝜏𝑒^𝑞1𝜏(𝑞₂(𝑠)+𝑞₁𝑝)𝑑𝑠>(1−𝑝𝑒^𝑞1𝜏𝑚)²/𝑚,where  𝑚  is defined as in Theorem 3. Then all solutions of (1) are oscillatory.

Proof. Assume, for the sake of contradiction, that (1) has a nonoscillatory solution 𝑥(𝑡). Without loss of generality, assume that (𝑡)>0∀𝑡≥𝑡₀>0. Let 𝑥(𝑡)=𝑒^−𝑞1𝑡𝑦(𝑡), (32) which is oscillation invariant transformation. Then 𝑦(𝑡) is a positive solution of the equation 𝑦(𝑡)−𝑝₁𝑦(𝑡−𝜏)^′+𝑞(𝑡)𝑦(𝑡−𝜏)=0, (33) where  𝑝₁=𝑝𝑒^𝑞1𝜏and𝑞(𝑡)=𝑒^𝑞1𝜏(𝑞₂(𝑡)+𝑞₁𝑝)   is periodic with period 𝜏.
Let 𝑧(𝑡)=𝑦(𝑡)−𝑝₁𝑦(𝑡−𝜏). (34)
Then  𝑧(𝑡) is decreasing positive solution of the equation 𝑧(𝑡)−𝑝₁𝑧(𝑡−𝜏)^′+𝑞(𝑡)𝑧(𝑡−𝜏)=0. (35)
Set 𝑤(𝑡)=𝑧(𝑡−𝜏)𝑧(𝑡). (36)
This implies that  𝑤(𝑡)≥1, since  𝑧(𝑡−𝜏)≥𝑧(𝑡).
Dividing both sides of (33) by 𝑧(𝑡) and then integrating from 𝑡−𝜏 to  𝑡, we obtain that (𝑡)=ln𝑤_{𝑡𝑡−𝜏}𝑞(𝑠)𝑦(𝑠−𝜏)−𝑝₁𝑦(𝑠−2𝜏)+𝑝₁𝑦(𝑠−2𝜏)𝑦(𝑠)−𝑝₁𝑦(𝑠−𝜏)𝑑𝑠. (37)
Hence (𝑡)=ln𝑤_{𝑡𝑡−𝜏}𝑞(𝑠)𝑤(𝑠)𝑑𝑠+𝑝₁^𝑡_𝑡−𝜏𝑞(𝑠)𝑦(𝑠−2𝜏)𝑦(𝑠)−𝑝₁𝑦(𝑠−𝜏)𝑑𝑠. (38)
Since 𝑞(𝑡) is periodic with period  𝜏, then we obtain 𝑞(𝑡)()𝑦()=𝑞𝑡−𝜏=−𝑡−𝜏−𝑝₁𝑦(𝑡−2𝜏)^′𝑦().𝑡−2𝜏 (39)
Substituting in (38) we find, for all 𝑡≥𝑡₀, ln𝑤(𝑡)=^𝑡_𝑡−𝜏𝑞(𝑠)𝑤(𝑠)𝑑𝑠−𝑝₁^𝑡_𝑡−𝜏𝑤(𝑠)𝑑ln𝑦(𝑠−𝜏)−𝑝₁𝑦(𝑠−2𝜏)𝑑𝑠. (40)
Now, we want to prove that 𝑤(𝑡)  is bounded.
Applying the assumption (iv), we can find 𝑡^∗∈(𝑡−𝜏,𝑡) such that ^𝑡∗_𝑡−𝜏𝑞(𝑠)𝐹(𝑑𝑠>𝑚)2,^𝑡_𝑡∗𝑞(𝑠)𝐹(𝑑𝑠>𝑚)2, (41) where 𝐹(𝑚) is similar as in the proof of Theorem 3.
Integrating (33) from 𝑡^∗ to 𝑡 we obtain 𝑦𝑡^∗−𝑝₁𝑦𝑡^∗−𝜏≥^𝑡_𝑡∗𝑞(𝑠)𝑦()≥𝑠−𝜏𝑑𝑠_𝑡𝑡∗𝑞(𝑠)𝑦(𝑠−𝜏)−𝑝₁𝑦(𝑠−2𝜏)𝑑𝑠, (42)
Using Bonnet’s Theorem and in particular (as 𝑧^′(𝑡−𝜏)<𝑜), we get 𝑦𝑡^∗−𝑝₁𝑦𝑡^∗≥𝑦()−𝜏𝑡−𝜏−𝑝₁𝑦(𝑡−2𝜏)⋅^𝑡_𝑡∗𝑞(𝑠)𝑑𝑠. (43)
Integrating (33) from 𝑡−𝜏 to 𝑡^∗, we get 𝑦(𝑡−𝜏)−𝑝₁𝑦()≥𝑡−2𝜏_{𝑡∗𝑡−𝜏}𝑞(𝑠)𝑦()≥𝑠−𝜏𝑑𝑠_{𝑡∗𝑡−𝜏}𝑞(𝑠)𝑦(𝑠−𝜏)−𝑝₁𝑦(𝑠−2𝜏)𝑑𝑠. (44)
Using Bonnet’s Theorem and in particular (as 𝑧^′(𝑡−𝜏)<𝑜), we get 𝑦(𝑡−𝜏)−𝑝₁𝑦()≥𝑦𝑡𝑡−2𝜏_∗−𝜏−𝑝₁𝑦𝑡^∗−2𝜏⋅^𝑡∗_𝑡−𝜏𝑞(𝑠)𝑑𝑠. (45)
Combining (43) and (45), we conclude 𝑦𝑡^∗−𝑝₁𝑦𝑡^∗≥𝑦𝑡−𝜏_∗−𝜏−𝑝₁𝑦𝑡^∗−2𝜏𝐹(𝑚)2², (46) or 𝑤𝑡^∗=𝑦𝑡^∗−𝜏−𝑝₁𝑦𝑡^∗−2𝜏𝑦(𝑡^∗)−𝑝₁𝑦(𝑡^∗)≤4−𝜏(𝐹(𝑚))². (47)
Then  𝑤(𝑡)  is bounded.
Now, let 𝑚=lim_𝑡→∞inf𝑤(𝑡). (48)
But we have proved that 𝑤(𝑡)  is bounded; that is, 𝑚  is finite.
From (40), we obtain ln𝑚≥𝑝₁𝑚ln𝑚+lim_𝑡→∞inf𝑚^𝑡_𝑡−𝜏𝑞(𝑠)𝑑𝑠. (49)
Therefore, we get lim_𝑡→∞inf𝑚^𝑡_𝑡−𝜏𝑞(𝑠)𝑑𝑠≤1−𝑝₁𝑚𝑚ln𝑚. (50)
Hence lim_𝑡→∞inf^𝑡_𝑡−𝜏𝑞(𝑠)𝑑𝑠≤1−𝑝₁𝑚²𝑚. (51)
This contradicts our assumption (iv) and then completes the proof.

