# New Oscillation Criteria for Second-Order Neutral Delay Differential Equations with Positive and Negative Coefficients

Authors: Yuzhen Bai; Lihua Liu
Journal: Abstract and Applied Analysis
 (2010)

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## Abstract

We establish the oscillation and asymptotic criteria for the second-order neutral delay differential equations with positive and negative coefficients having the forms[x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=0 and [x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=f(t). The obtained new oscillation criteria extend and improve the recent results given in the paperof B. Karpuz et al. (2009).


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## Body

## 1. Introduction

In this paper, we consider the oscillation of all solutions of the second-order neutral delay differential equations with positive and negative coefficients having the forms(1.1)[x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=0,(1.2)[x(t)+∑i∈Rci(t)x(αi(t))]′′+r(t)[x(t)+∑i∈Rci(t)x(αi(t))]′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=f(t).We introduce the following class of functionsD([t0,∞)) equipped with the functions satisfying the following properties:(P1)
f∈C1([t0,∞),ℝ) is strictly increasing and limt→∞f(t)=∞ holds,(P1)
f(t)≤t holds for all t≥t0.In this paper, we make the following assumptions:(H1)
R,P,Q are bounded starting segments of positive integers; that is,
(1.3)R={1,2,…,R0},P={1,2,…,P0},Q={1,2,…,Q0},R0,P0,Q0∈ℕ;(H2)
ci∈C([t0,∞),ℝ+) for all i∈R,pi∈C([t0,∞),ℝ+) for all i∈P, and qi∈C([t0,∞),ℝ+) for all i∈Q;(H3)
αi∈D([t0,∞)) with liminft→∞αi′(t)>0 for all i∈R,βi∈D([t0,∞)) for all i∈P, and γi∈D([t0,∞)) for all i∈Q;(H4)
r∈C1([t0,∞),ℝ+), and r′(t)≤0;(H5)
f∈C([t0,∞),ℝ) and that there exists a function F∈C2([t0,∞),ℝ) which satisfies F′′=f and limt→∞F(t)=0.In order to establish our main results, we will assume that there exists a mappingψ:Q→P satisfying the following conditions:(A1)
γi(t)≥βψ(i)(t) for all t≥t0 and i∈Q;(A2)
hi∈C([t0,∞),ℝ+) for all i∈P, where
(1.4)hi(t)≔{pi(t)-∑j∈Q,ψ(j)=iρj′(t)qj(ρj(t)),i∈ψ(Q),pi(t),i∉ψ(Q),
and ρi(t):=γi-1(βψ(i)(t)) for all i∈Q and t≥t0;(A3)
there existsi0∈P such that liminft→∞hi0(t)>0 and limsupt→∞βi0′(t)<∞.A functionx is called a solution of (1.1) (or (1.2)) provided that x satisfies (1.1)(or (1.2)) identically on [t0,∞),x+∑i∈Rci(t)x∘αi∈C2([t0,∞),ℝ) and x∈C([t-1,∞),ℝ), where t-1:=min{α,β,γ},α:=min{αi(t0):i∈R},β:=min{βi(t0):i∈P}, and γ:=min{γi(t0):i∈Q}. We restrict our attention only to the nontrivial solution x, that is, to the solution x such that sup{|x(t)|:t≥t1}>0 for all t1≥t0. A nontrivial solution of (1.1) (or (1.2)) is called oscillatory if it has arbitrary large zeros, otherwise, it is called nonoscillatory.The oscillation and nonoscillation of solutions of second-order neutral delay differential equations have been studied by many authors; see [1–10]. However, to the best of our knowledge,there seem to be fewoscillation results for (1.1) and (1.2).Recently,Manojlovićet al. [4] andWeng and Sun[10] have studied oscillation and asymptotic behavior of all solutions of the following equations:(1.5)[x(t)+∑i=1lci(t)x(t-τi)]′′+∑i=1mpi(t)x(t-δi)-∑i=1nqi(t)x(t-σi)=0,t≥t0,[x(t)+∑i=1lci(t)x(t-τi)]′′+∑i=1mpi(t)x(t-δi)-∑i=1nqi(t)x(t-σi)=f(t),t≥t0,
and several well-known results have been obtained.By using weaker conditions than in [4, 10], Karpuz et al. [1] have established oscillation criteria for differential equation (1.6)[x(t)+∑i∈Rri(t)x(αi(t))]′′+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=f(t).In this paper, we shall continue in the direction to study the oscillatory properties of (1.1) and (1.2). We establish new oscillation criteria for (1.1) and (1.2), which extend and improve the corresponding results in [1, 4, 10]. We also give two examples to illustrate our main results.


