We study oscillatory behavior of solutions to a class of second-order nonlinear neutral differential equations under the
assumptions that allow applications to differential equations with delayed and advanced arguments. New theorems do not need several restrictive assumptions required in related results reported in the literature. Several examples are provided to show that the results obtained are sharp even for second-order ordinary differential equations and improve related contributions to the subject.

1. Introduction

This paper is concerned with the oscillation of a class of second-order nonlinear neutral functional differential equations
(1)(r(t)((x(t)+p(t)x(η(t)))′)γ)′+f(t,x(g(t)))=0,
where t≥t0>0. The increasing interest in problems of the existence of oscillatory solutions to second-order neutral differential equations is motivated by their applications in the engineering and natural sciences. We refer the reader to [1–21] and the references cited therein.

We assume that the following hypotheses are satisfied:

γ is a quotient of odd natural numbers, the functions r,p∈C([t0,∞),R), and r(t)>0;

the functions η,g∈C([t0,∞),R) and
(2)limt→∞η(t)=limt→∞g(t)=∞;

the function f(t,u)∈C([t0,∞)×R,R) satisfies
(3)uf(t,u)>0
for all u≠0 and there exists a positive continuous function q(t) defined on [t0,∞) such that
(4)|f(t,u)|≥q(t)|u|γ.

By a solution of (1), we mean a function x defined on [Tx,∞) for some Tx≥t0 such that x+p·x∘η and r((x+p·x∘η)′)γ are continuously differentiable and x satisfies (1) for all t≥Tx. In what follows, we assume that solutions of (1) exist and can be continued indefinitely to the right. Recall that a nontrivial solution x of (1) is said to be oscillatory if it is not of the same sign eventually; otherwise, it is called nonoscillatory. Equation (1) is termed oscillatory if all its nontrivial solutions are oscillatory.

Recently, Baculíková and Džurina [6] studied oscillation of a second-order neutral functional differential equation
(5)(r(t)(x(t)+p(t)x(η(t)))′)′+q(t)x(g(t))=0
assuming that the following conditions hold:

r,p,q∈C([t0,∞),R), r(t)>0, 0≤p(t)≤p0<∞, and q(t)>0;

g∈C1([t0,∞),R) and limt→∞g(t)=∞;

η∈C1([t0,∞),R), η′(t)≥η0>0, and η∘g=g∘η.

They established oscillation criteria for (5) through the comparison with associated first-order delay differential inequalities in the case where
(6)∫t0∞r-1(t)dt=∞.
Assuming that
(7)∫t0∞r-1(t)dt<∞,
Han et al. [9], Li et al. [15], and Sun et al. [20] obtained oscillation results for (5), one of which we present below for the convenience of the reader. We use the notation
(8)Q(t)∶=min{q(t),q(η(t))},ρ+′(t)∶=max{0,ρ′(t)},φ(t)∶=∫t∞r-1(s)ds.

Assume that conditions (H1)–(H3) and (7) hold. Suppose also that g(t)≤η(t)≤t and g′(t)>0 for all t≥t0. If there exists a function ρ∈C1([t0,∞),(0,∞)) such that
(9)limsupt→∞∫t0t[r(g(s))(ρ+′(s))24ρ(s)g′(s)ρ(s)Q(s)hhhhiihhhh-(1+p0η0)r(g(s))(ρ+′(s))24ρ(s)g′(s)]
d
s=∞,limsupt→∞∫t0t[φ(s)Q(s)-1+(p0/η0)4φ(s)r(s)]
d
s=∞,
then (5) is oscillatory.

Replacing (6) with the condition
(10)∫t0∞r-1/γ(t)dt=∞,
Baculíková and Džurina [7] extended results of [6] to a nonlinear neutral differential equation
(11)(r(t)((x(t)+p(t)x(τ(t)))′)γ)′+q(t)xβ(σ(t))=0,
where β and γ are quotients of odd natural numbers. Hasanbulli and Rogovchenko [10] studied a more general second-order nonlinear neutral delay differential equation
(12)(r(t)(x(t)+p(t)x(t-τ))′)′+q(t)f(x(t),x(σ(t)))=0
assuming that 0≤p(t)≤1, σ(t)≤t, σ′(t)>0, and (6) holds. To introduce oscillation results obtained for (1) by Erbe et al. [8], we need the following notation:
(13)D∶={(t,s):t≥s≥t0},D0∶={(t,s):t>s≥t0},h-(t,s)∶=max{0,-h(t,s)},θ(t,u)∶=∫ug(t)r-1/γ(s)ds∫utr-1/γ(s)ds.
We say that a continuous function H:D→[0,∞) belongs to the class H if

