Theoretical Mathematics & Applications, vol.3, no.1, 2013, 19-33
ISSN: 1792-9687 (print), 1792-9709 (online)
Scienpress Ltd, 2013
Oscillation and Asymptotic Behavior of
Solutions of Certain Third-Order Nonlinear
Delay Dynamic Equations
Da-Xue Chen1
Abstract
The paper deals with the oscillation and asymptotic behavior of solutions of the third-order nonlinear delay dynamic equation
(
Ã
! β )∆
h
¡ ∆ ¢α i∆
b(t) a(t) x (t)
+ f (t, x(τ (t))) = 0
on a time scale T, where α, β > 0 are quotients of odd positive integers.
We obtain some sufficient conditions which ensure that every solution
of the equation either oscillates or converges to zero. Our results extend
and improve some known results.
Mathematics Subject Classification: 34N05
Keywords: Oscillation; Asymptotic behavior; Third-order nonlinear delay
dynamic equation; Time scale
1
College of Science, Hunan Institute of Engineering, 88 East Fuxing Road,
Xiangtan 411104, Hunan, P. R. China, e-mail: cdx2003@163.com
Article Info: Received : October 29, 2012. Revised : December 28, 2012
Published online : April 15, 2013
20
1
Oscillation and Asymptotic Behavior of Solutions of Dynamic Equations
Introduction
In this paper, we are concerned with the oscillation and asymptotic behavior of solutions of the third-order nonlinear delay dynamic equation
(
!β )∆
Ã
h
¡ ∆ ¢α i∆
+ f (t, x(τ (t))) = 0
b(t) a(t) x (t)
(1)
on a time scale T. Throughout this paper we assume that the following conditions hold:
(S1 ) sup T = ∞, and α and β are quotients of odd positive integers;
(S2 ) t0 ∈ T, I := {t : t ∈ T, t ≥ t0 }, a, b ∈ Crd (I, R), a(t), b(t) > 0 for t ∈ I,
R ∞ −1/α
R∞
a
(t)∆t = ∞, and t0 b−1/β (t)∆t = ∞;
t0
(S3 ) τ ∈ Crd (T, T), τ (t) ≤ t for t ∈ I, and limt→∞ τ (t) = ∞;
(S4 ) f ∈ C(I × R, R), and there exists a positive rd-continuous function q
defined on I such that f (t, u)/(uγ ) ≥ q(t) for all t ∈ I and for all u 6= 0,
where γ := αβ;
(S5 ) τ ∆ (t) > 0 is rd-continuous on T, T̃ := τ (T) = {τ (t) : t ∈ T} ⊂ T is a
time scale, and (τ σ )(t) = (σ ◦ τ )(t) for all t ∈ T, where σ is the forward
jump operator on T and (τ σ )(t) := (τ ◦ σ)(t).
Recall that a solution of (1) is a nontrivial real function x such that
x ∈ C1rd [tx , ∞), a(x∆ )α ∈ C1rd [tx , ∞), b([a(x∆ )α ]∆ )β ∈ C1rd [tx , ∞) for a certain
tx ≥ t0 , and x satisfies (1) for t ≥ tx . Our attention is restricted to those
solutions of (1) which exist on the half-line [tx , ∞) and satisfy sup{|x(t)| : t >
t∗ } > 0 for any t∗ ≥ tx . A solution x of (1) is said to be oscillatory if it is neither eventually positive nor eventually negative, otherwise it is nonoscillatory.
Equation (1) is said to be oscillatory if all its solutions are oscillatory.
