Oscillation of Second-Order Neutral Differential Equations: Tongxing Li
Oscillation of Second-Order Neutral Differential Equations: Tongxing Li
Oscillation of Second-Order Neutral Differential Equations: Tongxing Li
201300029
Key words Oscillation, neutral differential equations, delayed argument, advanced argument
MSC (2010) 34K11
We study oscillatory behavior of a class of second-order neutral differential equations under the assumptions
that allow applications to differential equations with both delayed and advanced arguments, and not only. New
theorems complement and improve a number of results reported in the literature. Illustrative examples are
provided.
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1 Introduction
This paper is concerned with the oscillatory behavior of solutions to a nonlinear second-order neutral differential
equation
γ
a(t) [z (t)] + q(t)x β (σ (t)) = 0, (1.1)
where t ∈ I := [t0 , ∞), t0 ∈ R, z(t) := x(t) + p(t)x(τ (t)), the functions a, p, q, τ , σ ∈ C(I) and a, p, q take on
positive values. We also assume that β, γ ∈ R, where R is the set containing all ratios of odd natural numbers.
The analysis of qualitative properties of Eq. (1.1) is important for applications since, for instance, its particular
case, an Emden-Fowler type equation
γ
a(t) [x (t)] + q(t)x β (σ (t)) = 0,
has numerous applications in mathematical and theoretical physics; see [21].
By a solution of Eq. (1.1) we mean a function x ∈ C([Tx , ∞)), Tx ≥ t0 , such that a (z )γ ∈ C1 ([Tx , ∞))
and x(t) satisfies (1.1) on [Tx , ∞). We consider only those solutions x(t) of (1.1) that satisfy the condition
sup{|x(t)| : t ≥ T } > 0, for all T ≥ Tx , and we tacitly assume that (1.1) possesses such solutions. As usual, a
solution x(t) of (1.1) is called oscillatory if it has arbitrarily large zeros on [Tx , ∞); otherwise, it is termed
non-oscillatory. Eq. (1.1) is called oscillatory if all its solutions are oscillatory.
During the last three decades, conditions for oscillatory or non-oscillatory behavior of various classes of
differential equations have been frequently discussed in the literature; see, e.g., the papers [1]–[25] and the
references cited therein. In what follows, we briefly comment on related results that motivated our study. In one of
the first important contributions to the field, Grammatikopoulos et al. [8] proved that a linear second-order neutral
differential equation
(x(t) + p(t)x(t − τ )) + q(t)x(t − σ ) = 0 (1.2)
is oscillatory provided that 0 ≤ p(t) ≤ 1 and
∞
q(s)(1 − p(s − σ )) ds = ∞.
t0
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Math. Nachr. 288, No. 10 (2015) / www.mn-journal.com 1151
Oscillation criteria for Eq. (1.2) which require that q(t) ≥ q0 > 0, p1 ≤ p(t) ≤ p2 , and p(t) is not eventually
negative can be found in the monograph by Erbe et al. [7]. Later on, Xu and Xia [24] proved that the conditions
0 ≤ p(t) < ∞ and q(t) ≥ M > 0 ensure oscillation of (1.2). Recently, Baculı́ková and Džurina [3], Han et al.
[10], and Li et al. [17] studied asymptotic behavior of solutions to a neutral differential equation
(r (t)[x(t) + p(t)x(τ (t))] ) + q(t)x(σ (t)) = 0
under the assumptions that 0 ≤ p(t) ≤ p0 < ∞ and τ ◦ σ = σ ◦ τ ; further extensions of these results can be
found in another paper by Baculı́ková and Džurina [4].
For γ ≥ 1 and 0 ≤ p(t) ≤ p0 < ∞, Baculı́ková et al. [5] established oscillation results for a general class of
even-order neutral differential equations which also includes a second-order differential equation
γ
r (t) ([x(t) + p(t)x(τ (t))] ) + q(t)x γ (σ (t)) = 0 (1.3)
assuming that σ (t) ≤ τ (t) ≤ t, whereas Li et al. [16] deduced oscillation of Eq. (1.3) under the assumptions
that τ ◦ σ = σ ◦ τ, τ (t) ≥ t, and σ (t) ≥ t. In the case where 0 ≤ p(t) < 1, Eq. (1.3), its particular cases and
modifications have been studied by Dong [6], Hasanbulli and Rogovchenko [11], [12], Liu and Bai [18], and Xu
and Meng [22], [23]. For p(t) > 1, Li et al. [15] explored oscillatory properties of Eq. (1.3) assuming that either
τ (t) > t or τ (t) < t. One should also note that results reported by Baculı́ková et al. [5] and Li et al. [15], [16] do
not apply to Eq. (1.1) when γ
= β.