H(t,t)=0 for t≥t0 and H(t,s)>0 for (t,s)∈D0;

H has a nonpositive continuous partial derivative ∂H/∂s with respect to the second variable satisfying
(14)-∂∂sH(t,s)-H(t,s)δ′(s)δ(s)=h(t,s)δ(s)(H(t,s))γ/(γ+1)
for some h∈Lloc(D,R) and for some δ∈C1([t0,∞),(0,∞)).

Theorem 2 (see [<xref ref-type="bibr" rid="B8">8</xref>, Theorem 2.2, when <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M67"><mml:mi mathvariant="double-struck">T</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi></mml:math></inline-formula>]).

Let conditions (10) and (h1)–(h3) hold. Suppose that 0≤p(t)<1, η(t)≤t, and g(t)≥t for all t≥t0. If there exists a function H∈H such that, for all sufficiently large T≥t0,
(15)limsupt→∞1H(t,T)×∫Tt[r(s)(h-(t,s))γ+1(γ+1)γ+1δγ(s)δ(s)q(s)H(t,s)(1-p(g(s)))γhhhhhhh-r(s)(h-(t,s))γ+1(γ+1)γ+1δγ(s)]ds=∞,
then (1) is oscillatory.

Theorem 3 (see [<xref ref-type="bibr" rid="B8">8</xref>, Theorem 2.2, case <inline-formula><mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML" id="M76"><mml:mi mathvariant="double-struck">T</mml:mi><mml:mo>=</mml:mo><mml:mi mathvariant="double-struck">R</mml:mi></mml:math></inline-formula>]).

Let conditions (10) and (h1)–(h3) be satisfied. Suppose also that 0≤p(t)<1, η(t)≤t, and g(t)≤t for all t≥t0. If there exists a function H∈H such that, for all sufficiently large T*≥t0 and for some T>T*,
(16)limsupt→∞1H(t,T)×∫Tt[r(s)(h-(t,s))γ+1(γ+1)γ+1δγ(s)δ(s)θγ(s,T*)H(t,s)q(s)(1-p(g(s)))γhhhhhh-r(s)(h-(t,s))γ+1(γ+1)γ+1δγ(s)]ds=∞,
then (1) is oscillatory.

Assuming that
(17)∫t0∞r-1/γ(t)dt<∞,
Li et al. [16] extended results of [10] to a nonlinear neutral delay differential equation
(18)(r(t)((x(t)+p(t)x(t-τ))′)γ)′+q(t)f(x(t),x(σ(t)))=0,
where γ≥1 is a ratio of odd natural numbers. Han et al. [9, Theorems 2.1 and 2.2] established sufficient conditions for the oscillation of (1) provided that (17) is satisfied, 0≤p(t)<1, and
(19)η(t)=t-τ≤t,p′(t)≥0,g(t)≤t-τ.
Xu and Meng [21] studied (1) under the assumptions that (17) holds, 0≤p(t)<1, and
(20)η(t)=t-τ≤t,p′(t)≥0,limt→∞p(t)=A
obtaining sufficient conditions for all solutions of (1) either to be oscillatory or to satisfy limt→∞x(t)=0; see [21, Theorem 2.3]. Saker [17] investigated oscillatory nature of (1) assuming that (17) is satisfied,
(21)0≤p(t)<1,p′(t)≥0,g(t)≤η(t)≤t,η′(t)≥0,
and
(22)∫T∞(1r(s)∫Tsq(u)(1-p(u))γφγ(u)du)1/γds=∞
for some T≥t0, where φ(u)∶=∫u∞r-1/γ(t)dt. Li et al. [12] studied oscillation of (1) under the conditions that (17) holds, η and g are strictly increasing, p(t)>1, and
(23)eitherg(t)≥η(t)org(t)≤η(t).
Li et al. [13] investigated (1) in the case where (H1)–(H3) hold, γ≥1, η(t)≥t, and g(t)≥t. In particular, sufficient conditions for all solutions of (1) either to be oscillatory or to satisfy limt→∞x(t)=0 were obtained under the assumptions that (17) holds and 0≤p(t)≤p1<1; see [13, Theorem 3.8]. Sun et al. [19] established several oscillation results for (1) assuming that (h3), (H1)–(H3), (17), and (23) are satisfied. The following notation is used in the next theorem:
(24)Q(t)∶=min{q(t),q(η(t))},ρ+′(t)∶=max{0,ρ′(t)},φ(t)∶=∫τ(t)∞r-1/γ(s)ds.