## 2. Main Results

The following properties of the setL1([t0,∞)) in [1] are needed for our subsequent discussion.Property 1.
Iff∈L1([t0,∞)) and f∈C([t0,∞),ℝ+), then liminft→∞f(t)=0.Corollary 2.1.
Suppose thatf∈L1([t0,∞)) and limt→∞f(t) exists; then limt→∞f(t)=0.Property 2.
Iff∈C([t0,∞),ℝ) and f∈L1([t1,∞)), where t1≥t0, then we have f∈L1([t0,∞)).Property 3.
Lett1 be such that g(t1)≥t0. Suppose g∈D([t1,∞)) with lim supt→∞g'(t)<∞ and f∈C([t0,∞),ℝ+). If f∘g∈L1([t1,∞)) holds, then f∈L1([t0,∞)).Property 4.
Lett1 be such that g(t1)≥t0. Suppose g∈D([t1,∞)) with lim inft→∞g′(t)>0,f∘g∈C([t1,∞),ℝ+), and f∈C([t0,∞),ℝ+). If f∈L1([t0,∞)) holds, then f∘g∈L1([t1,∞)) holds.For simplicity, we denote the set of bounded functions by(2.1)B([t0,∞)):={f∈C([t0,∞),ℝ):∥f∥<∞},
where (2.2)∥f∥:=sup{|f(t)|,t≥t0}.
For an arbitrary function ψ:Q→P, which satisfies (A1)–(A3), we denote the function φ:[t0,∞)→ℝ+ by (2.3)φ(t):=∑i∈Q∫ρi(t)tqi(v)dv,t≥t0.In this section, for convenience, we suppose thatqi≡0 holds for all i∈Q on [t-1,t0).We start with the following Theorem.Theorem 2.2.
Assume that(H1)–(H4) hold and there exists a mapping ψ:Q→P which satisfies (A1)–(A3) and that ∑i∈Rci∈B([t0,∞)). If φ∈L1([t0,∞)), then every solution x of (1.1) is oscillatory.Proof.
Suppose thatx is a nonoscillatory solution of (1.1). Without loss of generality, we may assume that x(t)>0 for t≥t0. Therefore, we may assume existence of t1≥t0 such that
(2.4)∫t1∞φ(u)du<1,∀t≥t1,x(αi(t))>0,∀i∈R,x(βi(t))>0,∀i∈P.
Now, we set(2.5)w(t):=x(t)+∑i∈Rci(t)x(αi(t)),t≥t1,(2.6)z(t,s):=w(t)+∫str(u)w(u)du-∑i∈Q∫st∫ρi(u)uqi(v)x(γi(v))dvdu,t≥s≥t1.
By z′(t,s), we denote differential of functions with respect to the first component. Considering (2.5), we rewrite (1.1) in the form
(2.7)w′′(t)+r(t)w′(t)+∑i∈Ppi(t)x(βi(t))-∑i∈Qqi(t)x(γi(t))=0
on [t1,∞). By Leibnitz’s rule, (2.7) and (H4), we have
(2.8)z′′(t,t1)=w′′(t)+r′(t)w(t)+r(t)w′(t)-∑i∈Qqi(t)x(γi(t))+∑i∈Qρi′(t)qi(ρi(t))x(βψ(i)(t))≤w′′(t)+r(t)w′(t)-∑i∈Qqi(t)x(γi(t))+∑i∈Qρi′(t)qi(ρi(t))x(βψ(i)(t))=-∑i∈Ppi(t)x(βi(t))+∑i∈Qρi′(t)qi(ρi(t))x(βψ(i)(t))=-∑i∈ψ(Q)pi(t)x(βi(t))-∑i∉ψ(Q)pi(t)x(βi(t))+∑i∈ψ(Q)∑j∈Q,ψ(j)=iρj′(t)qj(ρj(t))x(βi(t))=-∑i∈ψ(Q)[pi(t)-∑j∈Q,ψ(j)=iρj′(t)qj(ρj(t))]x(βi(t))-∑i∉ψ(Q)pi(t)x(βi(t)).=-∑i∈Phi(t)x(βi(t))≤0,∀t≥t1,
which implies z′(·,t1) and z(·,t1) is eventually strictly monotonic on [t1,∞). Hence there exists t2≥t1 such that either z′(t,t1)<0 or z'(t,t1)>0 holds for all t≥t2.
We consider the following two possible cases:Case 1 (z′(t,t1)>0forallt≥t2).