Very recently, Zhong et al. [25] established oscillation results for Eq. (1.1) assuming that a (t) ≥ 0, p(t) =
p0
= 1, τ (t) = t − τ0 , τ0 ≥ 0, and σ (t) ≤ t − τ0 , although their results fail to apply to Eq. (1.1) for σ (t) ≥ t.
The problem of oscillation of Eq. (1.1) in the case where σ (t) ≤ t has been also discussed by Baculı́ková and
Džurina [4]. However, the latter study left without answers questions regarding oscillation of (1.1) in the following
two cases: (i) β ≥ γ ; (ii) τ (t) ≤ σ (t) ≤ t.
This study was strongly motivated by the research of Baculı́ková and Džurina [4] and Zhong et al. [25]. Its
purpose is not only to analyze oscillation of Eq. (1.1) in two cases mentioned above, but to derive also new
oscillation criteria for (1.1) without imposing the assumption σ (t) ≤ t. Main results in this paper are organized
into three parts in accordance with different assumptions on the coefficient p(t). In Section 2, new oscillation
results for Eq. (1.1) are established in the case where p(t) is bounded, 0 ≤ p(t) ≤ p0 < ∞. In Section 3,
different methods are used to derive new oscillation criteria for (1.1) in a very important particular case when
p(t) is constant, 0 ≤ p(t) = p0
= 1. Finally, in Section 4, we present oscillation theorems for (1.1) for p(t) ≥ 1
assuming that p(t) may be even unbounded.
As usual, all functional inequalities considered in this paper are supposed to be satisfied for all t large enough.
Without loss of generality, we deal only with positive solutions of Eq. (1.1) since under our assumptions −x(t) is
obviously a solution of this equation provided that x(t) is a solution.
In the sequel, we use the notation τ −1 for the function which is inverse to τ ; Q(t) := min{q(t), q(τ (t))},
∞ t
A(t) := a −1/γ (s) ds, B(t) := a −1/γ (s) ds,
t t1
α
Q α (t) := Q(t)[B(η(t))] , and Q β (t) := Q(t)[B(η(t))]β ,
where t1 is large enough and the choice of α is to be specified later.
We need the following auxiliary result due to Baculı́ková and Džurina [4, Lemma 3].
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1152 T. Li and Yu. V. Rogovchenko: Oscillation of second-order neutral differential equations
Lemma 2.1 Assume that x(t) is a positive solution of (1.1). Then, z(t) > 0, z (t) > 0, and (a(t) [z (t)]γ ) < 0,
for all t large enough.
Now we state and prove four theorems that ensure oscillation of Eq. (1.1).
Theorem 2.2 Let 0 < β ≤ 1 and η(t) < t ≤ τ (t). Suppose that conditions (H1 )−(H3 ) are satisfied, and there
exists a number α ∈ R such that α ≤ β and α < γ . If
∞
Q α (t) dt = ∞, (2.1)
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Then, by virtue of [19, Theorem 1], the associated delay differential equation
α/γ
τ∗
y (t) + M β−α Q α (t)y α/γ (η(t)) = 0 (2.11)
τ∗ + p0 β
also has a positive solution. However, [13, Theorem 2] implies that under the assumption (2.1) Eq. (2.11) is
oscillatory. Therefore, Eq. (1.1) cannot have positive solutions. This contradiction with our initial assumption
completes the proof.
Theorem 2.3 Suppose that 0 < β = γ ≤ 1 and η(t) ≤ t ≤ τ (t). If conditions (H1 )−(H3 ) are satisfied and
t
τ∗ 1
lim inf Q β (s) ds > , (2.12)
τ∗ + p0 β t→∞ η(t) e
Eq. (1.1) is oscillatory.
P r o o f . Assume that Eq. (1.1) is non-oscillatory, and let x(t) be a positive solution defined on I. Proceeding
as in the proof of Theorem 2.2, one comes to the conclusion that a delay differential equation
τ∗
y (t) + Q β (t)y(η(t)) = 0 (2.13)
τ∗ + p0 β
has a positive solution y(t). On the other hand, application of condition (2.12) along with [4, Lemma 4] implies
that Eq. (2.13) is oscillatory, a contradiction. The proof is complete.
In order to illustrate efficiency of our theorems, we recall a recent result due to Baculı́ková and Džurina [4,
Corollary 1] that reads as follows.