Theorem 4 (see [<xref ref-type="bibr" rid="B19">19</xref>, Theorem 4.1]).

Let conditions (h3), (H1)–(H3), and (17) be satisfied. Assume also that γ≥1, g(t)≤η(t)≤t, and g′(t)>0 for all t≥t0. Suppose further that there exist functions ρ∈C1([t0,∞),(0,∞)) and τ∈C1([t0,∞),R) such that τ(t)≥t, τ′(t)>0,
(25)limsupt→∞∫t0t[r(g(s))(ρ+′(s))γ+1(γ+1)γ+1(ρ(s)g′(s))γρ(s)Q(s)2γ-1-(1+p0γη0)hhhhhhhhhh×r(g(s))(ρ+′(s))γ+1(γ+1)γ+1(ρ(s)g′(s))γ]
d
s=∞,limsupt→∞∫t0t[γγ+1τ′(s)(γ+1)γ+1φ(s)r1/γ(τ(s))φγ(s)Q(s)2γ-1-(1+p0γη0)hhhhhhhhhh×γγ+1τ′(s)(γ+1)γ+1φ(s)r1/γ(τ(s))]
d
s=∞.
Then (1) is oscillatory.

Our principal goal in this paper is to analyze the oscillatory behavior of solutions to (1) in the case where (17) holds and without assumptions (H3), (19)–(23), and γ≥1.

2. Oscillation Criteria

In what follows, all functional inequalities are tacitly assumed to hold for all t large enough, unless mentioned otherwise. We use the notation
(26)z(t)∶=x(t)+p(t)x(η(t)),R(t)∶=∫t∞r-1/γ(s)ds.
A continuous function K:D→[0,∞) is said to belong to the class K if

K(t,t)=0 for t≥t0 and K(t,s)>0 for (t,s)∈D0;

K has a nonpositive continuous partial derivative ∂K/∂s with respect to the second variable satisfying
(27)∂∂sK(t,s)=-ζ(t,s)(K(t,s))γ/(γ+1)
for some ζ∈Lloc(D,R).

Theorem 5.

Let all assumptions of Theorem 2 be satisfied with condition (10) replaced by (17). Suppose that there exists a function m∈C1([t0,∞),(0,∞)) such that
(28)m(t)r1/γ(t)R(t)+m′(t)≤0,1-p(t)m(η(t))m(t)>0.
If there exists a function K∈K such that, for all sufficiently large T≥t0,
(29)limsupt→∞∫Tt[K(t,s)q(s)(1-p(g(s))m(η(g(s)))m(g(s)))γhhhhhhhhh×(m(g(s))m(s))γ-r(s)(ζ(t,s))γ+1(γ+1)γ+1]
d
s>0,
then (1) is oscillatory.

Proof.

Let x(t) be a nonoscillatory solution of (1). Without loss of generality, we may assume that there exists a t1≥t0 such that x(t)>0, x(η(t))>0, and x(g(t))>0 for all t≥t1. Then z(t)≥x(t)>0 for all t≥t1, and by virtue of
(30)(r(t)(z′(t))γ)′≤-q(t)xγ(g(t))<0,
the function r(t)(z′(t))γ is strictly decreasing for all t≥t1. Hence, z′(t) does not change sign eventually; that is, there exists a t2≥t1 such that either z′(t)>0 or z′(t)<0 for all t≥t2. We consider each of the two cases separately.

Case1. Assume first that z′(t)>0 for all t≥t2. Proceeding as in the proof of [8, Theorem 2.2, case T=R], we obtain a contradiction to (15).

