Oscillation of Second Order Sublinear Dynamic Equations with Damping on Isolated Time Scales

doi:10.1155/2010/103065

Research Article

Oscillation of Second-Order Sublinear Dynamic

Equations with Damping on Isolated Time Scales

Quanwen Lin

_{and Baoguo Jia}

1_{Department of Mathematics, Maoming University, Maoming 525000, China}

2_{School of Mathematics and Computer Science, Zhongshan University, Guangzhou 510275, China}

Correspondence should be addressed to Quanwen Lin,[email protected]

Received 8 October 2010; Accepted 27 December 2010

Academic Editor: M. Cecchi

Copyrightq2010 Q. Lin and B. Jia. 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.

This paper concerns the oscillation of solutions to the second sublinear dynamic equation with dampingxΔΔt qtxΔσt ptxα_{σt 0, on an isolated time scale}

Ìwhich is unbounded

above. In 0 < α < 1,αis the quotient of odd positive integers. As an application, we get the difference equationΔ2_{xn n}−γ_{Δxn1 1/nlnn}β _b−1n_/lnnβ _xα_{n1 0, where}

γ >0,β >0, andbis any real number, is oscillatory.

1. Introduction

During the past years, there has been an increasing interest in studying the oscillation of solution of second-order damped dynamic equations on time scale which attempts to harmonize the oscillation theory for continuousness and discreteness, to include them in one comprehensive theory, and to eliminate obscurity from both. We refer the readers to the

papers1–4 and the references cited therein.

In 5 , Bohner et al. consider the second-order nonlinear dynamic equation with

damping

xΔΔt qtxΔσt ptf◦xσt0, 1.1

wherepandqare real-valued, right-dense continuous functions on a time scale ⊂, with

sup ∞.f: → is continuously differentiable and satisfiesf

x /0. Whenfx xα_{, where 0< α <1,α >0 is the quotient of odd positive integers,1.1is}

the second-order sublinear dynamic equation with damping

xΔΔt qtxΔσt ptxασt 0. 1.2

Whenqt 0,1.2is the second-order sublinear dynamic equation

xΔΔt ptxασt 0. 1.3

When 0,1.3is the second-order sublinear difference equation

Δ2_{xn ptx}α_{n1 0. 1.4}

In6 , under the assumption of being an isolated time scale, we prove that, whenptis

allowed to take on negative values, _t1∞tα_{ptΔt ∞ is sufficient for the oscillation of the}

dynamic equation 1.3. As an application, we get that, when pn is allowed to take on

negative values,∞_n1nα_{pn ∞is sufficient for the oscillation of the dynamic equation}

1.4, which improves a result of Hooker and Patula7, Theorem 4.1 and Mingarelli8 .

In this paper, we extend the result of6 to dynamic equation1.1. As an application,

we get that the difference equation with damping

Δ2_{xn n}−γ_Δxn1

1

nlnnβb

−1n

lnnβ

xαn1 0, 1.5

where 0< α <1,γ >0,β >0, andbis any real number, is oscillatory.

For completenesssee9,10 for elementary results for the time scale calculus, we

recall some basic results for dynamic equations and the calculus on time scales. Let be a

time scalei.e., a closed nonempty subset ofwith sup ∞. The forward jump operator

is defined by

σt inf{s∈ :s > t}, 1.6

and the backward jump operator is defined by

ρt sup{s∈ :s < t}, 1.7

where sup∅ inf , where∅ denotes the empty set. Ifσt > t, we saytis right scattered,

while, ifρt< t, we saytis left scattered. Ifσt t, we saytis right dense, while, ifρt t

andt /inf , we saytis left-dense. Given a time scale intervalc, d Ì:{t∈ :c≤t≤d}in

the notationc, d κÌdenotes the intervalc, d Ìin caseρd dand denotes the interval

and for any functionf: → the notationf

σ_{tdenotesfσt. We say thatx: →}

is

differentiable att∈ provided

xΔt:lim

s→t

xt−xs

t−s 1.8

exists whenσt there, bys → t, it is understood thatsapproachestin the time scale

and whenxis continuous attandσt> t

xΔt:xσt−xt

μt . 1.9

Note that if , then the delta derivative is just the standard derivative, and when

the delta derivative is just the forward difference operator. Hence, our results contain the

discrete and continuous cases as special cases and generalize these results to arbitrary time scales e.g., the time scaleqÆ0

: {1, q, q2, . . .}which is very important in quantum theory

11 .

2. Lemmas

We will need the following second mean value theoremsee10, Theorem 5.45 .

