Hindawi Publishing Corporation
Abstract and Applied Analysis
Volume 2011, Article ID 719628, 14 pages
doi:10.1155/2011/719628
Research Article
Interval Oscillation Criteria for Second-Order
Dynamic Equations with Nonlinearities Given by
Riemann-Stieltjes Integrals
Yuangong Sun
School of Mathematics, University of Jinan, Jinan, Shandong 250022, China
Correspondence should be addressed to Yuangong Sun, sunyuangong@yahoo.cn
Received 18 May 2011; Accepted 30 July 2011
Academic Editor: Paul Eloe
Copyright q 2011 Yuangong Sun. 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.
By using a generalized arithmetic-geometric mean inequality on time scales, we study the forced
oscillation of second-order dynamic equations with nonlinearities given by Riemann-Stieltjes
σb
Δ
tΔ qtφα xτt a rt, sφγs xgt, sΔξs et,
integrals of the form ptφα x
T, T is a time scale which is unbounded from above; φ∗ u where t ∈ t0 , ∞T t0 , ∞
|u|∗ sgn u; γ : a, bT1 → R is a strictly increasing right-dense continuous function; p, q, e :
t0 , ∞T → R, r : t0 , ∞T × a, bT1 → R, τ : t0 , ∞T → t0 , ∞T , and g : t0 , ∞T × a, bT1 →
t0 , ∞T are right-dense continuous functions; ξ : a, bT1 → R is strictly increasing. Some interval
oscillation criteria are established in both the cases of delayed and advanced arguments. As a
special case, the work in this paper unifies and improves many existing results in the literature for
equations with a finite number of nonlinear terms.
1. Introduction
Following Hilger’s landmark paper 1, there have been plenty of references focused on the
theory of time scales in order to unify continuous and discrete analysis, where a time scale
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 and of difference
equations. The oscillation theory has been developed very rapidly since the discovery of time
scale calculus with this understanding. Throughout this paper, a knowledge and understanding of time scale calculus is assumed. For an introduction to time scale calculus and dynamic
equations, we refer to the seminal books by Bohner and Peterson 2, 3.
2
Abstract and Applied Analysis
In this paper, we consider the following second-order dynamic equation with the nonlinearity given by a Riemann-Stieltjes integral of the form
σb
Δ
Δ
ptφα x t
qtφα xτt rt, sφγs x gt, s Δξs et,
1.1
a
where t ∈ t0 , ∞T t0 , ∞ ∩ T, t0 ∈ T, T is a time scale a closed nonempty subset of
real numbers which is unbounded from above; φ∗ u |u|∗ sgn u; a, b ∈ T1 , b > a, T1 is
another time scale; γ : a, bT1 → R is a strictly increasing right-dense continuous function
satisfying 0 < γa < α < γb; p, q, e : t0 , ∞T → R are right-dense continuous with
p > 0; r : t0 , ∞T × a, bT1 → R is right-dense continuous; τ : t0 , ∞T → t0 , ∞T , g :
t0 , ∞T × a, bT1 → t0 , ∞T are right-dense continuous functions satisfying limt → ∞ τt σb
limt → ∞ gt, s ∞; ξ : a, bT1 → R is strictly increasing. Here a fsΔξs denotes the
Riemann-Stieltjes integral of the function f on a, σbT1 with respect to ξ, σ : t0 , ∞T →
t0 , ∞T is the forward jump operator.
We restrict our attention to those solutions of 1.1 which exist on the time scale halfline Tx , ∞T , where Tx ≥ t0 may depend on the particular solution, a nontrivial function in
any neighborhood of infinity. As usual, such a solution of 1.1 is said to be oscillatory if it is
neither eventually positive nor eventually negative. Equation 1.1 is said to be oscillatory if
every proper solution is oscillatory.