Case2. Assume now that z′(t)<0 for all t≥t2. For t≥t2, define a new function ω(t) by
(31)ω(t)∶=r(t)(z′(t))γzγ(t).
Then ω(t)<0 for all t≥t2, and it follows from (30) that
(32)z′(s)≤(r(t)r(s))1/γz′(t)
for all s≥t≥t2. Integrating (32) from t to l, l≥t≥t2, we have
(33)z(l)≤z(t)+r1/γ(t)z′(t)∫tldsr1/γ(s).
Passing to the limit as l→∞, we conclude that
(34)z(t)+r1/γ(t)z′(t)R(t)≥0,
which implies that
(35)z′(t)z(t)≥-1r1/γ(t)R(t).
Thus,
(36)(z(t)m(t))′=z′(t)m(t)-z(t)m′(t)m2(t)≥-z(t)m2(t)[m(t)r1/γ(t)R(t)+m′(t)]≥0.
Consequently, there exists a t3≥t2 such that, for all t≥t3,
(37)x(t)=z(t)-p(t)x(η(t))≥z(t)-p(t)z(η(t))≥z(t)-p(t)m(η(t))m(t)z(t)=(1-p(t)m(η(t))m(t))z(t),z(g(t))z(t)≥m(g(t))m(t).
Differentiating (31) and using (30), we have, for all t≥t3,
(38)ω′(t)≤-q(t)(1-p(g(t))m(η(g(t)))m(g(t)))γ(m(g(t))m(t))γ-r(t)(z′(t))γ(zγ(t))′z2γ(t)=-q(t)(1-p(g(t))m(η(g(t)))m(g(t)))γ(m(g(t))m(t))γ-γr(t)(z′(t))γ+1zγ+1(t).
Hence, by (31) and (38), we conclude that
(39)ω′(t)+q(t)(1-p(g(t))m(η(g(t)))m(g(t)))γ(m(g(t))m(t))γ+γr-1/γ(t)ω(γ+1)/γ(t)≤0
for all t≥t3. Multiplying (39) by K(t,s) and integrating the resulting inequality from t3 to t, we obtain
(40)∫t3tK(t,s)q(s)(1-p(g(s))m(η(g(s)))m(g(s)))γ×(m(g(s))m(s))γds≤K(t,t3)ω(t3)+∫t3t∂K(t,s)∂sω(s)ds-∫t3tγK(t,s)r-1/γ(s)ω(γ+1)/γ(s)ds=K(t,t3)ω(t3)-∫t3tζ(t,s)(K(t,s))γ/(γ+1)ω(s)ds-∫t3tγK(t,s)r-1/γ(s)(-ω(s))(γ+1)/γds.
In order to use the inequality
(41)γ+1γAB1/γ-A(γ+1)/γ≤1γB(γ+1)/γ,γ>0,A≥0,B≥0,
see Li et al. [16, Lemma 1 (ii)] for details; we let
(42)A(γ+1)/γ∶=γK(t,s)r-1/γ(s)(-ω(s))(γ+1)/γ,B1/γ∶=γζ(t,s)r1/(γ+1)(s)(γ+1)γγ/(γ+1).
Then, by virtue of (40), we conclude that
(43)∫t3t[K(t,s)q(s)(1-p(g(s))m(η(g(s)))m(g(s)))γhihh×(m(g(s))m(s))γ-r(s)(ζ(t,s))γ+1(γ+1)γ+1(1-p(g(s))m(η(g(s)))m(g(s)))γ]ds≤K(t,t3)ω(t3),
which contradicts (29). This completes the proof.

Theorem 6.

Let all assumptions of Theorem 3 be satisfied with condition (10) replaced by (17). Suppose further that there exists a function m∈C1([t0,∞),(0,∞)) such that (28) holds. If there exists a function K∈K such that, for all sufficiently large T≥t0,
(44)limsupt→∞∫Tt[K(t,s)q(s)(1-p(g(s))m(η(g(s)))m(g(s)))γhhhhihhhh-r(s)(ζ(t,s))γ+1(γ+1)γ+1(1-p(g(s))m(η(g(s)))m(g(s)))γ]
d
s>0,
then (1) is oscillatory.

Proof.

The proof is similar to that of Theorem 5 and hence is omitted.

Theorem 7.