In this work a knowledge and understanding of time scales and of time
scale notations is assumed; for an excellent introduction to the calculus on
time scales, see Bohner and Peterson [15, 16]. A time scale T is an arbitrary
nonempty closed subset of the reals, and the cases when this time scale is equal
to the reals or to the integers represent the classical theories of differential
equations and of difference equations. Many interesting time scales exist, and
Da-Xue Chen
21
they give rise to many applications (see [15]). The new theory of the so-called
“dynamic equations” not only can unify the theories of differential equations
and of difference equations, but also is able to extend these classical cases
to cases “in between,” e.g., to the so-called q-difference equations when T =
q N0 := {q k : k = 0, 1, 2, · · · , q > 1} (which has important applications in
quantum theory) and can be applied to different types of time scales like T =
hZ := {hk : k ∈ Z, h > 0}, T = N20 := {k 2 : k = 0, 1, 2, · · · } and T = Hn the
space of harmonic numbers. In the last few years, there has been an increasing
interest in obtaining sufficient conditions for the oscillation/nonoscillation and
asymptotic behavior of solutions of different classes of dynamic equations, and
we refer the reader to the papers [1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 17,
18, 19, 20, 21, 22, 23, 24] and the references cited therein.
Recently, Erbe et al. [11] considered the case when α = 1, β ≥ 1 is a
quotient of odd positive integers and τ (t) = t in (1) and established some
sufficient conditions which ensure that every solution of (1) either oscillates or
has a finite limit at ∞. Besides, Erbe et al. [14] studied the case when α = 1
and β is a quotient of odd positive integers in (1) and improved and extended
the results in [11]. Hassan [22] investigated the case when α = 1 and β ≥ 1 is
a quotient of odd positive integers in (1) and gave several oscillation criteria
for (1). Yu and Wang [24] were concerned with the case when α and β are
quotients of odd positive integers, αβ = 1 and τ (t) = t in (1) and obtained two
sufficient conditions for the asymptotic and oscillatory behavior of solutions of
(1).
It is clear that what the papers [11, 14, 22, 24] considered are some special
cases of (1) and that the results in [11, 14, 22, 24] cannot be applied to the
general cases of (1). For instance, all the results in [11, 14, 22] cannot be
applied to (1) when α 6= 1, the results in [11, 24] are invalid when τ (t) 6= t,
and the results in [24] fail to be applied to (1) when αβ 6= 1. Therefore, it
is of great interest to investigate the oscillation and asymptotic behavior of
solutions of (1) in the general cases. In this paper, for the case when α and
β are quotients of odd positive integers and τ (t) ≤ t, we establish several
sufficient conditions which ensure that every solution of (1) either oscillates or
tends to zero. Our results extend and improve some of the results presented
in [11, 14, 22, 24].
In what follows, for convenience, when we write a functional inequality
22
Oscillation and Asymptotic Behavior of Solutions of Dynamic Equations
without specifying its domain of validity we assume that it holds for all sufficiently large t.
2
Lemmas
Lemma 2.1 (Chen [7], Lemma 2.3). Suppose that (S5 ) holds. Let x :
T → R. If x∆ (t) exists for all sufficiently large t ∈ T, then (x ◦ τ )∆ (t) =
(x∆ ◦ τ )(t)τ ∆ (t) for all sufficiently large t ∈ T.
Lemma 2.2 (Chen [7], Lemma 2.4). Let ψ : T → R and λ > 0 be a
constant. Furthermore, assume ψ ∆ (t) > 0 and ψ(t) > 0 for all sufficiently
large t ∈ T. Then we have the following:
(i) If 0 < λ < 1, then (ψ λ )∆ (t) ≥ λ(ψ σ )λ−1 (t)ψ ∆ (t) for all sufficiently large
t ∈ T, where ψ σ := ψ ◦ σ;
(ii) If λ ≥ 1, then (ψ λ )∆ (t) ≥ λψ λ−1 (t)ψ ∆ (t) for all sufficiently large t ∈ T.
Lemma 2.3 (Hardy et al. [8]). If A and B are nonnegative, then
λAB λ−1 − Aλ ≤ (λ − 1)B λ
when λ > 1,
where the equality holds if and only if A=B.
Lemma 2.4. Suppose that (S1 )–(S4 ) and the following condition hold:
Z
∞
t0
½
Z
−1
a (s)
∞
·
Z
b (u)
s
¾1/α
¸1/β
∞
−1
q(v)∆v
∆u
∆s = ∞.
(2)
u
Furthermore, suppose that (1) has an eventually positive solution x. Then
h
¡
¢α i∆
> 0,
a(t) x∆ (t)
and either x∆ (t) > 0 or limt→∞ x(t) = 0.
