Electronic Journal of Differential Equations, Vol. 2007(2007), No. 10, pp. 1–11.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu (login: ftp)
OSCILLATION CRITERIA FOR SECOND-ORDER NEUTRAL
DIFFERENTIAL EQUATIONS WITH DISTRIBUTED
DEVIATING ARGUMENTS
GAIHUA GUI, ZHITING XU
Abstract. Using a class of test functions Φ(t, s, T ) defined by Sun [13] and a
generalized Riccati technique, we establish some new oscillation criteria for the
second-order neutral differential equation with distributed deviating argument
Z b
(r(t)ψ(x(t))Z 0 (t))0 +
q(t, ξ)f [x(g(t, ξ))]dσ(ξ) = 0, t ≥ t0 ,
a
where Z(t) = x(t) + p(t)x(t − τ ). The obtained results are different from most
known ones and can be applied to many cases which are not covered by existing
results.
1. Introduction and Preliminaries
Consider the second-order neutral differential equation with distributed deviating argument
Z b
(r(t)ψ(x(t))Z 0 (t))0 +
q(t, ξ)f [x(g(t, ξ))]dσ(ξ) = 0, t ≥ t0 ,
(1.1)
a
where Z(t) = x(t) + p(t)x(t − τ ), τ ≥ 0, and the following conditions are assumed
to hold without further mentioning:
R∞
(A1) r, p ∈ C(I, R) and 0 ≤ p(t) ≤ 1, r(t) > 0 for t ∈ I,
1/r(s)ds = ∞,
I = [t0 , ∞);
(A2) ψ ∈ C 1 (R, R), ψ(x) > 0 for x 6= 0;
(A3) f ∈ C(R, R), xf (x) > 0 for x 6= 0;
(A4) q ∈ C(I × [a, b], [0, ∞)) and q(t, ξ) is not eventually zero on any half-line
[tu , ∞) × [a, b], tu ≥ t0 ;
(A5) g ∈ C(I × [a, b], [0, ∞)), g(t, ξ) ≤ t for t ≥ t0 and ξ ∈ [a, b], g(t, ξ) has a
continuous and positive partial derivative on I × [a, b] with respect to the
first variable t and nondecreasing with respect to the second variable ξ,
respectively, and lim inf t→∞ g(t, ξ) = ∞ for ξ ∈ [a, b];
(A6) σ ∈ C([a, b], R) is nondecreasing, and the integral of (1.1) is in the sense of
Riemann-Stieltijes.
2000 Mathematics Subject Classification. 34K11, 34C10, 34K40.
Key words and phrases. Oscillation; neutral differential equation; second order;
distributed deviating argument; Riccati technique.
c
2007
Texas State University - San Marcos.
Submitted July 28, 2006. Published January 2, 2007.
1
2
G. GUI, Z. XU
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Let τ ∗ (t) = min{τ (t) = t − τ, δ(t) = min g(t, ξ) for ξ ∈ [a, b]} and let T ∗ =
∗
min{τ ∗ (t) : t ≥ 0} and τ−1
(t) = sup{s ≥ 0 : τ ∗ (s)] ≤ t} for t ≥ T ∗ . Clearly
∗
∗
∗
τ−1 (t) ≥ t for t ≥ T , τ−1 (t) is nondecreasing and coincides with the inverse of τ ∗ (t)
when latter exists. By a solution of (1.1) we means a nontrivial real-valued function
∗
x(t) which has the properties Z(t) ∈ C 1 ([τ−1
(t0 ), ∞), R), and r(t)ψ(x(t)Z 0 (t) ∈
1
∗
C ([τ−1 (t0 ), ∞), R). Our attention is restricted to those solutions x(t) of (1.1)
which exist on some half-line [tx , ∞) with sup{|x(t)| : t ≥ T } > 0 for any T ≥ tx ,
and satisfy (1.1). As usual, a solution x(t) of (1.1) is called oscillatory if the set
of its zeros is unbounded from above, otherwise, it is called nonoscillatory. (1.1) is
called oscillatory if all solutions are oscillatory.
