Electronic Journal of Differential Equations, Vol. 2013 (2013), No. 79, pp. 1–17.
ISSN: 1072-6691. URL: http://ejde.math.txstate.edu or http://ejde.math.unt.edu
ftp ejde.math.txstate.edu
OSCILLATION CRITERIA FOR FOURTH-ORDER NONLINEAR
DELAY DYNAMIC EQUATIONS
YUNSONG QI, JINWEI YU
Abstract. We obtain criteria for the oscillation of all solutions to a fourthorder nonlinear delay dynamic equation on a time scale that is unbounded
from above. The results obtained are illustrated with examples
1. Introduction
This article concerns the oscillation of all solutions to the fourth-order nonlinear
delay dynamic equation
4
x∆ (t) + p(t)xγ (τ (t)) = 0
(1.1)
on a time scale T, where γ is the ratio of positive odd integers, p is a positive
real-valued rd-continuous function defined on T, τ ∈ Crd (T, T), τ (t) ≤ t, and
limt→∞ τ (t) = ∞. As we are interested in oscillatory behavior, we assume throughout this paper that the given time scale T is unbounded above and is a time scale
interval of the form [t0 , ∞)T := [t0 , ∞) ∩ T with t0 ∈ T.
By a solution to (1.1) we mean a nontrivial real-valued function x ∈ C4rd [Tx , ∞)T ,
Tx ∈ [t0 , ∞)T which satisfies (1.1) on [Tx , ∞)T . The solutions vanishing in some
neighbourhood of infinity will be excluded from our consideration. A solution x
of (1.1) is said to be oscillatory if it is neither eventually positive nor eventually
negative, otherwise it is nonoscillatory. Equation (1.1) is called oscillatory if all its
solutions are oscillatory.
The theory of time scales, which has recently received a lot of attention, was
introduced by Hilger [20] in his PhD thesis in 1988 in order to unify continuous and
discrete analysis. The study of the oscillation of dynamic equations on time scales
is a new area of applied mathematics, and work in this topic is rapidly growing.
Recently, there has been an increasing interest in obtaining sufficient conditions for
oscillation and nonoscillation of solutions of various equations on time scales, we
refer the reader to the books [3, 4, 7, 8, 29] and the articles [1, 2, 5, 6, 9, 10, 11,
12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 30, 31], and the
references cited therein. Regarding the oscillation of first-order and second-order
dynamic equations, Agarwal and Bohner [1], Bohner et al. [6], Braverman and
2000 Mathematics Subject Classification. 34K11, 34N05, 39A10.
Key words and phrases. Oscillatory solution; delay dynamic equation; fourth-order; time scale.
c
2013
Texas State University - San Marcos.
Submitted November 7, 2012. Published March 22, 2013.
1
2
Y. QI, J. YU
EJDE-2013/79
Karpuz [9], Şahiner and Stavroulakis [26] examined a first-order delay dynamic
equation
x∆ (t) + p(t)x(τ (t)) = 0.
Agarwal et al. [2], Erbe et al. [13], Şahiner [27], Zhang and Zhu [30] considered a
second-order delay dynamic equation
2
x∆ (t) + p(t)x(τ (t)) = 0.
Akın-Bohner et al. [5] investigated a second-order Emden–Fowler dynamic equation
2
x∆ (t) + p(t)xγ (σ(t)) = 0.
Saker [28] studied a second-order dynamic equation
(rx∆ )∆ (t) + p(t)f (x(σ(t))) = 0.
For the oscillation of higher-order dynamic equations on time scales, Erbe et al.
[14] investigated a third-order dynamic equation
3
x∆ (t) + p(t)x(t) = 0.
Hassan [19] and Li et al. [21] considered a third-order nonlinear delay dynamic
equation
∆
a((rx∆ )∆ )γ (t) + f (t, x(τ (t))) = 0.
Grace et al. [16] studied a fourth-order dynamic equation
4
x∆ (t) + p(t)xγ (σ(t)) = 0.
Grace et al. [18] examined a fourth-order dynamic equation
4
x∆ (t) + p(t)xγ (t) = 0.
Zhang et al. [31] investigated a fourth-order dynamic equation
3
(rx∆ )∆ (t) + p(t)f (x(σ(t))) = 0.
Erbe et al. [12] considered a higher-order neutral delay dynamic equation
n
(x(t) + A(t)x(α(t)))∆ + B(t)x(β(t)) = 0.
Karpuz [23, 24] studied a higher-order neutral delay dynamic equation
n
(x(t) + A(t)x(α(t)))∆ + B(t)F (x(β(t))) = ϕ(t).
The Riccati transformation technique plays an important role in obtaining sufficient conditions for oscillation of dynamic equations. For instance, Erbe et al. [13],
Şahiner [27], and Saker [28] applied the Riccati substitution as
x∆
x
to the second-order dynamic equations, where x > 0, x∆ > 0, and δ is an optional
function. Hassan [19] used the Riccati transformation
ω := δ
ω := δ
a((rx∆ )∆ )γ
,
(x ◦ τ )γ
where x ◦ τ > 0, (rx∆ )∆ > 0, and δ is an optional function. Erbe et al. [14] utilized
the Riccati substitution
2
x∆
ω := δ ∆ ,
x
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OSCILLATION CRITERIA
3
2
where x∆ > 0, x∆ > 0, and δ is an optional function.
The aim of this paper is to give some new oscillation theorems for (1.1). This
article is organized as follows: In the next section, we present the basic definitions
and the theory of calculus on time scales. In the section 3, we will establish some
oscillation results for (1.1) by employing some different Riccati substitutions. In
the section 4, we shall give two examples to illustrate our main results.
2. Preliminaries
A time scale T is an arbitrary nonempty closed subset of the real numbers R.
Since we are interested in oscillatory behavior, we suppose that the time scale under
consideration is not bounded above and is a time scale interval of the form [t0 , ∞)T .
