INTERVAL OSCILLATION CRITERIA FOR HIGHER ORDER NEUTRAL NONLINEAR DIFFERENTIAL EQUATIONS WITH DEVIATING ARGUMENTS | Xu | Journal of the Indonesian Mathematical Society 41 139 1 PB
J. Indones. Math. Soc. (MIHMI)
Vol. 15, No. 1 (2009), pp. 21–35.
INTERVAL OSCILLATION CRITERIA
FOR HIGHER ORDER NEUTRAL
NONLINEAR DIFFERENTIAL EQUATIONS
WITH DEVIATING ARGUMENTS
Run Xu, Fanwei Meng
Abstract. Some oscillation criteria for nth order neutral differential equations with
deviating arguments of the form
[r(t)|(y(t) + p(t)y(τ (t)))
(n−1) α−1
|
(y(t) + p(t)y(τ (t)))
(n−1) ′
] +
m
X
qi (t)fi (y(σi (t))) = 0
i=1
n even are established. New oscillation criteria are different from most known ones in
the sense that they based on a class of new function H(t, s) defined in the sequel. The
results are sharper than some previous results which can be seen by the examples at the
end of this paper.
1. INTRODUCTION
In this paper we consider the oscillation behavior of solutions of the n-th
order neutral differential equations of the form
[r(t)|(y(t) + p(t)y(τ (t)))(n−1) |α−1 (y(t) + p(t)y(τ (t)))(n−1) ]′
m
X
qi (t)fi (y(σi (t))) = 0, (1)
+
i=1
Received 07-10-2008, Accepted 27-12-2009.
2000 Mathematics Subject Classification: 34C10.
Key words and Phrases: Oscillation; Even Order; Nonlinear Differential Equations; Neutral type.
This research was partial supported by the NNSF of China(10771118) and EDF of shandong Province,
China(J07yh05)
21
22
Run Xu and Fanwei Meng
where t ≥ t0 , n ≥ 2 is even integer, α > 0 are constant. In this paper, we assume
that
(I1 ) p(t) ∈ C([t0 , ∞); [0, 1)), qi (t) ∈ C([t0 , ∞); [0, ∞)), fi ∈ C(R; R), i = 1, 2, · · · , m,
t ∈ [t0 , ∞).
Rt
(I2 ) r(t) ∈ C 1 ([t0 , ∞); (0, ∞)), r′ (t) ≥ 0, R1 (t) := t0 1ds → ∞(t → ∞).
r α (s)
(I3 )
fi (x)
|x|α−1 x
≥ βi > 0 for x 6= 0, βi are constants, i = 1, 2, · · · , m.
(I4 ) τ (t), σi (t) ∈ C 1 ([t0 , ∞); [0, ∞)), τ (t) ≤ t, σi (t) ≤ t, σ ′ (t) > 0 for t ≥ t0
and lim σi (t) = lim σ(t) = lim τ (t) = ∞, i = 1, 2, · · · , m, where σ(t) ≤
t→∞
t→∞
t→∞
min{σ1 (t), σ2 (t), · · · , σm (t), 2t }.
By a solution of Eq. (1), we mean a function y(t) ∈ C n−1 ([Tx , ∞); R) for
some Tx ≥ t0 which has the property that
r(t)|(y(t) + p(t)y(τ (t)))(n−1) |α−1 (y(t) + p(t)y(τ (t)))(n−1) ∈ C 1 ([Tx , ∞); R)
and satisfies Eq. (1) on [Tx , ∞).
A nontrivial solution of Eq. (1) is called oscillatory if it has arbitrary large
zero. Otherwise, it is called nonoscillatory. Eq. (1) is called oscillatory if all of its
solutions are oscillatory.
If p(t) = 0, r(t) = 1, m = 1, then Eq. (1) becomes
(|x(n−1) (t)|α−1 x(n−1) (t))′ + q(t)f (x(σ(t))) = 0
(2)
and the related equations have been studied by Agarwal et. al. [2], Xu et. al. [15].
Eq. (1) with n = 2, p(t) = 0, m = 1, namely, the equation
[r(t)|x′ (t)|α−1 x′ (t)]′ + q(t)f (x(σ(t))) = 0
(3)
and related equations have been investigated by Dzurina and Stavroulakis [4], Sun
and Meng [14], Mirzov [10-12], Elbert [5,6] Agarwal et. al. [1], Chern et. al. [3],
Li [7], Zhuang and Li [19].
Recently, Xu and Meng [16-18] have studied the oscillation properties of Eq.
(1) for n = 2. Very recently Meng and Xu [8,9] have investigated the oscillation
properties for higher order neutral differential equations.
Motivated by the idea of Li [7], by using averaging functions and inequality,
in this paper we obtain several new interval criteria for oscillation, that is, criteria
are given by the behavior of Eq. (1) (or of r, p and qi ) only on a sequence of
subintervals of [t, ∞). Our results improve and extend the results of Li [7] and
Zhuang and Li [19]. In order to prove our Theorems, we use the function class X
to study the oscillatory of Eq. (1). We say that a function H = H(t, s) belongs to
the function class X, if H ∈ C(D; R+ ), where D = {(t, s) : t0 ≤ s ≤ t < ∞}, which
23
Interval Oscillation Criteria
∂H
∂H
and
satisfies H(t, t) = 0, H(t, s) > 0 for t > s, and has partial derivative
∂s
∂t
on D such that
p
p
∂H
∂H
= h1 (t, s) H(t, s),
= −h2 (t, s) H(t, s),
∂t
∂s
(4)
where h1 (t, s), h2 (t, s) are locally nonnegative continuous functions on D.
2. MAIN RESULTS
First, we give the following lemmas for our results.
Lemma 2.1. [13] Let u(t) ∈ C n ([t0 , ∞); R+ ). If u(n) (t) is eventually of one sign
for all large t, say t1 > t0 , then there exist a tx > t0 and an integer l, 0 ≤ l ≤ n,
with n + l even for u(n) (t) ≥ 0 or n + l odd for u(n) (t) ≤ 0 such that l > 0
implies that u(k) (t) > 0 for t > tx , k = 0, 1, 2, · · · , l − 1, and l ≤ n − 1 implies that
(−1)l+k u(k) (t) > 0 for t > tx , k = l, l + 1, · · · , n − 1.
Lemma 2.2. [13] If the function u(t) is as in Lemma 2.1 and u(n−1) (t)u(n) (t) ≤ 0
for t > tx , then there exists a constant θ, 0 < θ < 1, such that
u(t) ≥
θ
tn−1 u(n−1) (t) for all large t.
(n − 1)!
and
t
θ
u′ ( ) ≥
tn−2 u(n−1) (t) for all large t.
2
(n − 2)!
Lemma 2.3. Suppose that y(t) is an eventually positive solution of Eq. (1), let
z(t) = y(t) + p(t)y(τ (t)),
(5)
then there exists a number t1 ≥ t0 such that
z(t) > 0, z ′ (t) > 0, z (n−1) (t) > 0 and z (n) (t) ≤ 0, t ≥ t1 .
(6)
Proof. Since y(t) is an eventually positive solution of (1), from (I4 ), there exists a
number t1 ≥ t0 such that
y(t) > 0, y(τ (t)) > 0, y(σi (t)) > 0, t ≥ t1 .
(7)
24
Run Xu and Fanwei Meng
Noting that p(t) ≥ 0, we have z(t) > 0, t ≥ t1 and from (I1 ), (I3 ) we have
(r(t)|z (n−1) (t)|α−1 z (n−1) (t))′ = −
m
X
i=1
qi (t)fi (y(σi (t))) ≤ 0, t ≥ t1 .
So r(t)|z (n−1) (t)|α−1 z (n−1) (t) is decreasing and z (n−1) (t) is eventually of one sign.
we claim that
z (n−1) (t) ≥ 0 for t ≥ t1 .
(8)
Otherwise, if there exist a te1 ≥ t1 such that z (n−1) (te1 ) < 0, then for all t ≥ te1 ,
r(t)|z (n−1) (t)|α−1 z (n−1) (t) ≤ r(e
t1 )|z (n−1) (e
t1 )|α−1 z (n−1) (e
t1 ) = −C(C > 0),
then we have −z (n−1) (t) ≥
te1 to t, we have
C
r(t)
(9)
1
α
, t ≥ te1 , integrating the above inequality from
1
z (n−2) (t) ≤ z (n−2) (te1 ) − C α (R(t) − R(te1 )).
