doi:10.1155/2010/389109
Research Article
Oscillation of Second-Order Mixed-Nonlinear
Delay Dynamic Equations
M. ¨
Unal
1and A. Zafer
21Department of Software Engineering, Bahc¸es¸ehir University, Bes¸iktas¸, 34538 Istanbul, Turkey 2Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Correspondence should be addressed to M. ¨Unal,munal@bahcesehir.edu.tr
Received 19 January 2010; Accepted 20 March 2010 Academic Editor: Josef Diblik
Copyrightq 2010 M. ¨Unal and A. Zafer. 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.
New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique. No restriction is imposed on the coefficient functions and the forcing term to be nonnegative.
1. Introduction
In this paper we are concerned with oscillatory behavior of the second-order nonlinear delay dynamic equation of the form
rtxΔtΔ p0txτ0t n i1 pit|xτit|αi−1xτit et, t ≥ t0 1.1
on an arbitrary time scaleT, where
α1 > α2 > · · · > αm> 1 > αm1> · · · > αn > 0, n > m ≥ 1; 1.2
the functions r, pi, e: T → R are right-dense continuous with r > 0 nondecreasing; the delay
functions τi:T → T are nondecreasing right-dense continuous and satisfy τit ≤ t for t ∈ T
with τit → ∞ as t → ∞.
We assume that the time scaleT is unbounded above, that is, sup T ∞ and define the time scale interval t0, ∞Tby t0, ∞T: t0, ∞ ∩ T. It is also assumed that the reader is
already familiar with the time scale calculus. A comprehensive treatment of calculus on time scales can be found in 1–3.
By a solution of1.1 we mean a nontrivial real valued function x : T → R such that
x ∈ C1
rd T, ∞T and rxΔ ∈ C1rd T, ∞T for all T ∈ T with T ≥ t0, and that x satisfies 1.1.
A function x is called an oscillatory solution of 1.1 if x is neither eventually positive nor
eventually negative, otherwise it is nonoscillatory. Equation1.1 is said to be oscillatory if
and only if every solution x of 1.1 is oscillatory.
Notice that when T R, 1.1 is reduced to the second-order nonlinear delay
differential equation rtxt p0txτ0t n i1 pit|xτit|αi−1xτit et, t ≥ t0 1.3
while whenT Z, it becomes a delay difference equation
ΔrkΔxk p0kxτ0k
n
i1
pik|xτik|αi−1xτik ek, k ≥ k0. 1.4
Another useful time scale isT qN: {qm: m ∈ N and q > 1 is a real number}, which leads
to the quantum calculus. In this case,1.1 is the q-difference equation
Δq rtΔqxt p0txτ0t n i1 pit|xτit|αi−1xτit et, t ≥ t0, 1.5 whereΔqft fσt − ft/μt, σt qt, and μt q − 1t.
Interval oscillation criteria are more natural in view of the Sturm comparison theory since it is stated on an interval rather than on infinite rays and hence it is necessary to establish more interval oscillation criteria for equations on arbitrary time scales as inT R. As far as we know when T R, an interval oscillation criterion for forced second-order linear differential equations was first established by El-Sayed 4. In 2003, Sun 5 demonstrated
nicely how the interval criteria method can be applied to delay differential equations of the form
xt pt|xτt|α−1xτt et, α ≥ 1, 1.6
where the potential p and the forcing term e may oscillate. Some of these interval oscillation criteria were recently extended to second-order dynamic equations in 6–10. Further results
on oscillatory and nonoscillatory behavior of the second order nonlinear dynamic equations on time scales can be found in 11–23, and the references cited therein.
Therefore, motivated by Sun and Meng’s paper 24, using similar techniques
introduced in 17 by Kong and an arithmetic-geometric mean inequality, we give oscillation
criteria for second-order nonlinear delay dynamic equations of the form1.1. Examples are
2. Main Results
We need the following lemmas in proving our results. The first two lemmas can be found in 25, Lemma 1.
Lemma 2.1. Let {αi}, i 1, 2, . . . , n be the n-tuple satisfying α1 > α2 > · · · > αm> 1 > αm1> · · · >
αn> 0. Then, there exists an n-tuple {η1, η2, . . . , ηn} satisfying n i1 αiηi 1, n i1 ηi< 1, 0 < ηi < 1. 2.1
Lemma 2.2. Let {αi}, i 1, 2, . . . , n be the n-tuple satisfying α1 > α2 > · · · > αm> 1 > αm1> · · · >
αn> 0. Then there exists an n-tuple {η1, η2, . . . , ηn} satisfying n i1 αiηi 1, n i1 ηi 1, 0 < ηi < 1. 2.2
The next two lemmas are quite elementary via differential calculus; see 23,25.
