Hindawi Publishing Corporation
Advances in Difference Equations
Volume 2007, Article ID 70730, 16 pages
doi:10.1155/2007/70730
Research Article
Oscillation Criteria for Second-Order Delay Dynamic Equations
on Time Scales
Zhenlai Han, Bao Shi, and Shurong Sun
Received 4 September 2006; Revised 15 January 2007; Accepted 9 February 2007
Recommended by Martin J. Bohner
By means of Riccati transformation technique, we establish some new oscillation criteria
γ Δ
for the second-order nonlinear delay dynamic equations (p(t)(xΔ (t)) ) + q(t) f (x(τ(t)))
= 0 on a time scale T, here γ ≥ 1 is a quotient of odd positive integers with p and q realvalued positive rd-continuous functions defined on T.
Copyright © 2007 Zhenlai Han et al. This is an open access article distributed under the
Creative Commons Attribution License, which permits unrestricted use, distribution,
and reproduction in any medium, provided the original work is properly cited.
1. Introduction
The theory of time scales, which has recently received a lot of attention, was introduced by
Hilger in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis (see
Hilger [1]). Several authors have expounded on various aspects of this new theory; see the
survey paper by Agarwal et al. [2] and references cited therein. A book on the subject of
time scales, by Bohner and Peterson [3], summarizes and organizes much of the time scale
calculus, we refer also the last book by Bohner and Peterson [4] for advances in dynamic
equations on time scales. For the notions used below we refer to the next section that
provides some basic facts on time scales extracted from Bohner and Peterson [3].
A time scale T is an arbitrary 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
and of difference equations. Many other interesting time scales exist, and they give rise to
plenty of applications, among them the study of population dynamic models which are
discrete in season (and may follow a difference scheme with variable step-size or often
modeled by continuous dynamic systems), die out, say in winter, while their eggs are
incubating or dormant, and then in season again, hatching gives rise to a nonoverlapping
population (see Bohner and Peterson [3]).
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Advances in Difference Equations
In recent years, there has been much research activity concerning the oscillation and
nonoscillation of solutions of various equations on time scales, and we refer the reader
to Bohner and Saker [5], Erbe [6], Erbe et al. [7], Saker [8, 9]. However, there are few
results dealing with the oscillation of the solutions of delay dynamic equations on time
scales [10–17].
To the best of our knowledge, there are no results regarding the oscillation of the solutions of the following second-order nonlinear delay dynamic equations on time scales up
to now:
p(t) xΔ (t)
γ Δ
+ q(t) f x τ(t)
=0
for t ∈ T.
(1.1)
Zhang and Deng [16] (see also Bohner [12]) considered the first-order delay dynamic
equations on time scales
xΔ (t) + p(t)x τ(t) = 0 for t ∈ T,
(1.2)
and unified oscillation criteria of the first-order delay differential and difference equations. Agarwal et al. [10] considered the second-order delay dynamic equations on time
scales
xΔΔ (t) + p(t)x τ(t) = 0 for t ∈ T,
(1.3)
and established some sufficient conditions for oscillation of (1.3). Zhang and Zhu [17]
considered the second-order nonlinear delay dynamic equations on time scales
xΔΔ (t) + p(t) f x(t − τ) = 0 for t ∈ T,
(1.4)
and the second-order nonlinear dynamic equations on time scales
xΔΔ (t) + p(t) f x σ(t)
=0
for t ∈ T,
(1.5)
and established the equivalence of the oscillation of (1.4) and (1.5), from which obtained
some oscillation criteria and comparison theorems for (1.4). Sahiner [13] considered the
second-order nonlinear delay dynamic equations on time scales
xΔΔ (t) + p(t) f x τ(t)
=0
for t ∈ T,
(1.6)
and obtained some sufficient conditions for oscillation of (1.6) by means of Riccati transformation technique. Erbe et al. [18] considered the pair of second-order dynamic equations
r(t) xΔ
r(t) x
γ Δ
Δ γ Δ
+ p(t)xγ (t) = 0
γ
for t ∈ T,
+ p(t)x σ(t) = 0 for t ∈ T,
(1.7)
and established some necessary and sufficient conditions for nonoscillation of HilleKneser type. Han et al. [19] considered the second-order Emden-Fowler delay dynamic
equations on time scales
xΔΔ (t) + p(t)xγ τ(t) = 0
for t ∈ T,
(1.8)
Zhenlai Han et al. 3
and established some sufficient conditions for oscillation of (1.8). Agarwal et al. [11],
Saker [15] considered the second-order nonlinear neutral delay dynamic equations on
time scales
r(t) x(t) + p(t)x(t − τ)
Δ γ Δ
+ f t,x(t − δ) = 0
for t ∈ T,
(1.9)
and established some oscillation criteria of (1.9). Sahiner [14] considered the secondorder neutral delay and mixed-type dynamic equations on time scales
r(t) x(t) + p(t)x τ(t)
Δ γ Δ
+ f t,x δ(t)
=0
for t ∈ T,
(1.10)
and obtained some sufficient conditions for oscillation of (1.10).