On any time scale we define the forward and backward jump operators by
σ(t) := inf{s ∈ T : s > t},
and ρ(t) := sup{s ∈ T|s < t},
where inf ∅ := sup T and sup ∅ := inf T, ∅ denotes the empty set. A point t ∈ T is
said to be left-dense if ρ(t) = t and t > inf T, right-dense if σ(t) = t and t < sup T,
left-scattered if ρ(t) < t, and right-scattered if σ(t) > t. The graininess µ of the
time scale is defined by µ(t) := σ(t) − t.
For a function f : T → R (the range R of f may actually be replaced by any
Banach space), the (delta) derivative is defined by
f ∆ (t) =
f (σ(t)) − f (t)
,
σ(t) − t
if f is continuous at t and t is right-scattered. If t is not right-scattered then the
derivative is defined by
f ∆ (t) = lim+
s→t
f (σ(t)) − f (s)
f (t) − f (s)
= lim+
,
t−s
t−s
s→t
provided this limit exists.
A function f : T → R is said to be rd-continuous if it is continuous at each
right-dense point and if there exists a finite left limit in all left-dense points. The
set of rd-continuous functions f : T → R is denoted by Crd (T, R).
A function f is said to be differentiable if its derivative exists. The set of functions
f : T → R that are differentiable and whose derivative is rd-continuous function is
denoted by C1rd (T, R).
The derivative and the shift operator σ are related by the formula
f σ (t) := f (σ(t)) = f (t) + µ(t)f ∆ (t).
Let f be a real-valued function defined on an interval [a, b]T . We say that f is
increasing, decreasing, nondecreasing, and non-increasing on [a, b]T if t1 , t2 ∈ [a, b]T
and t2 > t1 imply f (t2 ) > f (t1 ), f (t2 ) < f (t1 ), f (t2 ) ≥ f (t1 ) and f (t2 ) ≤ f (t1 ),
respectively. Let f be a differentiable function on [a, b]T . Then f is increasing,
decreasing, nondecreasing, and non-increasing on [a, b]T if f ∆ (t) > 0, f ∆ (t) < 0,
f ∆ (t) ≥ 0, and f ∆ (t) ≤ 0 for all t ∈ [a, b)T , respectively.
We will use the following product and quotient rules for the derivative of the
product f g and the quotient f /g (where g(t)g(σ(t)) 6= 0) of two differentiable
functions f and g,
(f g)∆ (t) = f ∆ (t)g(t) + f (σ(t))g ∆ (t) = f (t)g ∆ (t) + f ∆ (t)g(σ(t)),
4
Y. QI, J. YU
f ∆
g
(t) =
EJDE-2013/79
f ∆ (t)g(t) − f (t)g ∆ (t)
.
g(t)g(σ(t))
For a, b ∈ T and a differentiable function f , the Cauchy integral of f ∆ is defined
by
b
Z
f ∆ (t)∆t = f (b) − f (a).
a
The integration by parts formula reads
Z b
Z
f ∆ (t)g(t)∆t = f (b)g(b) − f (a)g(a) −
a
b
f σ (t)g ∆ (t)∆t,
a
and infinite integrals are defined as
Z t
Z ∞
f (s)∆s.
f (s)∆s = lim
t→∞
a
a
3. Main results
Below, all occurring functional inequalities are assumed to hold for all sufficiently
large t. We begin with the following lemma.
Lemma 3.1. Assume that there exists T ∈ [t0 , ∞)T such that
y(t) > 0,
2
y ∆ (t) > 0,
y ∆ (t) < 0,
t ∈ [T, ∞)T .
Then, for each k ∈ (0, 1), there exists a constant Tk ∈ [T, ∞)T such that
τ (t) − T
τ (t)
y(τ (t))
≥
≥k
y(σ(t))
σ(t) − T
σ(t)
y(τ (t))
τ (t) − T
τ (t)
≥
≥k
y(t)
t−T
t
and
for t ∈ [Tk , ∞)T .
Proof. The proof is similar to that of [11, Lemma 2.4], and so is omitted.
The Taylor monomials (See [7, Section 1.6]) {hn (t, s)}∞
n=0 are defined recursively
by
t
Z
h0 (t, s) = 1,
hn+1 (t, s) =
hn (τ, s)∆τ,
t, s ∈ T, n ≥ 0.
s
For any time scale, h1 (t, s) = t − s, but simple formulas in general do not hold for
n ≥ 2.
Lemma 3.2 (See [14, Lemma 4]). Assume that y satisfies
y(t) > 0,
2
y ∆ (t) > 0,
y ∆ (t) > 0,
3
y ∆ (t) ≤ 0
for t ∈ [t1 , ∞)T . Then
lim inf
t→∞
ty(t)
≥ 1.
h2 (t, t0 )y ∆ (t)
Lemma 3.3. Assume that x is an eventually positive solution of (1.1). Then there
are only the following two cases eventually:
2
x∆ > 0,
2
x∆ > 0,
(1) x > 0,
x∆ > 0,
x∆ > 0,
(2) x > 0,
x∆ > 0,
x∆ < 0,
3
x∆ < 0,
4
3
x∆ < 0.
or
4
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OSCILLATION CRITERIA
5
Proof. Let x be an eventually positive solution of (1.1). Then there exists a t1 ∈
[t0 , ∞)T such that x(t) > 0 and x(τ (t)) > 0 for t ∈ [t1 , ∞)T . From (1.1), we have
4
x∆ (t) = −p(t)xγ (τ (t)) < 0,
∆
∆2
t ∈ [t1 , ∞)T .
∆3
(3.1)
∆3
Thus x , x , x each is of constant sign eventually. We claim that x (t) > 0
for t ∈ [t1 , ∞)T . If not, then there exist a constant c < 0 and t2 ∈ [t1 , ∞)T such
that
3
x∆ (t) ≤ c < 0, t ∈ [t2 , ∞)T .
Integrating the above inequality from t2 to t, we obtain
2
2
x∆ (t) − x∆ (t2 ) ≤ c(t − t2 ),
which implies that
2
lim x∆ (t) = −∞,
t→∞
and so there exist a constant c1 < 0 and t3 ∈ [t2 , ∞)T such that
2
x∆ (t) ≤ c1 < 0,
t ∈ [t3 , ∞)T .