Integrating (2.8) from t2 to ∞, we have
(2.9)∞>z′(t2,t1)≥z′(t2,t1)-limt→∞z′(t,t1)=-∫t2∞z′′(u,t1)du=∑i∈P∫t2∞hi(u)x(βi(u))du,
which implies that ∑i∈Phi(t)·(x∘βi)(t)∈L1[t2,∞). Therefore, for i0∈P for which (A3) holds, we have x∘βi0∈L1[t2,∞). Then we conclude that x∈L1[t0,∞) by Property 3. Hence ∑i∈Rci∈B([t0,∞)).(H3) and Property 4 imply that w∈L1[t1,∞). Since r(t) is bounded, b(t)=r(t)w(t) is also integrable in [t1,∞). So we obtain that there exists a constant M>0 such that
(2.10)∫t1tr(u)w(u)du≤M,∀t≥t1.
Let
(2.11)u(t)=w(t)+∫t1tr(u)w(u)du,∀t≥t2.
From (2.6), we have
(2.12)u′(t)=z′(t,t1)+∑i∈Q∫ρi(t)tqi(v)x(γi(v))dv>0.
Then u(t) is bounded and monotonous and limt→∞u(t) exists. We can suppose that limt→∞u(t)=μ>0 since u(t)>x(t)>0andu′(t)>0.
So there existst3≥t2 such that u(t)>μ-ɛ/2,t≥t3, for arbitrary ɛ∈(0,μ-M); by
(2.13)u(t)=w(t)+∫t3tr(u)w(u)du>μ-ɛ2,
we have
(2.14)w(t)>-∫t3tr(u)w(u)du+μ-ɛ2>μ-12(μ-M)-M=12(μ-M).
This implies that w(t)∉L1([t1,∞), which is a contradiction.Case 2 (z′(t,t1)<0forallt≥t2).
Sincez′(·,t1) is nonincreasing by (2.8), the inequality z′(t,t1)≤z′(t2,t1) implies that limt→∞z(t,t1)=-∞. Hence z(·,t1)∉B([t1,∞)). We claim that x∈B([t0,∞)). On contrary, there exists t3≥t2 such that z(t3,t1)<0 and x(t3)=sup{x(t):t∈[t0,t3)}. We get the following contradiction:
(2.15)0>z(t3,t1)=w(t3)+∫t1t3r(u)w(u)du-∑i∈Q∫t1t3∫ρi(u)uqi(v)x(γi(v))dvdu≥x(t3)+x(t3)∫t1t3r(u)du-∑i∈Q∫t1t3∫ρi(u)uqi(v)x(γi(v))dvdu≥x(t3)[1+∫t1t3r(u)du-∫t1t3φ(u)du]>0,
since
(2.16)∫t1t3φ(u)du<∫t1∞φ(u)du<1.
Thus ∥x∥<∞. Accordingly, by (2.4) and (2.6), it follows that
(2.17)z(t,t1)≥w(t)+∫t1tr(u)w(u)du-∥x∥∫t1tφ(u)du≥-∥x∥∫t1tφ(u)du>-∥x∥,∀t≥t2.
Therefore, ∥x∥≥∥z(·,t1)∥ holds and we see that z(·,t1)∈B([t1,∞)), which is a contradiction.
Therefore, we completed the proof by considering both possible cases.Remark 2.3.
Whenr(t)=0, and Theorem 2.2 reduces to Theorem 3.1 in [1]. So Theorem 2.2 extends and improves the corresponding results in [1, 4, 10].Theorem 2.4.
Assume that(H1)–(H5) hold and there exists a mapping ψ:Q→P which satisfies (A1)–(A3). Furthermore, assume that ∑i∈Rci∈B([t0,∞)). If φ∈L1([t0,∞)), then every solution x of (1.2) is oscillatory or tends to zero asymptotically.Proof.
Suppose thatx is a nonoscillatory solution of (1.2). Without loss of generality, we assume that x(t)>0 for t≥t0. Therefore, we may assume existence of a constant ɛ>0 and t1≥t0 such that (2.4) and F(t)≤ɛ hold for all t≥t1. Now, for t≥s≥t1, set