Recently, people have been interested in the combined effects of linear, superlinear,
sublinear terms, and a forced term in oscillation. For instance, Sun and Wong 4 investigated
the following forced differential equation with mixed nonlinearities
n
ptx t qtxt qj tφαj xt et,
1.2
j1
where p, q, qj , e ∈ Ct0 , ∞, and 0 < α1 < · · · < αm < 1 < αm1 < · · · < αn . The authors
obtained interval oscillation criteria for 1.2 by using an arithmetic-geometric inequality and
employing arguments developed earlier in 5–9. Sun and Meng 10 studied 1.2 again by
making use of some of the arguments developed by Kong 11. In 12, Agarwal and Zafer
extended the results in 4 to dynamic equations on time scales of the form
n
Δ
qtφα xt qj tφβj xt et,
ptφα xΔ t
1.3
j1
where t ∈ t0 , ∞T ,
β1 > β2 > · · · > βm > α > βm1 > · · · > βn .
1.4
Very recently, Agarwal et al. 13 further to extend the results in 12 to the case of
several delays of the form
n
ptφα xΔ t
qtφα xτt qj tφβj x τj t et,
j1
1.5
Abstract and Applied Analysis
3
where τt and τj t are right-dense continuous functions satisfying limt → ∞ τt limt → ∞ τj t ∞ for j 1, 2, . . . , n. Sun and Kong 14 studied the oscillation of the secondorder forced differential equation with the nonlinearity given by a Riemann-Stieltjes integral
of the form
ptx t qtxt b
rt, sφγs xτsdξs et,
1.6
0
where t ≥ 0, γ ∈ C0, b is a strictly increasing function satisfying 0 ≤ γ0 < 1 < γb−. Some
interval oscillation criteria of the El-Sayed type and the Kong type are established which unify
many existing results in the literature.
It is obvious that 1.2, 1.3, 1.5, and 1.6 are special cases of 1.1. Some other
particular cased of 1.1 can be found in 15–20. In this paper, we will establish interval
oscillation criteria for the more general 1.1. Clearly, our work is of significance because
1.1 allows an infinite number of nonlinear terms and even a continuum of nonlinearities
determined by the function ξ. Moreover, even for the special cases of 1.2, 1.3, 1.5, and
1.6, our results generalize many existing oscillation criteria in the literature.
This paper is organized as follows. We present some lemmas in Section 2 which play a
key role in the proof of the main results. The main results are given in Section 3. Two examples
are given to illustrate the main results in Section 4.
2. Preliminaries
We here present four lemmas which play a key role in the proof of the main results in the
next section. In the sequel, we denote by Lξ a, bT1 the set of Riemann-Stieltjes integrable
functions on a, σbT1 with respect to ξ. Assume that γ, γ −1 ∈ Lξ a, bT1 . Let h sup{s ∈
a, bT1 : γs ≤ α}.
We first present the following two Lemmas 2.1, and 2.2, which generalize Lemma 2.1
and Lemma 3.1 in 14.
Lemma 2.1. Let
σb
m1 α
−1
γ sΔξs
σb
σh
σh
σh
σh
m2 α
γ −1 sΔξs
a
−1
Δξs
,
2.1
−1
Δξs
.
a
Then for any δ ∈ m1 , m2 , there exists η ∈ Lξ a, bT1 such that ηs > 0 on a, bT1 ,
σb
γsηsΔξs α,
2.2
a
σb
a
ηsΔξs δ.
2.3
4
Abstract and Applied Analysis
Proof. By the choice of h and the definitions of m1 and m2 , we have that 0 < m1 < 1 < m2 . Set
⎧
−1
σb
⎪
⎪
⎨αγ −1 s
Δξs , s ∈ σh, bT ,
η1 s σh
⎪
⎪
⎩0,
s ∈ a, σhT ,
η2 s ⎧
0,
⎪
⎪
⎨
⎪
⎪
⎩αγ −1 s
σh
−1
Δξs
s ∈ σh, bT ,
2.4
, s ∈ a, σhT .
a
It is easy to see that ηi ∈ Lξ a, bT1 and
σb
γsηi sΔξs α,
2.5
i 1, 2.
a
Moreover,
σb
σb
η1 sΔξs m1 ,
a
η2 sΔξs m2 .
2.6
s ∈ a, bT1 , l ∈ 0, 1.