We note that second order neutral delay differential equations have various applications in problems dealing with vibrating masses attached to an elastic bar and
some variational problems, etc. For further applications and questions concerning
existence and uniqueness of solutions of neutral delay differential equations, see [8].
In the last decades, there has been an increasing interest in obtaining sufficient
conditions for the oscillation and/or nonoscillation of solutions of second order
linear and nonlinear neutral delay differential equations with distributed deviating
arguments (see, for example, [4, 10, 14, 16, 17] and the references therein). Very
recently, in [16], the results in [5, 11, 15] for second order differential equations have
been extended to (1.1).
For other oscillation results of various neutral functional differential equations
we refer the reader to the monographs [1, 2, 3, 6, 7].
The purpose of this paper is to establish some new oscillation criteria for (1.1)
by introducing a class of functions Φ(t, s, T ) defined in the recent paper [13] and
a generalized Riccati technique. Our results are different from most known ones
in the sense that they are given in the form that lim supt→∞ [·] is greater than a
constant, rather than in the form lim supt→∞ [·] = +∞. Thus, our results can be
applied to many cases, which are not covered by existing ones.
Following the idea of Sun [13], we say that a function Φ = Φ(t, s, T ) belongs
to a function class X, denoted by Φ ∈ X, if Φ ∈ C(E, R), where E = {(t, s, T ) :
t0 ≤ T ≤ s ≤ t < ∞}, which satisfies Φ(t, t, T ) = Φ(t, T, T ) = 0, Φ(t, s, T ) 6= 0 for
T < s < t, and has the partial derivative ∂Φ/∂s on E such that ∂Φ/∂s is locally
integrable with respect to s on E.
We now recall to introduce another class of functions defined by Philos [11] which
is used extensively. Let D0 = {(t, s) : t > s > t0 } and D = {(t, s) : t ≥ s ≥ t0 }. Say
that H ∈ C(D, R) belongs to a function class Y, denoted by H ∈ Y, if H(t, t) = 0
for t ≥ t0 , H(t, s) > 0 on D0 , H has continuous partial derivatives on D0 with
respect to t and s.
Let us state three sets of conditions commonly used as in literature; see for
example [12, 16], which we rely on:
(S1) f 0 (x) exists, f 0 (x) ≥ k1 and ψ(x) ≤ L−1 for x 6= 0;
(S2) f 0 (x) exists, f 0 (x)/ψ(x) ≥ k2 for x 6= 0;
(S3) f (x)/x ≥ k3 and ψ(x) ≤ L−1 for x 6= 0,
where k1 , k2 , k3 and L are positive real numbers.
In addition, we will use the following conditions as in [12, 16]:
(N1) There exists a positive real number M such that ±f (±uv) ≥ M f (u)f (v)
for uv ≥ 0;
(N2) uψ 0 (u) > 0 for u 6= 0.
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OSCILLATION CRITERIA
3
In order to prove our theorems we need the following three Lemmas whose proof
can be found in [16].
Lemma 1.1 ([16]). Suppose that (S1) and (N1) are satisfied. Let x(t) be an eventually positive solution of (1.1). Then there exists a T0 ≥ t0 such that
Z 0 (t) > 0
Z(t) > 0,
(r(t)ψ(x(t))Z 0 (t))0 ≤ 0,
for t ≥ T0 .
(1.2)
Moreover, for t ≥ T0 ,
0
Z
0
b
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ) ≤ 0.
(r(t)ψ(x(t))Z (t)) + M f [Z(g(t, a))]
(1.3)
a
Lemma 1.2 ([16]). Suppose that (S2), (N1) and (N2) are satisfied. Let x(t) be an
eventually positive solution of (1.1). Then there exists a T0 ≥ t0 such that (1.2)
and (1.3) hold.