Clearly, (1.3), (1.4), and (1.6) are the special cases of (1.1), and (1.9) is different from
(1.1). To develop the qualitative theory of delay dynamic equations on time scales, in this
paper, we consider the second-order nonlinear delay dynamic equation on time scales
(1.1).
As we are interested in oscillatory behavior, we assume throughout this paper that the
given time scale T is unbounded above, that is, it is a time scale interval of the form [a, ∞)
with a ∈ T.
We assume that γ ≥ 1 is a quotient of odd positive integer, p and q are positive, realvalued rd-continuous functions defined on T, τ : T → T is an rd-continuous function
such that τ(t) ≤ t and τ(t) → ∞ as t → ∞, f ∈ C(R, R) such that satisfies for some positive
constant L, f (x)/xγ ≥ L, for all nonzero x. We will also consider the two cases
∞
a
∞
a
1
p(t)
1/γ
1
p(t)
Δt = ∞,
1/γ
(1.11)
Δt < ∞.
(1.12)
1
[tx , ∞), tx ≥ a,
By a solution of (1.1), we mean a nontrivial real-valued function x ∈ Crd
1
Δ
γ
which has the property p(x ) ∈ Crd [tx , ∞) and satisfying (1.1) for t ≥ tx . A solution x
of (1.1) is called oscillatory if it is neither eventually positive nor eventually negative;
otherwise it is called nonoscillatory. Equation (1.1) is called oscillatory if all solutions are
oscillatory. Our attention is restricted to those solutions x of (1.1) which exist on some
half line [tx , ∞) with sup{|x(t)| : t ≥ t0 } > 0 for any t0 ≥ tx .
In this paper we intend to use the Riccati transformation technique for obtaining several oscillation criteria for (1.1) when (1.11) or (1.12) holds.
The paper is organized as follows: in the next section we present the basic definitions
and the theory of calculus on time scales. In Section 3, we apply a simple consequence of
Keller’s chain rule, and the inequality
λAB λ−1 − Aλ ≤ (λ − 1)B λ ,
λ ≥ 1,
(1.13)
where A and B are nonnegative constants, devoted to the proof of the sufficient conditions
for oscillation of all solutions of (1.1). In Section 4, we present some examples to illustrate
our main results.
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Advances in Difference Equations
We note that if T = R, then σ(t) = 0, μ(t) = 0, xΔ (t) = x (t) and (1.1) becomes the
second-order nonlinear delay differential equation
p(t) x (t)
γ + q(t) f x τ(t)
=0
for t ∈ R.
(1.14)
If T = Z, then σ(t) = t + 1, μ(t) = 1, xΔ (t) = Δx(t) = x(t + 1) − x(t) and (1.1) becomes
the second-order nonlinear delay difference equation
Δ p(t) Δx(t)
γ + q(t) f x τ(t)
=0
for t ∈ Z.
(1.15)
Numerous oscillation and nonoscillation criteria have been established for the forms
of (1.14) and (1.15); see, for example, [20–26] and references therein.
2. Some preliminaries
A time scale T is an arbitrary nonempty closed subset of the real numbers R. On any time
scale we define the forward and backward jump operators by
σ(t) := inf {s ∈ T | s > t },
ρ(t) := sup{s ∈ T | s < t }.
(2.1)
A point t ∈ T is said to be left-dense if ρ(t) = t, right-dense if σ(t) = 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 σ(t) − f (t)
,
f (t) =
σ(t) − t
Δ
(2.2)
if f is continuous at t and t is right-scattered. If t is right-dense, then derivative is defined
by
f σ(t) − f (s)
f (t) − f (s)
= lim+
,
f (t) = lim+
s→t
s→t
t−s
t−s
Δ
(2.3)
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 at all left-dense points. The set of rd-continuous
functions f : T → R is denoted by Crd (T, R).
f is said to be differentiable if its derivative exists. The set of functions f : T → R that
1
(T, R).
are differentiable and whose derivative is rd-continuous function is denoted by Crd
The derivative and the shift operator σ are related by the formula
f σ = f + μ f Δ,
where f σ := f ◦ σ.