Integrating the above inequality from t3 to t, we obtain
x∆ (t) − x∆ (t3 ) ≤ c(t − t3 ).
This gives
lim x∆ (t) = −∞,
t→∞
2
which yields limt→∞ x(t) = −∞ from x∆ < 0 and x∆ < 0. This is a contradiction.
Hence
3
x∆ (t) > 0, t ∈ [t1 , ∞)T .
If
2
x∆ > 0,
3
then x∆ > 0 due to x∆ > 0. If
2
x∆ < 0,
then x∆ > 0 due to x > 0. The proof is complete.
Lemma 3.4. Assume that x is an eventually positive bounded solution of (1.1).
Then x only satisfies Case (2) of Lemma 3.3.
Proof. Suppose that x is an eventually positive solution of (1.1). Proceeding as
in the proof of Lemma 3.3, x satisfies Case (1) or Case (2). It is easy to see that
limt→∞ x(t) = ∞ when Case (1) holds. Thus, x only satisfies Case (2) of Lemma
3.3. The proof is complete.
∆
Next we give the main results. For simplification, we let d∆
+ (t) := max{0, d (t)}.
Theorem 3.5. Let γ ≥ 1. Assume that there exist positive functions α, β ∈
C1rd ([t0 , ∞)T , R) such that, for some k ∈ (0, 1), for all constants M, P ∈ (0, ∞)
and sufficiently large t1 , for t2 > t1 , and for t3 > t2 , one has τ (t) > t2 for t ≥ t3 ,
Z th
t2 − t1 τ (s) γ
lim sup
ασ (s)p(s) kh2 (τ (s), t2 )
τ (s) − t1 σ(s)
t→∞
t3
(3.2)
i
∆
2
γ
(α+ (s))
σ(s)
−
∆s = ∞,
4γM γ−1 ασ (s) ks
6
Y. QI, J. YU
EJDE-2013/79
and
lim sup
t→∞
Z th
ξ γ
∆
σ γ (ξ)(β+
(ξ))2 i
k 2γ β σ (ξ)
f (ξ) −
∆ξ = ∞,
σ(ξ)
4γk γ P γ−1 β σ (ξ)ξ γ
t1
where
Z
∞
(3.3)
∞
Z
τ (v) γ
p(v)
∆v∆s
v
s
ξ
is well defined. Then (1.1) is oscillatory.
f (ξ) =
Proof. Suppose that (1.1) has a nonoscillatory solution x on [t0 , ∞)T . We may
assume without loss of generality that there exists a t1 ∈ [t0 , ∞)T such that x(t) > 0
and x(τ (t)) > 0 for t ∈ [t1 , ∞)T . Proceeding as in the proof of Lemma 3.3, we get
(3.1) and then x satisfies either Case (1) or Case (2).
Assume Case (1) holds. Define
3
x∆ (t)
,
ω(t) := α(t) ∆2
(x (t))γ
t ∈ [t1 , ∞)T .
(3.4)
Then ω(t) > 0 for t ∈ [t1 , ∞)T , and
ω ∆ (t) = α∆ (t)
3
x∆3 (t) ∆
x∆ (t)
σ
+
α
(t)
,
(x∆2 (t))γ
(x∆2 (t))γ
which implies that
3
ω ∆ (t) = α∆ (t)
4
3
2
x∆ (t)
x∆ (t)
x∆ (t)((x∆ )γ )∆ (t)
σ
σ
+α
(t)
−α
(t)
. (3.5)
2
2
(x∆ (t))γ
(x∆ (σ(t)))γ
(x∆2 (t))γ (x∆2 (σ(t)))γ
By Pötzsche chain rule [7, Theorem 1.90], we see that
Z 1
∆2
γ−1
3
2
((x∆∆ )γ )∆ (t) = γx∆ (t)
hx (σ(t)) + (1 − h)x∆ (t)
dh
≥ γ(x
∆2
0
γ−1 ∆3
(t))
x
(3.6)
(t).
Substituting (3.6) into (3.5), we have
3
4
x∆ (t)
x∆ (t)
σ
+
α
(t)
ω (t) ≤ α (t) ∆2
(x (t))γ
(x∆2 (σ(t)))γ
2
3
(x∆ (t))2 x∆ (t) γ ∆2
− γασ (t) ∆2
(x (t))γ−1 .
(x (t))2γ x∆2 (σ(t))
∆
∆
In view of the above inequality, (3.1), and (3.4), we obtain
ω ∆ (t) ≤
Note that
2
α∆ (t)
xγ (τ (t))
ω(t) − ασ (t)p(t) ∆2
α(t)
(x (σ(t)))γ
2
∆2
ω (t)
x (t) γ ∆2
− γασ (t) 2
(x (t))γ−1 .
α (t) x∆2 (σ(t))
x(τ (t)) x∆2 (τ (t)) γ
xγ (τ (t))
.
=
2
(x∆ (σ(t)))γ
x∆2 (τ (t)) x∆2 (σ(t))
4
From x∆ (t1 ) > 0 and x∆ (t) < 0, we have
Z t
2
3
3
x∆ (s)∆s ≥ (t − t1 )x∆ (t).
x∆ (t) >
t1
(3.7)
(3.8)
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OSCILLATION CRITERIA
Then
7
3
2
x∆2 ∆
(t − t1 )x∆ (t) − x∆ (t)
(t) =
< 0,
h1 (·, t1 )
(t − t1 )(σ(t) − t1 )
2
which implies that x∆ /h1 (·, t1 ) is decreasing. Using Taylor’s formula [7, Theorem
1.113] and choosing any t2 ∈ (t1 , ∞)T , we have
Z ρn−1 (t)
n−1
X
k
n
x(t) =
hk (t, t2 )x∆ (t2 ) +
hn−1 (t, σ(η))x∆ (η)∆η.
t2
k=0
i
Substituting n = 3 into the above equality and using x∆ > 0, i = 0, 1, 2, 3, we
obtain
2
t 2 − t 1 ∆2
x (t).
x(t) ≥ h2 (t, t2 )x∆ (t2 ) ≥ h2 (t, t2 )
t − t1
Hence
x(τ (t))
t2 − t1
≥ h2 (τ (t), t2 )
.