Let conditions (10) and (h1)–(h3) be satisfied, 0≤p(t)<1, η(t)≥t, and g(t)≥t. Assume that there exists a function m∈C1([t0,∞),(0,∞)) such that, for all sufficiently large T*≥t0,
(45)m(t)r1/γ(t)∫T*tr-1/γ(s)
d
s-m′(t)≤0,1-p(t)m(η(t))m(t)>0.
If there exists a function H∈H such that, for all sufficiently large T≥t0,
(46)limsupt→∞1H(t,T)∫Tt[(1-p(g(s))m(η(g(s)))m(g(s)))γδ(s)q(s)H(t,s)hhhhhihhhhhhhhhh×(1-p(g(s))m(η(g(s)))m(g(s)))γhhhhhihhhhhhhhhh-r(s)(h-(t,s))γ+1(γ+1)γ+1δγ(s)(1-p(g(s))m(η(g(s)))m(g(s)))γ]
d
s=∞,
then (1) is oscillatory.

Proof.

Without loss of generality, assume again that (1) possesses a nonoscillatory solution x(t) such that x(t)>0, x(η(t))>0, and x(g(t))>0 on [t1,∞) for some t1≥t0. Then, for all t≥t1, (30) is satisfied and z(t)≥x(t)>0. It follows from (10) that there exists a T*≥t1 such that z′(t)>0 for all t≥T*. By virtue of (30), we have
(47)z(t)=z(T*)+∫T*t(r(s)(z′(s))γ)1/γr1/γ(s)ds≥r1/γ(t)z′(t)∫T*tr-1/γ(s)ds.
Since
(48)(z(t)m(t))′=z′(t)m(t)-z(t)m′(t)m2(t)≤z(t)m2(t)[m(t)r1/γ(t)∫T*tr-1/γ(s)ds-m′(t)]≤0,
we conclude that
(49)x(t)=z(t)-p(t)x(η(t))≥z(t)-p(t)z(η(t))≥(1-p(t)m(η(t))m(t))z(t).
For t≥T*, define a new function u(t) by
(50)u(t)∶=δ(t)r(t)(z′(t))γzγ(t).
Then u(t)>0 for all t≥T*, and the rest of the proof is similar to that of [8, Theorem 2.2, case T=R]. This completes the proof.

Theorem 8.

Let conditions (10) and (h1)–(h3) be satisfied. Suppose also that 0≤p(t)<1, η(t)≥t, g(t)≤t, and there exists a function m∈C1([t0,∞),(0,∞)) such that (45) holds for all sufficiently large T*≥t0. If there exists a function H∈H such that, for some T>T*,
(51)limsupt→∞1H(t,T)∫Tt[(1-p(g(s))m(η(g(s)))m(g(s)))γr(s)(h-(t,s))γ+1(γ+1)γ+1δγ(s)δ(s)θγ(s,T*)q(s)H(t,s)hhhhhhhhhhhhhhhh×(1-p(g(s))m(η(g(s)))m(g(s)))γhhhhhhhhhhhihhhh-r(s)(h-(t,s))γ+1(γ+1)γ+1δγ(s)(1-p(g(s))m(η(g(s)))m(g(s)))γ]
d
s=∞,
then (1) is oscillatory.

Proof.

The proof runs as in Theorem 7 and [8, Theorem 2.2, case T=R] and thus is omitted.

Theorem 9.

Let all assumptions of Theorem 7 be satisfied with condition (10) replaced by (17). Suppose that there exist a function K∈K and a function ϕ∈C1([t0,∞),(0,∞)) such that
(52)ϕ(t)r1/γ(t)R(t)+ϕ′(t)≤0,
and, for all sufficiently large T≥t0,
(53)limsupt→∞∫Tt[(ϕ(g(s))ϕ(s))γ-r(s)(ζ(t,s))γ+1(γ+1)γ+1K(t,s)q(s)(1-p(g(s)))γhhhhhhhhh×(ϕ(g(s))ϕ(s))γ-r(s)(ζ(t,s))γ+1(γ+1)γ+1]
d
s>0.
Then (1) is oscillatory.

Proof.