(2.4)
Let f be a real-valued function defined on an interval [a,b]. We say that f is increasing, decreasing, nondecreasing, and nonincreasing on [a,b] if t1 ,t2 ∈ [a,b] 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
Zhenlai Han et al. 5
a differentiable function on [a,b]. Then f is increasing, decreasing, nondecreasing, and
nonincreasing on [a,b] if f Δ (t) > 0, f Δ (t) < 0, f Δ (t) ≥ 0, and f Δ (t) ≤ 0 for all t ∈ [a,b),
respectively.
We will make use of the following product and quotient rules for the derivative of the
product f g and the quotient f /g of two differentiable functions f and g:
( f g)Δ (t) = f Δ (t)g(t) + f σ(t) g Δ (t) = f (t)g Δ (t) + f Δ (t)g σ(t) ,
Δ
f
g
(t) =
f Δ (t)g(t) − f (t)g Δ (t)
.
g(t)g σ(t)
(2.5)
(2.6)
For a,b ∈ T and a differentiable function f , the Cauchy integral of f Δ is defined by
b
f Δ (t)Δt = f (b) − f (a).
a
(2.7)
The integration by parts formula reads
b
a
f Δ (t)g(t)Δt = f (b)g(b) − f (a)g(a) −
b
a
f σ (t)g Δ (t)Δt,
(2.8)
and infinite integrals are defined as
∞
a
f (s)Δs = lim
t
t →∞ a
f (s)Δs.
(2.9)
In case T = R we have
σ(t) = ρ(t) = t,
Δ
μ(t) ≡ 0,
f =f ,
b
a
f (t)Δt =
b
a
f (t)dt,
(2.10)
and in case T = Z we have
σ(t) = t + 1,
ρ(t) = t − 1,
μ(t) ≡ 1,
fΔ =Δf,
b
a
f (t)Δt =
b
−1
f (t).
(2.11)
t =a
3. Main results
In this section we give some new oscillation criteria for (1.1). In order to prove our main
results, we will use the formula
x(t)
γ Δ
=γ
1
0
hxσ + (1 − h)x
γ−1
xΔ (t)dh,
(3.1)
which is a simple consequence of Keller’s chain rule (see Bohner and Peterson [3, Theorem 1.90]). Also, we need the following auxiliary result.
Lemma 3.1 (Sahiner [13]). Suppose that the following conditions hold:
2
(I, R) where I = [t∗ , ∞) ⊂ T for some t∗ > 0,
(H1 ) u ∈ Crd
(H2 ) u(t) > 0, uΔ (t) > 0 and uΔΔ (t) ≤ 0 for t ≥ t∗ .
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Advances in Difference Equations
Then, for each k ∈ (0,1), there exists a constant tk ∈ T, tk ≥ t∗ , such that
σ(t) u τ(t)
kτ(t)
u σ(t) ≤
for t ≥ tk .
(3.2)
1
([a, ∞), R), pΔ ≥ 0 and
Lemma 3.2. Assume (1.11) holds. Furthermore, assume that p ∈ Crd
x is an eventually positive solution of (1.1). Then, there exists a t1 ≥ a such that
xΔ (t) > 0,
xΔΔ (t) < 0,
p(t) xΔ (t)
γ Δ
< 0 for t ≥ t1 .
(3.3)
Proof. Since x(t) is an eventually positive solution of (1.1), there exists a number t0 ≥ a
such that x(t) > 0 and x(τ(t)) > 0 for all t ≥ t0 > a. In view of (1.1), we have
p(t) xΔ (t)
γ Δ
γ
= −q(t) f x τ(t) ≤ −Lq(t) x τ(t)
<0
for t ≥ t0 ,
(3.4)
and so p(t)(xΔ (t))γ is an eventually decreasing function. We first show that p(t)(xΔ (t))γ
is eventually positive. Indeed, the decreasing function p(t)(xΔ (t))γ is either eventually
positive or eventually negative. Suppose that there exists an integer t1 ≥ t0 such that
p(t1 )(xΔ (t1 ))γ = c < 0, then from (3.4) we have p(t)(xΔ (t))γ ≤ p(t1 )(xΔ (t1 ))γ = c for t ≥
t1 , hence
Δ
x (t) ≤ c
1/γ
1
p(t)
1/γ
,
(3.5)
which implies by (1.11) that
x(t) ≤ x t1 + c1/γ
t t1
1
p(s)
1/γ
Δs −→ −∞ as t −→ ∞,
(3.6)
and this contradicts the fact that x(t) > 0 for all t ≥ t0 . Hence p(t)(xΔ (t))γ is eventually
positive. So xΔ (t) is eventually positive. Then x(t) is eventually increasing.