(3.9)
τ (t) − t1
x∆2 (τ (t))
2
Letting y := x∆ , we have
y ∆ > 0,
y > 0,
2
y ∆ < 0.
Then from Lemma 3.1, for each k ∈ (0, 1),
τ (t)
y(τ (t))
≥k
y(σ(t))
σ(t)
and
y(t)
t
≥k
.
y(σ(t))
σ(t)
That is,
2
2
x∆ (τ (t))
τ (t)
x∆ (t)
t
≥k
and
≥k
.
2
2
∆
∆
σ(t)
σ(t)
x (σ(t))
x (σ(t))
It follows from (3.8), (3.9), and (3.10) that
t2 − t1 τ (t) γ
xγ (τ (t))
≥
kh
(τ
(t),
t
)
2
2
2
τ (t) − t1 σ(t)
(x∆ (σ(t)))γ
(3.10)
(3.11)
for each k ∈ (0, 1). On the other hand, there exists a constant M > 0 such that
2
(x∆ (t))γ−1 ≥ M γ−1
due to x
(3.12)
∆3
> 0 and γ ≥ 1. From (3.7), (3.10), (3.11), and (3.12), we obtain
∆
(t)
t2 − t1 τ (t) γ α+
+
ω ∆ (t) ≤ −ασ (t)p(t) kh2 (τ (t), t2 )
ω(t)
τ (t) − t1 σ(t)
α(t)
ασ (t) t γ 2
− γM γ−1 2
k
ω (t).
α (t) σ(t)
Thus
t2 − t1 τ (t) γ
ω ∆ (t) ≤ −ασ (t)p(t) kh2 (τ (t), t2 )
τ (t) − t1 σ(t)
∆
(α+
(t))2 σ(t) γ
+
.
4γM γ−1 ασ (t) kt
Integrating the above inequality from t3 (τ (t) > t2 when t ≥ t3 ) to t, we obtain
Z th
∆
(α+
(s))2 σ(s) γ i
t2 − t1 τ (s) γ
ασ (s)p(s) kh2 (τ (s), t2 )
−
∆s
τ (s) − t1 σ(s)
4γM γ−1 ασ (s) ks
t3
≤ ω(t3 ) − ω(t) ≤ ω(t3 ),
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Y. QI, J. YU
EJDE-2013/79
which contradicts (3.2).
Assume Case (2) holds. Define the function
u(t) := β(t)
x∆ (t)
,
xγ (t)
t ∈ [t1 , ∞)T .
(3.13)
Then u(t) > 0 for t ∈ [t1 , ∞)T , and
2
u∆ (t) = β ∆ (t)
x∆ (t)
x∆ (t)
x∆ (t)(xγ )∆ (t)
σ
σ
+
β
(t)
−
β
(t)
.
xγ (t)
xγ (σ(t))
xγ (t)xγ (σ(t))
(3.14)
It follows from Pötzsche chain rule [7, Thm. 1.90] that (xγ )∆ (t) ≥ γxγ−1 (t)x∆ (t).
Hence by (3.13) and (3.14), we have
2
u∆ (t) ≤
x∆ (t)
β σ (t) x(t) γ γ−1
β ∆ (t)
x
u(t) + β σ (t) γ
−γ 2
(t)u2 (t).
β(t)
x (σ(t))
β (t) x(σ(t))
(3.15)
2
Since x > 0, x∆ > 0, and x∆ < 0, we obtain
t
x(t)
≥k
x(σ(t))
σ(t)
for each k ∈ (0, 1)
(3.16)
due to Lemma 3.1. From x∆ > 0, there exists a constant P > 0 such that xγ−1 (t) ≥
P γ−1 . Thus, by (3.15), we see that
2
β ∆ (t)
x∆ (t)
β σ (t) t γ 2
u (t) ≤ β (t) γ
+ + u(t) − γk γ P γ−1 2
u (t).
(
x (σ(t))
β(t)
β (t) σ(t)
∆
σ
(3.17)
On the other hand, by (1.1), we calculate
Z z
∆3
∆3
x (z) − x (t) +
p(s)xγ (τ (s))∆s = 0.
t
Let y := x. By Lemma 3.1, we have
τ (t)
x(τ (t))
≥k
x(t)
t
for any k ∈ (0, 1). Thus, from x∆ > 0, we have
Z z
τ (s) γ
3
3
x∆ (z) − x∆ (t) + k γ xγ (t)
∆s ≤ 0.
p(s)
s
t
Letting z → ∞ in the above inequality, we obtain
γ
Z ∞
τ (s)
∆3
γ γ
−x (t) + k x (t)
p(s)
∆s ≤ 0
s
t
3
due to limz→∞ x∆ (z) ≥ l ≥ 0. Therefore,
Z zZ
∆2
∆2
γ γ
−x (z) + x (t) + k x (t)
t
s
∞
τ (v) γ
p(v)
∆v∆s ≤ 0.
v
2
Letting z → ∞ in the last inequality and using limz→∞ (−x∆ (z)) ≥ l1 ≥ 0, we
have
Z ∞Z ∞
τ (v) γ
∆2
γ γ
∆v∆s ≤ 0.
x (t) + k x (t)
p(v)
v
t
s
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OSCILLATION CRITERIA
9
Thus by (3.16), we have
Z ∞Z ∞
2
τ (v) γ
γ
x∆ (t)
γ x (t)
p(v)
∆v∆s
≤
−k
xγ (σ(t))
xγ (σ(t)) t
v
s
Z
Z
τ (v) γ
t γ ∞ ∞
p(v)
∆v∆s.