2.7
a
Let
ηs, l 1 − lη1 s lη2 s,
Then we have that
σb
ηs, 0Δξs a
σb
a
σb
η1 sΔξs m1 < 1,
a
ηs, 1Δξs σb
2.8
η2 sΔξs m2 > 1.
a
By the continuous dependence of ηs, l on l, there exists l∗ ∈ 0, 1 such that ηs : ηs, l∗ σb
σb
satisfies a ηsΔξs δ ∈ m1 , m2 . Note that ηs > 0 on a, bT1 and a γsηsΔξs α. This completes the proof of Lemma 2.1.
The next lemma is a generalized arithmetic-geometric mean inequality on time scales.
Lemma 2.2. Assume that u : a, bT1 → R is right-dense continuous, η ∈ Lξ a, bT , u > 0, η > 0
σb
on a, bT1 and a ηsΔξs 1. Then
σb
a
ηsusΔξs ≥ exp
σb
ηs ln usΔξs .
a
2.9
Abstract and Applied Analysis
5
Proof. Define an operator L as follows:
L f σb
ηsfsΔξs.
2.10
a
It is obvious that L is a linear operator satisfying that L1 1 and Lu > 0. To derive inequality 2.9 it suffices to show that
Lu ≥ expLln u.
2.11
Note that ln t ≤ t − 1 for t > 0. Thus, for any s ∈ a, bT1 we have
us
ln
Lu
≤
us
− 1,
Lu
2.12
which follows that
ln us − ln Lu ≤
us
− 1.
Lu
2.13
Taking the operator L on both sides of 2.13, we get
Lln u − ln Lu Lln u − ln Lu
u
≤L
− L1
Lu
2.14
1 − 1 0,
which implies 2.11. This completes the proof.
The following two lemmas generalize Lemma 2.4 and Lemma 6.1 in 13.
Lemma 2.3. Let τ : t0 , ∞T → t0 , ∞T be a right-dense continuous function satisfying 0 ≤ τt ≤
t, c, d ∈ t0 , ∞T with c < d, and τcd min{τt : t ∈ c, dT }. Assume x : τcd , dT → R is a
positive right-dense continuous function such that ptφα xΔ t is nonincreasing on τcd , dT . Then
xτt P τt, τcd ≥
,
xσt P σt, τcd where P t, s t
s
p−1/α sΔs.
t ∈ c, d,
2.15
6
Abstract and Applied Analysis
Proof. Set zt p1/α txΔ t. It is not difficult to verify that zt is nonincreasing on τcd , dT
since ptφα xΔ t is nonincreasing on τcd , dT . Then we have
xt xτcd t
xΔ sΔs
τcd
xτcd t
p−1/α szsΔs
2.16
τcd
≥ zt
t
p−1/α s Δs
τcd
p
1/α
tP t, τcd xΔ t,
t ∈ τcd , bT .
Next, for s ∈ τt, σtT , and t ∈ c, dT we define
s : xs − p1/α sP s, τcd xΔ s.
2.17
Then 2.16 yields that s ≥ 0 for s ∈ τt, σtT and t ∈ c, dT . Consequently, for t ∈
c, dT , we have
0≤
σt
τt
p−1/α ss
Δs xsxσ s
σt τt
P s, τcd xs
Δ
Δs
P σt, τcd P τt, τcd −
.
xσt
xτt
2.18
This implies 2.15. The proof of Lemma 2.3 is complete.
Similar to the proof of Lemma 2.3, we can get the following result.
Lemma 2.4. Let τ : t0 , ∞T → t0 , ∞T be a right-dense continuous function satisfying τt > t,
c, d ∈ t0 , ∞T with c < d, and τ cd max{τt : t ∈ c, dT }. If x : c, τ cd T → R is a positive
right-dense continuous function for which ptφα xΔ t is nonincreasing on c, τ cd T , then
xτt P τ cd , τt
≥ ,
xσt P τ cd , σt
t ∈ c, dT ,
2.19
where P t, s is defined as in Lemma 2.3.