By (2.5), we get
p(t) xΔ (t)
γ Δ
γ
γ Δ
= pΔ (t) xΔ (t) + p σ(t) xΔ (t)
.
(3.7)
From (3.4), (3.7) and pΔ (t) ≥ 0, we can easily verify that
xΔ (t)
γ Δ
< 0.
(3.8)
Using (3.1), we get
xΔ (t)
1
γ Δ
=γ
1
0
h xΔ
σ
+ (1 − h)xΔ
γ−1
xΔΔ (t)dh.
(3.9)
From (3.8), (3.9), and 0 [h(xΔ )σ + (1 − h)xΔ ]γ−1 dh > 0, we have xΔΔ (t) is eventually negative. Therefore, we see that there is some t1 ≥ t0 such that (3.3) holds. The proof is com
plete.
Zhenlai Han et al. 7
1
([a, ∞), R), and pΔ ≥ 0. Furthermore, assume
Theorem 3.3. Assume (1.11) holds, p ∈ Crd
1
that there exists a positive function δ ∈ Crd ([a, ∞), R) such that for some positive constant
k ∈ (0,1),
t
limsup
t →∞
a
τ(s)
Lk q(s)δ(s)
σ(s)
γ
γ
γ+1 p(s) δ Δ (s) +
γ Δs = ∞,
−
(γ + 1)γ+1 δ(s)
(3.10)
where (δ Δ (s))+ = max{0,δ Δ (s)}. Then (1.1) is oscillatory on [a, ∞).
Proof. Suppose that (1.1) has a nonoscillatory solution x(t). We may assume without loss
of generality that x(t) > 0 and x(τ(t)) > 0 for all t ≥ t1 > a. We will consider only this case,
since the proof when x(t) is eventually negative is similar. In view of Lemmas 3.1 and 3.2,
for each positive constant k ∈ (0,1), there exists a t2 = max{tk ,t1 } such that
x(t) ≤ x σ(t) ≤
σ(t) σ(t)
x τ(t) ≤
x(t) for t ≥ t2 .
kτ(t)
kτ(t)
(3.11)
We get (3.3), (3.4), and (3.7). Define the function ω(t) by
ω(t) = δ(t)
p(t) xΔ (t)
γ
x(t)
γ
for t ≥ t2 .
(3.12)
Then ω(t) > 0, and using (2.5) and (2.6) we get
γ Δ
δ(t) γ p(t) xΔ (t)
ωΔ (t) = x(t)
+ p σ(t) x
Δ
γ
γ Δ
γ x(t) δ Δ (t) − δ(t) x(t)
γ γ
σ(t)
.
x(t)
(3.13)
x σ(t)
In view of (3.4), (3.11), and (3.12), we obtain
ωΔ (t) ≤ −Lkγ q(t)δ(t)
τ(t)
σ(t)
γ
δ Δ (t) ω σ(t)
+ δ σ(t)
γ δ(t)p σ(t) xΔ σ(t)
x(t)
γ γ
−
x(t) x σ(t)
(3.14)
γ Δ
.
Using (3.3) we have x(σ(t)) ≥ x(t), and then from (3.1) that
ωΔ (t) ≤ −Lkγ q(t)δ(t)
τ(t)
σ(t)
γ
δ Δ (t) ω σ(t)
+ δ σ(t)
γ γδ(t)p σ(t) xΔ σ(t)
x(t)
γ γ
−
x(t) x σ(t)
γ−1
xΔ (t)
(3.15)
.
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Advances in Difference Equations
So,
τ(t)
ω (t) ≤ −Lk q(t)δ(t)
σ(t)
Δ
γ
−
γ
δ Δ (t) ω σ(t)
+ δ σ(t)
γδ(t)p σ(t) xΔ σ(t)
x σ(t)
γ
γ+1
xΔ (t)
(3.16)
.
From (3.3), since (p(t)(xΔ (t))γ )Δ < 0, we have
p σ(t)
Δ
x (t) >
p(t)
1/γ
Δ
1/γ x σ(t) .
(3.17)
Substituting (3.17) in (3.16) we find that
ωΔ (t) ≤ −Lkγ q(t)δ(t)
τ(t)
σ(t)
−
γδ(t) p σ(t)
p(t)
γ
δ Δ (t) ω σ(t)
+ δ σ(t)
(γ+1)/γ xΔ σ(t)
1/γ x σ(t)
γ+1
(3.18)
.