(3)
23
Da-Xue Chen
Proof. Since x is an eventually positive solution of (1), form (S3 ) there exists
t1 ≥ t0 such that
x(t) > 0 and x(τ (t)) > 0 for t ∈ [t1 , ∞).
Therefore, from (1) and (S4 ) we have for t ∈ [t1 , ∞),
(
Ã
!β )∆
h
¡ ∆ ¢α i∆
b(t) a(t) x (t)
= −f (t, x(τ (t))) ≤ −q(t)xγ (τ (t)) < 0,
(4)
(5)
³h
¡
¢α i∆ ´β
which implies that b(t) a(t) x∆ (t)
is strictly decreasing on [t1 , ∞).
h
i
h
¡
¢α ∆
¡
¢α i∆
Thus, a(t) x∆ (t)
is eventually of one sign, i.e., a(t) x∆ (t)
is eventually positive or eventually negative.
We now claim
h
¡
¢α i ∆
a(t) x∆ (t)
> 0 for t ∈ [t1 , ∞).
(6)
If not, then there exists t2 ≥ t1 such that
h
¡
¢α i ∆
a(t) x∆ (t)
< 0 for t ∈ [t2 , ∞).
(7)
³h
¡
¢α i∆ ´β
Since b(t) a(t) x∆ (t)
is strictly decreasing on [t1 , ∞), we get
³h
³h
¡
¢α i ∆ ´β
¡
¢α i∆ ´β
b(t) a(t) x∆ (t)
≤ b(t2 ) a(t2 ) x∆ (t2 )
:= c1 < 0
h
¡
¢α i∆
1/β
for t ∈ [t2 , ∞). Therefore, we conclude a(t) x∆ (t)
≤ c1 b−1/β (t) for
t ∈ [t2 , ∞). Integrating both sides of the last inequality from t2 to t, we obtain
Z t
¡ ∆ ¢α
¡ ∆
¢α
1/β
a(t) x (t) ≤ a(t2 ) x (t2 ) + c1
b−1/β (s)∆s for t ∈ [t2 , ∞).
t2
¡
¢α
Letting t → ∞ and using (S2 ), we get limt→∞ a(t) x∆ (t) = −∞. Thus,
¡
¢α
there exists t3 ≥ t2 such that a(t3 ) x∆ (t3 ) < 0. It follows from (7) that
¡
¢α
¡
¢α
a(t) x∆ (t) is strictly decreasing on [t2 , ∞). Hence, we obtain a(t) x∆ (t) ≤
¡
¢α
1/α
a(t3 ) x∆ (t3 ) := c2 < 0 and x∆ (t) ≤ c2 a−1/α (t) for t ∈ [t3 , ∞). Integrating
both sides of the last inequality from t3 to t, we find
Z t
1/α
a−1/α (s)∆s for t ∈ [t3 , ∞).
x(t) ≤ x(t3 ) + c2
t3
24
Oscillation and Asymptotic Behavior of Solutions of Dynamic Equations
Letting t → ∞ and using (S2 ), we get limt→∞ x(t) = −∞, which contradicts
the fact that x is an eventually positive solution of (1). Therefore, (6) holds.
¡
¢α
From (6) we conclude that a(t) x∆ (t) is strictly increasing on [t1 , ∞).
¡
¢α
¡
¢α
Thus, a(t) x∆ (t) as well as x∆ (t) is eventually of one sign, i.e., a(t) x∆ (t)
as well as x∆ (t) is eventually positive or eventually negative. If x∆ (t) is eventually negative, then we obtain limt→∞ x(t) := l1 ≥ 0 and
¡
¢α
lim a(t) x∆ (t) := l2 ≤ 0,
(8)
t→∞
and we conclude that there exists t4 ≥ t1 such that x(t) ≥ l1 for t ∈ [t4 , ∞).
Thus, from (S3 ) there exists t5 ∈ [t4 , ∞) such that
x(τ (t)) ≥ l1
for t ∈ [t5 , ∞).
