# Oscillation Theorems for Second-Order Nonlinear Neutral Delay Differential Equations

Authors: Tongxing Li; Yuriy V. Rogovchenko
Journal: Abstract and Applied Analysis
 (2014)

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## Abstract

We analyze the oscillatory behavior of solutions to a class of second-order nonlinear neutral delay differential equations. Our theorems improve a number of related results reported in the literature.


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## Body

## 1. Introduction

In this paper, we study the oscillatory behavior of a class of second-order nonlinear neutral delay differential equations(1)(r(t)|z′(t)|α-1z′(t))′+q(t)f(x(σ(t)))=0,
where t∈I≔[t0,∞),  t0>0, z(t)≔x(t)+p(t)x(τ(t)), and α>0 is a constant. We assume that the following conditions hold:(A1)
r, p, q∈C(I,R),  r(t)>0,  p(t)≥0,  q(t)≥0, and q(t) is not identically zero for large t;(A2)
f∈C(R,R),  uf(u)>0, for all u≠0, and there exists a positive constant k such that
(2)f(u)|u|α-1u≥k,∀u≠0;(A3)
σ∈C(I,R), σ(t)≤t, and lim⁡t→∞σ(t)=∞;(A4)
τ∈C1(I,R), τ(t)≤t, and lim⁡t→∞τ(t)=∞;(A5)
τ′(t)≥τ0>0 and τ∘σ=σ∘τ.By a solution of (1) we mean a function x∈C([Tx,∞),R),  Tx≥t0, such that r|z′|α-1z′∈C1([Tx,∞),R) and x satisfies (1) on [Tx,∞). We consider only those solutions of (1) which satisfy sup⁡⁡{|x(t)|:t≥T}>0, for all T≥Tx, and assume that (1) possesses such solutions. As customary, we say that a solution of (1) is oscillatory if it has arbitrarily large zeros on the interval [Tx,∞); otherwise, it is called nonoscillatory. Equation (1) is termed oscillatory if all its solutions are oscillatory.An increasing interest in oscillation of solutions to functional differential equations during the last few decades has been stimulated by applications arising in engineering and natural sciences; see Hale [1]. This resulted in publication of several monographs [2–5] and numerous research articles [6–20]; see also the references cited there. Prior to presenting our oscillation criteria, we briefly comment on a number of closely related results for (1) and its particular cases which motivated the present study. In the sequel, the following notation is frequently used:
(3)f+(t):=max⁡⁡{0,f(t)},Q(t):=min⁡⁡{q(t),q(τ(t))}.Grace and Lalli [10] studied a second-order nonlinear neutral delay differential equation
(4)(r(t)(x(t)+p(t)x(t-τ))′)′+q(t)f(x(t-σ))=0
under the assumptions that
(5)0≤p(t)<1,f(u)u≥k>0,∀u≠0,∫t0∞r-1(t)dt=∞.
They proved that (4) is oscillatory if there exists a function ρ∈C1(I,(0,∞)) such that
(6)∫t0∞[ρ(t)q(t)(1-p(t-σ))-(ρ′(t))2r(t-σ)4kρ(t)]dt=∞.
Hasanbulli and Rogovchenko [11] obtained several oscillation criteria for a nonlinear neutral differential equation
(7)(r(t)(x(t)+p(t)x(t-τ))′)′+q(t)f(x(t),x(σ(t)))=0
in the case where 0≤p(t)<1. Ye and Xu [18, Theorem 2.1] proved the following result for (1).Theorem 1.
Suppose that0≤p(t)<1, σ∈C1(I,R), and σ′(t)>0. Assume also that conditions (A1)–(A4) are satisfied and
(8)∫t0∞r-1/α(t)dt=∞.
If there exists a function ρ∈C1(I,(0,∞)) such that
(9)∫t0∞[-1k(α+1)α+1(ρ+′(t))α+1r(σ(t))(ρ(t)σ′(t))α]ρ(t)q(t)(1-p(σ(t)))α-1k(α+1)α+1(ρ+′(t))α+1r(σ(t))(ρ(t)σ′(t))α]dt=∞,
then (1) is oscillatory.In a special casef(u):=|u|α-1u, (1) reduces to a quasilinear neutral differential equation
(10)(r(t)|z′(t)|α-1z′(t))′+q(t)|x(σ(t))|α-1x(σ(t))=0.
Equation (10) was studied by Sun et al. [17] and Zhong et al. [20] who established the following results.Theorem 2 (see [17, Theorem 3.4]).
Suppose thatα≥1, σ(t)≥τ(t), and 0≤p(t)≤p0<∞. Assume also that conditions (A1), (A3)–(A5), and (8) are satisfied. If there exists a function ρ∈C1(I,(0,∞)) such that
(11)∫t0∞[ρ(t)Q(t)2α-1-1+(p0α/τ0)(α+1)α+1(ρ+′(t))α+1r(τ(t))(τ0ρ(t))α]dt=∞,
then (10) is oscillatory.Theorem 3 (see [20, Theorem 3.1]).
Assume thatr′(t)≥0, τ(t)=t-τ≤t, 0≤p(t)=p0<∞, p0≠1, σ∈C1(I,R), and σ′(t)>0. Suppose also that conditions (A1), (A3), and (8) are satisfied. If there exist an ε∈(0,1) and a function ρ∈C1(I,(0,∞)) such that
(12)∫t0∞[-1(α+1)α+1(ρ+′(t))α+1r(σ(t))(ρ(t)σ′(t))α](1-ε)α(1+p0(1+ε))αρ(t)q(t)-1(α+1)α+1(ρ+′(t))α+1r(σ(t))(ρ(t)σ′(t))α]dt=∞,
then (10) is oscillatory.The purpose of this note is to refine Theorems1–3 in some cases. In what follows, all functional inequalities are assumed to hold for all t large enough. Without loss of generality, we can deal only with positive solutions of (1).