Letting t → ∞, from (I2 ), we get lim z (n−2) (t) = −∞, which implies z (n−1) (t)
t→∞
and z (n−2) (t) are negative for all large t, from Lemma 2.1, no two consecutive
derivatives can be eventually negative, for this would imply that lim z(t) = −∞,
t→∞
a contradiction. Hence z (n−1) (t) ≥ 0 for t ≥ t1 . from Eq. (1) and (I1 ), (I2 ) we have
αr(t)(z (n−1) (t))α−1 z (n) (t) = [r(t)(z (n−1) (t))α ]′ − r′ (t)(z (n−1) (t))α
m
X
qi (t)fi (y(σi (t))) − r′ (t)(z (n−1) (t))α ≤ 0, t ≥ t1 ,
= −
i=1
this implies that z (n) (t) ≤ 0, t ≥ t1 . From Lemma 2.1 again (note n is even), we
have z ′ (t) > 0, t ≥ t1 . This completes the proof.
Theorem 2.1. Assume that there exist a positive, nondecreasing function ρ(t) ∈
C 1 ([t0 , ∞)) such that for any constant M > 0, some H ∈ X and for each sufficient large T0 ≥ t0 , there exist increasing divergent sequences of positive numbers
{an }, {bn }, {cn } with T0 ≤ an < cn < bn such that
Z cn
Z bn
1
1
H(s, an )ρ(s)C1 (s)ds +
H(bn , s)ρ(s)C1 (s)ds
H(cn , an ) an
H(bn , cn ) cn
α+1
Z cn
p
C2 (s)
ρ′ (s)
1
ds
>
h1 (s, an ) + H(s, an )
α−1
H(cn , an ) an
ρ(s)
H 2 (s, an )
α+1
Z bn
p
1
ρ′ (s)
C2 (s)
h2 (bn , s) + H(bn , s)
ds,
(10)
+
α−1
H(bn , cn ) cn
ρ(s)
H 2 (bn , s)
Interval Oscillation Criteria
25
where
C1 (t) =
m
X
i=1
βi qi (t)(1 − p(σi (t)))α , C2 (t) =
r(t)ρ(t)
,
M α (α + 1)α+1 (σ ′ (t)σ n−2 (t))α
then every solution of Eq. (1) is oscillatory.
Proof. Suppose the contrary, let y(t) is a nonoscillatory solution of Eq. (1), without
loss of generality we assume
y(t) > 0, y(τ (t)) > 0 for t ≥ t1 ≥ t0 .
Then
z(t) = y(t) + p(t)y(τ (t)) > 0 for t ≥ t1 ≥ t0 .
(11)
From Lemma 2.3, there exists t2 ≥ t1 such that
z(t) > 0, z ′ (t) > 0, z (n−1) (t) > 0 and z (n) (t) ≤ 0, t ≥ t2 .
(12)
It is easy to check that we can apply Lemma 2.2 and conclude that there exist
0 < θ < 1 and t3 > t2 such that
z ′ (σ(t)) ≥
≥
θ
(2σ(t))n−2 z (n−1) (2σ(t))
(n − 2)!
θ
2n−2 σ n−2 (t)z (n−1) (t) = M σ n−2 (t)z (n−1) (t), t ≥ t3 ,
(n − 2)!
(13)
θ
2n−2 .
(n − 2)!
From (5), we have
where M =
y(t) = z(t) − p(t)y(τ (t)) ≥ z(t) − p(t)z(τ (t)) ≥ z(t)(1 − p(t)), t ≥ t3 .
(14)
Since lim σ(t) = ∞, there exists t4 ≥ t3 such that σ(t) ≥ t3 , t ≥ t4 , so
t→∞
y(σ(t)) ≥ z(σ(t))(1 − p(σ(t))), t ≥ t4 .
(15)
By (I3 ) and (15) we get
fi (y(σi (t))) ≥ βi y α (σi (t)) ≥ βi z α (σi (t))(1 − p(σi (t)))α , t ≥ t4 .
From (1), (16), we get
(16)
26
Run Xu and Fanwei Meng
0 = [r(t)(z (n−1) (t))α ]′ +
≥ [r(t)(z (n−1) (t))α ]′ +
≥ [r(t)(z (n−1) (t))α ]′ +
m
X
i=1
m
X
i=1
m
X
i=1
βi qi (t)fi (y(σi (t)))
βi qi (t)z α (σi (t))(1 − p(σi (t)))α
βi qi (t)z α (σ(t))(1 − p(σi (t)))α , t ≥ t4 .
(17)
Let
w(t) = ρ(t)
r(t)(z (n−1) (t))α
, t ≥ t4 ,
z α (σ(t))
(18)
clearly, w(t) > 0, from (13), (17) and (18) we get
w′ (t) =
ρ′ (t)
[r(t)(z (n−1) (t))α ]′
w(t) + ρ(t)
ρ(t)
z α (σ(t))
r(t)(z (n−1) (t))α αz α−1 (σ(t))z ′ (σ(t))σ ′ (t)
z 2α (σ(t))
m
X
ρ′ (t)
βi qi (t)(1 − p(σi (t)))α
w(t) − ρ(t)
≤
ρ(t)
i=1
− ρ(t)
r(t)(z (n−1) (t))α z ′ (σ(t))
,
z α+1 (σ(t))
m
X
ρ′ (t)
≤
βi qi (t)(1 − p(σi (t)))α
w(t) − ρ(t)
ρ(t)
i=1
− ασ ′ (t)ρ(t)
− αM σ ′ (t)σ n−2 (t)
w
α+1
α
(t)
1
(r(t)ρ(t)) α
α+1
≤
w α (t)
ρ′ (t)
w(t) − ρ(t)C1 (t) − αM σ ′ (t)σ n−2 (t)
1 .
ρ(t)
(r(t)ρ(t)) α
Then from above inequality we have
α+1
w α (t)
ρ′ (t)
w(t) − αM σ ′ (t)σ n−2 (t)
ρ(t)C1 (t) ≤ −w (t) +
1 .
ρ(t)
(r(t)ρ(t)) α
′
(19)
Multiplying (19) by H(s, t), integrating it with respect s from t to cn and
27
Interval Oscillation Criteria
using (4) we get that
Z
Z cn
H(s, t)ρ(s)C1 (s)ds ≤ −
cn
′
w (s)H(s, t)ds +
t
t
Z
Z
cn
H(s, t)
t
cn
ρ′ (s)
w(s)ds
ρ(s)
α+1
α
(s)
1 ds
α
(r(s)ρ(s))
t
Z cn
p
w(s)h1 (s, t) H(s, t)ds
= −H(cn , t)w(cn ) +
t
Z cn
ρ′ (s)
+
H(s, t)
w(s)ds
ρ(s)
t
Z cn
α+1
w α (s)
′
n−2
H(s, t)σ (s)σ
(s)
− αM
1 ds
(r(s)ρ(s)) α
t
= −H(cn , t)w(cn )
Z cn
p
ρ′ (s)
h1 (s, t) H(s, t) + H(s, t)
+
w(s)ds
ρ(s)
t
Z cn
α+1
w α (s)
′
n−2
H(s, t)σ (s)σ
(s)
− αM
1 ds
(r(s)ρ(s)) α
t
− αM
Set
H(s, t)σ ′ (s)σ n−2 (s)
w
α+1
√
w α
ρ′
F (w) = h1 H + H
w − αM Hσ ′ σ n−2
1 ,
ρ
(rρ) α
by simple calculate, we can get that when
α
√
ρ′
h1 H + H
rρ
ρ
w=
,
[M (α + 1)Hσ ′ σ n−2 ]α
F (w) has the maximum value
α+1
ρ′
rρ
h1 H + H
ρ
,
[M (α + 1)Hσ ′ σ n−2 ]α (α + 1)
√
that is
F (w) ≤ Fmax (w) =
h1 +
√
ρ′
H
ρ
α+1
H
1−α
2
C2 (s),
from above inequality, we get
Z cn
H(s, t)ρ(s)C1 (s)ds ≤ −H(cn , t)w(cn )
t
+
Z
t
cn
h1 (s, t) +
p
ρ′ (s)
H(s, t)
ρ(s)
α+1
H
1−α
2
(s, t)C2 (s)ds.
28
Run Xu and Fanwei Meng
Letting t → a+
n in the above, we obtain
1
H(cn , an )
Z
cn
an
H(s, an )ρ(s)C1 (s)ds ≤ −w(cn )
1
+
H(cn , an )
Z
cn
an
α+1
p
ρ′ (s)
C2 (s)
h1 (s, an ) + H(s, an )
ds.
