# Oscillation Criteria for Linear Neutral Delay Differential Equations of First Order

Authors: Fatima N. Ahmed; Rokiah Rozita Ahmad; Ummul Khair Salma Din; Mohd Salmi Md Noorani
Journal: Abstract and Applied Analysis
 (2013)

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## 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.


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## Body

## 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 timet 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(1)(x(t)-px(t-τ))′+q1x(t)+q2(t)x(t-σ)=0;t≥t0,
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
(2)λ-λpe-λτ+qe-λσ=0.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,
(3)(x(t)-px(t-τ))′+q1x(t)+q2x(t-σ)=0,t≥0.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(4)(x(t)+px(t-τ))′+∑i=1nqix(t-σi)=0,t≥t0,
where  τ≥0,qi>0,   and  σi≥0foralli=1,2,…,n.
Letx(t)   be a positive solution of (4). Set
(5)z(t)=x(t)-px(t-τ).
Ifp≥-1, then z(t) is a positive and decreasing solution of (4); that is,
(6)z′(t)+pz′(t-τ)+∑i=1nqiz(t-σi)=0,t≥t0.Lemma 2.
Letp and τ  be positive constants. Let x(t)   be an eventually positive solution of the delay differential inequality
(7)x′(t)+px(t-τ)≤0.
Then fort sufficiently large,
(8)x(t-τ)≤Bx(t),
where
(9)B=4(pτ)2.Our main results can now be given as follows.Theorem 3.
Consider NDDE (3). Assume that (i)
q2∈[0,∞),σ≥τ,  0<peq1τ<1 and(ii)
τ(q1peq1τ+q2eq1σ)>(1-peq1τm-)2/m-,
wherem- is the unique real root of the equation
(10)1-peq1τm=lnm,1≤m≤1pe-q1τ.
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
(11)x(t)=e-q1ty(t).
So that y(t) is also a positive solution of (3).
That is,(12)(y(t)-p1y(t-τ))′+∑i=12aiy(t-τi)=0,
where
(13)p1=peq1τ,a1=q1peq1τ,a2=q2eq1σ,τ1=τ,τ2=σ.
Set fort≥t0+2τ(14)z(t)=y(t)-p1y(t-τ),w(t)=z(t-τ)z(t).
Thus it follows from Lemma1 that z(t)   is a positive and decreasing solution of
(15)z′(t)-p1z′(t-τ)+∑i=12aiz(t-τi)=0,
and in particular (as σ>τ implies that t-τi≤t-τ,i=1,2.), it follows that
(16)z′(t)-p1z′(t-τ)+(a1+a2)z(t-τ)≤0.
But we have(17)z′(t-τ)<0.
This implies that(18)z′(t)+(a1+a2)z(t-τ)≤0.
Applying Lemma2 with (18) we get
(19)z(t-τ)≤Bz(t);B=4τ2(a1+a2)2.
Thenw(t)  is bounded.
Dividing (16) by  z(t)>0  and integrating from  t-τ  to  t, we get
(20)lnw(t)≥(a1+a2)∫t-τtw(s)ds-p1∫t-τtw(s)dds(lnz(s-τ))ds.
Letm=limt→∞infw(t).
Then, it follows from (20) that for ε>0 and sufficiently small,
(21)ln(m+ε)≥(a1+a2)(m-ε)τ+p1(m-ε)ln(m-ε).
Asε is arbitrary, so we have
(22)(a1+a2)τ≤(1-p1m)lnmm.
Let(23)F(m)=(1-p1m)lnmm,1≤m≤B.
Then(24)F′(m)=1-p1m-lnmm2,1≤m≤B.
Letm- be the unique real root of the equation
(25)1-p1m=lnm,m∈[1,1p1].
Then(26)maxm≥1F(m)=F(m-)=(1-p1m-)2m-.
Hence(27)(q1peq1τ+q2eq1σ)τ≤(1-peq1τm-)2m-.
This contradicts condition (ii) and then completes the proof.Example 4.
Consider the NDDE(28)(x(t)-19x(t-π2))′+19x(t)+x(t-5π2)=0,t≥0.
We note that(29)p=19,q1=19,q2=1,τ=π2,σ=5π2.
Then we have(i)
0<peq1τ=(1/9)eπ/18<1andσ≥τ,(ii)
(30)τ(q1peq1τ+q2eq1σ)=π2(181eπ/18+19e5π/18)>(1-peq1τm-)2m-=(1-(1/9)eπ/18)22,where  m-=2  is the unique real root of the equation
(31)1-19eπ/18m=lnm,m∈[1,9e-π/18].
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<peq1τ<1,σ=τ,and q2(t)∈C[[t0,∞),(0,∞)]  is periodic with period  τ,(iv)
limt→∞inf∫t-τteq1τ(q2(s)+q1p)ds>(1-peq1τm-)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