Lemma 2.3. Let u, A, and B be nonnegative real numbers. Then
Auγ B ≥ γγ − 11/γ−1A1/γB1−1/γu, γ > 1. 2.3
Lemma 2.4. Let u, A, and B be nonnegative real numbers. Then
Cu − Duγ ≥γ − 1γγ/1−γCγ/γ−1D1/1−γ, 0 < γ < 1. 2.4 The last important lemma that we need is a special case of the one given in 6. For
completeness, we provide a proof.
Lemma 2.5. Let τ : T → T be a nondecreasing right-dense continuous function with τt ≤ t, and
a, b ∈ T with a < b. If x ∈ C1rd τa, bTis a positive function such that rtxΔt is nonincreasing on τa, bTwith r > 0 nondecreasing, then
xτt xσt ≥
τt − τa
σt − τa, t ∈ a, bT. 2.5 Proof. By the Mean Value Theorem 2, Theorem 1.14
xt − xτa ≥ xΔηt − τa, 2.6
for some η ∈ τa, tT, for any t ∈ τa, bT. Since rtxΔt is nonincreasing and rt is
nondecreasing, we have
and so xΔt ≤ xΔη, t ≥ η. Now
xt − xτa ≥ xΔtt − τa, t ∈ τa, bT. 2.8
Define
μs : xs − s − τaxΔs, s ∈ τt, σtT, t ∈ a, bT. 2.9
It follows from2.8 that μs ≥ xτa > 0 for s ∈ τt, σtTand t ∈ a, bT. Thus, we have 0 < σt τt μs xsxσsΔs σt τt s − τa xs Δ Δs σt − τa xσt − τt − τa xτt , 2.10
which completes the proof.
In what follows we say that a function Ht, s : T2 → R belongs to H
Tif and only if
H is right-dense continuous function on {t, s ∈ T2: t ≥ s ≥ t
0} having continuous Δ-partial
derivatives on{t, s ∈ T2: t > s ≥ t0}, with Ht, t 0 for all t and Ht, s / 0 for all t / s. Note
that in caseHR, theΔ-partial derivatives become the usual partial derivatives of Ht, s. The partial derivatives for the casesHZandHNwill be explicitly given later.
Denoting theΔ-partial derivatives HΔtt, s and HΔst, s of Ht, s with respect to t
and s by H1t, s and H2t, s, respectively, the theorems below extend the results obtained
in 5 to nonlinear delay dynamic equation on arbitrary time scales and coincide with them
when H2t, s is replaced by Ht, s. Indeed, if we set Ht, s Ut, s, then it follows that
H1t, s U1t, s
Uσt, s Ut, s, H2t, s
U2t, s
Ut, σs Ut, s. 2.11
WhenT R, they become
∂Ht, s ∂t ∂Ut, s/∂t 2Ut, s , ∂Ht, s ∂s ∂Ut, s/∂s 2Ut, s 2.12
as in 5. However, we prefer using H2t, s instead of Ut, s for simplicity.
Theorem 2.6. Suppose that for any given (arbitrarily large) T ∈ T there exist subintervals a1, b1T
and a2, b2Tof T, ∞T, where a1< b1and a2< b2such that
pit ≥ 0 for t ∈ a1, b1T∪ a2, b2T, i 0, 1, 2, . . . , n, −1l et > 0 for t ∈ al, blT, l 1, 2, 2.13 where al min τjal : j 0, 1, 2, . . . , n 2.14
hold. Let{η1, η2, . . . ηn} be an n-tuple satisfying 2.1 ofLemma 2.1. If there exist a function H ∈ HT
and numbers cν∈ aν, bνTsuch that
1 H2cν, aν cν aν QtH2σt, aν − rtH12t, aν Δt 1 H2bν, cν bν cν QtH2bν, σt − rtH22bν, t Δt > 0 2.15 for ν 1, 2, where Qt p0tτ0t − τ0aν σt − τ0aν k0|et| η0 n i1 pit ηi τit − τiaν σt − τiaν αiηi , k0 n i0 η−ηi i , η0 1 − n i1 ηi, 2.16 then1.1 is oscillatory.