3. Main Results
We note from the definition of m1 and m2 that 0 < m1 < 1 < m2 . In the following we will use
the values of δ in the interval m1 , 1 to establish interval criteria for oscillation of 1.1. For
1
c, dT : uc 0 c, d ∈ t0 , ∞T with c < d, we define the function class Uc, d {u ∈ Crd
1
ud, u /
≡ 0}, where Crd c, dT denotes the set of right-dense continuously Δ—differentiable
Abstract and Applied Analysis
7
functions on c, dT . In the following, let τcd and τ cd be defined as in Lemmas 2.3 and 2.4.
Set
gcd min
t,s∈c,dT ×a,bT1
gt, s,
g cd max
t,s∈c,dT ×a,bT1
gt, s.
3.1
Theorem 3.1. Assume that τt, gt, s ≤ t for t ∈ t0 , ∞T and s ∈ a, bT1 . Suppose also that for
any T ≥ t0 , there exist subintervals ci , di T of T, ∞, i 1, 2, such that ci , di ∈ T, di > ci , and
rt, s ≥ 0,
t, s ∈ hi , di T × a, bT1 ,
−1i et ≥ 0,
t ∈ hi , di T ,
3.2
where hi min{τci di , gci di }. For each δ ∈ m1 , 1, let η ∈ Lξ a, bT1 be defined as in Lemma 2.1. If
there exists ui ∈ Uci , di for i 1, 2 such that
sup
δ∈m1 ,1
di α1 Qi t|ui σt|α1 − ptuΔ
Δt ≥ 0,
t
i
3.3
ci
where
P τt, τci di α
|et| 1−δ
Qi t qt
P σt, τci di 1−δ
⎞
⎛
⎛
γs ⎞
σb
P
gt,
s,
g
rt,
s
c
d
i
i
⎠Δξs⎠.
× exp⎝
ηs ln⎝
ηs P σt, gci di
a
3.4
Here we use the convention that ln 0 −∞, e−∞ 0, and 01−δ 0 and 1 − δ1−δ 1 for δ 1 due
to the fact that limt → 0 tt 1. Then 1.1 is oscillatory.
Proof. We prove this result by the contradiction method. Assume the contrary. Then 1.1 has
an extendible solution xt which is eventually positive or negative. Without loss of generality, we may assume that xt > 0 for all t ∈ t0 , ∞T . When xt is eventually negative, the
proof is in the same way except that the interval c2 , d2 T , instead of c1 , d1 T , is used. Define
ptφα xΔ t
wt −
,
φα xt
t ∈ c1 , d1 T .
3.5
It follows that
φγs x gt, s
Δξs
rt, s
φα xσt
a
Δ
φα xΔ t φα xt
et
−
pt
.
φα xσt
φα xtφα xσt
φα xτt
w t qt
φα xσt
Δ
σb
3.6
8
Abstract and Applied Analysis
It is obvious that the conditions in Lemma 2.3 are satisfied with τ replaced by gt, s. By 2.15
we have for t ∈ c1 , d1 T
xτt P τt, τc1 d1 ≥
,
xσt P σt, τc1 d1 P gt, s, gc1 d1
x gt, s
≥
.
xσt
P σt, gc1 d1
3.7
By 3.2, 3.6, and 3.7, and the fact that φ∗ is increasing, we get
P τt, τc1 d1 α
et
wΔ t ≥ qt
−
P σt, τc1 d1 φα xσt
γs
σb
P gt, s, gc1 d1
rt, s
xσtγs−α Δξs
P σt, gc1 d1
a
pt
3.8
Δ
φα xΔ t φα xt
.
φα xtφα xσt
I We first consider the case where the supremum in 3.3 is assumed at δ 1. From
3.2 and 3.8 we have that for t ∈ c1 , d1 T
Δ
φα xΔ t φα xt
P τt, τc1 d1 α
w t ≥ qt
pt
P σt, τc1 d1 φα xtφα xσt
γs
σb
P gt, s, gc1 d1
rt, s
xσtγs−α Δξs.