α−1
ρ(s)
H 2 (s, an )
(20)
Multiplying (19) by H(t, s), integrating it with respect s from cn to t, using (4)
and by simple calculate we get that
Z
t
cn
H(t, s)ρ(s)C1 (s)ds ≤ −
Z
t
w′ (s)H(t, s)ds +
cn
− αM
Z
t
H(t, s)
cn
t
cn
t
Z
t
cn
ρ′ (s)
w(s)ds
ρ(s)
α+1
α
(s)
1 ds
(r(s)ρ(s)) α
p
w(s)h2 (t, s) H(t, s)ds
H(t, s)σ ′ (s)σ n−2 (s)
= H(t, cn )w(cn ) −
Z
Z
w
ρ′ (s)
w(s)ds
H(t, s)
+
ρ(s)
cn
Z t
α+1
w α (s)
′
n−2
− αM
H(t, s)σ (s)σ
(s)
1 ds
(r(s)ρ(s)) α
cn
≤ H(t, cn )w(cn )
Z t
p
ρ′ (s)
h2 (t, s) H(t, s) + H(t, s)
w(s)ds
+
ρ(s)
cn
Z t
α+1
w α (s)
H(t, s)σ ′ (s)σ n−2 (s)
− αM
1 ds
(r(s)ρ(s)) α
cn
≤ H(t, cn )w(cn )
α+1
Z t
p
1−α
ρ′ (s)
h2 (t, s) + H(t, s)
+
H 2 (t, s)C2 (s)ds.
ρ(s)
cn
Letting t → b+
n in the above, we obtain
1
H(bn , cn )
Z
bn
cn
H(bn , s)ρ(s)C1 (s)ds ≤ w(cn )
1
+
H(bn , cn )
Z
bn
cn
α+1
p
C2 (s)
ρ′ (s)
h2 (bn , s) + H(bn , s)
ds. (21)
α−1
ρ(s)
H 2 (bn , s)
29
Interval Oscillation Criteria
Adding (20) and (21) we have the inequality
Z bn
1
H(bn , s)ρ(s)C1 (s)ds
H(bn , cn ) cn
an
α+1
Z cn
p
1
C2 (s)
ρ′ (s)
ds
h1 (s, an ) + H(s, an )
α−1
H(cn , an ) an
ρ(s)
H 2 (s, an )
α+1
Z bn
p
C2 (s)
1
ρ′ (s)
ds, t ≥ t4 .
h2 (bn , s) + H(bn , s)
α−1
H(bn , cn ) cn
ρ(s)
H 2 (bn , s)
(22)
1
H(cn , an )
≤
+
Z
cn
H(s, an )ρ(s)C1 (s)ds +
Which contradict the assumption (10). Thus, the claim holds, i.e., no nontrivial
solution of Eq. (1) can be eventually positive. Therefore, every solution of Eq. (1)
is oscillatory.
We can easily see that the following result is true.
Theorem 2.2. If there exists a positive, nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞)),
such that for any constant M > 0,
lim sup
t→∞
Z
t
l
"
H(s, l)ρ(s)C1 (s) − h1 (s, l) +
p
ρ′ (s)
H(s, l)
ρ(s)
p
ρ′ (s)
H(t, s)
ρ(s)
α+1
C2 (s)
H
α−1
2
(s, l)
#
ds > 0
(23)
and
lim sup
t→∞
Z
t
l
"
H(t, s)ρ(s)C1 (s) − h2 (t, s) +
α+1
C2 (s)
H
α−1
2
(t, s)
#
ds > 0
(24)
hold, where C1 (t), C2 (t) is defined as in Theorem 2.1, then every solution of Eq.
(1) is oscillatory.
Proof. For any T ≥ t0 , let an = T, in (23) we choose l = an , then there exist
cn > an such that
Z
cn
an
"
H(s, an )ρ(s)C1 (s) − h1 (s, an ) +
p
ρ′ (s)
H(s, an )
ρ(s)
α+1
C2 (s)
H
α−1
2
(s, an )
#
ds > 0.
(25)
In (24) we choose l = cn , then there exist bn > cn such that
Z
bn
cn
"
H(bn , s)ρ(s)C1 (s) − h2 (bn , s) +
p
ρ′ (s)
H(bn , s)
ρ(s)
α+1
C2 (s)
H
α−1
2
(bn , s)
#
ds > 0.
(26)
30
Run Xu and Fanwei Meng
Combining (25) and (26) we obtain (10). The conclusion thus comes from Theorem
2.1. the proof is complete.
Remark If we take p(t) = 0, n = 2, m = 1, f (x) = |x|α−1 x, then Theorem 2.1
and Theorem 2.2 reduce to Theorem 2.1 and Theorem 2.2 of Li [7], respectively.
If r(t) = 1, n = 2, α = 1, then Theorem 2.1 and Theorem 2.2 reduce to Theorem
2.1 and Theorem 2.2 of Zhuang and Li [19], respectively. For the case where
H := H(t − s) ∈ X, we have h1 (t − s) = h2 (t − s) and denote them by h(t − s).
The subclass of X containing such H(t − s) is denoted by X1 , applying Theorem
2.1 to X1 we obtain the following theorem.
Theorem 2.3. If for each T ≥ t0 and any constant M > 0, there exists a positive,
nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞)), H ∈ X1 and an , cn ∈ R such that T ≤
an < cn and
Z cn
H(s − an ) [ρ(s)C1 (s) + ρ(2cn − s)C1 (2cn − s)] ds
an
cn
α+1
p
ρ′ (s)
C2 (s)
ds
h(s − an ) + H(s − an )
>
α−1
ρ(s)
H 2 (s − an )
an
α+1
Z cn
p
C2 (2cn − s)
ρ′ (2cn − s)
h(s − an ) + H(s − an )
+
ds
α−1
ρ(2c
−
s)
n
H 2 (s − an )
an
Z
(27)
hold, where C1 (t), C2 (t) is defined as in Theorem 2.1, then Eq. (1) is oscillatory.
Proof. Let bn = 2cn − an , then H(bn − cn ) = H(cn − an ) = H(
Rc
Rb
any g ∈ L[an , bn ],we have cnn g(s)ds = ann g(2cn − s)ds, hence,
Z
bn
cn
H(bn − s)ρ(s)C1 (s)ds =
Z
bn
h2 (bn − s) +
p
Z
bn − an
) and for
2
cn
an
H(s − an )ρ(2cn − s)C1 (2cn − s)ds,
ρ′ (s)
H(bn − s)
ρ(s)
α+1
C2 (s)
ds
(bn − s)
H
α+1
Z cn
p
ρ′ (2cn − s)
C2 (2cn − s)
h(s − an ) + H(s − an )
=
ds.
α−1
ρ(2c
−
s)
n
H 2 (s − an )
an
cn
α−1
2
So that (27) holds implies that (10) holds for H ∈ X1 , and therefore, Eq. (1) is
oscillatory by Theorem 2.1.
From above oscillation criteria, we can obtain different sufficient conditions
for oscillation of all solutions of Eq. (1) by different choices of H(t, s). Now we
31
Interval Oscillation Criteria
choose H(t, s) = (t − s)λ , t ≥ s ≥ t0 , where λ > α is a constant. Then H ∈ X1 and
λ
h(t − s) = λ(t − s) 2 −1 , based on the above results we obtain the following corollary.
Corollary 2.1. Every solution of Eq. (1) is oscillatory provided that for any
constant M > 0, there exist a positive, nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞))
such that for each l ≥ t0 and for some λ > α, the following two inequalities hold:
α+1 #
ρ′ (s)
lim sup λ−α
ds > 0,
(s − l) ρ(s)C1 (s) − C2 (s)(s − l)
(s − l)
λ+
ρ(s)
t→∞ t
l
α+1 #
Z t"
ρ′ (s)
1
λ
λ−α−1
lim sup λ−α
(t − s)
ds > 0.
(t − s) ρ(s)C1 (s) − C2 (s)(t − s)
λ+
ρ(s)
t→∞ t
l
Z
1
t
"
λ
λ−α−1
where C1 (t), C2 (t) is defined as in Theorem 2.1.
Define
R(t) =
Z
l
t
ds
1
r α (s)
, t ≥ l ≥ t0
and let
H(t, s) = [R(t) − R(s)]λ , t ≥ t0 ,
where λ > α is constant.
If we take ρ(t) = 1, then by Theorem 2.2 we have the following important
oscillation criterion.
Theorem 2.1. Assume that lim R(t) = ∞, then every solution of Eq.(1.1) is
t→∞
oscillatory provided that for any constant M > 0, there exist a positive, nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞)) such that for each l ≥ t0 and for some λ > α,
the following two inequalities hold:
lim sup
t→∞
1
Rλ−α (t)
Z
t
[(R(s) − R(l)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds >
l
M α (α
λα+1
,
+ 1)α+1 (λ − α)
(28)
and
lim sup
t→∞
1
Rλ−α (t)
Z
t
[(R(t) − R(s)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds >
l
λα+1
,
M α (α + 1)α+1 (λ − α)
(29)
where C1 (t), C2 (t) is defined as in Theorem 2.1.