(9)
Next, we prove l1 = 0. From (5) and (9) we obtain
(
Ã
!β )∆
h
¡ ∆ ¢α i∆
b(t) a(t) x (t)
≤ −q(t)xγ (τ (t)) ≤ −l1γ q(t) for t ∈ [t5 , ∞),
Integrating both sides of the last inequality from t to u, we get
Ã
!β
h
¡ ∆ ¢α i ∆
− b(t) a(t) x (t)
Ã
!β
Ã
!β
h
h
¡ ∆ ¢α i ∆
¡ ∆ ¢α i∆
< b(u) a(u) x (u)
− b(t) a(t) x (t)
Z
≤
−l1γ
u
q(v)∆v
for u ≥ t ≥ t5 .
t
³h
¡ ∆ ¢α i∆ ´β
R∞
Letting u → ∞, we have −b(t) a(t) x (t)
≤ −l1γ t q(v)∆v and
·
¸1/β
h
¡ ∆ ¢α i ∆
R∞
γ/β
−1
for t ∈ [t5 , ∞). Integrating
≤ −l1 b (t) t q(v)∆v
− a(t) x (t)
both sides of the last inequality from t to ∞, we conclude for t ∈ [t5 , ∞)
¸1/β
Z ∞·
Z ∞
¡ ∆ ¢α
¡ ∆ ¢α
γ/β
−1
∆u,
b (u)
q(v)∆v
a(t) x (t) ≤ −l2 + a(t) x (t) ≤ −l1
t
where l2 is defined as in (8). Hence, we have
½
¸1/β ¾1/α
Z ∞·
Z ∞
∆
−1
−1
∆u
x (t) ≤ −l1 a (t)
b (u)
q(v)∆v
t
u
u
for t ∈ [t5 , ∞).
25
Da-Xue Chen
Integrating both sides of the last inequality from t5 to t, we obtain for t ≥ t5 ,
¸1/β ¾1/α
Z t½
Z ∞·
Z ∞
−1
−1
x(t) ≤ x(t5 ) − l1
a (s)
b (u)
q(v)∆v
∆u
∆s.
t5
s
u
Assume l1 > 0. Letting t → ∞ and using (2), we see limt→∞ x(t) = −∞, which
contradicts the fact that x is an eventually positive solution of (1). Therefore,
we have l1 = 0, which implies limt→∞ x(t) = 0. The proof is complete.
Lemma 2.5. Suppose that (S1 )–(S4 ) hold and that x is an eventually positive
solution of (1). Furthermore, assume that there exists T∗ ∈ [t0 , ∞) such that
h
¡
¢α i∆
a(t) x∆ (t)
> 0 and x∆ (t) > 0 for t ∈ [T∗ , ∞). Then there exists T ≥ T∗
such that
³h
¡
¢α i∆ ´1/α
x∆ (t) > g1 (t, T )b1/γ (t) a(t) x∆ (t)
for t ∈ [T, ∞),
(10)
³R
´1/α
t
where g1 (t, T ) := a−1/α (t) T b−1/β (s)∆s
, here γ is defined as in (S4 ).
Proof. Proceeding as in the proof of Lemma 2.4, we obtain (4) and (5). Let
T := max{t1 , T∗ }. Since x∆ (t) > 0 for t ∈ [T, ∞), we have for t ∈ [T, ∞),
¡
¢α
¡
¢α
¡
¢α
a(t) x∆ (t) > a(t) x∆ (t) − a(T ) x∆ (T )
¾
Z t½
h
¡ ∆ ¢α i∆ −1/β
1/β
=
b (s) a(s) x (s)
b
(s)∆s.
(11)
T
³h
¡ ∆ ¢α i ∆ ´β
From (5) we obtain that b(t) a(t) x (t)
is strictly decreasing on [T, ∞).