P
σt,
g
c1 d1
a
Δ
3.9
Let η ∈ Lξ a, bT1 be defined as in Lemma 2.1 with δ 1. Then η satisfies 2.2 and 2.3 with
δ 1. This follows that
σb
ηs γs − α Δξs 0.
a
Therefore, by Lemma 2.2 we have that for t ∈ c1 , d1 T
σb
a
γs
P gt, s, gc1 d1
rt, s
xσtγs−α Δξs
P σt, gc1 d1
γs
P gt, s, gc1 d1
ηsη srt, s
xσtγs−α Δξs
P σt, gc1 d1
a
⎞
⎛
⎛
⎞
γs
σb
P
gt,
s,
g
rt, s
c1 d1
≥ exp⎝
ηs ln⎝
xσtγs−α ⎠Δξs⎠
ηs
P
σt,
g
c1 d1
a
σb
−1
3.10
Abstract and Applied Analysis
⎞
⎛
⎛
γs ⎞
σb
P gt, s, gc1 d1
rt,
s
⎠Δξs⎠
exp⎝
ηs ln⎝
ηs
P σt, gc1 d1
a
× exp ln xσt
σb
9
ηs γs − α Δξs
a
⎛
exp⎝
σb
a
⎞
γs ⎞
P gt, s, gc1 d1
rt,
s
⎠Δξs⎠.
ηs ln⎝
ηs
P σt, gc1 d1
⎛
3.11
Substituting 3.11 into 3.9 we obtain
Δ
φα xΔ t φα xt
w t ≥ Q1 t pt
,
φα xtφα xσt
Δ
t ∈ c1 , d1 T ,
3.12
where Q1 t is defined by 3.4 with i 1 and δ 1. Multiplying both sides of the above
inequality by |u1 σt|α1 and proceeding as in the proof of Theorem 3.1 in 13, we can get
a contradiction with 3.3.
II Now we consider the case where the supremum in 3.3 is assumed at δ ∈ m1 , 1.
Let ηs
δ−1 ηs. Then from 2.2 and 2.3, we get
σb
ηsΔξs
1,
a
σb
ηs
δγs − α Δξs 0.
3.13
a
Hence for t ∈ c1 , d1 T
et
−
φα xσt
σb
a
γs
σb
P gt, s, gc1 d1
rt, s
ηsΩt,
sΔξs,
xσtγs−α Δξs P σt, gc1 d1
a
3.14
where
γs
rt, s P gt, s, gc1 d1
|et|
Ωt, s δ
.
xσtγs−α α
ηs
x σt
P σt, gc1 d1
3.15
On the other hand, by the basic arithmetic-geometric mean inequality, we have that
rt, s
Ωt, s ≥
ηs
δγs δ P gt, s, gc1 d1
|et| 1−δ
xσtδγs−α .
1−δ
P σt, gc1 d1
3.16
10
Abstract and Applied Analysis
Substituting 3.16 into 3.14, using Lemma 2.2 and similar to the computation in I, for
t ∈ c1 , d1 T we can get
et
−
φαxσt
≥
σb
a
σb
a
γs
P gt, s, gc1 d1
rt, s
xγs−α σtΔξs
P σtgc1 d1
rt, s
ηs
ηs
|et|
≥
1−δ
1−δ
δγs δ P gt, s, gc1 d1
|et| 1−δ
xσtδγs−α Δξs
1−δ
P σt, gc1 d1
3.17
⎞
⎛
⎛
γs ⎞
σb
P gt, s, gc1 d1
rt,
s
⎠Δξs⎠,
exp⎝
ηs ln⎝
ηs
P σt, gc1 d1
a
which also implies 3.12 for t ∈ c1 , d1 T . The rest of the proof is similar to Part I and hence
is omitted. This completes the proof of Theorem 3.1.
For the case when τt, gt, s > t for t ∈ t0 , ∞T and s ∈ a, bT1 , using Lemma 2.4
and following the proof of Theorem 3.1, we have the following oscillation result for 1.1
immediately.