Lemma 1.3 ([16]). Suppose that (S3) is satisfied. Let x(t) be an eventually positive
solution of (1.1). Then there exists a T0 ≥ t0 such that (1.2) holds. Moreover,
Z b
0
0
(r(t)ψ(x(t))Z (t)) + k3 Z[g(t, a)]
q(t, ξ)[1 − p(g(t, ξ))]dσ(ξ) ≤ 0, t ≥ T0 . (1.4)
a
2. Kamenev-type oscillation criteria
Theorem 2.1. Let (S1) and (N1) hold. If there exist functions ϕ ∈ C 1 (I, R+ ),
φ ∈ C 1 (I, R) and Φ ∈ X such that for each T ≥ t0 ,
Z tn
2 o
1
ds > 0, (2.1)
lim sup
Φ2 (t, s, T )Θ1 (s) − γ1 (s) Φ0s (t, s, T ) + l1 (s)Φ(t, s, T )
2
t→∞
T
where
r[g(t, a)]ϕ(t)
ϕ0 (t) 2k1 Lg 0 (t, a)φ(t)
+
, γ1 (t) =
,
ϕ(t)
r[g(t, a)]
k1 Lg 0 (t, a)
n Z b
o
k1 Lg 0 (t, a)φ2 (t)
Θ1 (t) = ϕ(t) M
− φ0 (t) ,
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ) +
r[g(t, a)]
a
l1 (t) =
then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of (1.1) on I. Without loss of generality
we assume that x(t) > 0 for t ≥ t0 , (The case of x(t) < 0 can be considered
similarly). By Lemma 1.1, there exists a T0 ≥ t0 such that (1.2) and (1.3) hold.
Define
h r(t)ψ(x(t))Z 0 (t)
i
v(t) = ϕ(t)
+ φ(t) for all t ≥ T0 .
(2.2)
f [Z(g(t, a))]
Differentiating (2.2) and using (1.3), it follows that
h Z b
ϕ0 (t)
0
v t) ≤
v(t) − ϕ(t) M
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ)
ϕ(t)
a
i
r(t)ψ(x(t))Z 0 (t) 0
0
0
0
+
f
[Z(g(t,
a))]Z
[g(t,
a)]g
(t,
a)
−
φ
(t)
.
f 2 [Z(g(t, a))]
Since g(t, a) ≤ t and (r(t)ψ(x(t))Z 0 (t))0 ≤ 0, we have
r(t)ψ(x(t))Z 0 (t) ≤ r[g(t, a)]ψ[x(g(t, a))]Z 0 [g(t, a)].
4
G. GUI, Z. XU
EJDE-2007/10
Thus,
i
h Z b
ϕ0 (t)
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ) − φ0 (t)
v(t) − ϕ(t) M
ϕ(t)
a
r(t)ψ(x(t))Z 0 (t) 2
k1 ϕ(t)g 0 (t, a)
−
r[g(t, a)]ψ[x(g(t, a))]
f [Z(g(t, a))]
Z b
h
i
0
ϕ (t)
≤
v(t) − ϕ(t) M
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ) − φ0 (t)
ϕ(t)
a
2
k1 L ϕ(t)g 0 (t, a) v(t)
− φ(t) .
−
r[g(t, a)]
ϕ(t)
v 0 (t) ≤
So that
v 0 (s) ≤ −Θ1 (s) + l1 (s)v(s) −
1 2
v (s).
γ1 (s)
(2.3)
Multiplying (2.3) by Φ2 (t, s, T0 ), and integrating from T0 to t, we get
Z
t
Φ2 (t, s, T0 )Θ1 (s)ds
T0
Z
t
2
≤
0
Z
t
Φ (t, s, T0 )[−v (s) + l1 (s)v(s)]ds −
T0
T0
1
Φ2 (t, s, T0 )v 2 (s)ds.