Proof. Suppose on the contrary that x is a nonoscillatory solution of 1.1. First assume that
xt and xτjt j 0, 1, 2 . . . , n are positive for all t ≥ t1for some t1 ∈ t0, ∞T. Choose a1
sufficiently large so that τjτja1 ≥ t1. Let t ∈ a1, b1T.
Define
wt −rtx
Δt
xt , t ≥ t1. 2.17
Using the delta quotient rule, we have
wΔt − rtxΔtΔxt − rtxΔt2 xtxσt − rtxΔtΔ xσt rtxΔt2 xtxσt . 2.18 Notice that xtxσt xt xt μtxΔt x2t 1− μtwt rt x2t rt rt − μtwt 2.19 which implies rt − μtwt rtx σt xt > 0. 2.20 Hence, we obtain wΔt − rtxΔtΔ xσt w2t rt − μtwt. 2.21
Substituting2.21 into 1.1 yields wΔt p0txτ0t xσt w2t rt − μtwt n i1 pit|xτit|αi−1xτit xσt − et xσt. 2.22
By assumption, we can choose a1, b1 ≥ t1such that pit ≥ 0 i 1, 2, 3 . . . , n and et ≤ 0
for all t ∈ a1, b1T, where a1 is defined as in2.14. Clearly, the conditions ofLemma 2.5are
satisfied when, τ replaced with τjfor each fixedj 0, 1, 2, . . . , n. Therefore, from 2.5, we
have xτjt xσt ≥ τjt − τja1 σt − τja1, t ∈ a1, b1T 2.23
and taking into account2.22 yields
wΔt ≥ p0tτ0t − τ0a1 σt − τ0a1 w2t rt − μtwt n i1 pkt τit − τia1 σt − τia1 αi xσtαi−1 |et| xσt. 2.24 Denote Q∗0t : p0tτ0t − τ0a1 σt − τ0a1, Q ∗ it : pit τit − τia1 σt − τia1 αi . 2.25 From2.24, we have wΔt ≥ Q∗0t w 2t rt − μtwt n i1 Q∗itxσtαi−1 |et| xσt. 2.26
Now recall the well-known arithmetic-geometric mean inequality, see 26,
n i0 uiηi≥ n i0 uηii , 2.27
where η0 1 −ni1ηiand ηi> 0, i 1, 2, . . . , n. Setting
u0η0 : |et| xσt, uiηi: Q∗itxσtαi−1 2.28 in2.26 yields wΔt ≥ Q∗0t w2t rt − μtwt n i1 uiηi u0η0 Q∗0t w2t rt − μtwt n i0 uiηi. 2.29
From2.29 and taking into account 2.27, we get wΔt ≥ Q∗0t w2t rt − μtwt n i0 uηi i 2.30 and hence, wΔt ≥ Q∗0t w2t rt − μtwt η −η0 0 |et|η0 xσtη0 n i1 η−ηi i Qi∗t ηixσtαi−1ηi Q∗ 0t w2t rt − μtwt η −η0 0 |et| η0 n i1 η−ηi i Qi∗tηixσt−η0nj1αjηj−ηj Q∗ 0t w2t rt − μtwt η −η0 0 |et|η0 n i1 η−ηi i Qi∗tηi 2.31 which yields wΔt ≥ Q∗0t w 2t rt − μtwt η −η0 0 |et|η0 n i1 η−ηi i pit ηiτit − τia1 σt − τia1 αiηi Qt w2t rt − μtwt, 2.32 where Qt Q0∗t η−η0 0 |et|η0 n i1 η−ηi i pit ηiτit − τia1 σt − τia1 αiηi . 2.33
Multiplying both sides of2.32 by H2σt, a
1 and integrating both sides of the resulting
inequality from a1to c1, a1< c1< b1yield
c1 a1 wΔtH2σt, a1Δt ≥ c1 a1 QtH2σt, a1Δt c1 a1 w2tH2σt, a 1 rt − μtwt Δt. 2.34
Fix s and note that
wtH2t, sΔt H2σt, swΔt H2t, sΔtwt
H2σt, swΔt H
1t, sHσt, swt Ht, sH1t, swt,
2.35
from which we obtain
H2σt, swΔt wtH2t, sΔt− H
Therefore, c1 a1 wΔtH2σt, a1Δt c1 a1 wtH2t, a1 Δt Δt − c1 a1 H1t, a1Hσt, a1wt Ht, a1H1t, a1wtΔt. 2.37 Notice that c1 a1 wtH2t, a1 Δt Δt wc1H2c1, a1 − wa1H2a1, a1 wc1H2c1, a1 2.38
since Ha1, a1 0 and hence, we obtain from 2.34 that