≤ −k 2γ
σ(t)
v
s
t
(3.18)
Substituting (3.18) into (3.17), we obtain
τ (v) γ
t γ Z ∞ Z ∞
∆
2γ σ
p(v)
u (t) ≤ −k β (t)
∆v∆s
σ(t)
v
s
t
β ∆ (t)
β σ (t) t γ 2
+ + u(t) − γk γ P γ−1 2
u (t),
β(t)
β (t) σ(t)
which implies that
t γ Z ∞ Z ∞
τ (v) γ
p(v)
∆v∆s
σ(t)
v
t
s
∆
(t))2
σ γ (t)(β+
.
+
4γk γ P γ−1 β σ (t)tγ
u∆ (t) ≤ −k 2γ β σ (t)
Integrating the last inequality from t1 to t, we have
Z th
ξ γ h Z ∞ Z ∞
i
τ (v) γ
∆
(ξ))2 i
σ γ (ξ)(β+
k 2γ β σ (ξ)
∆v∆s −
∆ξ
p(v)
γ
γ−1
σ(ξ)
v
4γk P
β σ (ξ)ξ γ
t1
ξ
s
≤ u(t1 ) − u(t) ≤ u(t1 ),
which contradicts (3.3). The proof is complete.
Combining Theorem 3.5 with Lemma 3.4, we obtain the following criterion for
oscillation of all bounded solutions of (1.1).
Corollary 3.6. Let γ ≥ 1. Assume that there exists a positive function β ∈
C1rd ([t0 , ∞)T , R) such that, for some k ∈ (0, 1), for all constants P ∈ (0, ∞) and
sufficiently large t1 , one has (3.3). Then every bounded solution of (1.1) is oscillatory.
Next, we establish another oscillation result for (1.1) under the case when γ > 1.
Theorem 3.7. Let γ > 1. If for all sufficiently large t1 , for t2 > t1 , and for
t3 > t2 , one has τ (t) > t2 for t ≥ t3 ,
Z ∞
t2 − t1 τ (t) γ
σ(t)p(t) h2 (τ (t), t2 )
∆t = ∞,
(3.19)
τ (t) − t1 σ(t)
t3
and
Z
∞
t1
ξ γ h Z ∞ Z ∞
τ (v) γ
i
p(v)
σ(ξ)
∆v∆s ∆ξ = ∞,
σ(ξ)
v
ξ
s
where
Z
∞
Z
∞
p(v)
ξ
s
is well defined, then (1.1) is oscillatory.
τ (v) γ
v
∆v∆s
(3.20)
10
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Proof. Let x be a non-oscillatory solution of (1.1). Without loss of generality, we
may assume that there exists a t1 ∈ [t0 , ∞)T such that x(t) > 0 and x(τ (t)) > 0 for
t ∈ [t1 , ∞)T . Proceeding as in the proof of Lemma 3.3, we obtain (3.1) and then x
satisfies either Case (1) or Case (2).
Suppose that Case (1) holds. We define the function ω by
3
ω(t) :=
tx∆ (t)
,
(x∆2 (t))γ
t ∈ [t1 , ∞)T .
(3.21)
Then ω(t) > 0 for t ∈ [t1 , ∞)T , and
−γ
3
4
2
3
2
ω ∆ (t) = x∆ (t) + σ(t)x∆ (t) x∆ (σ(t))
+ tx∆ (t)((x∆ )−γ )∆ (t)
x(τ (t)) γ
3
2
≤ x∆ (t)(x∆ (σ(t)))−γ − σ(t)p(t) ∆2
x (σ(t))
(3.22)
2
due to (3.1) and ((x∆ )−γ )∆ ≤ 0 (see Pötzsche chain rule [7, Theorem 1.90]). On
the other hand, by Pötzsche chain rule [7, Theorem 1.90], we have
Z 1
∆2
−γ
2
∆2 1−γ ∆
∆3
((x )
) (t) = (1 − γ)x (t)
hx (σ(t)) + (1 − h)x∆ (t)
dh
0
3
≤ (1 − γ)x∆ (t)
= (1 − γ)x
∆3
Z
1
∆2
−γ
2
hx (σ(t)) + (1 − h)x∆ (σ(t))
dh
0
∆2
(t)(x
(σ(t)))−γ .
Then by (3.22), we see that
2
ω ∆ (t) ≤
((x∆ )1−γ )∆ (t)
− σ(t)p(t)
1−γ
x(τ (t))
x∆2 (σ(t))
γ
.
(3.23)
Similar as in the proof of Theorem 3.5, we have (3.11). Hence from (3.23), we
obtain
γ
2
t2 − t1 τ (t)
((x∆ )1−γ )∆ (t)
∆
− σ(t)p(t) kh2 (τ (t), t2 )
ω (t) ≤
1−γ
τ (t) − t1 σ(t)
for each k ∈ (0, 1) and t2 ∈ (t1 , ∞)T . Integrating the last inequality from t3
(τ (t) > t2 when t ≥ t3 ) to t, we get
Z t
t2 − t1 τ (s) γ
∆s
σ(s)p(s) kh2 (τ (s), t2 )
τ (s) − t1 σ(s)
t3
Z t
2
2
(x∆ )1−γ (t3 )
((x∆ )1−γ )∆ (s) ∆s ≤ ω(t3 ) +
,
≤−
ω ∆ (s) −
1−γ
γ−1
t3
which contradicts (3.19).
Assume Case (2) holds. We define the function u by
u(t) :=
tx∆ (t)
,
xγ (t)
t ∈ [t1 , ∞)T .
(3.24)
Then u(t) > 0 for t ∈ [t1 , ∞)T , and
2
u∆ (t) = (x∆ (t) + σ(t)x∆ (t))x−γ (σ(t)) + tx∆ (t)(x−γ )∆ (t)
2
∆
≤ x (t)x
−γ
x∆ (t)
(σ(t)) + σ(t) γ
x (σ(t))
(3.25)
EJDE-2013/79
OSCILLATION CRITERIA
11
due to (x−γ )∆ ≤ 0 (see Pötzsche chain rule [7, Theorem 1.90]). On the other hand,
by Pötzsche chain rule [7, Theorem 1.90], we get
Z 1
−γ
(x1−γ )∆ (t) = (1 − γ)x∆ (t)
hx(σ(t)) + (1 − h)x(t)
dh
0
≤ (1 − γ)x∆ (t)
Z
1
−γ
hx(σ(t)) + (1 − h)x(σ(t))
dh
0
= (1 − γ)x (t)(x(σ(t)))−γ .