³h
i
´
³h
∆
β
¡
¢α
¡
¢α i∆ ´β
Thus, we get b(s) a(s) x∆ (s)
≥ b(t) a(t) x∆ (t)
for t ≥ s ≥
T and
h
h
¡
¢α i ∆
¡
¢α i∆
b1/β (s) a(s) x∆ (s)
≥ b1/β (t) a(t) x∆ (t)
for t ≥ s ≥ T. (12)
It follows from (11) and (12) that
h
¡
¢α i ∆
¡
¢α
a(t) x∆ (t) > b1/β (t) a(t) x∆ (t)
Z
t
b−1/β (s)∆s
T
and
∆
x (t) > a
−1/α
(t)b
1/γ
µZ t
¶1/α
³h
¡ ∆ ¢α i∆ ´1/α
−1/β
(t) a(t) x (t)
b
(s)∆s
³h
¡
¢α i∆ ´1/α
= g1 (t, T )b1/γ (t) a(t) x∆ (t)
T
for t ∈ [T, ∞),
where g1 (t, T ) is defined as in Lemma 2.5. The proof is complete.
(13)
26
3
Oscillation and Asymptotic Behavior of Solutions of Dynamic Equations
Main Results
Theorem 3.1. Assume that (S1 )–(S5 ) and (2) hold. Furthermore, suppose
that, for all sufficiently large T ∈ [t0 , ∞), there exist T1 > T and a positive
function ϕ ∈ C1rd (I, R) such that τ (T1 ) > T and
¾
Z t½
γ+1
(ϕ∆
+ (s))
lim sup
ϕ(s)q(s) −
∆s = ∞, (14)
(γ + 1)γ+1 [τ ∆ (s)ϕ(s)g1 (τ (s), T )]γ
t→∞
T1
∆
where ϕ∆
+ (s) := max{ϕ (s), 0} and the function g1 is defined as in Lemma 2.5.
Then every solution of (1) either oscillates or tends to zero.
Proof. Let x be a nonoscillatory solution of (1). Without loss of generality,
we may assume that x is an eventually positive solution of (1). Proceeding as
in the proof of Lemma 2.4, we see that there exists t1 ∈ [t0 , ∞) such that (4)
and (5) hold. By Lemma 2.4, there exists t2 ∈ [t1 , ∞) such that (3) holds for
t ∈ [t2 , ∞) and either x∆ (t) > 0 for t ∈ [t2 , ∞) or limt→∞ x(t) = 0. Assume
x∆ (t) > 0 for t ∈ [t2 , ∞). Consider the generalized Riccati substitution
w(t) = Q(t)
ϕ(t)
xγ (τ (t))
for t ∈ [t2 , ∞),
(15)
³h
¡
¢α i ∆ ´β
where Q(t) := b(t) a(t) x∆ (t)
and γ is defined as in (S4 ). It is easy
to see that w(t) > 0 for t ∈ [t2 , ∞). By the following product and quotient
rules for the delta derivatives of the product F G and the quotient F/G of two
delta differentiable functions F and G:
³ F ´∆ F ∆ G − F G∆
F ∆ F G∆
(F G)∆ = F ∆ G + F σ G∆ and
=
=
−
,
G
GGσ
Gσ
GGσ
where F σ := F ◦ σ, Gσ := G ◦ σ and GGσ 6= 0, from (15) we get
h ϕ i∆
ϕ
∆
∆
σ
w =Q
+Q
(x ◦ τ )γ
(x ◦ τ )γ
h
ϕ∆
[(x ◦ τ )γ ]∆ i
ϕ
∆
σ
=Q
+Q
−ϕ
on [t2 , ∞). (17)
(x ◦ τ )γ
(x ◦ τ σ )γ
(x ◦ τ )γ (x ◦ τ σ )γ
Hence, from (5), (15) and (17) we have
ϕ∆ σ
Qσ [(x ◦ τ )γ ]∆
w
−
ϕ
ϕσ
(x ◦ τ )γ (x ◦ τ σ )γ
ϕ∆
Qσ [(x ◦ τ )γ ]∆
≤ −ϕq + +σ wσ − ϕ
ϕ
(x ◦ τ )γ (x ◦ τ σ )γ
w∆ ≤ −ϕq +
on [t2 , ∞),
(18)
27
Da-Xue Chen
where ϕ∆
+ is defined as in Theorem 3.1. From (S5 ) and Lemma 2.1, there exists
t3 ∈ [t2 , ∞) such that
(x ◦ τ )∆ = (x∆ ◦ τ )τ ∆ > 0 on [t3 , ∞).