wc1H2c1, a1 ≥ c1 a1 QtH2σt, a1Δt c1 a1 w2t rt − μtwtH 2σt, a 1Δt c1 a1 H1t, a1Hσt, a1wt Ht, a1H1t, a1wtΔt. 2.39
On the other hand,
w2tH2σt, s rt − μtwt wtHσt, sH1t, s Ht, sH1t, swt wtHσt, s rt − μtwt rt − μtwtH1t, s 2 −rt − μtwtH12t, s − wtHσt, sH1t, s Ht, sH1t, swt. 2.40
Taking into account that Hσt, s Ht, s μtH1t, s, we have
w2tH2σt, a 1 rt − μtwt wtHσt, a1H1t, a1 Ht, a1H1t, a1wt ≥ −rtH 2 1t, a1. 2.41 Using this inequality in2.39, we have
wc1H2c1, a1 ≥ c1 a1 QtH2σt, a1 − rtH12t, a1 Δt. 2.42
Similarly, by following the above calculation step by step, that is, multiplying both sides of2.32 this time by H2b
1, σs after taking into account that
H2t, σswΔs wsH2t, sΔs− H2t, sHt, σsws − Ht, sH2t, sws, 2.43
one can easily obtain
−wc1H2b1, c1 ≥ b1 c1 QsH2b1, σs − rsH22b1, s Δs. 2.44 Adding up2.42 and 2.44, we obtain
0≥ 1 H2c1, a1 c1 a1 QtH2σt, a1 − rtH12t, a1 Δt 1 H2b 1, c1 b1 c1 QtH2b1, σt − rsH22b1, t Δt. 2.45
This contradiction completes the proof when xt is eventually positive. The proof when xt is eventually negative is analogous by repeating the above arguments on the interval a2, b2T
instead of a1, b1T.
Corollary 2.7. Suppose that for any given (arbitrarily large) T ≥ t0there exist subintervals a1, b1
and a2, b2 of T, ∞ such that
pit ≥ 0 for t ∈ a1, b1 ∪ a2, b2, i 0, 1, 2, . . . , n,
−1let ≥ 0 for t ∈ a
l, bl, l 1, 2,
2.46
where al min{τjal : j 0, 1, 2, . . . , n} holds. Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.1
ofLemma 2.1. If there exist a function H ∈ HRand numbers cν∈ aν, bν such that
1 H2c ν, aν cν aν QtH2t, a ν − rtH12t, aν dt 1 H2b ν, cν bν cν QtH2bν, t − rtH22bν, t dt > 0 2.47 for ν 1, 2, where Qt p0tτ0t − τ0aν t − τ0aν k0|et| η0 n i1 pit ηi τit − τiaν t − τiaν αiηi , k0 n i0 η−ηi i , η0 1 − n i1 ηi, 2.48 then1.3 is oscillatory.
Corollary 2.8. Suppose that for any given (arbitrarily large) T ≥ t0there exist a1, b1, a2, b2∈ Z with
T ≤ a1< b1and T ≤ a2< b2such that for each i 0, 1, 2, . . . , n,
pit ≥ 0 for t ∈ {a1, a1 1, a1 2, . . . , b1} ∪ {a2, a2 1, a2 2, . . . , b2},
−1l
et ≥ 0 for t ∈ {al, al 1, al 2, . . . , bl} l 1, 2,
2.49
where al min{τjal : j 0, 1, 2, . . . , n} holds. Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.1 of Lemma 2.1. If there exist a function H ∈ HZand numbers cν∈ {aν 1, aν 2, . . . , bν− 1} such that
1 H2c ν, aν cν−1 taν QtH2t 1, a ν − rtH12t, aν 1 H2b ν, cν bν−1 tcν QtH2bν, t 1 − rtH22bν, t > 0 2.50 for ν 1, 2, where H1t, aν : Ht 1, aν − Ht, aν, H2bν, t : Hbν, t 1 − Hbν, t, Qt p0tτ0t − τ0aν t 1 − τ0aν k0|et| η0 n i1 pit ηi τit − τiaν t 1 − τiaν αiηi , k0 n i0 η−ηi i , η0 1 − n i1 ηi, 2.51 then1.4 is oscillatory.