Corollary 2.4 Suppose that 0 < β ≤ 1, β ≤ γ , and σ (t) ≤ t ≤ τ (t). Assume also that conditions (H1 )−(H3 )
are satisfied and
β/γ t β
σ (s)
τ∗ −1/γ 1
lim inf Q(s) a (v) dv ds > .
τ∗ + p0 β t→∞ σ (t) t1 e
Then Eq. (1.1) is oscillatory.
Remark 2.5 Note that for β = γ , Theorem 2.3 extends [4, Corollary 1] because in our result the assumption
σ (t) ≤ t is no longer required.
Example 2.6 Consider a second-order nonlinear neutral differential equation
b
(x(t) + 2x(θ t)) + x 1/3 (λt) = 0, (2.14)
t 4/3
where θ ≥ 1, b > 0, and λ ∈ (0, ∞). An application of Corollary 2.4 yields oscillation of Eq. (2.14) for any
λ ∈ (0, 1) provided that
θ −1 1 1
bλ1/3 ln > .
(θ + 21/3 )1/3 λ e
However, choosing in Theorem 2.2 γ = 1, α = β = 1/3, λ1 ∈ (0, 1), η(t) = λ1 t ≤ λt, one concludes that
Eq. (2.14) is oscillatory for all λ ∈ (0, ∞) and b > 0, which demonstrates superiority of Theorem 2.2 over
Corollary 2.4.
Theorem 2.7 Let 0 < β ≤ 1 and η(t) < τ (t) ≤ t. Assume also that there exists a number α ∈ R such that
α ≤ β and α < γ . If conditions (H1 )−(H3 ) and (2.1) are satisfied, Eq. (1.1) is oscillatory.
P r o o f . As above, let x(t) be a positive solution of Eq. (1.1). It has been established in the proof of Theorem
2.2 that the function ω(t) defined by (2.6) is positive, decreasing, and satisfies the inequality (2.8). Introducing
again y(t) by (2.9) and using the monotonicity of ω(t), we conclude that
p0 β
y(t) ≤ ω(τ (t)) 1 + . (2.15)
τ∗
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1154 T. Li and Yu. V. Rogovchenko: Oscillation of second-order neutral differential equations
Substitution of (2.15) into (2.8) implies that, for sufficiently large t, y(t) is a positive solution of a delay differential
inequality
α/γ
β−α τ∗
y (t) + M Q α (t)y α/γ τ −1 (η(t)) ≤ 0. (2.16)
τ∗ + p0 β
By virtue of [19, Theorem 1], a delay differential equation
α/γ
β−α τ∗
y (t) + M Q α (t)y α/γ τ −1 (η(t)) = 0 (2.17)
τ∗ + p0 β
associated with (2.16) also has a positive solution. Using condition (2.1) in [13, Theorem 2], one concludes that
Eq. (2.17) is oscillatory, which contradicts the fact that y(t) is a positive solution of (2.17). The proof is complete
now.
Theorem 2.8 Suppose that 0 < β = γ ≤ 1 and η(t) ≤ τ (t) ≤ t. If conditions (H1 )−(H3 ) are satisfied and
t
τ∗ 1
lim inf Q β (s) ds > , (2.18)
τ∗ + p0 β t→∞ τ −1 (η(t)) e
τ∗
y (t) + Q β (t)y τ −1 (η(t)) = 0 (2.19)
τ∗ + p0 β
has an eventually positive solution y(t). On the other hand, using condition (2.18) in [4, Lemma 4], one deduces
that Eq. (2.19) is oscillatory. This contradiction completes the proof.
Remark 2.9 In this section, assuming as Baculı́ková and Džurina [4] that τ ◦ σ = σ ◦ τ and using a modified
technique, we established new oscillation criteria that provide answers to two open problems formulated in the
cited paper. We also improved some of results reported there, cf. Example 2.6. Note that oscillation theorems can
be also obtained for β ≥ 1; in this case, one simply has to replace Q(t) := min{q(t), q(τ (t))} in [4, Lemma 1]
with a function Q(t) := 21−β min{q(t), q(τ (t))} and proceed as above.
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Consequently,
l1
lim x(t) = >0
t→∞ 1 + p0
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1156 T. Li and Yu. V. Rogovchenko: Oscillation of second-order neutral differential equations
and
x(σ (t) − τ 0 )
lim = 1.
t→∞ x(σ (t))
The latter condition means that, for any ∈ (0, 1) and for t large enough,
x(σ (t) − τ0 )
1− < < 1 + .
x(σ (t))
Then,
x(σ (t)) x(σ (t)) 1 1
= = ≥ . (3.10)
z(σ (t)) x(σ (t)) + p0 x(σ (t) − τ0 ) x(σ (t)−τ0 )
1 + p0 x(σ (t)) 1 + p 0 (1 + )
Combining (1.1) with (3.10) and using the fact that η(t) ≤ σ (t), we observe that z(t) is a positive solution of a
delay differential inequality
γ q(t)
a(t) [z (t)] + z β (η(t)) ≤ 0.