(19)
If 0 < γ < 1, then by taking ψ = x ◦ τ and by Lemma 2.2 (i) and (19)
there exists t4 ∈ [t3 , ∞) such that
[(x ◦ τ )γ ]∆ ≥ γ(x ◦ τ σ )γ−1 (x ◦ τ )∆ = γ(x ◦ τ σ )γ−1 (x∆ ◦ τ )τ ∆
on [t4 , ∞). (20)
It follows from (18) and (20) that
ϕ∆
Qσ · γ(x ◦ τ σ )γ−1 (x∆ ◦ τ )τ ∆
+ σ
w
−
ϕ
ϕσ
(x ◦ τ )γ (x ◦ τ σ )γ
ϕ∆
Qσ
(x ◦ τ σ )γ ∆
= −ϕq + +σ wσ − γτ ∆ ϕ
·
(x ◦ τ )
ϕ
(x ◦ τ σ )γ+1 (x ◦ τ )γ
w∆ ≤ −ϕq +
(21)
on [t4 , ∞). If γ ≥ 1, then by taking ψ = x ◦ τ and by Lemma 2.2 (ii) and (19)
there exists t5 ≥ t4 such that
[(x ◦ τ )γ ]∆ ≥ γ(x ◦ τ )γ−1 (x ◦ τ )∆ = γ(x ◦ τ )γ−1 (x∆ ◦ τ )τ ∆
on [t5 , ∞). (22)
It follows from (18) and (22) that
ϕ∆
Qσ · γ(x ◦ τ )γ−1 (x∆ ◦ τ )τ ∆
+ σ
w ≤ −ϕq + σ w − ϕ
ϕ
(x ◦ τ )γ (x ◦ τ σ )γ
ϕ∆
Qσ
(x ◦ τ σ ) ∆
= −ϕq + +σ wσ − γτ ∆ ϕ
·
(x ◦ τ )
ϕ
(x ◦ τ σ )γ+1 (x ◦ τ )
∆
(23)
on [t5 , ∞). From (S5 ) we see τ (t) is increasing on T. Since t ≤ σ(t) for t ∈ T,
we have τ (t) ≤ τ σ (t) for t ∈ T. In view of x∆ (t) > 0 for t ∈ [t2 , ∞), we have
(x ◦ τ )(t) ≤ (x ◦ τ σ )(t) for t ∈ [t2 , ∞). Therefore, for all γ > 0, from (21) and
(23) we get
w∆ ≤ −ϕq +
ϕ∆
Qσ
+ σ
∆
w
−
γτ
ϕ
(x∆ ◦ τ ) on [t5 , ∞).
ϕσ
(x ◦ τ σ )γ+1
(24)
From (10) and the definition of the function Q, there exists T ∈ [t5 , ∞)
such that x∆ (t) > g1 (t, T )Q1/γ (t) for t ∈ [T, ∞). Take t6 ∈ (T, ∞) such that
τ (t) > T for t ∈ [t6 , ∞). Then we get (x∆ ◦ τ )(t) > g1 (τ (t), T )Q1/γ (τ (t)) for
t ∈ [t6 , ∞). Since τ (t) ≤ t ≤ σ(t) for t ∈ T and (5) implies that Q(t) is
28
Oscillation and Asymptotic Behavior of Solutions of Dynamic Equations
decreasing on [t1 , ∞), we have Q(τ (t)) ≥ Qσ (t) for t ∈ [t1 , ∞). Therefore, we
get (x∆ ◦ τ )(t) > g1 (τ (t), T )(Qσ )1/γ (t) for t ∈ [t6 , ∞). Hence, from (24) we
obtain for t ∈ [t6 , ∞),
ϕ∆
Qσ (t)g1 (τ (t), T )(Qσ )1/γ (t)
+ (t) σ
∆
w
(t)
−
γτ
(t)ϕ(t)
.