Corollary 2.9. Suppose that for any given (arbitrarily large) T ≥ t0there exist a1, b1, a2, b2∈ N with
T ≤ a1< b1and T ≤ a2< b2such that for each i 0, 1, 2, . . . , n,
pit ≥ 0 for t ∈ qa1, qa11, . . . , qb1∪qa2, qa21, . . . , qb2, −1l et ≥ 0 for t ∈qal, qal1, . . . , qbl , l 1, 2 2.52
where qal min{τjqal : j 0, 1, 2, . . . , n} holds. Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.1 ofLemma 2.1. If there exist a function H ∈ Hqand numbers qcν ∈ {qaν1, qaν2, . . . , qbν−1} such that
1 H2qcν, qaν cν−1 maν qm QqmH2qm1, qaν − rqmH12qm, qaν 1 H2qbν, qcν bν−1 mcν qm QqmH2qbν, qm1 − rqmH22qbν, qm > 0 2.53
for ν 1, 2, where H1 qm, qaν: H qm1, qaν− Hqm, qaν q − 1qm , H2 qbν, qm : H qbν, qm1− Hqbν, qm q − 1qm , Qt p0t τ0t − τ0 qaν qt − τ0 qaν k0|et| η0 n i1 pit ηi τit − τiqaν qt − τiqaν αiηi , k0 n i0 η−ηi i , η0 1 − n i1 ηi, 2.54 then1.5 is oscillatory.
Notice thatTheorem 2.6does not apply if there is no forcing term, that is, et ≡ 0. In this case we have the following theorem.
Theorem 2.10. Suppose that for any given (arbitrarily large) T ∈ T there exists a subinterval a, bT
of T, ∞T, where a < b such that
pit ≥ 0 for t ∈ a, bT, i 0, 1, 2, . . . , n, 2.55
where a min{τja : j 0, 1, 2, . . . , n} holds. Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.2 in Lemma 2.2. If there exist a function H ∈ HTand a number c ∈ a, bTsuch that
1 H2c, a c a QtH2σt, a − rtH2 1t, a Δt 1 H2b, c b c QtH2b, σt − rsH22b, t2 Δt > 0, 2.56 where Qt p0tτ0t − τ0a σt − τ0a k0 n i1 pit ηi τit − τia σt − τia αiηi , k0 n i1 η−ηi i , 2.57
then1.1 with et ≡ 0 is oscillatory.
Proof. We will just highlight the proof since it is the same as the proof ofTheorem 2.6. We should remark here that taking et ≡ 0 and η0 0 in proof ofTheorem 2.6, we arrive at
wΔt ≥ Q0∗t w2t rt − μtwt n i1 uiηi. 2.58
The arithmetic-geometric mean inequality we now need is n i1 uiηi≥ n i1 uηi i , 2.59
where 1ni1ηiand ηi > 0, i 1, 2, . . . , n are as inLemma 2.2.
Corollary 2.11. Suppose that for any given (arbitrarily large) T ≥ t0there exists a subinterval a, b
of T, ∞, where T ≤ a < b with a, b ∈ R such that
pit ≥ 0 for t ∈ a, b, i 0, 1, 2, . . . , n, 2.60
where a min{τja : j 0, 1, 2, . . . , n} holds. Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.2 in Lemma 2.2. If there exist a function H ∈ HRand a number c ∈ a, b such that
1 H2c, a c a QtH2t, a − rtH12t, adt 1 H2b, c b c QsH2b, t − rtH22b, tdt > 0, 2.61 where Qt p0tτ0t − τ0a t − τ0a k0 n i1 pit ηi τit − τia t − τia αiηi , k0 n i1 η−ηi i , 2.62
then1.3 with et ≡ 0 is oscillatory.