(1 + p0 (1 + ))β
Repetition of an argument similar to the one used above leads to the conclusion that the associated delay differential
equation
M β−α
u (t) + q(t)[B(η(t))]α u α/γ (η(t)) = 0 (3.11)
(1 + p0 β )α/γ
also has a positive solution. Using condition (3.1) in [13, Theorem 2], we conclude that (3.11) is oscillatory. This
contradiction completes the proof.
Theorem 3.2 Let 0 < β = γ ≤ 1 and τ0 ≤ 0. Suppose that assumptions (H3 )−(H5 ) are satisfied. If, for some
sufficiently large t1 ≥ t0 ,
t
1 1
lim inf R(s)[B(η(s))]β ds > , (3.12)
1 + p0 β t→∞ η(t) e
Eq. (1.1) is oscillatory.
P r o o f . As in the proof of Theorem 3.1, one can show that the differential equation
1
y (t) + R(t)[B(η(t))]β y(η(t)) = 0 (3.13)
1 + p0 β
has a positive solution y(t) and thus, equation
q(t)
u (t) + [B(η(t))]β u(η(t)) = 0 (3.14)
(1 + p0 (1 + ))β
also has a positive solution u(t). On the other hand, using (3.12) in [4, Lemma 4], we conclude that Eq. (3.13) is
oscillatory, and so is (3.14). This contradiction completes the proof.
Example 3.3 Consider a second-order nonlinear neutral differential equation
1/3
b
t 1/4 (x(t) + 2x(t − τ0 )) + 13/12 x 1/3 (λt) = 0, (3.15)
t
where τ0 < 0, b > 0, and λ ∈ [1, ∞). Let η(t) = λ1 t, for some λ1 ∈ (0, 1). Using Theorem 3.2, we conclude that
Eq. (3.15) is oscillatory provided that
41/3 1 1
bλ1 1/12 ln > .
1 + 21/3 λ1 e
Observe that neither results reported in [4] nor those in Section 2 can be applied to Eq. (3.15) since, for λ
= 1,
one has λ(t − τ0 )
= λt − τ0 . Furthermore, oscillation criteria reported by Zhong et al. [25] also fail to apply to
Eq. (3.15) since λ ≥ 1.
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Theorem 3.4 Let 0 < β ≤ 1, τ0 ≥ 0, and η(t) < t − τ0 . Assume also that there exists an α ∈ R such that
α ≤ β and α < γ . If, for some sufficiently large t1 ≥ t0 , conditions (H3 )−(H5 ) are satisfied and (3.1) holds,
Eq. (1.1) is oscillatory.
P r o o f . Let x(t) be a positive solution of (1.1). It has been established in Theorem 3.1 that the function ω(t)
defined by (2.6) is decreasing and satisfies (3.6). Introducing a new function u(t) by (3.7), one observes that, by
virtue of (3.7),
u(t) ≤ ω(t − τ0 ) 1 + p0 β . (3.16)
Substituting (3.16) into (3.6), we conclude that u(t) is a positive solution of a delay differential inequality
M β−α
u (t) + R(t)[B(η(t))]α u α/γ (η(t) + τ0 ) ≤ 0.
(1 + p0 β )α/γ
It follows then from [19, Theorem 1] that the associated delay differential equation
M β−α
u (t) + R(t)[B(η(t))]α u α/γ (η(t) + τ0 ) = 0 (3.17)
(1 + p0 β )α/γ
also has a positive solution. On the other hand, using condition (3.1) in [13, Theorem 2], we conclude that Eq.
(3.17) is oscillatory. This contradiction completes the proof.
Theorem 3.5 Let 0 < β = γ ≤ 1, τ0 ≥ 0, and η(t) ≤ t − τ0 . If, for some sufficiently large t1 ≥ t0 , assumptions
(H3 )−(H5 ) are satisfied and
t
1 1
lim inf R(s)[B(η(s))]β ds > , (3.18)
1 + p0 β t→∞ η(t)+τ0 e
Eq. (1.1) is oscillatory.