Example 6. Consider the NDDE 𝑥(𝑡)−19𝑥(𝑡−𝜋)^′+1𝑥(𝑡)+()𝑥()181+cos2𝑡𝑡−𝜋=0,𝑡≥0, (52) where 1𝑝=9,𝑞₁=1𝑞18,𝜏=𝜎=𝜋,₂(𝑡)=1+cos2𝑡. (53)
Then we have(1)0<𝑝𝑒^𝑞1𝜏=𝑒^𝜋/18/9<1; (2)𝑞₂(𝑡)=1+cos2𝑡∈𝐶[[0,∞),(0,∞)] is periodic with period 𝜋 and satisfies lim_𝑡→∞inf^𝑡_𝑡−𝜏𝑒^𝑞1𝜏𝑞₂(𝑠)+𝑞₁𝑝𝑑𝑠=lim_𝑡→∞inf^𝑡_𝑡−𝜋𝑒^𝜋/18()+11+cos2𝑠162𝑑𝑠=𝑒^𝜋/18lim_𝑡→∞𝑠inf𝑠++11622sin2𝑠^𝑡_𝑡−𝜋=∞>1−𝑝𝑒^𝑞1𝜏𝑚²𝑚=1−(2/9)𝑒^𝜋/1822, (54)
where 𝑚=2 is the unique real root of the equation 11−9𝑒^𝜋/18𝑚=ln𝑚;1≤𝑚≤9𝑒^−𝜋/18. (55)
Therefore (52) satisfies all the hypotheses of Theorem 5. Hence every solution of this equation is oscillatory.

Theorem 7. Suppose that condition (iii) holds. If (v)  lim_𝑡→∞inf∫^𝑡_𝑡−𝜏𝑒^𝑞1𝜏[𝑞₂(𝑠)+𝑞₁𝑝]𝑑𝑠>(1−𝑝𝑒^𝑞1𝜏)/𝑒,
then every solution of (1) is oscillatory.

Proof. Proceeding as in the proof of Theorem 5, we get (49) which implies that ln𝑚≥𝑝₁ln𝑚+𝑚lim_𝑡→∞inf^𝑡_𝑡−𝜏𝑞(𝑠)𝑑𝑠. (56)
Hence lim_𝑡→∞inf^𝑡_𝑡−𝜏𝑞(𝑠)𝑑𝑠≤1−𝑝₁𝑚ln𝑚≤1−𝑝₁𝑒. (57)
But this is a contradiction of assumption (v), and then the proof is complete.

Example 8. Consider the NDDE 𝑥(𝑡)−1𝑥𝜋5𝑒𝑡−2^′+2𝜋𝑥(𝑡)+()𝑥𝜋𝑒+sin4𝑡𝑡−2=0,𝑡≥0. (58)
Here we have 1𝑝=5𝑒,𝑞₁=2𝜋𝜋,𝜏=𝜎=2,𝑞₂(𝑡)=𝑒+sin4𝑡. (59)
Note that 𝑞₂(𝑡)=𝑒+sin4𝑡  is positive and periodic with period 𝜋/2, and also(1)0<𝑝𝑒^𝑞1𝜏=1/5<1, (2)lim_𝑡→∞inf^𝑡_𝑡−𝜏𝑒^𝑞1𝜏𝑞₂(𝑠)+𝑞₁𝑝𝑑𝑠=lim_𝑡→∞inf^𝑡_𝑡−𝜏𝑒)+2(𝑒+sin4𝑡>5𝑒𝜋𝑑𝑠=∞1−𝑝𝑒^𝑞1𝜏𝑒=4.5𝑒 (60)
Then (58) satisfies hypotheses of Theorem 7, and so all its solutions are oscillatory.

Funding

This research has been completed with the support of these Grants: ukm-DLP-2011-049, DIP-2012-31 and FRGS/1/2012/SG04/ukm/01/1.