γ+1
So,
τ(t)
ω (t) ≤ −Lk q(t)δ(t)
σ(t)
Δ
γ
−
γδ(t)
p(t)
γ
δ Δ (t) + ω σ(t)
+ δ σ(t)
λ −1 δ σ(t)
λ ω σ(t)
λ
(3.19)
,
where λ = (γ + 1)/γ, (δ Δ (t))+ = max{0,δ Δ (t)}. Set
γδ(t)
A= λ λ −1
δ σ(t)
p(t)
⎡
B=⎣
δ Δ (t)
+
λδ σ(t)
1/λ
γδ(t)
δ σ(t)
λ ω σ(t) ,
p(t)
λ−1
(3.20)
−1/λ ⎤1/(λ−1)
⎦
.
Using the inequality (1.13) we have
λ
δ Δ (t) + γδ(t)
ω σ(t) − λ λ−1 ω σ(t)
δ σ(t)
δ σ(t)
p(t)
≤ (λ − 1)λ−λ/(λ−1)
λ/(λ−1) δ Δ (t) +
δ σ(t)
γδ(t)
δ σ(t)
λ p(t)
(3.21)
−1/(λ−1)
,
λ −1
then
γ+1
λ
δ Δ (t) + p(t) δ Δ (t)
γδ(t)
ω σ(t) − γ
≤C
λ λ−1 ω σ(t)
δ σ(t)
δ(t)
δ σ(t)
p(t)
+
,
(3.22)
Zhenlai Han et al. 9
where C = (λ − 1)λλ/(λ−1) γ−1/(λ−1) = 1/(γ + 1)γ+1 . Thus, from (3.19) and (3.22) we obtain
τ(t)
ω (t) ≤ −Lk q(t)δ(t)
σ(t)
Δ
γ
γ
γ+1
p(t) δ Δ (t) +
γ .
+
(γ + 1)γ+1 δ(t)
(3.23)
Integrating the inequality (3.23) from t2 to t we obtain
−ω t2 ≤ ω(t) − ω t2 ≤ −
t t2
τ(s)
Lk q(s)δ(s)
σ(s)
γ
γ
γ+1 p(s) δ Δ (s) +
γ Δs,
−
(γ + 1)γ+1 δ(s)
(3.24)
which yields
t t2
τ(s)
Lk q(s)δ(s)
σ(s)
γ
γ
γ+1 p(s) δ Δ (s) +
γ Δs ≤ ω t2
−
γ+1
(γ + 1)
δ(s)
for all large t, which contradicts (3.10). The proof is complete.
(3.25)
From Theorem 3.3, we can obtain different conditions for oscillation of all solutions
of (1.1) with different choices of δ(t).
For example, let δ(t) = t, t ≥ a. Now, Theorem 3.3 yields the following result.
1
([a, ∞), R), pΔ ≥ 0. Furthermore, assume
Corollary 3.4. Assume (1.11) holds and p ∈ Crd
that for some positive constant k ∈ (0,1),
t
lim sup
t →∞
a
Lkγ sq(s)
τ(s)
σ(s)
γ
−
p(s)
Δs = ∞,
(γ + 1)γ+1 sγ
(3.26)
then (1.1) is oscillatory on [a, ∞).
Let δ(t) = 1, t ≥ a. Now, Theorem 3.3 yields the following well-known result (LeightonWintner theorem).
1
([a, ∞), R), pΔ ≥ 0.
Corollary 3.5 (Leighton-Wintner). Assume (1.11) holds and p ∈ Crd
If
t
lim sup
t →∞
a
q(s)
τ(s)
σ(s)
γ
Δs = ∞,
(3.27)
then (1.1) is oscillatory on [a, ∞).
Let γ = 1 and p(t) = 1 for t ≥ a. Now, Theorem 3.3 yields the following result.
1
([a, ∞), R) such that
Corollary 3.6. Assume that there exists a positive function δ ∈ Crd
for some positive constant k ∈ (0,1),
t
lim sup
t →∞
a
2 δ Δ (s) +
τ(s)
Lkq(s)δ(s)
−
Δs = ∞,
σ(s)
4δ(s)
where (δ Δ (s))+ = max{0,δ Δ (s)}. Then every solution of (1.1) is oscillatory on [a, ∞).
(3.28)
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Advances in Difference Equations
Remark 3.7. From Theorem 3.3, we can give some special sufficient conditions for oscillation of (1.1) on different type of time scales, for example, we can deduce that if there
exists a positive function δ ∈ C 1 ([a, ∞), R) such that for some positive constant k ∈ (0,1),
∞
a
1
p(t)
t
1/γ
dt = ∞,
limsup
t →∞
a
τ(s)
Lk q(s)δ(s)
s
γ
γ
γ+1 p(s) δ (s) +
γ ds = ∞,
−
(γ + 1)γ+1 δ(s)
p (t) ≥ 0,
(3.29)
where (δ (s))+ = max{0,δ (s)}, are sufficient conditions for oscillation of (1.14).