P r o o f . As in the proof of Theorem 3.4, application of [19, Theorem 1] yields that the differential equation
1
y (t) + R(t)[B(η(t))]β y(η(t) + τ0 ) = 0 (3.19)
1 + p0 β
has a positive solution. However, by virtue of condition (3.18), [4, Lemma 4] implies that Eq. (3.19) is oscillatory.
This contradiction completes the proof.
Remark 3.6 Results presented in this section complement and improve several oscillation criteria reported
in [3], [4], [5], [16], [17], [25] because our theorems apply even if σ (t) − τ0
≡ σ (t − τ0 ) and σ (t) ≥ t − τ0 .
Oscillation criteria for Eq. (1.1) for β ≥ 1 can be deduced in a similar manner. To this end, while using [4, Lemma
1], one simply has to define R(t) := 21−β min{q(t), q(t − τ0 )}. Note that as in the paper by Zhong et al. [25],
methods used in this section require that τ (t) = t − τ0 and p(t) = p0 .
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1158 T. Li and Yu. V. Rogovchenko: Oscillation of second-order neutral differential equations
Theorem 4.2 Assume that there exist α, λ ∈ R such that α ≤ β ≤ λ and α < γ < λ. Suppose further that
conditions (H6 )−(H8 ) hold, and there exist four functions δ∗ , δ ∗ , η∗ , η∗ ∈ C(I) such that δ∗ (t) ≤ t ≤ δ ∗ (t),
η∗ (t) < t < η∗ (t),
τ (δ∗ (t)) ≤ t ≤ τ (δ ∗ (t)), τ (η∗ (t)) ≤ σ (t) ≤ τ (η∗ (t)), (4.1)
and
lim η∗ (t) = lim δ∗ (t) = ∞.
t→∞ t→∞
and
∞
q(t)( p∗ (σ (t)))β Aλ (η∗ (t)) dt = ∞, (4.3)
where
∗ 1B τ −1 (δ ∗ (t)) 1
p (t) := 1−
p(τ −1 (t)) B(τ −1 (t)) p(τ −1 (τ −1 (t)))
and
1 A τ −1 (δ∗ (t)) 1
p∗ (t) := 1− .
p (τ −1 (t)) A(τ −1 (t)) p(τ −1 (τ −1 (t)))
Then (1.1) is oscillatory.
P r o o f . Let x(t) be a positive solution of (1.1). By Lemma 4.1, z (t) does not change sign eventually.
Case I. Assume first that z (t) > 0. Using the definition of z(t) and (4.1), we obtain
1 −1
x(t) = z τ (t) − x τ −1 (t)
p(τ −1 (t))
1 −1 z τ −1 τ −1 (t) − x τ −1 τ −1 (t)
= z τ (t) −
p(τ −1 (t)) p (τ −1 (τ −1 (t)))
1 −1 z τ −1 τ −1 (t)
≥ z τ (t) −
p(τ −1 (t)) p (τ −1 (τ −1 (t)))
1 −1 z τ −1 (δ ∗ (t))
≥ z τ (t) − . (4.4)
p(τ −1 (t)) p (τ −1 (τ −1 (t)))
By Lemma 4.1, (a(t) [z (t)]γ ) < 0. Therefore,
t t
γ 1/γ −1/γ
z(t) ≥ a(s) [z (s)] a (s) ds ≥ a 1/γ (t)z (t) a −1/γ (s) ds. (4.5)
t1 t1
Consequently,
z(t)
≤ 0. (4.6)
B(t)
Using (4.6) and the condition δ ∗ (t) ≥ t, we conclude that
−1 ∗ B τ −1 (δ ∗ (t)) −1
z τ (δ (t)) ≤ z τ (t) .
B(τ −1 (t))
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other hand, our results complement those by Li et al. [10], [16] since our conditions allow unbounded functions
p(t).
Remark 4.6 We point out versatility of the obtained results with respect to the behavior of the functions τ and
σ . Our theorems apply in the cases where τ (t) ≤ t, τ (t) ≥ t, as well as when τ (t) − t oscillates. Similarly, σ (t)
can be a delayed argument, an advanced argument, and σ (t) − t can also oscillate.
Remark 4.7 As in the paper by Agarwal et al. [1], in the case where p(t) ≥ 1, we require τ to be strictly
increasing. The question regarding oscillatory behavior of Eq. (1.1) without monotonicity assumption on τ remains
open at the moment.
Acknowledgements The authors thank three anonymous referees for careful reading of the manuscript and useful comments
that helped to correct several inaccuracies in the original submission. The research of the first author is supported by the AMEP
of Linyi University, P. R. China.
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