Proof. By assumption, we have
h1 (t, s) = h2 (t, s) = λ[(R(t) − R(s)]
λ−2
2
1
1
r α (t)
,
32
Run Xu and Fanwei Meng
noting that
Z
t
h1 (s, l)α+1 C2 (s)
α−1
2
H
(s, l)
α+1
λ
[(R(t) − R(l)]λ−α
,
(λ − α)M α (α + 1)α+1
l
=
(σ ′ (s)σ n−2 (s))α ds =
Z
t
λα+1 [(R(s) − R(l)]λ−α−1 r(s)
r
l
α+1
α
(s)M α (α + 1)α+1
and
Z
t
h2 (t, s)α+1 C2 (s)
α−1
2
(t, s)
H
λα+1 [(R(t) − R(l)]λ−α
,
(λ − α)M α (α + 1)α+1
l
=
(σ ′ (s)σ n−2 (s))α ds =
Z
t
λα+1 [(R(t) − R(s)]λ−α−1 r(s)
l
r
α+1
α
(s)M α (α + 1)α+1
in view of lim R(t) = ∞, we have
t→∞
1
lim
t→∞ Rλ−α (t)
Z
t
h1 (s, l)α+1 C2 (s)
H
l
α−1
2
(s, l)
(σ ′ (s)σ n−2 (s))α ds =
M α (α
λα+1
,
+ 1)α+1 (λ − α)
(30)
and
1
lim
t→∞ Rλ−α (t)
Z
t
h2 (t, s)α+1 C2 (s)
H
l
α−1
2
(t, s)
(σ ′ (s)σ n−2 (s))α ds =
M α (α
λα+1
.
+ 1)α+1 (λ − α)
(31)
From (28) and (30), we have that
1
Z
t
"
λ
h1 (s, l)α+1 C2 (s)
#
(R(s) − R(l)) C1 (s) −
(σ ′ (s)σ n−2 (s))α ds =
α−1
Rλ−α (t) l
H 2 (s, l)
Z t
λα+1
1
(R(s)−R(l))λ C1 (s)(σ ′ (s)σ n−2 (s))α ds− α
> 0,
lim sup λ−α
(t) l
M (α + 1)α+1 (λ − α)
t→∞ R
(32)
lim sup
t→∞
i.e., (23) holds. Similarly, (29) and (31) imply that (24) holds. By Theorem 2.2,
every solution of Eq. (1) is oscillatory.
This complete the proof.
Example Consider the following equation :
[|(x(t) + (1 − e−µt )x(t − π))(n−1) |α−1 (x(t) + (1 − e−µt )x(t − π))(n−1) ]′
β
+ α(n−1)+1 eγαµt |x(γt)|α−1 x(γt) = 0, t ≥ 1,
(33)
t
33
Interval Oscillation Criteria
where n ≥ 2 is even and α > 0, β > 0, µ ≥ 0, 0 < γ ≤ 1.
βeγαµt
Here p(t) = 1 − e−µt , q(t) = α(n−1)+1 , m = 1, σ1 (t) = γt.
t
Rt
1
Then R(t) = 1 dt = t − 1, R′ (t) = 1, lim R(t) = ∞, σ1′ (t) = γ, if 0 < γ ≤ ,
t→∞
2
1
t
then σ(t) = γt. if < γ ≤ 1, then σ(t) = .
2
2
1
For ρ(t) ≡ 1, λ > α. If 0 < γ ≤ , then σ(t) = σ1 (t) = γt,
2
Z t
1
lim λ−α
[(R(s) − R(l)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds
t→∞ R
(t) l
Z t
βeλαµs
1
(s − l)λ α(n−1)+1 (1 − p(γs))α (γ(γs)n−2 )α ds
= lim
λ−α
t→∞ (t − 1)
s
l
Z t
βγ α(n−1)
1
(s − l)λ
ds
= lim
λ−α
t→∞ (t − 1)
sα+1
l
=
(t − l)λ
1
βγ α(n−1)
α(n−1)
βγ
=
.
t→∞ (λ − α)(t − 1)λ−α−1 tα+1
λ−α
lim
(34)
Next, we will prove that
Z
t
[(R(t) − R(s)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds
l
≥
Z
l
t
[(R(s) − R(l)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds.
(35)
Let
Z
t
{[(R(t) − R(s)]λ − [(R(s) − R(l)]λ }C1 (s)(σ ′ (s)σ n−2 (s))α ds
G(t) =
l
Z t
β
{(t − s)λ − (s − l)λ } α(n−1)+1 µαγs γ α(n−1) sα(n−2) ds
=
s
e
l
Z t
1
{(t − s)λ − (s − l)λ } α+1 µαγs ds,
= βγ α(n−1)
s
e
l
then G(l) = 0, and for t ≥ l,
Z
t
1
1
λ(t − s)λ−1 α+1 µαγs ds − (t − l)λ α+1 µαγt
s
e
t
e
l
Z t
1
≥ βγ α(n−1) α+1 µαγt
λ(t − s)λ−1 ds − (t − l)λ
t
e
l
βγ α(n−1)
= α+1 µαγt −(t − s)λ |tl − (t − l)λ = 0.
t
e
G′ (t) = βγ α(n−1)
34
Run Xu and Fanwei Meng
Hence G(t) ≥ G(l) = 0 for t ≥ l, i.e., (2.31) holds. By (34) and (35), we have
1
t→∞ (t − 1)λ−α
lim
Z
l
t
[(R(t) − R(s)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds >
βγ α(n−1)
.
λ−α
α α+1
)
α
1
, there exists λ > α such that
Then for β > α+ α(n−1)
M γ
(
λα+1
αα+1
M α γ α(n−1) β
>
>
,
α+1
λ−α
(λ − α)(α + 1)
(λ − α)(α + 1)α+1
this means that
γ α(n−1) β
λα+1
> α
,
λ−α
M (λ − α)(α + 1)α+1
so that (28) and (29) hold for the same λ. Applying Theorem 2.4, we fined (33) is
α α+1
)
(
t
1
α
1
oscillatory for β > α+ α(n−1)
. If < γ ≤ 1, then σ(t) = , use the same method
2
2
M γ
α α+1
)
(
α+1
. However, the main
above, we can get (33) is oscillatory for β >
1
M α ( )α(n−1)
2
results of [2, 15] fail to apply to (33), since µ 6= 0.
REFERENCES
1. R.P. Agarwal, S.H. Shieh, C.C. Yeh, “Oscillation criteria for second Order retarded
differential equations”, Math.Comput. Modelling 26 (1997), 1–11.
2. R.P. Agarwal, S.R. Grace and D. O’Regan, “Oscillation criteria for certain n-th
order differential equations with deviating arguments”, J. Math. Anal. Appl. 262
(2001), 601–622.
3. J.L. Chern, W.Ch. Lian, C.C. Yeh, “Oscillation criteria for second order half-linear
differential equations with functional arguments”, Publ.Math.Ddbrecen 48 (1996), 209–
216.
4. J. Dzurina and I.P. Stavroulakis, “Oscillation criteria for second-order delay differential equations”, Applied Mathematics and Computation 140 (2003), 445–453.
5. A. Elbert, “A half-linear second order differential equation”, Colloquia Math. Soc.
Janos Bolyai, Qualitative Theory of differential equations 30 (1979), 153–180.
6. A. Elbert, “Oscillation and nonoscillation theorems for some nonlinear differential
equations, in: Ordinary and Partial Differential Equations”, Lecture Notes in Mathematics 964 (1982), 187–212.
7. W.T. Li, “Interval oscillation of second order half-linear functional differential equations”, Appl. Math. Comput. 155 (2004), 451–468.
Interval Oscillation Criteria
35
8. F. Meng and R. Xu, “Oscillation Criteria for Certain Even Order Quasi-Linear Neutral Differential Equations with Deviating Arguments”, Appl. Math. Comput. 190
(2007), 458–464.
9. F. Meng and R. Xu, “Kamenev-type Oscillation Criteria for Even Order Neutral
Differential Equations with Deviating Arguments”, Appl. Math. Comput. 190 (2007),
1402–1408.
10. D.D. Mirzov, “On the oscillation of system of nonlinear differential equations”, Diferencianye Uravnenija 9 (1973), 581–583.
11. D.D. Mirzov, “On some analogs of Sturm’s and Kneser’s theorems for nonlinear systems”, J. Math. Anal. Appl. 53 (1976), 418–425.