Corollary 2.12. Suppose that for any given (arbitrarily large) T ≥ t0 there exists a, b ∈ Z with
T ≤ a < b such that
pit ≥ 0 for t ∈ {a, a 1, . . . , b}, i 0, 1, 2, . . . , n, 2.63
where a min{τja : j 0, 1, 2, . . . , n} holds. Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.2 in Lemma 2.2. If there exist a function H ∈ HZand a number c ∈ {a 1, a 2, . . . , b − 1} such that
1 H2c, a c−1 ta QtH2t 1, a − rtH2 1t, a 1 H2b, c b−1 tc QtH2b, t 1 − rtH22b, t> 0, 2.64
where H1t, a : Ht 1, a − Ht, a, H2b, t : Hb, t 1 − Hb, t, Qt p0tτ0t − τ0a t 1 − τ0a k0 n i1 pit ηi τit − τia t 1 − τia αiηi , k0 n i1 η−ηi i , 2.65
then1.4 with et ≡ 0 is oscillatory.
Corollary 2.13. Suppose that for any given (arbitrarily large) T ≥ t0 there exist a, b ∈ N with
T ≤ a < b such that
pit ≥ 0 for t ∈
qa, qa1, . . . , qb, i 0, 1, 2, . . . , n 2.66
where qa min{τ
jqa : j 0, 1, 2, . . . , n} holds. Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.2
inLemma 2.2. If there exist a function H ∈ HqN and a number qc∈ {qa, qa1, . . . , qb} such that
1 H2qc, qa c−1 ma qm QqmH2qm1, qa− rqmH1qm, qa 2 1 H2qb, qc b−1 mc qm QqmH2qb, qm1− rqmH2qb, qm 2 > 0, 2.67 where H1 qm, qa: H qm1, qa− Hqm, qa q − 1qm , H2 qb, qm: H qb, qm1− Hqb, qm q − 1qm , Qt p0t τ0t − τ0 qa qt − τ0 qa k0 n i1 pit ηi τit − τiqa qt − τiqa αiηi , k0 n i1 η−ηi i , 2.68
then1.5 with et ≡ 0 is oscillatory.
It is obvious thatTheorem 2.6is not applicable if the functions pit are nonpositive
for i m 1, m 2, . . . , n. In this case the theorem below is valid.
Theorem 2.14. Suppose that for any given (arbitrarily large) T ∈ T there exist subintervals a1, b1T
and a2, b2Tof T, ∞T, where a1< b1and a2< b2such that
pit ≥ 0 for t ∈ a1, b1T∪ a2, b2T, i 0, 1, 2, . . . , n,
−1l
et > 0 for t ∈ al, blT, l 1, 2,
where al min{τjal : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ HT, positive numbers λiand νisatisfying m i1 λi n im1 νi 1, 2.70
and numbers cν∈ aν, bνTsuch that
1 H2c ν, aν cν aν QtH2σt, a ν − rtH12t, aν Δt 1 H2b ν, cν bν cν QtH2bν, σt − rtH22bν, t Δt > 0 2.71 for ν 1, 2, where Qt p0tτ0t − τ0aν σt − τ0aν m i1 μiλi|et|1−1/αip1/αi it τit − τiaν σt − τiaν − n im1 βiνi|et|1−1/αipi1/αit τit − τiaν σt − τiaν , 2.72 with μi αiαi− 11/αi−1, βi αi1 − αi1/αi−1, pi max −pit, 0 , 2.73 then1.1 is oscillatory.
Proof. Suppose that1.1 has a nonoscillatory solution. Without losss of generality, we may
assume that xt and xτit i 0, 1, 2, . . . , n are eventually positive on a1, b1Twhen a1is
sufficiently large. If xt is eventually negative, one may repeat the same proof step by step on the interval a2, b2T. Rewriting1.1 for t ∈ a1, b1Tas rtxΔtΔ p0txτ0t m i1 pitxαiτit λi|et| n im1 pitxαiτit νi|et| 0 2.74 and applyingLemma 2.3to each term in the first sum, we obtain
rtxΔtΔ p0txτ0t m i1 μiλi|et|1−1/αip1/αi itxτit n im1 pitxαiτit νi|et| ≤ 0, 2.75
where μi αiαi− 11/αi−1for i 1, 2, . . . , m. Setting wt −rtx Δt xt 2.76 yields wΔt − rtxΔtΔ xσt w2t rt − μtwt. 2.77