∆
Then from (3.25), we obtain
2
u∆ (t) ≤
(x1−γ )∆ (t)
x∆ (t)
+ σ(t) γ
.
1−γ
x (σ(t))
(3.26)
As in the proof of Theorem 3.5, we obtain (3.18). It follows from (3.18) and (3.26)
that
t γ Z ∞ Z ∞
τ (v) γ
(x1−γ )∆ (t)
− k 2γ σ(t)
u∆ (t) ≤
∆v∆s
p(v)
1−γ
σ(t)
v
t
s
for each k ∈ (0, 1). Integrating the last inequality from t1 to t, we have
Z t
i
ξ γ h Z ∞ Z ∞
τ (v) γ
2γ
∆v∆s ∆ξ
k σ(ξ)
p(v)
σ(ξ)
v
ξ
t1
s
Z t
1−γ ∆
x1−γ (t1 )
(x
) (s)
≤−
∆s ≤ u(t1 ) +
,
u∆ (s) −
1−γ
γ−1
t1
which contradicts (3.20). This completes the proof.
Combining Theorem 3.7 with Lemma 3.4, we obtain the following result for
oscillation of all bounded solutions of (1.1).
Corollary 3.8. Let γ > 1. Suppose that (3.20) holds for all sufficiently large t1 .
Then every bounded solution of (1.1) is oscillatory.
Next, we give a new oscillation criterion for (1.1) by using a different class of
Riccati substitution.
Theorem 3.9. Let γ ≥ 1. Suppose that τ ∈ C1rd ([t0 , ∞)T , T), τ ∆ > 0, and
τ ([t0 , ∞)T ) := [τ (t0 ), ∞)T . Assume also that there exist positive functions α, β ∈
C1rd ([t0 , ∞)T , R) such that, for some k ∈ (0, 1), for all constants M, P ∈ (0, ∞) and
sufficiently large t1 , for t2 > t1 , one has (3.3) and
Z th
i
∆
σ(s)(α+
(s))2
lim sup
α(s)p(s) − 2
∆s = ∞. (3.27)
4k γM γ−1 τ ∆ (s)α(s)h2 (τ (s), t0 )τ (s)
t→∞
t2
Then (1.1) is oscillatory.
Proof. Suppose that (1.1) has a nonoscillatory solution x on [t0 , ∞)T . We may
assume without loss of generality that there exists a t1 ∈ [t0 , ∞)T such that x(t) > 0
and x(τ (t)) > 0 for t ∈ [t1 , ∞)T . Proceeding as in the proof of Lemma 3.3, we get
(3.1) and then x satisfies either Case (1) or Case (2).
Assume Case (1) holds. Define the function
ω(t) :=
3
α(t)
x∆ (t),
xγ (τ (t))
t ∈ [t1 , ∞)T .
(3.28)
12
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Clearly, ω(t) > 0 for t ∈ [t1 , ∞)T , and
∆
3
4
α(t)
α(t)
ω ∆ (t) =
x∆ (σ(t)) + γ
x∆ (t),
xγ (τ (t))
x (τ (t))
which yields
3
ω ∆ (t) =
4
α(t)
α∆ (t)x∆ (σ(t))
x∆ (t) +
γ
x (τ (t))
xγ (τ (σ(t)))
3
x∆ (σ(t))(xγ (τ (t)))∆
− α(t) γ
.
x (τ (t))xγ (τ (σ(t)))
From chain rules [7, Theorem 1.90 and Theorem 1.93], we have
Z 1
γ−1
(xγ (τ (t)))∆ = γx∆ (τ (t))τ ∆ (t)
hx(τ (σ(t))) + (1 − h)x(τ (t))
dh
0
γ−1 ∆
≥ γ(x(τ (t)))
(3.29)
(3.30)
∆
x (τ (t))τ (t).
Substituting (3.30) into (3.29), we find that
3
3
4
α∆ (t)x∆ (σ(t))
x∆ (σ(t))x∆ (τ (t))τ ∆ (t)
α(t)
x∆ (t) +
−
γα(t)
.
ω (t) ≤ γ
x (τ (t))
xγ (τ (σ(t)))
x(τ (t))xγ (τ (σ(t)))
∆
In view of (3.1), (3.28), and the above inequality, we obtain
α∆ (t) σ
ω (t)
ασ (t)
α(t) xγ (τ (σ(t))) x∆ (τ (t)) σ
(ω (t))2 .
− γτ ∆ (t) σ
(α (t))2 x(τ (t)) x∆3 (σ(t))
ω ∆ (t) ≤ −α(t)p(t) +
(3.31)
Let y := x∆ . Then from Lemma 3.2, we see that
x∆ (t)
h2 (t, t0 )
≥k
t
x∆2 (t)
for each k ∈ (0, 1). Since
2
3
x∆ (t) > 0,
4
x∆ (t) > 0,
x∆ (t) < 0,
t ∈ [t1 , ∞)T ,
we have
2
3
3
x∆ (t) > (t − t1 )x∆ (t) ≥ ktx∆ (t).
Thus
2
x∆ (t) x∆ (t)
x∆ (t)
=
≥ k 2 h2 (t, t0 ).