If there exists a positive sequence {δn } such that for some positive constant k ∈ (0,1),
∞ 1 1/γ
i=a
p(i)
= ∞,
limsup
t →∞
n
−1
i =a
τ(i)
Lk q(i)δ(i)
i+1
γ
γ
γ+1 p(i) Δδ(i) +
γ
−
(γ + 1)γ+1 δ(i)
= ∞,
Δp(n) ≥ 0,
(3.30)
where (Δδ(i))+ = max{0,Δδ(i)}, are sufficient conditions for oscillation of (1.15).
1
([a, ∞), R), pΔ ≥ 0. Furthermore, assume
Theorem 3.8. Assume (1.11) holds and p ∈ Crd
1
that there exists a positive function δ ∈ Crd ([a, ∞), R) such that for some positive constant
k ∈ (0,1), and m ≥ 1,
1
lim sup m
t →∞
t
t
a
(t − s)
m
τ(s)
Lk q(s)δ(s)
σ(s)
γ
γ
γ+1 p(s) δ Δ (s) +
γ Δs = ∞,
−
(γ + 1)γ+1 δ(s)
(3.31)
where (δ Δ (s))+ = max{0,δ Δ (s)}. Then (1.1) is oscillatory on [a, ∞).
Proof. Suppose that (1.1) has a nonoscillatory solution x(t). We may assume without loss
of generality that x(t) > 0 and x(τ(t)) > 0 for all t ≥ t1 > a. We proceed as in the proof of
Theorem 3.3 and we get (3.23). Then from (3.23) we have
τ(t)
Lk q(t)δ(t)
σ(t)
γ
γ
γ+1
p(t) δ Δ (t) +
−
≤ −ωΔ (t).
(γ + 1)γ+1 δ γ (t)
(3.32)
Therefore,
t
t2
(t − s)m Lkγ q(s)δ(s)
τ(s)
σ(s)
γ
−
γ+1 p(s) δ Δ (s) +
Δs ≤ −
(γ + 1)γ+1 δ γ (s)
t
t2
(t − s)m ωΔ (s)Δs.
(3.33)
An integration by parts formula (2.8) the right-hand side leads to
t
t2
t
(t − s)m ωΔ (s)Δs = (t − s)m ω(s)t2 −
t
t2
(t − s)m
Δs ω σ(s) Δs.
(3.34)
Zhenlai Han et al.
11
Note that since ((t − s)m )Δs ≤ −m(t − σ(s))m−1 ≤ 0 for t ≥ σ(s), m ≥ 1 (see Saker [15]).
Then from (3.33) we have
t
t2
(t − s)
m
γ
Lk q(s)δ σ(s)
τ(s) γ
σ(s)
γ+1 m p(s) δ Δ (s) +
−
Δs ≤ t − t2 ω t2 .
γ+1
γ
(γ + 1) δ (s)
(3.35)
Then
1
tm
t
t2
(t − s)
m
γ
Lk q(s)δ σ(s)
τ(s) γ
σ(s)
γ+1 p(s) δ Δ (s) +
t − t2
−
Δs ≤
(γ + 1)γ+1 δ γ (s)
t
m
ω t2 ,
(3.36)
which contradicts (3.31). The proof is complete.
From Theorem 3.8, we have the following oscillation criteria for (1.14) and (1.15).
Corollary 3.9. If there exists a positive function δ ∈ C 1 ([a, ∞), R) such that for some
positive constant k ∈ (0,1), m ≥ 1,
∞
a
1
p(t)
1/γ
1
limsup m
t →∞ t
dt = ∞,
t
a
(t − s)
m
τ(s)
Lk q(s)δ(s)
s
γ
γ
γ+1 p(s) δ (s) +
γ ds = ∞,
−
(γ + 1)γ+1 δ(s)
p (t) ≥ 0,
(3.37)
where (δ (s))+ = max{0,δ (s)}, then (1.14) is oscillatory.
Corollary 3.10. If there exists a positive sequence {δ(n)} such that for some positive constant k ∈ (0,1), m ≥ 1,
∞ 1 1/γ
i =a
p(i)
= ∞,
γ+1
n −1
p(i) Δδ(i) +
1 τ(i) γ
m
γ
γ
−
limsup m (n − i) Lk q(i)δ(i)
i+1
(γ + 1)γ+1 δ(i)
n→∞ n i=a
= ∞,
(3.38)
Δp(n) ≥ 0,
where (Δδ(i))+ = max{0,Δδ(i)}, then (1.15) is oscillatory.
