Third-Order Differential Equation with Piecewise Constant Argument

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Journal of Applied Mathematics Volume 2012, Article ID 498073,18pages doi:10.1155/2012/498073

Research Article

Oscillation Criteria of Certain

Third-Order Differential Equation with Piecewise Constant Argument

Haihua Liang and Gen-qiang Wang

Department of Computer Science, Guangdong Polytechnic Normal University, Guangzhou 510665, Guangdong, China

Correspondence should be addressed to Haihua Liang,[email protected] Received 17 August 2012; Revised 27 October 2012; Accepted 20 November 2012 Academic Editor: Samir H. Saker

Copyrightq2012 H. Liang and G.-q. Wang. 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.

We study the oscillation and asymptotic behavior of third-order nonlinear delay diﬀerential equation with piecewise constant argument of the formr2tr1txtptxtft, xt 0. We establish several suﬃcient conditions which insure that any solution of this equation oscillates or converges to zero. Some examples are given to illustrate the importance of our results.

1. Introduction

Let · denote the greatest-integer function. Consider the following third-order nonlinear delay diﬀerential equation with piecewise constant argument:

r₂t

r₁txt

ptxt ft, xt 0, t≥0, 1.1

where r₁t, r2t are continuous on 0,∞ with r₁t, r2t > 0, pt is continuously diﬀerentiable on0,∞withpt≥0. We will show that every solutionxtof1.1oscillates or converges to zero, provided appropriate conditions are imposed.

Throughout this paper, we assume thatxft, x≥0 and that there exist functionsqt andφxsuch that

iqtis continuous on0,∞withqt>0,

iiφxis continuously diﬀerentiable and nondecreasing on−∞,∞,φx/x≥K >

0 forx /0,

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iii|ft, x| ≥qt|φx|, x /0, t≥0,

ivKqt−pt≥0 and is not identically zero in any subinterval of0,∞.

The delay functional differential equations provide a mathematical model for a physical or biological system in which the rate of change of the system depends upon its past history. In recent years, the oscillation theory and asymptotic behavior of delay functional differential equations and their applications have been and still are receiving intensive attention. In fact, in the last few years several monographs and hundreds of research papers have been written, see, for example1–4. In particular case, determining oscillation criteria for second-order delay differential equations has received a great deal of attention, while the study of oscillation and asymptotic behavior of the third-order delay differential equations has received considerably less attention in the literature.

The delay differential equations with piecewise continuous arguments can be looked as a special kind of delay functional differential equations. Since the delays of such equations are discontinuous, it need to be investigated individually. As is shown in5, the solutions of differential equations with piecewise continuous arguments are determined by a finite set of initial data, rather than by an initial function as in the case of general functional differential equations. Moreover, the strong interest in such equations is motivated by the fact that they represent a hybrid of continuous and discrete dynamical systems and combine the properties of both differential and difference equation.

In the last few decades, there has been increasing interest in obtaining sufficient conditions for the oscillatory of solutions of different classes of the first-order differential equations with piecewise constant arguments, see6–9and the references therein. It is found that the presence of piecewise constant arguments plays an important role in the oscillation of the solution. For instant, all solution ofx2xt 0 are oscillatorysee6. But the corresponding ordinary differential equationx2x0 has nonoscillatory solutionxe^−2t. However, there are few results about the oscillation of higher order equations. As mentioned in 5, 10, there are reasons for investigating the higher order equations with piecewise constant arguments. For example, suppose a moving particle with time variable massrtis subjected to a restoring controller−φxtwhich acts at sampled timet, then the second law of motion asserts that

rtxt

φxt 0. 1.2

In 10, the authors study a slightly more general second-order delay diﬀerential equations of the form:

rtxt

ft, xt 0, t≥0. 1.3

Two suﬃcient conditions which insure that any solution of1.3oscillates are obtained.

However, as far as we know, there are not works studying the oscillation and asymptotic behavior of third-order delay diﬀerential equations with piecewise constant argument. Motivated by this fact, in the present paper, we will investigate the oscillatory and asymptotic behavior of a certain class of third-order equation1.1with damping. The main ideas we used here are based on the paper3,4,10.

The rest of this paper is organized as follows. InSection 2we give the definition of the solution of1.1and establish some lemmas which are useful in the proof of our main results.