3
x∆ (t)
x∆2 (t) x∆3 (t)
Then
3
x∆ (τ (t)) x∆ (τ (t))
h2 (τ (t), t0 )τ (t)
x∆ (τ (t))
=
≥ k2
3
σ(t)
x∆ (σ(t))
x∆3 (τ (t)) x∆3 (σ(t))
(3.32)
due to
x∆3 (t) ∆
t
4
=
3
tx∆ (t) − x∆ (t)
< 0.
tσ(t)
∆
On the other hand, from x > 0 and τ ∆ > 0, we have
x(τ (σ(t)))
≥ 1,
x(τ (t))
(3.33)
EJDE-2013/79
OSCILLATION CRITERIA
13
and there exists a constant M > 0 such that
xγ−1 (τ (σ(t))) ≥ M γ−1 .
(3.34)
Substituting (3.32), (3.33), and (3.34) into (3.31), we obtain
ω ∆ (t) ≤ −α(t)p(t) +
∆
α+
(t) σ
α(t) h2 (τ (t), t0 )τ (t) σ
ω (t) − k 2 γM γ−1 τ ∆ (t) σ
(ω (t))2 .
ασ (t)
(α (t))2
σ(t)
Therefore,
ω ∆ (t) ≤ −α(t)p(t) +
∆
σ(t)(α+
(t))2
.
4k 2 γM γ−1 τ ∆ (t)α(t)h2 (τ (t), t0 )τ (t)
Integrating the above inequality from t2 (t2 > t1 ) to t, we have
Z th
i
∆
σ(s)(α+
(s))2
∆s ≤ ω(t2 ) − ω(t) ≤ ω(t2 ),
α(s)p(s) − 2
γ−1
∆
4k γM
τ (s)α(s)h2 (τ (s), t0 )τ (s)
t2
which contradicts (3.27). The proof of Case (2) is the same as that of Case (2) in
Theorem 3.5, and so is omitted. This finishes the proof.
In the following, we will establish some oscillation results for (1.1) in the case
when γ ≤ 1.
Theorem 3.10. Let γ ≤ 1. Assume that there exist positive functions α, β ∈
C1rd ([t0 , ∞)T , R) such that, for some k ∈ (0, 1), for all constants M, P ∈ (0, ∞) and
sufficiently large t1 , for t2 > t1 , and for t3 > t2 , one has τ (t) > t2 for t ≥ t3 ,
Z th
t2 − t1 τ (s) γ
lim sup
ασ (s)p(s) kh2 (τ (s), t2 )
τ (s) − t1 σ(s)
t→∞
t3
(3.35)
i
2
∆
γ
(α+ (s))
σ(s)
∆s = ∞,
−
4γ(M σ(s))γ−1 ασ (s) ks
and
lim sup
t→∞
Z th
t1
k 2γ β σ (ξ)
i
ξ γ
∆
(ξ))2
σ γ (ξ)(β+
f (ξ) −
∆ξ = ∞,
γ
γ−1
σ
γ
σ(ξ)
4γk (P σ(s))
β (ξ)ξ
where
Z
∞
Z
f (ξ) =
ξ
s
∞
(3.36)
τ (v) γ
∆v∆s
p(v)
v
is well defined. Then (1.1) is oscillatory.
Proof. Suppose that (1.1) has a nonoscillatory solution x on [t0 , ∞)T . We may
assume without loss of generality that there exists a t1 ∈ [t0 , ∞)T such that x(t) > 0
and x(τ (t)) > 0 for t ∈ [t1 , ∞)T . Proceeding as in the proof of Lemma 3.3, we obtain
(3.1) and then x satisfies either Case (1) or Case (2).
Assume Case (1) holds. Define a Riccati substitution as in (3.4). Then we have
(3.5). From Pötzsche chain rule [7, Theorem 1.90], we find that
Z 1
∆2
γ−1
2
3
2
((x∆ )γ )∆ (t) = γx∆ (t)
hx (σ(t)) + (1 − h)x∆ (t)
dh
(3.37)
0
2
3
≥ γ(x∆ (σ(t)))γ−1 x∆ (t).
14
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Substituting (3.37) into (3.5), we have
3
4
x∆ (t)
x∆ (t)
σ
ω (t) ≤ α (t) ∆2
+
α
(t)
(x (t))γ
(x∆2 (σ(t)))γ
3
2
(x∆ (t))2 x∆ (t) γ ∆2
− γασ (t) ∆2
(x (σ(t)))γ−1 .
(x (t))2γ x∆2 (σ(t))
∆
∆
By (3.1), (3.4), and the above inequality, we obtain
ω ∆ (t) ≤
α∆ (t)
xγ (τ (t))
ω(t) − ασ (t)p(t) ∆2
α(t)
(x (σ(t)))γ
2
ω 2 (t) x∆ (t) γ ∆2
− γασ (t) 2
(x (σ(t)))γ−1 .
α (t) x∆2 (σ(t))
(3.38)
As in the proof of Theorem 3.5, we have (3.10) and (3.11) for each k ∈ (0, 1). On
the other hand, there exists a constant M > 0 such that
Z t
3
∆2
∆2
x (t) = x (t1 ) +
x∆ (s)∆s ≤ M t.
(3.39)
t1
It follows from (3.10), (3.11), (3.38), and (3.39) that
γ
α∆ (t)
t2 − t1 τ (t)
∆
σ
+ + ω(t)
ω (t) ≤ −α (t)p(t) kh2 (τ (t), t2 )
τ (t) − t1 σ(t)
α(t)
σ
γ
α (t)
t
− γ(M σ(t))γ−1 2
ω 2 (t).
k
α (t) σ(t)
Then
t2 − t1 τ (t) γ
ω ∆ (t) ≤ −ασ (t)p(t) kh2 (τ (t), t2 )
τ (t) − t1 σ(t)
∆
2
(α+ (t))
σ(t) γ
+ .
γ−1
kt
σ
4γ M σ(t)
α (t)
Integrating the last inequality from t3 (τ (t) > t2 when t ≥ t3 ) to t, we obtain
Z th
σ(s) γ i
∆
(α+
(s))2
t2 − t1 τ (s) γ
−
∆s
ασ (s)p(s) kh2 (τ (s), t2 )
τ (s) − t1 σ(s)
4γ(M σ(s))γ−1 ασ (s) ks
t3
≤ ω(t3 ) − ω(t) ≤ ω(t3 ),
which contradicts (3.35).