γ1 (s)
Integrating by parts, we obtain
Z t
Φ2 (t, s, T0 )Θ1 (s)ds
T0
Z
t
1
Φ(t, s, T0 ) Φ0s (t, s, T0 ) + l1 (s)Φ(t, s, T0 ) v(s)ds
2
T0
Z t
1
−
Φ2 (t, s, T0 )v 2 (s)ds
γ
(s)
T0 1
Z t
2
1
=
γ1 (s) Φ0s (t, s, T0 ) + l1 (s)Φ(t, s, T0 ) ds
2
T0
Z t
2
1
1
−
Φ(t, s, T0 )v(s) − γ1 (s) Φ0s (t, s, T0 ) + l1 (s)Φ(t, s, T0 ) ds,
2
T0 γ1 (s)
≤2
which implies
Z t
2
2 1
Φ (t, s, T0 )Θ1 (s) − γ1 (s) Φ0s (t, s, T0 ) + l1 (s)Φ(t, s, T0 ) ds ≤ 0.
2
T0
This contradicts (2.1) and completes the proof.
Theorem 2.2. Let (S2), (N1), (N2) hold. If there exist functions ϕ ∈ C 1 (I, R+ ),
φ ∈ C 1 (I, R) and Φ ∈ X such that for each T ≥ t0 ,
Z tn
2 o
1
lim sup
Φ2 (t, s, T )Θ2 (s) − γ2 (s) Φ0s (t, s, T ) + l2 (s)Φ(t, s, T )
ds > 0, (2.4)
2
t→∞
T
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OSCILLATION CRITERIA
5
where
ϕ0 (t) 2k2 g 0 (t, a)φ(t)
r[g(t, a)]ϕ(t)
+
, γ2 (t) =
,
ϕ(t)
r[g(t, a)]
k2 g 0 (t, a)
o
n Z b
k2 g 0 (t, a)φ2 (t)
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ) +
− φ0 (t) ,
Θ2 (t) = ϕ(t) M
r[g(t, a)]
a
l2 (t) =
then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of (1.1) on I, say x(t) > 0 for t ≥
t0 . Then, by Lemma 1.2, there exists a T0 ≥ t0 such that (1.2) and (1.3) hold.
We consider the function v(t) defined by (2.2), and proceeding as in the proof of
Theorem 2.1 to get
i
h Z b
ϕ0 (t)
0
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ) − φ0 (t)
v (t) ≤
v(t) − ϕ(t) M
ϕ(t)
a
2
0
ϕ(t)g (t, a)
f 0 [Z(g(t, a))] r(t)ψ(x(t))Z 0 (t)
−
.
r[g(t, a)]ψ[x(g(t, a))] ψ[x(g(t, a))]
f [Z(g(t, a))]
Now, we use x[g(t, a)] ≤ Z[g(t, a)] and (N 2) to obtain
f 0 [Z(g(t, a))]
f 0 [Z(g(t, a))]
≥
≥ k2 .
ψ[x(g(t, a))]
ψ[Z(g(t, a))]
Therefore,
h Z b
i
ϕ0 (t)
v (t) ≤
v(t) − ϕ(t) M
q(t, ξ)f [1 − p(g(t, ξ))]dσ(ξ) − φ0 (t)
ϕ(t)
a
2
k2 ϕ(t)g 0 (t, a) r(t)ψ(x(t))Z 0 (t)
−
r[g(t, a)]
f [Z(g(t, a))]
1 2
= −Θ2 (t) + l2 (t)v(t) −
v (t).
γ2 (t)
0
The rest of the proof is as in Theorem 2.1.
Theorem 2.3. Let (S3) holds. If there exist functions ϕ ∈ C 1 (I, R+ ), φ ∈ C 1 (I, R)
and Φ ∈ X such that for each T ≥ t0 ,
Z tn
2 o
1
lim sup
Φ2 (t, s, T )Θ3 (s) − γ3 (s) Φ0s (t, s, T ) + l3 (s)Φ(t, s, T )
ds > 0, (2.5)
2
t→∞
T
where
r[g(t, a)]ϕ(t)
ϕ0 (t) 2Lg 0 (t, a)φ(t)
+
, γ3 (t) =
,
ϕ(t)
r[g(t, a)]
Lg 0 (t, a)
n Z b
o
Lg 0 (t, a)φ2 (t)
Θ3 (t) = ϕ(t) k3
q(t, ξ)[1 − p(g(t, ξ))]dσ(ξ) +
− φ0 (t) ,
r[g(t, a)]
a
l3 (t) =
then (1.1) is oscillatory.