12. D.D. Mirzov, “On the oscillation of solutions of a system of differential equations”,
Mat. Zametki 78 (1978), 401–404.
13. CH.G. Philos, “A new criteria for the oscillatory and asymptotic behavior of delay
differential equations”, Bull. Acad. Pol. Sci. Ser. Mat. 39 (1981), 61–64.
14. Y.G. Sun and F.W. Meng, “Note on the paper of Dzurina and Stavroulakis”, Appl.
Math. Comput. 174 (2006), 1634–1641.
15. Z. Xu, Y. Xia, “Integral averaging technique and oscillation of certin even order delay
differential equations”, J. Math. Anal. Appl. 292 (2004), 238–246.
16. R. Xu, F. Meng, “Some New Oscillation Criteria For Second Order Quasi-Linear
Neutral Delay Differential Equations”, Appl. Math. Comput. 182 (2006), 797–803.
17. R. Xu, F. Meng, “New Kamenev-type oscillation criteria for second order neutral
nonlinear differential equations”, Appl. Math. Comput. 188 (2007), 1364–1370.
18. R. Xu, F. Meng, “Oscillation Criteria for Second Order Quasi-Linear Neutral Delay
Differential Equations”, Appl. Math. Comput. 192 (2007), 216–222.
19. R.K. Zhang, W.T. Li, “Interval oscillation criteria for second order quasilinear perturbed differential equations”, Int.J.Differential Equations Appl. 7:2 (2003), 165–179.
Run Xu: Department of Mathematics, Qufu Normal University, Qufu, 273165, Shangdong, China.
Fanwei Meng: Department of Mathematics, Qufu Normal University, Qufu, 273165,
Shangdong, China.
Vol. 15, No. 1 (2009), pp. 21–35.
INTERVAL OSCILLATION CRITERIA
FOR HIGHER ORDER NEUTRAL
NONLINEAR DIFFERENTIAL EQUATIONS
WITH DEVIATING ARGUMENTS
Run Xu, Fanwei Meng
Abstract. Some oscillation criteria for nth order neutral differential equations with
deviating arguments of the form
[r(t)|(y(t) + p(t)y(τ (t)))
(n−1) α−1
|
(y(t) + p(t)y(τ (t)))
(n−1) ′
] +
m
X
qi (t)fi (y(σi (t))) = 0
i=1
n even are established. New oscillation criteria are different from most known ones in
the sense that they based on a class of new function H(t, s) defined in the sequel. The
results are sharper than some previous results which can be seen by the examples at the
end of this paper.
1. INTRODUCTION
In this paper we consider the oscillation behavior of solutions of the n-th
order neutral differential equations of the form
[r(t)|(y(t) + p(t)y(τ (t)))(n−1) |α−1 (y(t) + p(t)y(τ (t)))(n−1) ]′
m
X
qi (t)fi (y(σi (t))) = 0, (1)
+
i=1
Received 07-10-2008, Accepted 27-12-2009.
2000 Mathematics Subject Classification: 34C10.
Key words and Phrases: Oscillation; Even Order; Nonlinear Differential Equations; Neutral type.
This research was partial supported by the NNSF of China(10771118) and EDF of shandong Province,
China(J07yh05)
21
22
Run Xu and Fanwei Meng
where t ≥ t0 , n ≥ 2 is even integer, α > 0 are constant. In this paper, we assume
that
(I1 ) p(t) ∈ C([t0 , ∞); [0, 1)), qi (t) ∈ C([t0 , ∞); [0, ∞)), fi ∈ C(R; R), i = 1, 2, · · · , m,
t ∈ [t0 , ∞).
Rt
(I2 ) r(t) ∈ C 1 ([t0 , ∞); (0, ∞)), r′ (t) ≥ 0, R1 (t) := t0 1ds → ∞(t → ∞).
r α (s)
(I3 )
fi (x)
|x|α−1 x
≥ βi > 0 for x 6= 0, βi are constants, i = 1, 2, · · · , m.
(I4 ) τ (t), σi (t) ∈ C 1 ([t0 , ∞); [0, ∞)), τ (t) ≤ t, σi (t) ≤ t, σ ′ (t) > 0 for t ≥ t0
and lim σi (t) = lim σ(t) = lim τ (t) = ∞, i = 1, 2, · · · , m, where σ(t) ≤
t→∞
t→∞
t→∞
min{σ1 (t), σ2 (t), · · · , σm (t), 2t }.
By a solution of Eq. (1), we mean a function y(t) ∈ C n−1 ([Tx , ∞); R) for
some Tx ≥ t0 which has the property that
r(t)|(y(t) + p(t)y(τ (t)))(n−1) |α−1 (y(t) + p(t)y(τ (t)))(n−1) ∈ C 1 ([Tx , ∞); R)
and satisfies Eq. (1) on [Tx , ∞).
A nontrivial solution of Eq. (1) is called oscillatory if it has arbitrary large
zero. Otherwise, it is called nonoscillatory. Eq. (1) is called oscillatory if all of its
solutions are oscillatory.
If p(t) = 0, r(t) = 1, m = 1, then Eq. (1) becomes
(|x(n−1) (t)|α−1 x(n−1) (t))′ + q(t)f (x(σ(t))) = 0
(2)
and the related equations have been studied by Agarwal et. al. [2], Xu et. al. [15].
Eq. (1) with n = 2, p(t) = 0, m = 1, namely, the equation
[r(t)|x′ (t)|α−1 x′ (t)]′ + q(t)f (x(σ(t))) = 0
(3)
and related equations have been investigated by Dzurina and Stavroulakis [4], Sun
and Meng [14], Mirzov [10-12], Elbert [5,6] Agarwal et. al. [1], Chern et. al. [3],
Li [7], Zhuang and Li [19].
Recently, Xu and Meng [16-18] have studied the oscillation properties of Eq.
(1) for n = 2. Very recently Meng and Xu [8,9] have investigated the oscillation
properties for higher order neutral differential equations.
Motivated by the idea of Li [7], by using averaging functions and inequality,
in this paper we obtain several new interval criteria for oscillation, that is, criteria
are given by the behavior of Eq. (1) (or of r, p and qi ) only on a sequence of
subintervals of [t, ∞). Our results improve and extend the results of Li [7] and
Zhuang and Li [19]. In order to prove our Theorems, we use the function class X
to study the oscillatory of Eq. (1). We say that a function H = H(t, s) belongs to
the function class X, if H ∈ C(D; R+ ), where D = {(t, s) : t0 ≤ s ≤ t < ∞}, which
23
Interval Oscillation Criteria
∂H
∂H
and
satisfies H(t, t) = 0, H(t, s) > 0 for t > s, and has partial derivative
∂s
∂t
on D such that
p
p
∂H
∂H
= h1 (t, s) H(t, s),
= −h2 (t, s) H(t, s),
∂t
∂s
(4)
where h1 (t, s), h2 (t, s) are locally nonnegative continuous functions on D.
2. MAIN RESULTS
First, we give the following lemmas for our results.
Lemma 2.1. [13] Let u(t) ∈ C n ([t0 , ∞); R+ ). If u(n) (t) is eventually of one sign
for all large t, say t1 > t0 , then there exist a tx > t0 and an integer l, 0 ≤ l ≤ n,
with n + l even for u(n) (t) ≥ 0 or n + l odd for u(n) (t) ≤ 0 such that l > 0
implies that u(k) (t) > 0 for t > tx , k = 0, 1, 2, · · · , l − 1, and l ≤ n − 1 implies that
(−1)l+k u(k) (t) > 0 for t > tx , k = l, l + 1, · · · , n − 1.
Lemma 2.2. [13] If the function u(t) is as in Lemma 2.1 and u(n−1) (t)u(n) (t) ≤ 0
for t > tx , then there exists a constant θ, 0 < θ < 1, such that
u(t) ≥
θ
tn−1 u(n−1) (t) for all large t.
(n − 1)!
and
t
θ
u′ ( ) ≥
tn−2 u(n−1) (t) for all large t.
2
(n − 2)!
Lemma 2.3. Suppose that y(t) is an eventually positive solution of Eq. (1), let
z(t) = y(t) + p(t)y(τ (t)),
(5)
then there exists a number t1 ≥ t0 such that
z(t) > 0, z ′ (t) > 0, z (n−1) (t) > 0 and z (n) (t) ≤ 0, t ≥ t1 .
(6)
Proof. Since y(t) is an eventually positive solution of (1), from (I4 ), there exists a
number t1 ≥ t0 such that
y(t) > 0, y(τ (t)) > 0, y(σi (t)) > 0, t ≥ t1 .
