Oscillation of Second-Order Mixed-Nonlinear Delay Dynamic Equations

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doi:10.1155/2010/389109

Research Article

Oscillation of Second-Order Mixed-Nonlinear

Delay Dynamic Equations

M. ¨

Unal

1

and A. Zafer

2

1_{Department of Software Engineering, Bahc¸es¸ehir University, Bes¸iktas¸, 34538 Istanbul, Turkey} 2_{Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey}

Correspondence should be addressed to M. ¨Unal,[email protected]

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.

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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

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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 ∈ C1_rd τa, b_Tis a positive function such that rtxΔt is nonincreasing on τa, b_Twith r > 0 nondecreasing, then

xτt xσ_t ≥

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

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and so xΔt ≤ xΔη, t ≥ η. Now

xt − xτa ≥ xΔtt − τa, t ∈ τa, b_T. 2.8

Define

μs : xs − s − τaxΔs, s ∈ τt, σt_T, t ∈ a, b_T. 2.9

It follows from2.8 that μs ≥ xτa > 0 for s ∈ τt, σt_Tand t ∈ a, b_T. Thus, we have 0 < σt τt μs xsxσ_sΔs  σt τt s − τa xs Δ Δs  σt − τa xσ_t − 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 caseH_R, 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 H2_{t, 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 H2_{t, 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, bl_T, l 1, 2, 2.13 where al min τjal : j 0, 1, 2, . . . , n 2.14

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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 H2c_ν_{, a}_ν cν aν QtH2σt, aν − rtH₁2t, aν Δt 1 H2_b_ν_{, c}_ν _b_ν 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, b1_T.

Define

wt −rtx

Δ_t

xt , t ≥ t1. 2.17

Using the delta quotient rule, we have

wΔt − rtxΔtΔxt − rtxΔt2 xtxσ_t − rtxΔtΔ xσ_t rtxΔt2 xtxσ_t . 2.18 Notice that xtxσt xt xt μtxΔt x2t 1− μtwt rt x2t rt rt − μtwt 2.19 which implies rt − μtwt rtx σ_t xt > 0. 2.20 Hence, we obtain wΔt − rtxΔtΔ xσ_t w2_t rt − μtwt. 2.21

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Substituting2.21 into 1.1 yields wΔt  p0txτ0t xσ_t w2_t rt − μtwt n i1 pit|xτit|αi−1xτit xσ_t − et xσ_t. 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, b1_T, 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 xσ_t ≥ τjt − τja1 σt − τja1, t ∈ a1, b1T 2.23

and taking into account2.22 yields

wΔt ≥ p0tτ0t − τ0a1 σt − τ0a1 w2_t rt − μtwt n i1 pkt τit − τia1 σt − τia1 αi xσ_tαi−1 |et| xσ_t. 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 2_t rt − μtwt n i1 Q∗_itxσ_tαi−1 |et| xσ_t. 2.26

Now recall the well-known arithmetic-geometric mean inequality, see 26,

n i0 uiηi≥ n i0 uηi_i , 2.27

where η0 1 −ni1ηiand ηi> 0, i 1, 2, . . . , n. Setting

u0η0 : |et| xσ_t, uiηi: Q∗itxσtαi−1 2.28 in2.26 yields wΔt ≥ Q∗0t w2_t rt − μtwt n i1 uiηi u0η0 Q∗0t w2_t rt − μtwt n i0 uiηi. 2.29

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From2.29 and taking into account 2.27, we get wΔt ≥ Q∗0t w2_t 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 ηi_xσ_tαi−1ηi  Q∗ 0t w2_t rt − μtwt η −η0 0 |et| η0 n i1 η−ηi i Q_i∗tηi_xσ_t−η0nj1αjηj−ηj  Q∗ 0t w2_t rt − μtwt η −η0 0 |et|η0 n i1 η−ηi i Q_i∗tηi 2.31 which yields wΔt ≥ Q∗₀t  w 2_t 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 H2_{σt, 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 w2_tH2_{σt, a} 1 rt − μtwt Δt. 2.34

Fix s and note that

wtH2t, sΔt H2σt, swΔt H2t, sΔtwt

 H2_{σt, sw}Δ_{t H}

1t, sHσt, swt Ht, sH1t, swt,

2.35

from which we obtain

H2_{σt, sw}Δ_t_wtH2_{t, s}Δt_{− H}

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Therefore, _c₁ a1 wΔtH2σt, a1Δt _c₁ a1 wtH2t, a1 _Δ_t Δt − _c₁ a1 H1t, a1Hσt, a1wt Ht, a1H1t, a1wtΔt. 2.37 Notice that _c₁ a1 wtH2t, a1 _Δ_t 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 w2_t rt − μtwtH 2_{σt, a} 1Δt _c₁ a1 H1t, a1Hσt, a1wt Ht, a1H1t, a1wtΔt. 2.39

On the other hand,

w2_tH2_{σt, 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

w2_tH2_{σt, 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

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Similarly, by following the above calculation step by step, that is, multiplying both sides of2.32 this time by H2_b

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_{, a}₁ c1 a1 QtH2σt, a1 − rtH₁2t, a1 Δt 1 H2_b 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, b1_T.

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,

−1l_e_{t ≥ 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 H2_c ν, aν _c_ν aν QtH2_{t, a} ν − rtH12t, aν dt 1 H2_b 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.