(2.18)W(t):=w(t)-F(t),(2.19)Z(t,s):=W(t)+∫str(u)w(u)du-∑i∈Q∫st∫ρi(u)uqi(v)x(γi(v))dvdu+ɛ,
where w(t) is defined on the interval [t1,∞) as in (2.5). Then as in (2.8), we have
(2.20)Z′′(t,t1)≤-∑i∈Phi(t)x(βi(t))≤0,∀t≥t1.
Thus there exists t2≥t1 satisfying either Z′(t,t1)>0 or Z′(t,t1)<0 for all t≥t2.
We consider the following two possible cases.Case 1 (Z′(t,t1)>0forallt≥t2).
In this case, one can show thatw∈L1([t1,∞)) as shown in above proofs. Since r(t) is bounded, r(t)w(t) is also integrable in [t1,∞). Let
(2.21)v(t)=W(t)+∫t1tr(u)w(u)du,t≥t2.
By (2.19), we have
(2.22)v′(t)=Z′(t,s)+∑i∈Q∫ρi(t)tqi(v)x(γi(v))dv>0,t≥t2;
then v(t) is bounded and monotonous and limt→∞v(t) exists. By (2.21), we can obtain that limt→∞w(t) exists. Letting
(2.23)a(t)=w(t)+∫t1tr(u)w(u)du,
we can obtain that limt→∞a(t) exists. Suppose that limt→∞a(t)=θ, where θ∈[0,∞). We claim θ=0. Suppose that θ∈(0,∞). By r(t)w(t)∈L1([t1,∞)), we see that there exists N>0, such that
(2.24)∫t1tr(u)w(u)du≤N.
Because limt→∞a(t)=θ>0, there exists t3≥t2, such that a(t)>θ-(1/2)ϵ,t≥t3, for arbitrary ϵ∈(0,θ-N). But by
(2.25)w(t)+∫t3tr(u)w(u)du>θ-ϵ2,
we have
(2.26)w(t)>-∫t3tr(u)w(u)du+θ-ϵ2>θ-12(θ-N)-N=12(θ-N);
this implies that w(t)∉L1([t2,∞), which is a contradiction. Therefore, limt→∞a(t)=0. Since 0<x(t)<a(t) for t≥t2, we have that limt→∞x(t)=0.Case 2 (Z′(t,t1)<0forallt≥t2).
Then we have thatZ(·,t1)∉B([t1,∞)) by (2.20). We claim that x∈B([t0,∞)). On contrary, there exists t3≥t2 such that z(t3,t1)<0 and x(t3)=sup{x(t):t∈[t0,t3)} hold and limt→∞Z(t,t1)<0.
Taking (2.4), (2.5), (2.18), and (2.19) into account, we get the following contradiction:
(2.27)0>Z(t3,t1)=W(t3)+∫t1t3r(u)w(u)du-∑i∈Q∫t1t3∫ρi(u)uqi(v)x(γi(v))dvdu+ϵ≥x(t3)[1+∫t1t3r(u)du-∫t1t3φ(u)du]>0.
Hence ∥x∥<∞. Accordingly, using (2.18), (2.19), and the fact that w(t)>0 on [t2,∞), we have that
(2.28)Z(t,t2):=W(t)+∫t2tr(u)w(u)du-∑i∈Q∫st∫ρi(u)uqi(v)x(γi(v))dvdu+ϵ≥w(t)+∫t2tr(u)w(u)du-∥x∥∫t2tφ(u)du≥-∥x∥,∀t≥t2.
Thus ∥x∥≥∥Z(·,t1)∥ and Z(·,t1)∈B([t1,∞)), which is a contradiction.Therefore, we completed the proof by considering both possible cases.Remark 2.5.
If there existsF∈C2([t0,∞),ℝ) such that F′′=f on [t0,∞) and l0:=limt→∞F(t) exists and is finite, then G:=F-l0 on [t0,∞) satisfies (H5).Corollary 2.6.
If all conditions of Theorem2.4 hold, then every nonoscillatory solution of (1.2) converges to zero at infinity.Remark 2.7.
Whenr(t)=0, Theorem 2.4 reduces to Theorem 3.3 in [1]. So Theorem 2.4 extends and improves the corresponding results in [1, 4, 10].