Substituting the above last equality into2.75, we have
wΔt ≥ p0txτ0t xσt m i1 μiλi|et|1−1/αip1/αi it xτit xσt 1 xσt n im1 pitxαiτit νi|et| w2t rt − μtwt. 2.78
It follows from2.5 that
xτ0t xσt ≥ τ0t − τ0a1 σt − τ0a1, 2.79 xτit xσt ≥ τit − τia1 σt − τia1, 2.80 xαiτ it xσt ≥ x αi−1τ itτit − τia1 σt − τia1. 2.81
Notice that the second sum in2.78 can be written as
1 xσt n im1 pitxαiτit νi|et| n im1 pitx αiτ it xσt νi|et| xσt n im1 τit − τia1 σt − τia1 νi|et| 1 xτit− pit 1 xτit 1−αi , 2.82
and hence applying theLemma 2.4yields
n im1 τit − τia1 σt − τia1 νi|et| 1 xτit− pit 1 xτit 1−αi ≥ − n im1 τit − τia1 σt − τia1 βiνi|et|1−1/αipi1/αit, 2.83
where βi αi1 − αi1/αi−1 and pi max{−pit, 0} for i m 1, m 2, . . . , n. Using 2.79,
2.80, and 2.78 into 2.78, we obtain
wΔt ≥ p0tτ0t − τ0a1 σt − τ0a1 m i1 τit − τia1 σt − τia1 μiλi|et|1−1/αip1/αi it − n im1 τit − τia1 σt − τia1 βiνi|et|1−1/αipi1/αit w 2t rt − μtwt. 2.84 Setting Qt p0tτ0t − τ0a1 σt − τ0a1 m i1 μiλi|et|1−1/αip1/αi it τit − τia1 σt − τia1 − n im1 βiνi|et|1−1/αipi1/αit τit − τia1 σt − τia1 , 2.85 we have wΔt ≥ Qt w 2t rt − μtwt. 2.86
The rest of the proof is the same as that ofTheorem 2.6and hence it is omitted.
Corollary 2.15. Suppose that for any given (arbitrarily large) T ≥ t0there exist subintervals a1, b1
and a2, b2 of T, ∞, where T ≤ a1< b1and T ≤ a2< b2such that
pit ≥ 0 for t ∈ a1, b1 ∪ a2, b2, i 0, 1, 2, . . . , n,
−1let > 0 for t ∈ a
l, bl, l 1, 2,
2.87
where al min{τjal : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ HR, positive numbers
λiand νisatisfying m i1 λi n im1 νi 1, 2.88
and numbers cν∈ aν, bν such that
1 H2cν, aν cν aν QtH2t, aν − rtH12t, aν dt 1 H2bν, cν bν cν QtH2bν, t − rtH22bν, t dt > 0 2.89
for ν 1, 2, where Qt p0tτ0t − τ0aν t − τ0aν m i1 μiλi|et|1−1/αip1/αi it τit − τiaν t − τiaν − n im1 βiνi|et|1−1/αipi1/αit τit − τiaν t − τiaν 2.90 with μi αiαi− 11/αi−1, βi αi1 − αi1/αi−1, pi max −pit, 0 , 2.91 then1.3 is oscillatory.
Corollary 2.16. Suppose that for any given (arbitrarily large) T ≥ t0 there exist a1, b1, a2, b2 ∈ Z
with T ≤ a1< b1and T ≤ a2< b2such that for each i 0, 1, 2, . . . , n,
pit ≥ 0 for t ∈ {a1, a1 1, . . . , b1} ∪ {a2, a2 1, . . . , b2}
−1l
et > 0 for t ∈ {al, al 1, . . . , bl}, l 1, 2,
2.92
where al min{τjal : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ HZ, positive numbers
λiand νisatisfying m i1 λi n im1 νi 1, 2.93
and numbers cν∈ {aν 1, aν 2, . . . , bν− 1} such that
1 H2cν, aν cν−1 taν QtH2t 1, aν − rtH12t, aν 1 H2b ν, cν bν−1 tcν QtH2bν, t 1 − rtH22bν, t > 0 2.94 for ν 1, 2, where H1t, aν : Ht 1, aν − Ht, aν, H2bν, t : Hbν, t 1 − Hbν, t, Qt p0tτ0t − τ0aν t 1 − τ0aν m i1 μiλi|et|1−1/αip1/αi it τit − τiaν t 1 − τiaν − n im1 βiνi|et|1−1/αipi1/αit τit − τiaν t 1 − τiaν 2.95
with μi αiαi− 11/αi−1, βi αi1 − αi1/αi−1, pi max −pit, 0 , 2.96 then1.4 is oscillatory.