Proof. Let x(t) be a nonoscillatory solution of (1.1) on I, say x(t) > 0 for t ≥ t0 .
Then, by Lemma 1.3, there exists a T0 ≥ t0 such that (1.2) and (1.4) hold. We
define the function
h r(t)ψ(x(t))Z 0 (t)
i
v(t) = ϕ(t)
+ φ(t) for t ≥ T0 .
(2.6)
Z[g(t, a)]
6
G. GUI, Z. XU
EJDE-2007/10
Differentiating (2.6) and using (1.4), we obtain
n Z b
o
ϕ0 (t)
0
v (t) ≤
v(t) − ϕ(t) k3
q(t, ξ)[1 − p(g(t, ξ))]dσ(ξ) − φ0 (t)
ϕ(t)
a
2
ϕ(t)g 0 (t, a)
r(t)ψ(x(t))Z 0 (t)
−
r[g(t, a)]ψ[x(g(t, a))]
Z(g(t, a))
Z b
n
o
0
ϕ (t)
v(t) − ϕ(t) k3
q(t, ξ)[1 − p(g(t, ξ))]dσ(ξ) − φ0 (t)
≤
ϕ(t)
a
2
Lϕ(t)g 0 (t, a) v(t)
−
− φ(t)
r[g(t, a)]
ϕ(t)
1 2
v (t).
= −Θ3 (t) + l3 (t)v(t) −
γ3 (t)
The rest of the proof follows the same lines as that of Theorem 2.1.
p
Let Φ(t, s, T ) = H1 (t, s)H2 (s, T ), where H1 , H2 ∈ Y . By Theorems 2.1-2.3,
we have the following interesting results.
Corollary 2.4. Let (S1), (N1) hold. If there exist functions ϕ ∈ C 1 (I, R+ ), φ ∈
C 1 (I, R), Φ ∈ X and H1 , H2 ∈ Y such that for each T ≥ t0 ,
Z t
n
o
1
lim sup
H1 (t, s)H2 (s, T ) Θ1 (s) − γ1 (s)[h1 (t, s) + h2 (s, T ) + l1 (s)]2 ds > 0,
4
t→∞
T
(2.7)
where h1 (t, s) and h2 (s, T ) are defined by
∂H1 (t, s)
= h1 (t, s)H1 (t, s)
∂s
and
∂H2 (s, T )
= h2 (s, T )H2 (s, T ),
∂s
(2.8)
then (1.1) is oscillatory.
Corollary 2.5. Let (S2), (N1), (N2) hold. If there exist functions ϕ ∈ C 1 (I, R+ ),
φ ∈ C 1 (I, R), Φ ∈ X and H1 , H2 ∈ Y such that for each T ≥ t0 ,
Z t
n
o
1
lim sup
H1 (t, s)H2 (s, T ) Θ2 (s) − γ2 (s)[h1 (t, s) + h2 (s, T ) + l2 (s)]2 ds > 0,
4
t→∞
T
(2.9)
where h1 (t, s) and h2 (s, T ) are defined by (2.8), then (1.1) is oscillatory.
Corollary 2.6. Let (S3) hold. If there are functions ϕ ∈ C 1 (I, R+ ), φ ∈ C 1 (I, R),
Φ ∈ X and H1 , H2 ∈ Y such that for each T ≥ t0 ,
Z t
o
n
1
lim sup
H1 (t, s)H2 (s, T ) Θ3 (s) − γ3 (s)[h1 (t, s) + h2 (s, T ) + l3 (s)]2 ds > 0,
4
t→∞
T
(2.10)
where h1 (t, s) and h2 (s, T ) are defined by (2.8), then (1.1) is oscillatory.
Let Φ(t, s, T ) = (t − s)(s − T )α for α > 1/2. By Theorems 2.1-2.3, we can
establish the following important results.