ϕσ (t)
(x ◦ τ σ )γ+1 (t)
(25)
From (15) and (25) we have for t ∈ [t6 , ∞),
w∆ (t) < −ϕ(t)q(t) +
µ σ ¶λ
ϕ∆
w (t)
+ (t) σ
∆
w (t) < −ϕ(t)q(t) + σ w (t) − γτ (t)ϕ(t)g1 (τ (t), T )
,
ϕ (t)
ϕσ (t)
∆
(26)
£
¤1/λ wσ (t)
where λ = 1 + 1/γ. For t ∈ [t6 , ∞), taking A = γτ ∆ (t)ϕ(t)g1 (τ (t), T )
ϕσ (t)
½
¾γ
∆
ϕ+ (t)
and B =
£
¤1/λ , by Lemma 2.3 and (26) we obtain
λ γτ ∆ (t)ϕ(t)g1 (τ (t),T )
w∆ (t) < −ϕ(t)q(t) +
γ+1
(ϕ∆
+ (t))
(γ + 1)γ+1 [τ ∆ (t)ϕ(t)g1 (τ (t), T )]γ
(27)
for t ∈ [t6 , ∞). Integrating both sides of the last inequality from t6 to t, we
obtain for t ∈ [t6 , ∞),
¾
Z t½
γ+1
(ϕ∆
+ (s))
w(t) − w(t6 ) ≤ −
∆s.
ϕ(s)q(s) −
(γ + 1)γ+1 [τ ∆ (s)ϕ(s)g1 (τ (s), T )]γ
t6
Since w(t) > 0 for t ∈ [t2 , ∞), we have for t ∈ [t6 , ∞),
¾
Z t½
γ+1
(ϕ∆
+ (s))
∆s < w(t6 ).
ϕ(s)q(s) −
(γ + 1)γ+1 [τ ∆ (s)ϕ(s)g1 (τ (s), T )]γ
t6
½
¾
γ+1
Rt
(ϕ∆
+ (s))
Thus, we get lim supt→∞ t6 ϕ(s)q(s) − (γ+1)γ+1 [τ ∆ (s)ϕ(s)g1 (τ (s),T )]γ ∆s ≤ w(t6 )
< ∞, which contradicts (14). Hence, the proof is complete.
We now introduce a function class R to present our next theorem. Let
D := {(t, s) ∈ T × T : t ≥ s ≥ t0 } and D0 := {(t, s) ∈ T × T : t > s ≥ t0 .
A function H ∈ Crd (D, R) is said to belong to the class R if H(t, t) = 0 for
t ≥ t0 , H(t, s) > 0 for (t, s) ∈ D0 , and H has a rd-continuous delta partial
derivative H ∆s (t, s) on D0 with respect to the second variable.
29
Da-Xue Chen
Theorem 3.2. Assume that (S1 )–(S5 )and (2) hold. Furthermore, suppose
that, for all sufficiently large T ∈ [t0 , ∞), there exist T1 > T , a positive function
ϕ ∈ C1rd (I, R), a function H ∈ R and a function h ∈ Crd (D, R) such that
τ (T1 ) > T ,
H ∆s (t, s) + H(t, s)
γ
ϕ∆
h(t, s) γ+1
+ (s)
=
H
(t, s) f or
ϕσ (s)
ϕσ (s)
(t.s) ∈ D
(28)
and
1
lim sup
t→∞ H(t, T1 )
Z t·
T1
¸
hγ+1
+ (t, s)
H(t, s)ϕ(s)q(s) −
∆s = ∞,
(γ + 1)γ+1 Ψγ (s, T )
(29)
∆
where ϕ∆
+ (s) is defined as in Theorem 3.1, Ψ(s, T ) := τ (s)ϕ(s)g1 (τ (s), T ) and
h+ (t, s) := max{0, h(t, s)}, here g1 is defined as in Lemma 2.5. Then every
solution of (1) either oscillates or tends to zero.