## 2. Main Results

For a more compact presentation of conditions in our results, we use the notation(13)R(l,t):=(∫lσ(t)r-1/α(s)ds)(∫ltr-1/α(s)ds)-1.Theorem 4.
Let0<α≤1 and 0≤p(t)≤p0<∞. Assume also that conditions (A1)–(A5) and (8) are satisfied. If there exists a function ρ∈C1(I,(0,∞)) such that
(14)∫t**∞[-(ρ+′(t))α+1k(α+1)α+1ρα(t)(r(t)+p0αr(τ(t))τ0α+1)]ρ(t)Q(t)Rα(t*,t)-(ρ+′(t))α+1k(α+1)α+1ρα(t)(r(t)+p0αr(τ(t))τ0α+1)]dt=∞,
for all sufficiently large t* and for some t**≥t*≥t0, then (1) is oscillatory.Proof.
Letx(t) be a nonoscillatory solution of (1); we assume that it is eventually positive. Then there exists a t1≥t0 such that x(t)>0, x(τ(t))>0, and x(σ(t))>0, for all t≥t1. It follows from (1) that
(15)(r(t)|z′(t)|α-1z′(t))′≤-kq(t)xα(σ(t))≤0,hhhhhhhhhhhhhhhhhhhhhhhhhhhhh∀t≥t1.
Using condition (8), we conclude that there exists a t2≥t1 such that z′(t)>0, for all t≥t2. Hence, for all t≥t2, inequality (15) reduces to
(16)(r(t)(z′(t))α)′≤-kq(t)xα(σ(t))≤0,
and there exists a t3≥t2 such that, for all t≥t3,
(17)p0α(r(τ(t))(z′(τ(t)))α)′τ′(t)≤-kp0αq(τ(t))xα(σ(τ(t))).
Using the assumption τ′(t)≥τ0>0, we have, for all t≥t3,
(18)p0ατ0(r(τ(t))(z′(τ(t)))α)′≤-kp0αq(τ(t))xα(σ(τ(t))).
Combining inequalities (16) and (18), using the condition τ  ∘  σ = σ  ∘  τ and an auxiliary result due to Baculíková and Džurina [7, Lemma 2], we conclude that
(19)(r(t)(z′(t))α)′+p0ατ0(r(τ(t))(z′(τ(t)))α)′≤-k[q(t)xα(σ(t))+p0αq(τ(t))xα(τ(σ(t)))]≤-kmin⁡⁡{q(t),q(τ(t))}[xα(σ(t))+p0αxα(τ(σ(t)))]≤-kQ(t)zα(σ(t)),
for all t≥t3. Define a new function ω(t) by
(20)ω(t):=ρ(t)r(t)(z′(t))αzα(t).
Then ω(t)>0, for all t≥t3. Differentiation of (20) yields
(21)ω′(t)=ρ′(t)r(t)(z′(t))αzα(t)+ρ(t)(r(t)(z′(t))α)′zα(t)-αρ(t)r(t)(z′(t))α+1zα+1(t)≤ρ(t)(r(t)(z′(t))α)′zα(t)+ρ+′(t)ρ(t)ω(t)-α(ρ(t)r(t))1/αω(α+1)/α(t).
Let
(22)A:=ρ+′(t)ρ(t),B:=α(ρ(t)r(t))1/α,u:=ω(t).
Using the inequality
(23)Au-Bu(α+1)/α≤αα(α+1)α+1Aα+1Bα,B>0,
we deduce from (21) that
(24)ω′(t)≤ρ(t)(r(t)(z′(t))α)′zα(t)+1(α+1)α+1(ρ+′(t))α+1r(t)ρα(t).
Define another function ν(t) by
(25)ν(t):=ρ(t)r(τ(t))(z′(τ(t)))αzα(τ(t)).
Observe that ν(t)>0, for all t≥t3. Differentiation of (25) yields
(26)ν′(t)=ρ′(t)r(τ(t))(z′(τ(t)))αzα(τ(t))+ρ(t)(r(τ(t))(z′(τ(t)))α)′zα(τ(t))-αρ(t)r(τ(t))(z′(τ(t)))αz′(τ(t))τ′(t)zα+1(τ(t))≤ρ(t)(r(τ(t))(z′(τ(t)))α)′zα(τ(t))+ρ+′(t)ρ(t)ν(t)-ατ′(t)(ρ(t)r(τ(t)))1/αν(α+1)/α(t).
Let
(27)A:=ρ+′(t)ρ(t),B:=ατ′(t)(ρ(t)r(τ(t)))1/α,u:=ν(t).
Using the inequalities (23) and (26) along with the fact that z′(t)>0, we have
(28)ν′(t)≤ρ(t)(r(τ(t))(z′(τ(t)))α)′zα(t)+1(α+1)α+1(ρ+′(t))α+1r(τ(t))(ρ(t)τ′(t))α.
Combining (24) and (28) and using the inequality (19), we obtain
(29)ω′(t)+p0ατ0ν′(t)≤ρ(t)(r(t)(z′(t))α)′+(p0α/τ0)(r(τ(t))(z′(τ(t)))α)′zα(t)+(ρ+′(t))α+1(α+1)α+1ρα(t)(r(t)+p0αr(τ(t))τ0α+1)≤-kρ(t)Q(t)zα(σ(t))zα(t)+(ρ+′(t))α+1(α+1)α+1ρα(t)(r(t)+p0αr(τ(t))τ0α+1).
Since (r(t)(z′(t))α)′≤0, we have
(30)z(t)≥r1/α(t)z′(t)∫t2tr-1/α(s)ds,
and thus
(31)(z(t)∫t2tr-1/α(s)ds)′≤0.
Consequently,
(32)zα(σ(t))zα(t)≥Rα(t2,t).
Substitution of (32) in (29) yields
(33)ω′(t)+p0ατ0ν′(t)≤-kρ(t)Q(t)Rα(t2,t)+(ρ+′(t))α+1(α+1)α+1ρα(t)(r(t)+p0αr(τ(t))τ0α+1).
Integrating (33) from t3 to t, we have
(34)∫t3t[-(ρ+′(s))α+1k(α+1)α+1ρα(s)(r(s)+p0αr(τ(s))τ0α+1)]ρ(s)Q(s)Rα(t2,s)-(ρ+′(s))α+1k(α+1)α+1ρα(s)(r(s)+p0αr(τ(s))τ0α+1)]ds≤ω(t3)k+p0αkτ0ν(t3).
Passing in (34) to the limit as t→∞, we obtain contradiction with condition (14). Therefore, (1) is oscillatory.Proceeding as in the proof of Theorem4 and using another result by Baculíková and Džurina [7, Lemma 1], we obtain the following oscillation criterion for (1), for α≥1.Theorem 5.
Assume thatα≥1 and 0≤p(t)≤p0<∞. Let conditions (A1)–(A5) and (8) be satisfied. If there exists a function ρ∈C1(I,(0,∞)) such that
(35)∫t**∞[-(ρ+′(t))α+1k(α+1)α+1ρα(t)(r(t)+p0αr(τ(t))τ0α+1)]21-αρ(t)Q(t)Rα(t*,t)-(ρ+′(t))α+1k(α+1)α+1ρα(t)(r(t)+p0αr(τ(t))τ0α+1)]dt=∞,
for all sufficiently large t* and for some t**≥t*≥t0, then (1) is oscillatory.