Corollary 2.7. Let (S1) and (N1) hold. If there exist functions ϕ ∈ C 2 (I, R+ ),
φ ∈ C 1 (I, R) and constants α > 1/2, m1 > 0 such that γ1 (t) ≤ m1 and for each
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OSCILLATION CRITERIA
7
T ≥ t0 ,
lim sup
t→∞
t
Z
1
t2α+1
(t−s)2 (s−T )2α
T
h 1
i
1
1
α
Θ1 (s)+ l10 (s)− l12 (s) ds >
,
m1
2
4
(2α − 1)(2α + 1)
(2.11)
then (1.1) is oscillatory.
Proof. Suppose that (1.1) has a nonoscillatory solution x(t) > 0. By using the
same arguments as in the proof of Theorem 2.1 and denoting (t − s)(s − T0 )α by
Φ(t, s, T0 ), we have
Z t
Φ2 (t, s, T0 )Θ1 (s)ds
T0
Z
t
2
1
γ1 (s) Φ0s (t, s, T0 ) + l1 (s)Φ(t, s, T0 ) ds
2
T0
Z tn
o
1
≤ m1
Φ0 2s (t, s, T0 ) + Φ0s (t, s, T0 )Φ(t, s, T0 )l1 (s) + Φ2 (t, s, T0 )l12 (s) ds.
4
T0
≤
Noting that
Z
t
Φ0s (t, s, T0 )Φ(t, s, T0 )l1 (s)ds = −
T0
1
2
Z
t
Φ2 (t, s, T0 )l10 (s)ds,
T0
we get
Z t
1
Φ2 (t, s, T0 )Θ1 (s)ds
m1 T0
Z tn
o
1
1
≤
Φ0 2s (t, s, T0 ) − Φ2 (t, s, T0 )l10 (s) + Φ2 (t, s, T0 )l12 (s) ds
2
4
T0
Z t
Z t
1
1
=
Φ2 (t, s, T0 ) − l10 (s) + l12 (s) ds +
[α(t − s)(s − T0 )α−1 − (s − T0 )α ]2 ds
2
4
T0
T0
Z t
1
1
α
(t − T0 )2α+1 .
=
Φ2 (t, s, T0 ) − l10 (s) + l12 (s) ds +
2
4
(2α
−
1)(2α
+
1)
T0
Therefore,
lim sup
1
Z
t
(t − s)2 (s − T0 )2α
t2α+1 T0
α
≤
,
(2α − 1)(2α + 1)
t→∞
i
h 1
1
1
Θ1 (s) + l10 (s) − l12 (s) ds
m1
2
4
which contradicts (2.11) and completest the proof.
Similar to the proof of Corollary 2.7, by Theorems 2.2 and 2.3, we can easily
obtain the following results.
Corollary 2.8. Let (S2), (N1), (N2) hold. If there exist functions ϕ ∈ C 2 (I, R+ ),
φ ∈ C 1 (I, R) and constants α > 1/2, m2 > 0 such that γ2 (t) ≤ m2 and for each
T ≥ t0 ,
Z t
h 1
i
1
1
1
α
lim sup 2α+1
(t−s)2 (s−T )2α
Θ2 (s)+ l20 (s)− l22 (s) ds >
,
m2
2
4
(2α − 1)(2α + 1)
t→∞ t
T
(2.12)
then (1.1) is oscillatory.
8
G. GUI, Z. XU
EJDE-2007/10
Corollary 2.9. Let (S3) holds. If there exist functions ϕ ∈ C 2 (I, R+ ), φ ∈
C 1 (I, R) and constants α > 1/2, m3 > 0 such that γ3 (t) ≤ m3 and for each
T ≥ t0 ,
Z t
h 1
i
1
1
1
α
(t−s)2 (s−T )2α
lim sup 2α+1
Θ3 (s)+ l30 (s)− l32 (s) ds >
,
m3
2
4
(2α − 1)(2α + 1)
t→∞ t
T
(2.13)
then (1.1) is oscillatory.
3. Interval oscillation criteria
We can easily see that the results in Section 2 involve the integral of the coefficients p, q and r, and hence, requires the information of the coefficients on the
entire half-line [t0 , ∞). In this section, we will establish several interval oscillation
criteria for (1.1).