Proof. Assume that x is a nonoscillatory solution of (1). Without loss of
generality, assume that x is an eventually positive solution of (1). Proceeding
as in the proof of Theorem 3.1, we see that (26) holds. Multiplying (26) by
H(t, s) and then integrating from t6 to t, we find for t ∈ [t6 , ∞),
Z
Z
t
t
H(t, s)ϕ(s)q(s)∆s ≤ −
t6
Z
∆
t
H(t, s)w (s)∆s +
Z
t6
t
−
t6
H(t, s)
t6
ϕ∆
+ (s) σ
w (s)∆s
ϕσ (s)
µ σ ¶1+1/γ
w (s)
H(t, s)γΨ(s, T )
∆s,
ϕσ (s)
(30)
where Ψ(s, T ) is defined as in Theorem 3.2. Applying the integration by parts
formula
Z
d
h
id Z
F (s)G (s)∆s = F (s)G(s) −
d
∆
c
c
F ∆ (s)G(σ(s))∆s,
c
we get for t ∈ [t6 , ∞),
Z
t
−
t6
Z t
is=t
H(t, s)w (s)∆s = − H(t, s)w(s)
+
H ∆s (t, s)wσ (s)∆s
s=t6
t6
Z t
= H(t, t6 )w(t6 ) +
H ∆s (t, s)wσ (s)∆s.
(31)
∆
h
t6
30
Oscillation and Asymptotic Behavior of Solutions of Dynamic Equations
Substituting (31) in (30) and then using (28), we obtain for t ∈ [t6 , ∞)
Z
t
H(t, s)ϕ(s)q(s)∆s
Z t ½h
ϕ∆ (s) i σ
≤ H(t, t6 )w(t6 ) +
H ∆s (t, s) + H(t, s) +σ
w (s)
ϕ (s)
t6
µ σ ¶1+1/γ )
w (s)
−H(t, s)γΨ(s, T )
∆s
ϕσ (s)
t6
Z th
γ
h(t, s) γ+1
= H(t, t6 )w(t6 ) +
H (t, s)wσ (s)
σ
ϕ (s)
t6
µ σ ¶1+1/γ i
w (s)
∆s
− H(t, s)γΨ(s, T )
ϕσ (s)
Z th
γ
h+ (t, s) γ+1
≤ H(t, t6 )w(t6 ) +
H
(t, s)wσ (s)
σ (s)
ϕ
t6
µ σ ¶1+1/γ i
w (s)
∆s,
− H(t, s)γΨ(s, T )
ϕσ (s)
(32)
where h+ (t, s) is defined as in Theorem 3.2. Taking λ = 1 + 1/γ,
£
¤1/λ wσ (t)
A = H(t, s)γΨ(s, T )
ϕσ (t)
½
and B =
¾γ
£
h+ (t,s)
¤1/λ
λ γΨ(s,T )
Z
for t ≥ s ≥ t6 , by Lemma 2.3 and (32) we have
Z
t
t
H(t, s)ϕ(s)q(s)∆s ≤ H(t, t6 )w(t6 ) +
t6
for t ≥ t6 and
t6
1
H(t,t6 )
·
H(t, s)ϕ(s)q(s) −
t6
Rt
hγ+1
+ (t, s)
∆s
(γ + 1)γ+1 Ψγ (s, T )
hγ+1
+ (t,s)
(γ+1)γ+1 Ψγ (s,T )
¸
∆s ≤ w(t6 ) for t ∈
(t6 , ∞). Hence, we get
1
lim sup
t→∞ H(t, t6 )
Z t·
t6
¸
hγ+1
+ (t, s)
H(t, s)ϕ(s)q(s) −
∆s ≤ w(t6 ) < ∞,
(γ + 1)γ+1 Ψγ (s, T )
which implies a contradiction to (29). Thus, this completes the proof.
Da-Xue Chen
31
Remark 3.1. The results obtained in this paper are very general. From Theorems 3.1 and 3.2, we can get many different sufficient conditions for the
oscillation and asymptotic behavior of solutions of (1) with different choices of
the functions ϕ and H.
Acknowledgements. This work was supported by the Natural Science
Foundation of Hunan of P. R. China (Grant No. 11JJ3010).
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