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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 H2_c ν, aν cν−1 taν QtH2_{t 1, a} ν − rtH12t, aν 1 H2_b 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_{, q}a11_{, . . . , q}b1∪_qa2_{, q}a21_{, . . . , q}b2_, −1l et ≥ 0 for t ∈qal_{, q}al1_{, . . . , q}bl , l 1, 2 2.52

where qal  min{τ_jqal : j 0, 1, 2, . . . , n} holds. Let {η₁_{, η}₂_{, . . . , η}_n} 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 H2_qcν_{, q}aν cν−1 maν qm QqmH2qm1, qaν − rqmH₁2qm, qaν 1 H2_qbν_{, q}cν bν−1 mcν qm QqmH2qbν_{, q}m1 − rqmH₂2qbν_{, q}m > 0 2.53

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for ν 1, 2, where H1 qm, qaν: H qm1_{, q}aν− H_qm_{, q}aν q − 1qm , H2 qbν_{, q}m : H qbν_{, q}m1− H_qbν_{, q}m 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, b_T

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 H2_{c, a} _c a QtH2_{σt, a − rtH}2 1t, a Δt 1 H2_{b, 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  w2_t rt − μtwt n i1 uiηi. 2.58

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The arithmetic-geometric mean inequality we now need is n i1 uiηi≥ n i1 uηi i , 2.59

where 1n_i1ηiand 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 H2_{c, a} c a QtH2t, a − rtH₁2t, adt 1 H2_{b, 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

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 H2_{c, a} c−1 ta QtH2_{t 1, a − rtH}2 1t, a 1 H2_{b, c} b−1 tc QtH2b, t 1 − rtH₂2b, t> 0, 2.64

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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

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 H2_qc_{, q}a c−1 ma qm QqmH2qm1, qa− rqmH1qm, qa 2 1 H2_qb_{, q}c b−1 mc qm QqmH2qb, qm1− rqmH2qb, qm 2 > 0, 2.67 where H1 qm, qa: H qm1_{, q}a_{− H}_qm_{, q}a q − 1qm , H2 qb, qm: H qb_{, q}m1_{− H}_qb_{, q}m 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

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, bl_T, l 1, 2,

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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 H2_c ν, aν cν aν QtH2_{σt, a} ν − rtH12t, aν Δt 1 H2_b ν, cν _b_ν 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/α_i 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, b1_Twhen 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

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where μi αiαi− 11/αi−1for i 1, 2, . . . , m. Setting wt −rtx Δ_t xt 2.76 yields wΔt − rtxΔtΔ xσ_t w2t rt − μtwt. 2.77

Substituting the above last equality into2.75, we have

wΔt ≥ p0txτ0t xσ_t m i1 μiλi|et|1−1/αip1/αi it xτit xσ_t 1 xσ_t n im1 pitxαiτit νi|et| w2t rt − μtwt. 2.78

It follows from2.5 that

xτ0t xσ_t ≥ τ0t − τ0a1 σt − τ0a1, 2.79 xτit xσ_t ≥ τit − τia1 σt − τia1, 2.80 xαiτ it xσ_t ≥ x αi−1τ itτit − τia1 σt − τia1. 2.81

Notice that the second sum in2.78 can be written as

1 xσ_t n im1 pitxαiτit νi|et| n im1 pitx αiτ it xσ_t νi|et| xσ_t n im1 τit − τia1 σt − τia1 νi|et| 1 xτit− pit 1 xτit _1−α_i , 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

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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/α_i it − n im1 τit − τia1 σt − τia1 βiνi|et|1−1/αipi1/αit  w 2_t 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 2_t 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,

−1l_e_{t > 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 H2_c_ν_{, a}_ν _c_ν aν QtH2t, aν − rtH12t, aν dt 1 H2_b_ν_{, c}_ν bν cν QtH2bν, t − rtH₂2bν, t dt > 0 2.89

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for ν 1, 2, where Qt p0tτ0t − τ0aν t − τ0aν m i1 μiλi|et|1−1/αip1/α_i 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 H2_c_ν_{, a}_ν cν−1 taν QtH2t 1, aν − rtH₁2t, aν 1 H2_b 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

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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_{, q}a11_{, . . . , q}b1 ∪qa2_{, q}a21_{, . . . , q}b2 , −1l_e_{t > 0 for t ∈}_qal_{, q}al1_{, . . . , q}bl , l 1, 2, 2.97

where qal  min{τ_jqal : j 0, 1, 2, . . . , n} holds. If there exist a function H ∈ H_q_{, positive numbers} λiand νisatisfying m i1 λi n im1 νi 1, 2.98

and numbers qcν ∈ {qaν1_{, q}aν2_{, . . . , q}bν−1} such that

1 H2_qcν, qaν cν−1 maν qm QqmH2qm1, qaν − rtH2 1 qm, qaν 1 H2_qbν, qcν bν−1 mcν qm QqmH2qbν_{, q}m1 − rtH2 2 qbν_{, q}m > 0 2.99 for ν 1, 2, where H1 qm_{, q}aν: H qm1_{, q}aν− H_qm_{, q}aν q − 1qm , H2 qbν_{, q}m : H qbν_{, q}m1− H_qbν_{, q}m q − 1qm , Qt p0t τ0t − τ0 qaν qt − τ0 qaν m i1 μiλi|et|1−1/αip_i1/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.

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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 1/3_C2/3 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

Δ2_x_{t 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

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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 − 210j2/3_{16t − 2}10j1/3. 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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