## 3. Examples and Discussion

Example 1.
Fort≥1, consider a second-order neutral differential equation
(36)(x(t)+13x(t-2))′′+γt2x(t)=0,
where γ>0 is a constant. We have α=1,  r(t)=1,  p(t)=p0=1/3,  τ(t)=t-2,  q(t)=γ/t2,  σ(t)=t,  f(u)=u, and k=1. Choose ρ(t)=t and denote the left hand side of (14) by ψ(t**). Then
(37)ψ(t**)=(γ-13)∫t**∞dtt=∞,providedthatγ>13.
Hence, (36) is oscillatory by Theorem 4 for any γ>1/3. On the other hand, an application of Theorem 1 yields oscillation of (36) for γ>3/8, whereas Theorem 3 implies that (36) is oscillatory if γ>(4+ε)/(3(4-4ε)), for some ε∈(0,1). Therefore, we observe that our Theorem 4 improves Theorems 1 and 3.Example 2.
Fort≥1, consider a second-order neutral differential equation
(38)(x(t)+13x(t3))′′+γt2x(t)=0,
where γ>0 is a constant. We have α=1,  r(t)=1,  p(t)=p0=1/3,  τ(t)=t/3,  q(t)=γ/t2,  σ(t)=t,  f(u)=u, and k=1. Let ρ(t)=t and let ψ be defined as in Example 1. Then
(39)ψ(t**)=(γ-1)∫t**∞‍dtt=∞,
provided that γ>1. Therefore, by Theorem 4, (38) is oscillatory for any γ>1, whereas an application of Theorem 2 yields oscillation of (38) for all γ>3/2. Hence, Theorem 4 improves Theorem 2.



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*Source: 594190-2014-06-17.xml*