Theorem 3.1. Let (S1) and (N1) hold. If for each T ≥ t0 , there exist functions
ϕ ∈ C 1 (I, R+ ), φ ∈ C 1 (I, R), Φ ∈ X and two constants d > c ≥ T such that
Z dn
2 o
1
ds > 0,
(3.1)
Φ2 (d, s, c)Θ1 (s) − γ1 (s) Φ0s (d, s, c) + l1 (s)Φ(d, s, c)
2
c
where Θ1 , γ1 , l1 are defined as in Theorem 2.1, then (1.1) is oscillatory.
Proof. With the proof of Theorem 2.1, where t and T are replaced by d and c,
respectively, we can easily see that every solution of (1.1) has at least one zero
in (c, d); i.e., every solution of (1.1) has arbitrarily large zeros on [t0 , ∞). This
completes the proof of Theorem 3.1.
Similar to the proof of Theorem 3.1, we can establish the following theorems.
Theorem 3.2. Let (S2), (N1) and (N2) hold. If for each T ≥ t0 , there exist
functions ϕ ∈ C 1 (I, R+ ), φ ∈ C 1 (I, R), Φ ∈ X and two constants d > c ≥ T such
that
Z dn
2 o
1
Φ2 (d, s, c)Θ2 (s) − γ2 (s) Φ0s (d, s, c) + l2 (s)Φ(d, s, c)
ds > 0,
(3.2)
2
c
where Θ2 , γ2 , l2 are defined as in Theorem 2.2, then (1.1) is oscillatory.
Theorem 3.3. Let (S3) holds. If for each T ≥ t0 , there exist functions ϕ ∈
C 1 (I, R+ ), φ ∈ C 1 (I, R), Φ ∈ X and two constants d > c ≥ T such that
Z dn
2 o
1
ds > 0,
(3.3)
Φ2 (d, s, c)Θ3 (s) − γ3 (s) Φ0s (d, s, c) + l3 (s)Φ(d, s, c)
2
c
where Θ3 , γ3 , l3 are defined as in Theorem 2.3, then (1.1) is oscillatory.
Corollary 3.4. Let (S1) and (N1) hold. If for each T ≥ t0 , there exist functions
H1 , H2 ∈ Y and two constants d > c ≥ T such that
Z d
n
2 o
1
H1 (d, s)H2 (s, c) Θ1 (s) − γ1 (s) h1 (d, s) + h2 (s, c) + l1 (s)
ds > 0, (3.4)
4
c
where h1 (d, s) and h2 (s, c) are defined by (2.8), then (1.1) is oscillatory.
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OSCILLATION CRITERIA
9
Corollary 3.5. Let (S2), (N1) and (N2) hold. If for each T ≥ t0 , there exist
functions H1 , H2 ∈ Y and two constants d > c ≥ T such that
Z d
n
o
1
H1 (d, s)H2 (s, c) Θ2 (s) − γ2 (s)[h1 (d, s) + h2 (s, c) + l2 (s)]2 ds > 0, (3.5)
4
c
where h1 (d, s) and h2 (s, c) are defined by (2.8), then (1.1) is oscillatory.
Corollary 3.6. Let (S3) holds. If for each T ≥ t0 , there exist functions H1 , H2 ∈ Y
and two constants d > c ≥ T such that
Z d
o
n
1
H1 (d, s)H2 (s, c) Θ3 (s) − γ3 (s)[h1 (d, s) + h2 (s, c) + l3 (s)]2 ds > 0, (3.6)
4
c
where h1 (d, s) and h2 (s, c) are defined by (2.8), then (1.1) is oscillatory.
4. Examples
Example 4.1. Consider the equation
Z 1
0 0
1
ξx(ln tξ)
−t
x(t)
+
(1
−
e
)x(t
−
1)
dξ = 0,
+
µ
t
2
1
e (1 + x (t))
t
2
t ≥ 2, (4.1)
where r(t) = e−t , ψ(x) = 1/(1 + x2 ), p(t) = 1 − e−t , µ > 1, g(t, ξ) = ln tξ, f (x) = x
and q(t, ξ) = µξ/t.