Without loss of generality, assume as above that (1) possesses a nonoscillatory solution x(t) such that x(t)>0, x(η(t))>0, and x(g(t))>0 on [t1,∞) for some t1≥t0. Then, for all t≥t1, (30) is satisfied and z(t)≥x(t)>0. Therefore, the function r(t)(z′(t))γ is strictly decreasing for all t≥t1, and so there exists a T*≥t1 such that either z′(t)>0 or z′(t)<0 for all t≥T*. Assume first that z′(t)>0 for all t≥T*. As in the proof of Theorem 7, we obtain a contradiction with (46). Assume now that z′(t)<0 for all t≥T*. For t≥T*, define ω(t) by (31). By virtue of η(t)≥t,
(54)x(t)=z(t)-p(t)x(η(t))≥z(t)-p(t)z(η(t))≥(1-p(t))z(t).
The rest of the proof is similar to that of Theorem 5 and hence is omitted.

Theorem 10.

Let all assumptions of Theorem 8 be satisfied with condition (10) replaced by (17). Suppose that there exists a function K∈K such that, for all sufficiently large T≥t0,
(55)limsupt→∞∫Tt[r(s)(ζ(t,s))γ+1(γ+1)γ+1K(t,s)q(s)(1-p(g(s)))γhhhhhhhihh-r(s)(ζ(t,s))γ+1(γ+1)γ+1]
d
s>0.
Then (1) is oscillatory.

Proof.

The proof resembles those of Theorems 5 and 9.

Remark 11.

One can obtain from Theorems 5 and 6 various oscillation criteria by letting, for instance,
(56)m(t)=R(t).
Likewise, several oscillation criteria are obtained from Theorems 7–10 with
(57)m(t)=∫T*tdsr1/γ(s),ϕ(t)=R(t).

3. Examples and Discussion

The following examples illustrate applications of theoretical results presented in this paper.

Example 1.

For t≥1, consider a neutral differential equation
(58)(t2(x(t)+p0x(t2))′)′+q0x(2t)=0,
where p0∈(0,1/2) and q0>0 are constants. Here, γ=1, r(t)=t2, p(t)=p0, η(t)=t/2, q(t)=q0, and g(t)=2t. Let m(t)=t-1 and K(t,s)=s-1(t-s)2. Then ζ(t,s)=2s-1/2+s-3/2(t-s) and, for all sufficiently large T≥1 and for all q0 satisfying q0(1-2p0)>1/2, we have
(59)limsupt→∞∫Tt[(m(g(s))m(s))γr(s)(ζ(t,s))γ+1(γ+1)γ+1K(t,s)q(s)(1-p(g(s))m(η(g(s)))m(g(s)))γhhhhhhhhh×(m(g(s))m(s))γ-r(s)(ζ(t,s))γ+1(γ+1)γ+1]ds=limsupt→∞∫Tt[q0(1-2p0)2(t-s)2s-shhhhhhhhhhhhh-(t-s)24s-(t-s)]ds>0.
On the other hand, letting H(t,s)=s-1(t-s)2 and δ(t)=1, we observe that condition (15) is satisfied for q0(1-2p0)>1/2. Hence, by Theorem 5, we conclude that (58) is oscillatory provided that q0(1-2p0)>1/2. Observe that results reported in [9, 12, 17, 21] cannot be applied to (58) since p(t)<1 and conditions (19)–(22) fail to hold for this equation.

Example 2.

For t≥1, consider a neutral differential equation
(60)(t3(x(t)+18x(t2))′)′+q0tx(t3)=0,
where q0>0 is a constant. Here, γ=1, r(t)=t3, p(t)=1/8, η(t)=t/2, q(t)=q0t, and g(t)=t/3. Let m(t)=t-2/2 and K(t,s)=s-2(t-s)2. Then ζ(t,s)=2s-1+2s-2(t-s). Hence,
(61)limsupt→∞∫Tt[r(s)(ζ(t,s))γ+1(γ+1)γ+1K(t,s)q(s)(1-p(g(s))m(η(g(s)))m(g(s)))γhhhhhhhhh-r(s)(ζ(t,s))γ+1(γ+1)γ+1]ds=limsupt→∞∫Tt[q0(t-s)22s-s-(t-s)2s-2(t-s)]ds>0
whenever q0>2. Let H(t,s)=s-2(t-s)2 and δ(t)=1. Then (16) is satisfied for q0>2. Therefore, using Theorem 6, we deduce that (60) is oscillatory if q0>2, whereas Theorems 1 and 4 yield oscillation of (60) for q0>5/2, so our oscillation result is sharper.