## 3. Examples

In this section, we provide two examples to illustrate our main results.Example 3.1.
Consider the following equation:(3.1)[x(t)+3x(t-π)]′′+11+t2[x(t)+3x(t-π)]′+(t+1)x(t-2π)-e-t-2πx(t-π)=0,t≥1.
Here, we have
(3.2)c1(t)=3,α1=t-π,β1=t-2π,γ1=t-π,p1(t)=t+1,q1(t)=e-t-2π,r(t)=11+t2.
Set the function ψ with ψ(1)=1, then ρ1(t)=t-π.
By simple calculation, we have(3.3)h1(t)=t+1-e-t-π,φ(t)=e-t-π-e-t-2π,liminft→∞h1(t)=∞,∫1∞φ(u)du=e-1-π-e-1-2π.
Therefore, according to Theorem 2.2, every solution x of (3.1) is oscillatory. Clearly, the known results in [1–10] cannot be applied to (3.1).Example 3.2.
Consider the following equation:(3.4)[x(t)+e-tx(t2)]′′+11+t2[x(t)+e-tx(t2)]′+x(t2-2π)+x(t2-π)-e-tx(t2-3π2)-34e-tx(t2-π)=e-t-1t3sin1t,t≥0.
Here, we have(3.5)R={1},P={1,2},Q={1,2},c1(t)=e-t,p1(t)=p2(t)=1,q1(t)=e-t,q2(t)=-34e-t,α1(t)=t2,β1(t)=t2-2π,β2(t)=t2-π,γ1(t)=t2-3π2,γ2(t)=t2-π,f(t)=e-t+1t3sin1t,r(t)=11+t2.
Set the function ψ:Q→P with ψ(i)=1 for i=1,2; then ρ1(t)=t-πandρ2(t)=t-2π.
By simple calculation, we have(3.6)F(t)=e-t+tsin1t,h1(t)=1-e-t+π-34e-t+2π,h2(t)=1,φ(t)=∫t-πte-vdu+34∫t-2πte-vdv,liminft→∞h2(t)=1,∫0∞φ(u)du=34(e2π-1)+eπ-1,limt→∞F(t)=1.
Therefore, according to Theorem 2.4 and Remark 2.5, every solution x of (3.4) is oscillatory or tends to zero asymptotically. Clearly, the known results in [1–10] cannot be applied to (3.4).



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