Lemma 2.1. Let h be a bounded function that is integrable on a, b Ì. Let mH and MH be the infimum and supremum, respectively, of the functionHt : _athsΔs ona, b Ì. Suppose that

gis nonincreasing withgt≥0ona, b Ì. Then, there is some numberΛwithmH≤Λ≤MHsuch

that

_b

a

htgtΔtgaΛ. 2.1

Lemmas 2.2 and 2.4 give two lower bounds of definite integrals on time scale,

respectively.

Lemma 2.2. Assume that {ti}∞_i0, where1 t0 < t1 <· · ·< ti· · ·. If there exists a real number K≥1such thatti1−ti/ti−ti−1 K≥1, for alli≥1, then, forut>0,t≥T, one has

_t

T

uΔσs

uα_σsΔs≥ −

u1−α_σT

K1−α . 2.2

Remark 2.3. It is easy to know that, when ,K1 and, when q

Æ0

3. Main Theorem

Theorem 3.1. Assume that {t0, t1, . . . , tk. . .}, where1t0< t1 <· · ·< ti· · ·. Suppose that

ithere exists a real numberK≥1such thatti1−ti/ti−ti−1 K ≥1, for alli≥1;

iithere exists aC1

rdfunctionatsuch that fort∈T,∞,

at>0, aΔt≥0, aΔΔt≤0, qt≥0, qtaσtΔ≤0, _∞

T

aασspsΔs _∞

T

1

asΔs∞.

3.1

Then,1.1is oscillatory.

Proof. For the sake of contradiction, assume that1.1is nonoscillatory. Then, without loss of

generality, there is a solutionxtof1.1and aT ∈ withxt>0, for allt∈T,∞Ì. Making the substitutionxt atutin1.1and noticing that

xΔt aΔtuσt atuΔt,

xΔσt aΔσtuσ2taσtuΔσt,

xΔΔt aΔΔtuσt aΔσtuσtΔaΔtuΔt aσtuΔΔt,

3.2

we get that

aΔΔtuσt aΔσtuσtΔaΔtuΔt aσtuΔΔt

qtaΔσtuσ2tqtaσtuΔσt aασtptuασt 0.

3.3

Multiplying both sides of3.3by 1/uα_{σt, integrating fromTtot, and using an integration}

by parts formula, we get

_t

T

aΔΔsuσ_s uα_σs Δs

_t

T

aΔσssuσ_sΔ uα_σs Δs

_t

T

aΔsuΔs uα_σs Δs

asuΔs uα_s

t

T −

_t

T

as uα_s

Δ

uΔsΔs

_t

T

qsaΔσsuσ2_s

uα_σs Δs

_t

T

qsaσsuΔσs uα_σs Δs

t

T

aασspsΔs0.

sinceaΔt≥0, fort > T. So the first term of3.8is nonnegative. From3.8, we get that

Dividing both sides of this last inequality byatand integrating fromT1tot, we get, using

When ,qt 0, andat ρt, it is easy to get thata

Δ_{t 1,a}ΔΔ_{t 0. So we}

have the following corollarysee Corollary 2.4 of6 .Corollary 3.2shows that, with no sign

assumption onpn, the condition

∞

nαpn ∞ 3.14

is sufficient for the oscillation of the difference equation1.4.

Corollary 3.2. Assume that . If

∞

n1

nαpn ∞, 3.15

then1.4is oscillatory.

By using the idea inTheorem 3.1, we can also consider the differential equation

xt qtxt ptxαt 0, 0< α <1, 3.16

wherept, qt∈C1,∞,. It is easy to get the following.

Theorem 3.3. Suppose that there exists a functionat∈C2_{1,∞, Rsuch that}

at>0, qt≤0, at qtat≥0, at qtat≤0, _∞

1

1

atdt _∞

1

aαtptdt∞.

3.17

Then, the differential equation3.16is oscillatory.

Example 3.4. Consider the sublinear difference equation

Δ2_{xn n}−γ_Δxn1

1

nlnnβb

−1n

lnnβ

xαn1 0, 3.18

where 0< α <1,β >0,γ >0, andbis any real number. Takean lnn−1β/α,n >2. We have

an>0, Δan lnnβ/α−lnn−1β/α>0, forn >2,

Δ2_{an lnn1}β/α_−2lnnβ/α _lnn−1β/α_.

Letht lntβ/α. Then, we haveht<0, for larget. Sohtis concave for larget. Therefore,

3.18is oscillatory.

Example 3.5. Let qÆ

Example 3.6. Let 1,∞, and consider the differential equation

xt t−γ_xt

Acknowledgment

This work was supported by the National Natural Science Foundation of China no.

10971232.

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