Theorem 3.2. Assume that τt, gt, s > t for t ∈ t0 , ∞T and s ∈ a, bT1 . Suppose also that for
any T ≥ t0 , there exist subintervals ci , di T of T, ∞, i 1, 2, such that ci , di ∈ T, di > ci , and
3.2 holds for t ∈ ci , hi T and s ∈ a, bT , where hi max{τ ci di , g ci di }. For each δ ∈ m1 , 1, let
η ∈ Lξ a, bT1 be defined as in Lemma 2.1. If there exists ui ∈ Uci , di for i 1, 2 such that 3.4
holds, where
cd
α P τ 1 1 , τt
|et| 1−δ
Qi t qt c d
1−δ
P τ 1 1 , σt
⎛
× exp⎝
σb
a
⎞
cd
γs ⎞
P g 1 1 , gt, s
rt,
s
⎠Δξs⎠.
ηs ln⎝
ηs
P g c1 d1 , σt
⎛
3.18
Then 1.1 is oscillatory.
Remark 3.3. We see from the proof of Lemma 2.1 in Section 2 that for each δ ∈ m1 , 1, the
function η can be constructed explicitly for any nondecreasing function ξ, and hence the
functions Qi in Theorems 3.1 and 3.2 are explicitly given.
Remark 3.4. We observe that in Theorems 3.1 and 3.2, if the supremum in 3.3 is assumed
at δ 1, the effect of et is neglected in some extent. This implies that the magnitude of
et in ci , di T cannot be large. For otherwise, the supremum would have been taken at some
δ ∈ m1 , 1.
Abstract and Applied Analysis
11
Now, we interpret the results for 1.1 to the special case of 1.5. Set T1 N, a 1,
b n 1 for n ∈ N, and
ξs s, s 1, 2, . . . , n 1,
γs βs satisfying 1.4, s 1, 2, . . . , n,
rt, s qs t, s 1, 2, . . . , n,
3.19
gt, s τs t, s 1, 2, . . . , n.
Then 1.1 reduces to 1.5. By a straightforward computation, we have that
m1 n
α β−1 ,
n − m jm1 j
m2 m
α
β−1 .
m j1 j
3.20
Then Lemma 2.1 can be restated as the following: for any δ ∈ m1 , m2 , there exists a positive
n-tuple η1 , . . . , ηn satisfying
n
j1
αj ηj α,
n
ηj δ.
3.21
j1
Therefore, by Theorems 3.1 and 3.2, we obtain the following oscillation results for 1.5 which
generalize the results in 13.
Corollary 3.5. Assume that τt, τj t ≤ t for t ∈ t0 , ∞T and j 1, . . . , n. Suppose also that for
any T ≥ t0 , there exist subintervals ci , di T of T, ∞, i 1, 2, such that ci , di ∈ T, di > ci , and
qj t ≥ 0,
−1i et ≥ 0,
t ∈ θi , di T ,
3.22
where θi min{ωi , ωij : j 1, 2, . . . , n}, ωi min{τt : t ∈ ci , di T } and ωij min{τj t : t ∈
ci , di T }. For each δ ∈ m1 , 1, let η1 , . . . , ηn be defined by 3.21. We further assume that there
exists a function ui ∈ Uai , bi such that 3.4 holds, where
ηj βj ηj
P τj t, ωij
P τt, ωi α
|et| 1−δ n qj t
Qi t qt
.
P σt, ωi 1−δ
ηj
P σt, ωij
j1
3.23
Then 1.5 is oscillatory.
Corollary 3.6. Assume that τt, τj t > t for t ∈ t0 , ∞T and j 1, . . . , n. Suppose also that for
any T ≥ t0 , there exist subintervals ci , di T of T, ∞, i 1, 2, such that ci , di ∈ T, di > ci , and
qj t ≥ 0,
−1i et ≥ 0,
t ∈ ci , θ i ,
T
3.24
12
Abstract and Applied Analysis
where θi max{ωi , ωij : j 1, 2, . . . , n}, ωi max{τt : t ∈ ci , di T } and ωij max{τj t : t ∈
ci , di T }. For each δ ∈ m1 , 1, let η1 , . . . , ηn be defined by 3.21. We further assume there exists a
function ui ∈ Uai , bi such that 3.4 holds, where
P ωi , τt
Qi t qt
P ωi , σt
α
|et|
1−δ
1−δ
n
j1
qj t
ηj
ηj βj ηj
P ωij , τj t
.
P ωij , σt
3.25
Then 1.5 is oscillatory.