Now, we give some sufficient conditions when (1.12) holds, which guarantee that every
solution of (1.1) oscillates or converges to zero in [a, ∞).
1
([a, ∞), R). Furthermore, assume that
Theorem 3.11. Assume (1.12) holds and p ∈ Crd
1
there exists a positive function δ ∈ Crd ([a, ∞), R) such that for some positive constant k ∈
(0,1), (3.10) holds. If
∞
a
1
p(t)
t
a
1/γ
q(s)Δs
Δt = ∞,
then every solution of (1.1) is either oscillatory or converging to zero on [a, ∞).
(3.39)
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Advances in Difference Equations
Proof. We proceed as in Theorem 3.3, we assume that (1.1) has a nonoscillatory solution
such that x(t) > 0, and x(τ(t)) > 0, for all t ≥ t1 > a.
From the proof of Lemma 3.2, we see that there exist two possible cases for the sign
of xΔ (t). The proof when xΔ (t) is an eventually positive is similar to that of the proof of
Theorem 3.3 and hence it is omitted.
Next, suppose that xΔ (t) < 0 for t ≥ t1 > a. Then x(t) is decreasing and limt→∞ x(t) =
b ≥ 0. We assert that b = 0. If not, then x(τ(t)) > x(t) > x(σ(t)) > b > 0 for t ≥ t2 > t1 .
Since f (x(τ(t))) ≥ Lbγ , there exists a number t3 > t2 such that f (x(τ(t))) ≥ L(x(τ(t)))γ
for t ≥ t3 . Defining the function
γ
u(t) = p(t) xΔ (t) ,
(3.40)
we obtain from (1.1)
uΔ (t) = −q(t) f x τ(t)
γ
≤ −Lbγ q(t),
≤ −Lq(t) x τ(t)
for t ≥ t3 .
(3.41)
Hence, for t ≥ t3 , we have
u(t) ≤ u t3 − Lbγ
t
a
q(s)Δs ≤ −Lbγ
t
a
q(s)Δs,
(3.42)
because of u(t3 ) = p(t3 )(xΔ (t3 ))γ < 0. So, we have
t
t3
Δ
x (s)Δs ≤ −L b
1/γ
t t3
1
p(s)
s
t3
q(τ)Δτ
1/γ
Δs.
(3.43)
By condition (3.39) we get x(t) → −∞ as t → ∞, and this is a contradiction to the fact that
x(t) > 0 for t ≥ t1 . Thus b = 0 and then x(t) → 0 as t → ∞. The proof is complete.
Similar to that of the proof of Theorem 3.11, we omit the proof of the following theorem.
1
([a, ∞), R). Furthermore, assume that
Theorem 3.12. Assume (1.12) holds and p ∈ Crd
1
there exists a positive function δ ∈ Crd ([a, ∞), R) such that for some positive constant k ∈
(0,1), (3.31), and (3.39) hold. Then every solution of (1.1) is either oscillatory or converging
to zero on [a, ∞).
From Theorems 3.11 and 3.12, we have the following results for (1.14) and (1.15).
Corollary 3.13. If there exists a positive function δ ∈ C 1 ([a, ∞), R) such that for some
positive constant k ∈ (0,1),
p (t) ≥ 0,
∞
a
t
limsup
t →∞
a
1
p(t)
1/γ
dt < ∞,
τ(s)
Lk q(s)δ(s)
s
γ
γ
∞
a
1
p(t)
t
a
1/γ
q(s)ds
γ+1 dt = ∞,
p(s) δ (s) +
γ ds = ∞,
−
(γ + 1)γ+1 δ(s)
(3.44)
where (δ (s))+ = max{0,δ (s)}, then every solution of (1.14) is either oscillatory or converging to zero on [a, ∞).
Zhenlai Han et al.
13
Corollary 3.14. If there exists a positive sequence {δ(n)} such that for some positive constant k ∈ (0,1),
∞ 1 1/γ
Δp(n) ≥ 0,
i =a
limsup
p(i)
n
−1
n→∞
< ∞,
i=a
τ(i)
Lkγ q(i)δ(i)
i+1
i=a
∞
i −1
1 q( j)
p(i) j =a
= ∞,
(3.45)
γ+1 γ
1/γ
p(i) Δδ(i) +
γ
−
(γ + 1)γ+1 δ(i)
= ∞,
where (Δδ(i))+ = max{0,Δδ(i)}, then every solution of (1.15) is either oscillatory or converging to zero on [a, ∞).