(7)
24
Run Xu and Fanwei Meng
Noting that p(t) ≥ 0, we have z(t) > 0, t ≥ t1 and from (I1 ), (I3 ) we have
(r(t)|z (n−1) (t)|α−1 z (n−1) (t))′ = −
m
X
i=1
qi (t)fi (y(σi (t))) ≤ 0, t ≥ t1 .
So r(t)|z (n−1) (t)|α−1 z (n−1) (t) is decreasing and z (n−1) (t) is eventually of one sign.
we claim that
z (n−1) (t) ≥ 0 for t ≥ t1 .
(8)
Otherwise, if there exist a te1 ≥ t1 such that z (n−1) (te1 ) < 0, then for all t ≥ te1 ,
r(t)|z (n−1) (t)|α−1 z (n−1) (t) ≤ r(e
t1 )|z (n−1) (e
t1 )|α−1 z (n−1) (e
t1 ) = −C(C > 0),
then we have −z (n−1) (t) ≥
te1 to t, we have
C
r(t)
(9)
1
α
, t ≥ te1 , integrating the above inequality from
1
z (n−2) (t) ≤ z (n−2) (te1 ) − C α (R(t) − R(te1 )).
Letting t → ∞, from (I2 ), we get lim z (n−2) (t) = −∞, which implies z (n−1) (t)
t→∞
and z (n−2) (t) are negative for all large t, from Lemma 2.1, no two consecutive
derivatives can be eventually negative, for this would imply that lim z(t) = −∞,
t→∞
a contradiction. Hence z (n−1) (t) ≥ 0 for t ≥ t1 . from Eq. (1) and (I1 ), (I2 ) we have
αr(t)(z (n−1) (t))α−1 z (n) (t) = [r(t)(z (n−1) (t))α ]′ − r′ (t)(z (n−1) (t))α
m
X
qi (t)fi (y(σi (t))) − r′ (t)(z (n−1) (t))α ≤ 0, t ≥ t1 ,
= −
i=1
this implies that z (n) (t) ≤ 0, t ≥ t1 . From Lemma 2.1 again (note n is even), we
have z ′ (t) > 0, t ≥ t1 . This completes the proof.
Theorem 2.1. Assume that there exist a positive, nondecreasing function ρ(t) ∈
C 1 ([t0 , ∞)) such that for any constant M > 0, some H ∈ X and for each sufficient large T0 ≥ t0 , there exist increasing divergent sequences of positive numbers
{an }, {bn }, {cn } with T0 ≤ an < cn < bn such that
Z cn
Z bn
1
1
H(s, an )ρ(s)C1 (s)ds +
H(bn , s)ρ(s)C1 (s)ds
H(cn , an ) an
H(bn , cn ) cn
α+1
Z cn
p
C2 (s)
ρ′ (s)
1
ds
>
h1 (s, an ) + H(s, an )
α−1
H(cn , an ) an
ρ(s)
H 2 (s, an )
α+1
Z bn
p
1
ρ′ (s)
C2 (s)
h2 (bn , s) + H(bn , s)
ds,
(10)
+
α−1
H(bn , cn ) cn
ρ(s)
H 2 (bn , s)
Interval Oscillation Criteria
25
where
C1 (t) =
m
X
i=1
βi qi (t)(1 − p(σi (t)))α , C2 (t) =
r(t)ρ(t)
,
M α (α + 1)α+1 (σ ′ (t)σ n−2 (t))α
then every solution of Eq. (1) is oscillatory.
Proof. Suppose the contrary, let y(t) is a nonoscillatory solution of Eq. (1), without
loss of generality we assume
y(t) > 0, y(τ (t)) > 0 for t ≥ t1 ≥ t0 .
Then
z(t) = y(t) + p(t)y(τ (t)) > 0 for t ≥ t1 ≥ t0 .
(11)
From Lemma 2.3, there exists t2 ≥ t1 such that
z(t) > 0, z ′ (t) > 0, z (n−1) (t) > 0 and z (n) (t) ≤ 0, t ≥ t2 .
(12)
It is easy to check that we can apply Lemma 2.2 and conclude that there exist
0 < θ < 1 and t3 > t2 such that
z ′ (σ(t)) ≥
≥
θ
(2σ(t))n−2 z (n−1) (2σ(t))
(n − 2)!
θ
2n−2 σ n−2 (t)z (n−1) (t) = M σ n−2 (t)z (n−1) (t), t ≥ t3 ,
(n − 2)!
(13)
θ
2n−2 .
(n − 2)!
From (5), we have
where M =
y(t) = z(t) − p(t)y(τ (t)) ≥ z(t) − p(t)z(τ (t)) ≥ z(t)(1 − p(t)), t ≥ t3 .
(14)
Since lim σ(t) = ∞, there exists t4 ≥ t3 such that σ(t) ≥ t3 , t ≥ t4 , so
t→∞
y(σ(t)) ≥ z(σ(t))(1 − p(σ(t))), t ≥ t4 .
(15)
By (I3 ) and (15) we get
fi (y(σi (t))) ≥ βi y α (σi (t)) ≥ βi z α (σi (t))(1 − p(σi (t)))α , t ≥ t4 .
From (1), (16), we get
(16)
26
Run Xu and Fanwei Meng
0 = [r(t)(z (n−1) (t))α ]′ +
≥ [r(t)(z (n−1) (t))α ]′ +
≥ [r(t)(z (n−1) (t))α ]′ +
m
X
i=1
m
X
i=1
m
X
i=1
βi qi (t)fi (y(σi (t)))
βi qi (t)z α (σi (t))(1 − p(σi (t)))α
βi qi (t)z α (σ(t))(1 − p(σi (t)))α , t ≥ t4 .
(17)
Let
w(t) = ρ(t)
r(t)(z (n−1) (t))α
, t ≥ t4 ,
z α (σ(t))
(18)
clearly, w(t) > 0, from (13), (17) and (18) we get
w′ (t) =
ρ′ (t)
[r(t)(z (n−1) (t))α ]′
w(t) + ρ(t)
ρ(t)
z α (σ(t))
r(t)(z (n−1) (t))α αz α−1 (σ(t))z ′ (σ(t))σ ′ (t)
z 2α (σ(t))
m
X
ρ′ (t)
βi qi (t)(1 − p(σi (t)))α
w(t) − ρ(t)
≤
ρ(t)
i=1
− ρ(t)
r(t)(z (n−1) (t))α z ′ (σ(t))
,
z α+1 (σ(t))
m
X
ρ′ (t)
≤
βi qi (t)(1 − p(σi (t)))α
w(t) − ρ(t)
ρ(t)
i=1
− ασ ′ (t)ρ(t)
− αM σ ′ (t)σ n−2 (t)
w
α+1
α
(t)
1
(r(t)ρ(t)) α
α+1
≤
w α (t)
ρ′ (t)
w(t) − ρ(t)C1 (t) − αM σ ′ (t)σ n−2 (t)
1 .
ρ(t)
(r(t)ρ(t)) α
Then from above inequality we have
α+1
w α (t)
ρ′ (t)
w(t) − αM σ ′ (t)σ n−2 (t)
ρ(t)C1 (t) ≤ −w (t) +
1 .
ρ(t)
(r(t)ρ(t)) α
′
(19)
Multiplying (19) by H(s, t), integrating it with respect s from t to cn and
27
Interval Oscillation Criteria
using (4) we get that
Z
Z cn
H(s, t)ρ(s)C1 (s)ds ≤ −
cn
′
w (s)H(s, t)ds +
t
t
Z
Z
cn
H(s, t)
t
cn
ρ′ (s)
w(s)ds
ρ(s)
α+1
α
(s)
1 ds
α
(r(s)ρ(s))
t
Z cn
p
w(s)h1 (s, t) H(s, t)ds
= −H(cn , t)w(cn ) +
t
Z cn
ρ′ (s)
+
H(s, t)
w(s)ds
ρ(s)
t
Z cn
α+1
w α (s)
′
n−2
H(s, t)σ (s)σ
(s)
− αM
1 ds
(r(s)ρ(s)) α
t
= −H(cn , t)w(cn )
Z cn
p
ρ′ (s)
h1 (s, t) H(s, t) + H(s, t)
+
w(s)ds
ρ(s)
t
Z cn
α+1
w α (s)
′
n−2
H(s, t)σ (s)σ
(s)
− αM
1 ds
(r(s)ρ(s)) α
t
− αM
Set
H(s, t)σ ′ (s)σ n−2 (s)
w
α+1
√
w α
ρ′
F (w) = h1 H + H
w − αM Hσ ′ σ n−2
1 ,
ρ
(rρ) α
by simple calculate, we can get that when
α
√
ρ′
h1 H + H
rρ
ρ
w=
,
[M (α + 1)Hσ ′ σ n−2 ]α
F (w) has the maximum value
α+1
ρ′
rρ
h1 H + H
ρ
,
[M (α + 1)Hσ ′ σ n−2 ]α (α + 1)
√
that is
F (w) ≤ Fmax (w) =
h1 +
√
ρ′
H
ρ
α+1
H
1−α
2
C2 (s),
from above inequality, we get
Z cn
H(s, t)ρ(s)C1 (s)ds ≤ −H(cn , t)w(cn )
t
+
Z
t
cn
h1 (s, t) +
p
ρ′ (s)
H(s, t)
ρ(s)
α+1
H
1−α
2
(s, t)C2 (s)ds.