(32)x(t)=e-q1ty(t),
which is oscillation invariant transformation. Then y(t) is a positive solution of the equation
(33)(y(t)-p1y(t-τ))′+q(t)y(t-τ)=0,
where  p1=peq1τandq(t)=eq1τ(q2(t)+q1p)   is periodic with period τ.
Let(34)z(t)=y(t)-p1y(t-τ).
Thenz(t) is decreasing positive solution of the equation
(35)(z(t)-p1z(t-τ))′+q(t)z(t-τ)=0.
Set(36)w(t)=z(t-τ)z(t).
This implies thatw(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
(37)lnw(t)=∫t-τtq(s)(y(s-τ)-p1y(s-2τ)+p1y(s-2τ))y(s)-p1y(s-τ)ds.
Hence(38)lnw(t)=∫t-τtq(s)w(s)ds+p1∫t-τt(q(s)y(s-2τ)y(s)-p1y(s-τ))ds.
Sinceq(t) is periodic with period  τ, then we obtain
(39)q(t)=q(t-τ)=-(y(t-τ)-p1y(t-2τ))′y(t-2τ).
Substituting in (38) we find, for all t≥t0,
(40)lnw(t)=∫t-τtq(s)w(s)ds-p1∫t-τtw(s)dln(y(s-τ)-p1y(s-2τ))ds.
Now, we want to prove thatw(t)  is bounded.
Applying the assumption (iv), we can findt*∈(t-τ,t) such that
(41)∫t-τt*q(s)ds>F(m-)2,∫t*tq(s)ds>F(m-)2,
where F(m-) is similar as in the proof of Theorem 3.
Integrating (33) from  t* to  t we obtain
(42)y(t*)-p1y(t*-τ)≥∫t*tq(s)y(s-τ)ds≥∫t*tq(s)(y(s-τ)-p1y(s-2τ))ds,
Using Bonnet’s Theorem and in particular (asz′(t-τ)<o), we get
(43)y(t*)-p1y(t*-τ)≥[y(t-τ)-p1y(t-2τ)]·∫t*tq(s)ds.
Integrating (33) from  t-τ to  t*, we get
(44)y(t-τ)-p1y(t-2τ)≥∫t-τt*q(s)y(s-τ)ds≥∫t-τt*q(s)(y(s-τ)-p1y(s-2τ))ds.
Using Bonnet’s Theorem and in particular (asz′(t-τ)<o), we get
(45)y(t-τ)-p1y(t-2τ)≥[y(t*-τ)-p1y(t*-2τ)]·∫t-τt*q(s)ds.
Combining (43) and (45), we conclude
(46)y(t*)-p1y(t*-τ)≥(y(t*-τ)-p1y(t*-2τ))(F(m-)2)2,
or
(47)w(t*)=y(t*-τ)-p1y(t*-2τ)y(t*)-p1y(t*-τ)≤4(F(m-))2.
Thenw(t)  is bounded.
Now, let(48)m=limt→∞infw(t).
But we have proved thatw(t)  is bounded; that is, m  is finite.
From (40), we obtain
(49)lnm≥p1mlnm+limt→∞infm∫t-τtq(s)ds.
Therefore, we get(50)limt→∞infm∫t-τtq(s)ds≤1-p1mmlnm.
Hence(51)limt→∞inf∫t-τtq(s)ds≤(1-p1m-)2m-.
This contradicts our assumption (iv) and then completes the proof.Example 6.
Consider the NDDE(52)(x(t)-19x(t-π))′+118x(t)+(1+cos2t)x(t-π)=0,t≥0,
where
(53)p=19,q1=118,τ=σ=π,q2(t)=1+cos2t.
Then we have(1)
0<peq1τ=eπ/18/9<1;(2)
q2(t)=1+cos2t∈C[[0,∞),(0,∞)] is periodic with period π and satisfies
(54)limt→∞inf∫t-τteq1τ(q2(s)+q1p)ds=limt→∞inf∫t-πteπ/18((1+cos2s)+1162)ds=eπ/18limt→∞inf(s+s162+12sin2s|t-πt)=∞>(1-peq1τm-)2m-=(1-(2/9)eπ/18)22,
wherem-=2 is the unique real root of the equation
(55)1-19eπ/18m=lnm;1≤m≤9e-π/18.
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)
limt→∞inf∫t-τteq1τ[q2(s)+q1p]ds>(1-peq1τ)/e,
then every solution of (1) is oscillatory.Proof.
Proceeding as in the proof of Theorem5, we get (49) which implies that
(56)lnm≥p1lnm+mlimt→∞inf∫t-τtq(s)ds.
Hence(57)limt→∞inf∫t-τtq(s)ds≤1-p1mlnm≤1-p1e.
But this is a contradiction of assumption (v), and then the proof is complete.Example 8.
Consider the NDDE(58)(x(t)-15ex(t-π2))′+2πx(t)+(e+sin4t)x(t-π2)=0,t≥0.
Here we have(59)p=15e,q1=2π,τ=σ=π2,q2(t)=e+sin4t.
Note thatq2(t)=e+sin4t  is positive and periodic with period π/2, and also(1)
0<peq1τ=1/5<1,(2)
(60)limt→∞inf∫t-τteq1τ[q2(s)+q1p]ds=limt→∞inf∫t-τte[(e+sin4t)+25eπ]ds=∞>(1-peq1τ)e=45e.
Then (58) satisfies hypotheses of Theorem 7, and so all its solutions are oscillatory.



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