If we take k1 = L = M = 1, m1 = 2, φ(t) = t and ϕ(t) = 1, then
µ
1
Θ1 (t) = 2 + t2 − 1, l1 (t) = t, γ1 (t) = 2.
2t
2
Thus, the left-hand side of (2.11) takes the following from
Z t
1
µ
1
µ
1
lim sup
.
(t − s)2 (s − T )2α 2 ds =
4 t→∞ t2α+1 T
s
4 α(2α − 1)(2α + 1)
For any µ > 1, there exists a constant α > 1/2 such that µ/4 > α2 , i.e.,
µ
1
α
>
,
4 α(2α − 1)(2α + 1)
(2α − 1)(2α + 1)
i.e., (2.11) holds for µ > 1. By Corollary 2.7, (4.1) is oscillatory for µ > 1.
Example 4.2. Consider the equation
Z 1
0 0
1
2
+
t4 ξ 3 x3 (tξ)dξ = 0,
x (t) x(t) + (1 − )x(t − 1)
1
t
2
t ≥ 2,
(4.2)
where r(t) = 1, ψ(x) = x2 , p(t) = 1 − 1/t, g(t, ξ) = tξ, f (x) = x3 and q(t, ξ) = t4 ξ 3 .
If we take ϕ(t) = 1, φ(t) = 0, m2 = 2/3, k2 = 3 and M = 1, then
t
2
Θ2 (t) = , γ2 (t) = , l2 (t) = 0.
2
3
For any constant T ≥ 2, there exists n ∈ N0 = {1, 2, · · · } such that 2nπ ≥ T . Let
d = (2n + 1)π, c = 2nπ ≥ T and H1 (t, s)H2 (s, T ) = | sin(t − s) sin(s − T )|, we have
H1 ((2n + 1)π, s)H2 (s, 2nπ) = sin2 s for
c ≤ s ≤ d.
Thus, the left-hand side of (3.5) takes the from
Z (2n+1)π
s 4
(4n + 1)π 2
2π
sin2 s − cot2 s ds =
−
> 0.
2
3
8
3
2nπ
10
G. GUI, Z. XU
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Hence, by Corollary 3.5, (4.2) is oscillatory.
Example 4.3. Consider the equation
√
√
Z 1
0 0
1
1
ξ x3 ( t + ξ) + 3x( t + ξ)
√
x(t)+ x(t−1)
+µ
dξ = 0,
2
t(1 + x2 (t))
2
1 + x2 ( t + ξ)
0 t
where r(t) = 1/t, ψ(x) = 1/(1 + x2 ), p(t) = 1/2, g(t, ξ) =
3x)/(1 + x2 ), q(t, ξ) = µξ/t2 and µ > 2.
If we take k3 = L = 1, m3 = 2, ϕ(t) = 1, φ(t) = 1/t, then
√
t ≥ 1,
(4.3)
t + ξ, f (x) = (x3 +
µ+6
1
, l3 (t) = , γ3 (t) = 2.
4t2
t
Thus, the left-hand side of (2.13) takes the following from
Z t
1
1
µ
1
µ
lim sup
(t − s)2 (s − T )2α 2 ds =
.
8 t→∞ t2α+1 T
s
8 α(2α − 1)(2α + 1)
Θ3 (t) =
So, for any µ > 2, there exists a constant α > 1/2 such that
µ
1
α
>
.
8 α(2α − 1)(2α + 1)
(2α − 1)(2α + 1)
Therefore, by Corollary 2.9, Equation (4.3) is oscillatory for µ > 2.
Acknowledgments. The authors are grateful to the anonymous referee for their
valuable comments and suggestions which led to an improvement of the original
manuscript.
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Gaihua Gui
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631,
China
E-mail address: 1981hgg@163.com
Zhiting Xu
School of Mathematical Sciences, South China Normal University, Guangzhou, 510631,
China
E-mail address: xztxhyyj@pub.guangzhou.gd.cn