28
Run Xu and Fanwei Meng
Letting t → a+
n in the above, we obtain
1
H(cn , an )
Z
cn
an
H(s, an )ρ(s)C1 (s)ds ≤ −w(cn )
1
+
H(cn , an )
Z
cn
an
α+1
p
ρ′ (s)
C2 (s)
h1 (s, an ) + H(s, an )
ds.
α−1
ρ(s)
H 2 (s, an )
(20)
Multiplying (19) by H(t, s), integrating it with respect s from cn to t, using (4)
and by simple calculate we get that
Z
t
cn
H(t, s)ρ(s)C1 (s)ds ≤ −
Z
t
w′ (s)H(t, s)ds +
cn
− αM
Z
t
H(t, s)
cn
t
cn
t
Z
t
cn
ρ′ (s)
w(s)ds
ρ(s)
α+1
α
(s)
1 ds
(r(s)ρ(s)) α
p
w(s)h2 (t, s) H(t, s)ds
H(t, s)σ ′ (s)σ n−2 (s)
= H(t, cn )w(cn ) −
Z
Z
w
ρ′ (s)
w(s)ds
H(t, s)
+
ρ(s)
cn
Z t
α+1
w α (s)
′
n−2
− αM
H(t, s)σ (s)σ
(s)
1 ds
(r(s)ρ(s)) α
cn
≤ H(t, cn )w(cn )
Z t
p
ρ′ (s)
h2 (t, s) H(t, s) + H(t, s)
w(s)ds
+
ρ(s)
cn
Z t
α+1
w α (s)
H(t, s)σ ′ (s)σ n−2 (s)
− αM
1 ds
(r(s)ρ(s)) α
cn
≤ H(t, cn )w(cn )
α+1
Z t
p
1−α
ρ′ (s)
h2 (t, s) + H(t, s)
+
H 2 (t, s)C2 (s)ds.
ρ(s)
cn
Letting t → b+
n in the above, we obtain
1
H(bn , cn )
Z
bn
cn
H(bn , s)ρ(s)C1 (s)ds ≤ w(cn )
1
+
H(bn , cn )
Z
bn
cn
α+1
p
C2 (s)
ρ′ (s)
h2 (bn , s) + H(bn , s)
ds. (21)
α−1
ρ(s)
H 2 (bn , s)
29
Interval Oscillation Criteria
Adding (20) and (21) we have the inequality
Z bn
1
H(bn , s)ρ(s)C1 (s)ds
H(bn , cn ) cn
an
α+1
Z cn
p
1
C2 (s)
ρ′ (s)
ds
h1 (s, an ) + H(s, an )
α−1
H(cn , an ) an
ρ(s)
H 2 (s, an )
α+1
Z bn
p
C2 (s)
1
ρ′ (s)
ds, t ≥ t4 .
h2 (bn , s) + H(bn , s)
α−1
H(bn , cn ) cn
ρ(s)
H 2 (bn , s)
(22)
1
H(cn , an )
≤
+
Z
cn
H(s, an )ρ(s)C1 (s)ds +
Which contradict the assumption (10). Thus, the claim holds, i.e., no nontrivial
solution of Eq. (1) can be eventually positive. Therefore, every solution of Eq. (1)
is oscillatory.
We can easily see that the following result is true.
Theorem 2.2. If there exists a positive, nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞)),
such that for any constant M > 0,
lim sup
t→∞
Z
t
l
"
H(s, l)ρ(s)C1 (s) − h1 (s, l) +
p
ρ′ (s)
H(s, l)
ρ(s)
p
ρ′ (s)
H(t, s)
ρ(s)
α+1
C2 (s)
H
α−1
2
(s, l)
#
ds > 0
(23)
and
lim sup
t→∞
Z
t
l
"
H(t, s)ρ(s)C1 (s) − h2 (t, s) +
α+1
C2 (s)
H
α−1
2
(t, s)
#
ds > 0
(24)
hold, where C1 (t), C2 (t) is defined as in Theorem 2.1, then every solution of Eq.
(1) is oscillatory.
Proof. For any T ≥ t0 , let an = T, in (23) we choose l = an , then there exist
cn > an such that
Z
cn
an
"
H(s, an )ρ(s)C1 (s) − h1 (s, an ) +
p
ρ′ (s)
H(s, an )
ρ(s)
α+1
C2 (s)
H
α−1
2
(s, an )
#
ds > 0.
(25)
In (24) we choose l = cn , then there exist bn > cn such that
Z
bn
cn
"
H(bn , s)ρ(s)C1 (s) − h2 (bn , s) +
p
ρ′ (s)
H(bn , s)
ρ(s)
α+1
C2 (s)
H
α−1
2
(bn , s)
#
ds > 0.
(26)
30
Run Xu and Fanwei Meng
Combining (25) and (26) we obtain (10). The conclusion thus comes from Theorem
2.1. the proof is complete.
Remark If we take p(t) = 0, n = 2, m = 1, f (x) = |x|α−1 x, then Theorem 2.1
and Theorem 2.2 reduce to Theorem 2.1 and Theorem 2.2 of Li [7], respectively.
If r(t) = 1, n = 2, α = 1, then Theorem 2.1 and Theorem 2.2 reduce to Theorem
2.1 and Theorem 2.2 of Zhuang and Li [19], respectively. For the case where
H := H(t − s) ∈ X, we have h1 (t − s) = h2 (t − s) and denote them by h(t − s).
The subclass of X containing such H(t − s) is denoted by X1 , applying Theorem
2.1 to X1 we obtain the following theorem.
Theorem 2.3. If for each T ≥ t0 and any constant M > 0, there exists a positive,
nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞)), H ∈ X1 and an , cn ∈ R such that T ≤
an < cn and
Z cn
H(s − an ) [ρ(s)C1 (s) + ρ(2cn − s)C1 (2cn − s)] ds
an
cn
α+1
p
ρ′ (s)
C2 (s)
ds
h(s − an ) + H(s − an )
>
α−1
ρ(s)
H 2 (s − an )
an
α+1
Z cn
p
C2 (2cn − s)
ρ′ (2cn − s)
h(s − an ) + H(s − an )
+
ds
α−1
ρ(2c
−
s)
n
H 2 (s − an )
an
Z
(27)
hold, where C1 (t), C2 (t) is defined as in Theorem 2.1, then Eq. (1) is oscillatory.
Proof. Let bn = 2cn − an , then H(bn − cn ) = H(cn − an ) = H(
Rc
Rb
any g ∈ L[an , bn ],we have cnn g(s)ds = ann g(2cn − s)ds, hence,
Z
bn
cn
H(bn − s)ρ(s)C1 (s)ds =
Z
bn
h2 (bn − s) +
p
Z
bn − an
) and for
2
cn
an
H(s − an )ρ(2cn − s)C1 (2cn − s)ds,
ρ′ (s)
H(bn − s)
ρ(s)
α+1
C2 (s)
ds
(bn − s)
H
α+1
Z cn
p
ρ′ (2cn − s)
C2 (2cn − s)
h(s − an ) + H(s − an )
=
ds.
α−1
ρ(2c
−
s)
n
H 2 (s − an )
an
cn
α−1
2
So that (27) holds implies that (10) holds for H ∈ X1 , and therefore, Eq. (1) is
oscillatory by Theorem 2.1.
From above oscillation criteria, we can obtain different sufficient conditions
for oscillation of all solutions of Eq. (1) by different choices of H(t, s). Now we
31
Interval Oscillation Criteria
choose H(t, s) = (t − s)λ , t ≥ s ≥ t0 , where λ > α is a constant. Then H ∈ X1 and
λ
h(t − s) = λ(t − s) 2 −1 , based on the above results we obtain the following corollary.