Corollary 2.17. Suppose that for any given (arbitrarily large) T ≥ t0there exist a1, b1, a2, b2 ∈ N
with T ≤ a1< b1and T ≤ a2< b2such that for each i 0, 1, 2, . . . , n,
pit ≥ 0 for t ∈ qa1, qa11, . . . , qb1 ∪qa2, qa21, . . . , qb2 , −1let > 0 for t ∈qal, qal1, . . . , qbl , l 1, 2, 2.97
where qal min{τjqal : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ Hq, positive numbers λiand νisatisfying m i1 λi n im1 νi 1, 2.98
and numbers qcν ∈ {qaν1, qaν2, . . . , qbν−1} such that
1 H2qcν, qaν cν−1 maν qm QqmH2qm1, qaν − rtH2 1 qm, qaν 1 H2qbν, qcν bν−1 mcν qm QqmH2qbν, qm1 − rtH2 2 qbν, qm > 0 2.99 for ν 1, 2, where H1 qm, qaν: H qm1, qaν− Hqm, qaν q − 1qm , H2 qbν, qm : H qbν, qm1− Hqbν, qm q − 1qm , Qt p0t τ0t − τ0 qaν qt − τ0 qaν m i1 μiλi|et|1−1/αipi1/αit τit − τi qaν qt − τi qaν − n im1 βiνi|et|1−1/αipi1/αit τit − τi qaν qt − τi qaν 2.100 with μi αiαi− 11/αi−1, βi αi1 − αi1/αi−1, pi max −pit, 0 , 2.101 then1.5 is oscillatory.
3. Examples
In this section we give three examples when n 2, and α1 2, α2 1/2 in 1.1. That is, we
consider
xΔΔt p0txτ0t p1t|xτ1t|xτ1t p2t|xτ1t|−1/2xτ2t 0. 3.1
For simplicity we take Ht, s t − s, thus H1t, s −H2t, s 1. Note that η1 1/3 and
η2 2/3 byLemma 2.2.
Example 3.1. Let A ≥ 0 and B, C > 0 be constants. Consider the differential equation
xt Axt − 1 B|xt − 2|xt − 2 C|xt − 1|−1/2xt − 1 0. 3.2 Let a j, b j 2, and c j 1, j ∈ N. We calculate Qt A t − j t − j 1 √33 4B 1/3C2/3 t − j t − j 22/3t − j 11/3 3.3
and see that2.61 holds if
4A 9BC21/3> 27. 3.4
Since all conditions ofCorollary 2.11are satisfied, we conclude that3.2 is oscillatory when
3.4 holds.
Example 3.2. Let A ≥ 0 and B, C > 0 be constants. Define p0t A, p1t B, and p2t C
for t 10j k, k −3, −2, −1, 0, 1, 2, 3, j ≥ 1; otherwise, the functions are defined arbitrarily. Consider the difference equation
Δ2xt p
0txt − 1 p1t|xt − 2|xt − 2 p2t|xt − 1|−1/2xt − 1 0. 3.5
Let a 10j, b 10j 3, and c 10j 1. We derive
Qt A t − 10j t − 10j 2 3 3 √ 4 BC21/3 t − 10j t − 3j 32/3t − 10j 41/3 3.6
and see that positivity in2.64 satisfies if
A 9 BC21/3 4√3 5 > 48 5 . 3.7
Since all conditions ofCorollary 2.12are satisfied, we conclude that3.5 is oscillatory if 3.7
Example 3.3. Let A ≥ 0 and B, C > 0 be constants. Define p0t A, p1t B and p2t C
for t 210jk, k −3, −2, −1, 0, 1, 2, 3, j ≥ 1; otherwise, the functions are defined arbitrarily.
Consider the q-difference equation, q 2, Δ2 qxt p0tx t 2 p1t x4t x4t p2t x8t −1/2x t 8 0. 3.8 Let a 10j, b 10j 3, and c 10j 1. We have
Qt A t − 2 10j 4t − 210j 3 3 √ 4 BC21/3 t − 2 10j 8t − 210j2/316t − 210j1/3. 3.9
We see that2.67 holds for all A ≥ 0 and B, C > 0. Since all conditions ofCorollary 2.12are satisfied, we conclude that3.8 is oscillatory if A ≥ 0 and B, C > 0 are positive.
Acknowledgments
The paper is supported in part by the Scientific and Research Council of TurkeyTUBITAK under Contract 108T688. The authors would like to thank the referees for their valuable comments and suggestions.
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