If Case (2) holds, we define the function u by (3.13). Then, we have (3.14). By
Pötzsche chain rule [7, Theorem 1.90], (xγ )∆ (t) ≥ γxγ−1 (σ(t))x∆ (t). Hence from
(3.13) and (3.14), we have
2
u∆ (t) ≤
β ∆ (t)
x∆ (t)
β σ (t) x(t) γ γ−1
u(t) + β σ (t) γ
−γ 2
x
(σ(t))u2 (t). (3.40)
β(t)
x (σ(t))
β (t) x(σ(t))
Note that there exists a constant P > 0 such that
Z t
x(t) = x(t1 ) +
x∆ (s)∆s ≤ P t.
t1
(3.41)
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OSCILLATION CRITERIA
15
Thus, by (3.40) and (3.41), we see that
2
u∆ (t) ≤ β σ (t)
x(t) γ
∆
σ
β+
(t)
x∆ (t)
γ−1 β (t)
+
u(t)
−
γ(P
σ(t))
u2 (t). (3.42)
xγ (σ(t))
β(t)
β 2 (t) x(σ(t))
The rest of the proof is similar to that of Case (2) in Theorem 3.5, and we can
obtain a contradiction to (3.36). This completes the proof.
Combining Theorem 3.10 with Lemma 3.4, we give the following criterion for
oscillation of all bounded solutions of (1.1).
Corollary 3.11. Let γ ≤ 1. Assume that there exists a positive function β ∈
C1rd ([t0 , ∞)T , R) such that, for some k ∈ (0, 1), for all constants P ∈ (0, ∞) and
sufficiently large t1 , one has (3.36). Then every bounded solution of (1.1) is oscillatory.
4. Examples
In this section, we shall give two examples to illustrate the main results. Here we
set T := 2Z := 2Z ∪ {0} := {2k : k ∈ Z} ∪ {0}. To get the conditions for oscillation,
we will use the following facts; see [7, Example 1.104])
(t − s)(t − 2s)(t − 4s)
(t − s)(t − 2s)
and h3 (t, s) =
.
3
21
Example 4.1. Consider a fourth-order super-linear delay dynamic equation
4
λ
x∆ (t) +
xγ (2−k1 t) = 0, t ∈ [t0 , ∞)2Z ,
(4.1)
h3 (t, t0 )
h2 (t, s) =
where t0 > 0, γ > 1, λ > 0, and k1 is a positive integer. Let p(t) = λ/h3 (t, t0 ) and
τ (t) = 2−k1 t. Then
Z ∞
Z ∞
τ (v) γ
1
∆v = 2−k1 γ λ
∆v
p(v)
v
h3 (v, t0 )
s
s
Z ∞
1
≥ 21λ × 2−k1 γ
∆v
(v − t0 )3
s
≥
and
∞
∞
21λ × 2−(k1 γ+1)
(s − t0 )2
21λ × 2−(k1 γ+1)
.
v
ξ − t0
ξ
s
It is easy to see that all assumptions of Theorem 3.7 hold. Thus equation (4.1) is
oscillatory.
Z
Z
p(v)
τ (v) γ
∆v∆s ≥
Example 4.2. Consider a fourth-order linear delay dynamic equation
4
λh2 (t, t0 )
x(2−k1 t) = 0, t ∈ [t0 , ∞)2Z ,
x∆ (t) +
h3 (t, t0 )h3 (2t, t0 )
where t0 > 0, λ > 0, and k1 is a positive integer. We now let
p(t) = λh2 (t, t0 )/(h3 (t, t0 )h3 (2t, t0 ))
−k1
and τ (t) = 2
t. Then
Z ∞
τ (v) γ
h2 (v, t0 )
−k1
∆v = 2 λ
∆v
p(v)
v
h
(v,
t0 )h3 (2v, t0 )
3
s
s
Z
∞
(4.2)
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= 2−k1 λ
EJDE-2013/79
Z
∞
s
−
∆
1
∆v
h3 (v, t0 )
2−k1 λ
=
h3 (s, t0 )
and
Z
ξ
∞
Z
s
∞
τ (v) γ
21λ × 2−(k1 +1)
p(v)
.
∆v∆s ≥
v
(ξ − t0 )2
Note that
p(t) = λ
h2 (t, t0 )
147λ
.
≥
(h3 (t, t0 )h3 (2t, t0 ))
8t4
Let γ = 1, α(t) = t3 , and β(t) = t. If λ > 2(3+4k1 ) /(7k 2 ) for some k ∈ (0, 1), then
Z th
i
σ(s)((α∆ (s))+ )2
α(s)p(s) − 2
∆s = ∞.
lim sup
4k γM γ−1 τ ∆ (s)α(s)h2 (τ (s), t0 )τ (s)
t→∞
t2
If λ > 2(k1 −1) /(21k 3 ) for some k ∈ (0, 1), then (3.3) holds. Hence by Theorem
3.9, equation (4.2) oscillates if λ > max{2(3+4k1 ) /(7k 2 ), 2(k1 −1) /(21k 3 )} for some
k ∈ (0, 1).
The results obtained can be extended to a fourth-order neutral delay dynamic
equation
∆4
x(t) + p(t)x(δ(t))
(t) + q(t)xγ (τ (t)) = 0.
Moreover, similar methods can be applied to a fourth-order quasi-linear neutral
delay dynamic equation
h
i
3 γ ∆
(x(t) + p(t)x(δ(t)))∆
(t) + q(t)xγ (τ (t)) = 0.
The details are left to the reader.
Acknowledgements. The authors sincerely thank Professor Julio G. Dix and the
reviewers for their valuable suggestions and useful comments that have led to the
present improved version of the original manuscript.
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Yunsong Qi
Qingdao Technological University, School of Science, Qingdao, Shandong 266033, China
E-mail address: qys266520@163.com
Jinwei Yu
Qingdao Technological University, School of Science, Qingdao, Shandong 266033, China
E-mail address: yujwqdlg@163.com