Example 3.

For t≥1, consider the Euler differential equation
(62)(t2x′(t))′+q0x(t)=0,
where q0>0 is a constant. Here, γ=1, r(t)=t2, p(t)=0, q(t)=q0, and g(t)=t. Choose m(t)=t-1 and K(t,s)=s-1(t-s)2. Then ζ(t,s)=2s-1/2+s-3/2(t-s), and so
(63)limsupt→∞∫Tt[(m(g(s))m(s))γ-r(s)(ζ(t,s))γ+1(γ+1)γ+1K(t,s)q(s)(1-p(g(s))m(η(g(s)))m(g(s)))γhhhhhhhh×(m(g(s))m(s))γ-r(s)(ζ(t,s))γ+1(γ+1)γ+1]ds=limsupt→∞∫Tt[q0(t-s)2s-s-(t-s)24s-(t-s)]ds>0
provided that q0>1/4. Let H(t,s)=s-1(t-s)2 and δ(t)=1. Then (15) holds for q0>1/4. It follows from Theorem 5 that (62) is oscillatory for q0>1/4, and it is well known that this condition is the best possible for the given equation. However, results of Saker [17] do not allow us to arrive at this conclusion due to condition (22).

Remark 12.

In this paper, using an integral averaging technique, we derive several oscillation criteria for the second-order neutral equation (1) in both cases (10) and (17). Contrary to [9, 12, 15, 17, 19–21], we do not impose restrictive conditions (H3) and (19)–(23) in our oscillation results. This leads to a certain improvement compared to the results in the cited papers. However, to obtain new results in the case where (17) holds, we have to impose an additional assumption on the function p; that is, p(t)<m(t)/m(η(t)). The question regarding the study of oscillatory properties of (1) with other methods that do not require this assumption remains open at the moment.

Conflict of Interests

The authors declare that they have no competing interests.

Authors’ Contribution

Both authors contributed equally to this work and are listed in alphabetical order. They both read and approved the final version of the paper.

Acknowledgments

The authors thank the referees for pointing out several inaccuracies in the paper. The research of the first author was supported by the AMEP of Linyi University, China.

AgarwalR. P.BohnerM.LiT.ZhangC.Oscillation of second-order Emden-Fowler neutral delay differential equationsAgarwalR. P.BohnerM.LiW.-T.AgarwalR. P.GraceS. R.Oscillation theorems for certain neutral functional differential equationsAgarwalR. P.GraceS. R.O'ReganD.AgarwalR. P.GraceS. R.O'ReganD.BaculíkováB.DžurinaJ.Oscillation theorems for second order neutral differential equationsBaculíkováB.DžurinaJ.Oscillation theorems for second-order nonlinear neutral differential equationsErbeL.HassanT. S.PetersonA.Oscillation criteria for nonlinear functional neutral dynamic equations on time scalesHanZ.LiT.SunS.SunY.Remarks on the paper [Appl. Math. Comput. 207 (2009) 388-396]HasanbulliM.RogovchenkoYu. V.Oscillation criteria for second order nonlinear neutral differential equationsKarpuzB.ÖcalanÖ.ÖztürkS.Comparison theorems on the oscillation and asymptotic behaviour of higher-order neutral differential equationsLiT.AgarwalR. P.BohnerM.Some oscillation results for second-order neutral differential equationsLiT.HanZ.ZhangC.LiH.Oscillation criteria for second-order superlinear neutral differential equationsLiT.RogovchenkoYu. V.Asymptotic behavior of higher-order quasilinear neutral differential equationsLiT.RogovchenkoYu. V.ZhangC.Oscillation of second-order neutral differential equationsLiT.RogovchenkoYu. V.ZhangC.Oscillation results for second-order nonlinear neutral differential equationsSakerS. H.Oscillation criteria for a second-order quasilinear neutral functional dynamic equation on time scalesSakerS. H.O'ReganD.New oscillation criteria for second-order neutral functional dynamic equations via the generalized Riccati substitutionSunS.LiT.HanZ.LiH.Oscillation theorems for second-order quasilinear neutral functional differential equationsSunS.LiT.HanZ.ZhangC.On oscillation of second-order nonlinear neutral functional differential equationsXuR.MengF.Some new oscillation criteria for second order quasi-linear neutral delay differential equations