Corollary 2.1. Every solution of Eq. (1) is oscillatory provided that for any
constant M > 0, there exist a positive, nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞))
such that for each l ≥ t0 and for some λ > α, the following two inequalities hold:
α+1 #
ρ′ (s)
lim sup λ−α
ds > 0,
(s − l) ρ(s)C1 (s) − C2 (s)(s − l)
(s − l)
λ+
ρ(s)
t→∞ t
l
α+1 #
Z t"
ρ′ (s)
1
λ
λ−α−1
lim sup λ−α
(t − s)
ds > 0.
(t − s) ρ(s)C1 (s) − C2 (s)(t − s)
λ+
ρ(s)
t→∞ t
l
Z
1
t
"
λ
λ−α−1
where C1 (t), C2 (t) is defined as in Theorem 2.1.
Define
R(t) =
Z
l
t
ds
1
r α (s)
, t ≥ l ≥ t0
and let
H(t, s) = [R(t) − R(s)]λ , t ≥ t0 ,
where λ > α is constant.
If we take ρ(t) = 1, then by Theorem 2.2 we have the following important
oscillation criterion.
Theorem 2.1. Assume that lim R(t) = ∞, then every solution of Eq.(1.1) is
t→∞
oscillatory provided that for any constant M > 0, there exist a positive, nondecreasing function ρ(t) ∈ C 1 ([t0 , ∞)) such that for each l ≥ t0 and for some λ > α,
the following two inequalities hold:
lim sup
t→∞
1
Rλ−α (t)
Z
t
[(R(s) − R(l)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds >
l
M α (α
λα+1
,
+ 1)α+1 (λ − α)
(28)
and
lim sup
t→∞
1
Rλ−α (t)
Z
t
[(R(t) − R(s)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds >
l
λα+1
,
M α (α + 1)α+1 (λ − α)
(29)
where C1 (t), C2 (t) is defined as in Theorem 2.1.
Proof. By assumption, we have
h1 (t, s) = h2 (t, s) = λ[(R(t) − R(s)]
λ−2
2
1
1
r α (t)
,
32
Run Xu and Fanwei Meng
noting that
Z
t
h1 (s, l)α+1 C2 (s)
α−1
2
H
(s, l)
α+1
λ
[(R(t) − R(l)]λ−α
,
(λ − α)M α (α + 1)α+1
l
=
(σ ′ (s)σ n−2 (s))α ds =
Z
t
λα+1 [(R(s) − R(l)]λ−α−1 r(s)
r
l
α+1
α
(s)M α (α + 1)α+1
and
Z
t
h2 (t, s)α+1 C2 (s)
α−1
2
(t, s)
H
λα+1 [(R(t) − R(l)]λ−α
,
(λ − α)M α (α + 1)α+1
l
=
(σ ′ (s)σ n−2 (s))α ds =
Z
t
λα+1 [(R(t) − R(s)]λ−α−1 r(s)
l
r
α+1
α
(s)M α (α + 1)α+1
in view of lim R(t) = ∞, we have
t→∞
1
lim
t→∞ Rλ−α (t)
Z
t
h1 (s, l)α+1 C2 (s)
H
l
α−1
2
(s, l)
(σ ′ (s)σ n−2 (s))α ds =
M α (α
λα+1
,
+ 1)α+1 (λ − α)
(30)
and
1
lim
t→∞ Rλ−α (t)
Z
t
h2 (t, s)α+1 C2 (s)
H
l
α−1
2
(t, s)
(σ ′ (s)σ n−2 (s))α ds =
M α (α
λα+1
.
+ 1)α+1 (λ − α)
(31)
From (28) and (30), we have that
1
Z
t
"
λ
h1 (s, l)α+1 C2 (s)
#
(R(s) − R(l)) C1 (s) −
(σ ′ (s)σ n−2 (s))α ds =
α−1
Rλ−α (t) l
H 2 (s, l)
Z t
λα+1
1
(R(s)−R(l))λ C1 (s)(σ ′ (s)σ n−2 (s))α ds− α
> 0,
lim sup λ−α
(t) l
M (α + 1)α+1 (λ − α)
t→∞ R
(32)
lim sup
t→∞
i.e., (23) holds. Similarly, (29) and (31) imply that (24) holds. By Theorem 2.2,
every solution of Eq. (1) is oscillatory.
This complete the proof.
Example Consider the following equation :
[|(x(t) + (1 − e−µt )x(t − π))(n−1) |α−1 (x(t) + (1 − e−µt )x(t − π))(n−1) ]′
β
+ α(n−1)+1 eγαµt |x(γt)|α−1 x(γt) = 0, t ≥ 1,
(33)
t
33
Interval Oscillation Criteria
where n ≥ 2 is even and α > 0, β > 0, µ ≥ 0, 0 < γ ≤ 1.
βeγαµt
Here p(t) = 1 − e−µt , q(t) = α(n−1)+1 , m = 1, σ1 (t) = γt.
t
Rt
1
Then R(t) = 1 dt = t − 1, R′ (t) = 1, lim R(t) = ∞, σ1′ (t) = γ, if 0 < γ ≤ ,
t→∞
2
1
t
then σ(t) = γt. if < γ ≤ 1, then σ(t) = .
2
2
1
For ρ(t) ≡ 1, λ > α. If 0 < γ ≤ , then σ(t) = σ1 (t) = γt,
2
Z t
1
lim λ−α
[(R(s) − R(l)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds
t→∞ R
(t) l
Z t
βeλαµs
1
(s − l)λ α(n−1)+1 (1 − p(γs))α (γ(γs)n−2 )α ds
= lim
λ−α
t→∞ (t − 1)
s
l
Z t
βγ α(n−1)
1
(s − l)λ
ds
= lim
λ−α
t→∞ (t − 1)
sα+1
l
=
(t − l)λ
1
βγ α(n−1)
α(n−1)
βγ
=
.
t→∞ (λ − α)(t − 1)λ−α−1 tα+1
λ−α
lim
(34)
Next, we will prove that
Z
t
[(R(t) − R(s)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds
l
≥
Z
l
t
[(R(s) − R(l)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds.
(35)
Let
Z
t
{[(R(t) − R(s)]λ − [(R(s) − R(l)]λ }C1 (s)(σ ′ (s)σ n−2 (s))α ds
G(t) =
l
Z t
β
{(t − s)λ − (s − l)λ } α(n−1)+1 µαγs γ α(n−1) sα(n−2) ds
=
s
e
l
Z t
1
{(t − s)λ − (s − l)λ } α+1 µαγs ds,
= βγ α(n−1)
s
e
l
then G(l) = 0, and for t ≥ l,
Z
t
1
1
λ(t − s)λ−1 α+1 µαγs ds − (t − l)λ α+1 µαγt
s
e
t
e
l
Z t
1
≥ βγ α(n−1) α+1 µαγt
λ(t − s)λ−1 ds − (t − l)λ
t
e
l
βγ α(n−1)
= α+1 µαγt −(t − s)λ |tl − (t − l)λ = 0.
t
e
G′ (t) = βγ α(n−1)
34
Run Xu and Fanwei Meng
Hence G(t) ≥ G(l) = 0 for t ≥ l, i.e., (2.31) holds. By (34) and (35), we have
1
t→∞ (t − 1)λ−α
lim
Z
l
t
[(R(t) − R(s)]λ C1 (s)(σ ′ (s)σ n−2 (s))α ds >
βγ α(n−1)
.
λ−α
α α+1
)
α
1
, there exists λ > α such that
Then for β > α+ α(n−1)
M γ
(
λα+1
αα+1
M α γ α(n−1) β
>
>
,
α+1
λ−α
(λ − α)(α + 1)
(λ − α)(α + 1)α+1
this means that
γ α(n−1) β
λα+1
> α
,
λ−α
M (λ − α)(α + 1)α+1
so that (28) and (29) hold for the same λ. Applying Theorem 2.4, we fined (33) is
α α+1
)
(
t
1
α
1
oscillatory for β > α+ α(n−1)
. If < γ ≤ 1, then σ(t) = , use the same method
2
2
M γ
α α+1
)
(
α+1
. However, the main
above, we can get (33) is oscillatory for β >
1
M α ( )α(n−1)
2
results of [2, 15] fail to apply to (33), since µ 6= 0.
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2. R.P. Agarwal, S.R. Grace and D. O’Regan, “Oscillation criteria for certain n-th
order differential equations with deviating arguments”, J. Math. Anal. Appl. 262
(2001), 601–622.
3. J.L. Chern, W.Ch. Lian, C.C. Yeh, “Oscillation criteria for second order half-linear
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Run Xu: Department of Mathematics, Qufu Normal University, Qufu, 273165, Shangdong, China.
Fanwei Meng: Department of Mathematics, Qufu Normal University, Qufu, 273165,
Shangdong, China.