Corollary 3.15. If there exists a positive function δ ∈ C 1 ([a, ∞), R) such that for some
positive constant k ∈ (0,1), m ≥ 1,
p (t) ≥ 0,
1
limsup m
t
t →∞
t
a
∞
a
1
p(t)
(t − s)m
1/γ
dt < ∞,
∞
a
τ(s)
Lkγ q(s)δ(s)
s
γ
1
p(t)
t
a
1/γ
q(s)ds
dt = ∞,
γ+1 p(s) δ (s) +
γ ds = ∞,
−
(γ + 1)γ+1 δ(s)
(3.46)
where (δ (s))+ = max{0,δ (s)}, then every solution of (1.14) is either oscillatory or converging to zero on [a, ∞).
Corollary 3.16. If there exists a positive sequence {δ(n)} such that for some positive constant k ∈ (0,1), m ≥ 1,
Δp(n) ≥ 0,
∞ 1 1/γ
i=a
p(i)
< ∞,
∞
i =a
i −1
1 q( j)
p(i) j =a
1/γ
= ∞,
γ+1
n −1
p(i) Δδ(i) +
1 τ(i) γ
γ
limsup m (n − i)m Lkγ q(i)δ(i)
−
i+1
(γ + 1)γ+1 δ(i)
n→∞ n i=a
(3.47)
= ∞,
where (Δδ(i))+ = max{0,Δδ(i)}, then every solution of (1.15) is either oscillatory or converges to zero on [a, ∞).
In [8], Saker considered the second-order half-linear dynamic equations on time scales
p(t) xΔ (t)
γ Δ
+ q(t)xγ (t) = 0 for t ∈ [a,b]
(3.48)
and established following main oscillation criteria of (3.48).
Theorem A (Theorem 3.3, Saker [8]). Assume that
(H) p and q are positive, real-valued rd-continuous functions, and γ > 1 is an odd positive
integer, and
∞
a
1
p(t)
1/γ
Δt = ∞
(3.49)
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Advances in Difference Equations
hold. Furthermore, assume that there exists a positive Δ-differentiable function δ such that
t
limsup
t →∞
a
γ+1 p(s) δ Δ (s) +
δ(s)q(s) −
Δs = ∞,
(γ + 1)γ+1 δ γ (s)
(3.50)
where (δ Δ (t))+ = max{0,δ Δ (t)}. Then every solution of (3.48) is oscillatory on [a, ∞).
One can easily see that the result obtained in [8] cannot be applied for (1.1), so our results
are new ones.
4. Applications
In this section, we give some examples to illustrate our main results. To obtain the conditions for oscillation, we will use the fact
∞
a
Δt
= ∞ if 0 ≤ p ≤ 1.
tp
(4.1)
For more details we refer the reader to [4, Theorem 5.68].
Example 4.1. Consider the second-order delay dynamic equations on time scales
t γ−1 xΔ (t)
γ Δ
β σ(t)
t 2 τ(t)
+
γ
f x τ(t)
= 0,
t ∈ [1, ∞),
(4.2)
where p(t) = t γ−1 , q(t) = (β/t 2 )(σ(t)/τ(t))γ , f (x) = xγ , γ > 1, β > 1, k = 1/(γ + 1)(γ+1)/γ .
By Corollary 3.4, we have
∞ (γ−1)/γ
1
s
1
t
lim sup
t →∞
1
kγ s
β σ(s)
s2 τ(s)
γ τ(s)
σ(s)
γ
Δs = ∞,
−
sγ−1
Δs = lim sup
t →∞
(γ + 1)γ+1 sγ
t
1
β−1
Δs = ∞.
(γ + 1)γ+1 s
(4.3)
Then (4.2) is oscillatory on [1, ∞).
Acknowledgments
The authors thank the referees for their valuable suggestions and useful comments. This
research is supported by the Natural Science Foundation of China (10471077) and supported by Shandong Research Funds for Young Scientists (03BS094), also supported by
National Science Foundation of Educational Department of Shandong Province
(J04A60), and Jinan University Research Funds for Doctors (B0621).
Zhenlai Han et al.
15
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Zhenlai Han: Institute of Applied Mathematics, Naval Aeronautical Engineering Institute, Yantai,
Shandong 264001, China; School of Science, Jinan University, Jinan, Shandong 250022, China
Email address: hanzhenlai@163.com
Bao Shi: Institute of Applied Mathematics, Naval Aeronautical Engineering Institute, Yantai,
Shandong 264001, China
Email address: baoshi781@sohu.com
Shurong Sun: School of Science, Jinan University, Jinan, Shandong 250022, China
Email address: sshrong@163.com