Remark 3.7. Corollaries 3.5 and 3.6 generalize those results in 13 since the sufficient condid
tion for oscillation of 1.5 is given here in the form of supδ∈m1 ,1 cii ·Δt ≥ 0.
4. Examples
We will give two examples to illustrate Theorems 3.1 and 3.2 in the case when ξs s, T1 R,
and T RN.
Example 4.1. Consider on T R the following differential equation
π
k2 sin t
φ3/2 x t k1 sin tφ3/2 x t −
4
2
φs xtds −ft cos t,
4.1
1
where t ≥ 0, k1 , k2 > 0 are constants, f ∈ C0, ∞ is an arbitrary nonnegative function. Here
we have pt 1, qt k1 sin t, rt, s k2 sin t, et −ft cos t, a 1, b 2, γs s,
τt t − 2π, and gt, s t. For any T ≥ 0, we choose k large enough so that 2kπ ≥ T and let
c1 2kπ π/4, d1 2kτ π/2, c2 2kπ 3π/4 and d2 2kπ π. Then it is easy to verify
that 3.2 holds, and
Qi t k1 sin t
t − ci
t − ci π/4
3/2
k2 sin t,
i 1, 2.
4.2
A straightforward computation yields that m1 3 ln4/3 0.863 and m2 3 ln3/2 1.2164. By Lemma 2.1, for any δ ∈ m1 , m2 , there exists a positive Riemann integrable function on 1, 2 such that 2.2 and 2.3 hold. Particularly, we can choose δ 1 and hence
ηs 1. If we choose ui t t − ci di − t for i 1, 2, then we have
di
Qi t|ui t|5/2 dt
ci
π/2
π/4
di
ci
5/2
π π
t − π/44 π/2 − t5/2
dt,
−t
sin t k1
k2 t −
4
2
t3/2
5/2
u t dt i
π/2 5/2
7/2
3π
dt 2 π
−
2t
.
4
7 4
π/4
4.3
Abstract and Applied Analysis
13
If there exist positive constants k1 and k2 such that
di
Qi t|ui t|5/2 dt ≥
ci
2 π 7/2
,
7 4
4.4
then condition 3.3 holds. By Theorem 3.1, we have that 4.1 is oscillatory.
Example 4.2. Consider on T N the following difference equation
2
Δ
φ3/2 xΔ t
qtφ3/2 xt 1 rt φs xt 1ds et,
4.5
1
where t ∈ N,
qt rt ⎧
⎨k1 ,
t 8j, 8j 1, 8j 2, 8j 3, 8j 4, 8j 5,
⎩f t, t 8j 6, 8j 7,
1
⎧
⎨k2 ,
t 8j, 8j 1, 8j 2, 8j 3, 8j 4, 8j 5,
⎩f t,
2
t 8j 6, 8j 7,
⎧
⎪
f3 t,
⎪
⎪
⎨
et f4 t,
⎪
⎪
⎪
⎩
f5 t,
4.6
t 8j, 8j 1, 8j 2,
t 8j 3, 8j 4, 8j 5,
t 8j 6, 8j 7,
for j ∈ N, k1 and k2 are positive constants, fi t i 1, 2, 3, 4, 5 are arbitrary real-valued
functions with f3 t ≤ 0 and f4 t ≥ 0. For any N ∈ N, we can choose j large enough so that
8j ≥ N. Let c1 8j 1, d1 8j 3, c2 8j 4, and d2 8j 6. Then 3.2 is valid. Choose
ηs 1 and ui t t − ci di − t for i 1, 2. Then 2.2 and 2.3 hold with δ 1. By the
straightforward computation, we get
di 5/2 Qi t|ui σt|5/2 − uΔ
Δt k1 k2 − 1.
t
i
4.7
ci
By Theorem 3.2, 4.5 is oscillatory if k1 k2 ≥ 1.
Acknowledgments
The author thanks the reviewer for his/her valuable comments on this paper. This paper was
supported by the national natural science foundations of China 60704039, 61174217 and the
natural science foundations of Shandong Province ZR2010AL002, JQ201119.
14
Abstract and Applied Analysis
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