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Section 3is devoted to the presentation of several suﬃcient conditions for the oscillation of 1.1. InSection 4, several examples are given to illustrate the importance of our results.

2. Definitions and Preliminary Lemmas

Similar to11, we give the following definition.

Definition 2.1. A solution of1.1on0,∞is a functionxthat satisfies the conditions:

i r1txtis continuous on0,∞,

iir2tr1txtis diﬀerentiable at each pointt∈0,∞, with the possible exception of the pointst∈0,∞, where one-sided derivatives exist,

iiiEquation1.1is satisfied on each intervalk, k1⊂0,∞withk∈N.

Our attention is restricted to those solutions of1.1which exist on the half line0,∞ and satisfy sup{|xt| :T ≤ t < ∞} >0 for anyT > 0. We make a standing hypothesis that 1.1does possess such solutions. As usual, a solution of1.1is called oscillatory if it has arbitrarily large zeros and nonoscillatory otherwise. Equation1.1itself is called oscillatory if all its solutions are oscillatory.

Remark 2.2. Ifxis a solution of1.1, theny−xis a solution of the equation

r2t

r1tyt

ptyt f

t, yt

0, t≥0, 2.1

whereft, y −ft,−y. It is easy to check thatyft, y >0 and|ft, y| ≥ qt|φy|, where φy −φ−ywithφy/y ≥K. Thus, concerning nonoscillatory solutions of1.1, we can restrict our attention only to the positive ones.

For the sake of brevity, we denote the following two operators

L₁xt r₁txt, L₂xt r₂tL1xt. 2.2

Thus1.1can be written as

L2xtptxt ft, xt 0, t≥0. 2.3

Similar to12, we give the following definition.

Definition 2.3. Letxbe a solution of1.1. We say thatxhas propertyV₂onT,∞, T ≥0, if xtLkxt>0, i1,2 for everyt∈T,∞.

It is worth pointing out here that if x has property V2 on T,∞, then by 1.1, xtL2xtis eventually nonpositive.

Define the functions

Rit, T _t

T

ds

ris, i1,2. 2.4

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We assume that

R1t, T−→ ∞, ast−→ ∞, 2.5 R2t, T−→ ∞, ast−→ ∞. 2.6 To obtain our main results we need the following lemmas.

Lemma 2.4. Suppose that

r₁t

r₂tyt

ptyt 0 2.7

is nonoscillatory. Ifxis a nonoscillatory solution of1.1, thenxtdoes not change sign eventually.

Proof. Suppose that xt is a nonoscillatory solution of 1.1 on 0,∞. Without loss of generality, we may assume thatxt > 0 fort ≥ 0. Letytbe a nonoscillatory solution of 2.7. We will firstly consider the case thatytis eventually negative, that is, there exists a constantT₀such thatyt<0 fort≥T₀. By1.1and2.7, it is easy to see that

r₂t

r₁txt

yt−r₂tytr1txt

r₂t

r₁txt yt−

r₂tyt

r₁txt

−ytft, xk≥ −ytqtφxk, t∈k, k1, k∈N, k≥T0.

2.8

Suppose to the contrary that xt has arbitrarily large zeros, then there exist consecutive zeros ofxt, t1, andt2, such thatt2 > t1 > T0 and xt1 ≤ 0, xt2 ≥ 0. If there exists an integerk₁such that

k₁−1≤t₁< k₁< k₁1<· · ·< k₁m < t₂≤k₁m1, 2.9 seeFigure 1a, then integrating2.8we find

r₂t

r₁txt

yt

tt2

tt₁ r₂t

r₁txt

yt−r₂tr1txtyt tt2

tt₁

_t₂

t1

r₂t

r₁txt

yt−r₂tr1txtyt dt

_k₁

t1

r₂t

r₁txt

_k₁₁

k1

r2t

r1txt

yt−r2tr1txtyt dt

· · · _t₂

k1m

r₂t

r₁txt

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≥φxk1−1 _k₁

t1

ytqtdtφxk1 _k₁₁

k1

ytqtdt · · ·φxk1m

_t₂

k1m

ytqtdt >0.

2.10

Note thatr1txt|tti r1tixti i1,2, it follows fromxt1≤0, xt2≥0 that

r₂t

r₁txt

yt

tt2

tt₁ ≤0, 2.11

which is contrary to2.10.

If there exists an integer k₀ such thatk₀ ≤ t₁ < t₂ ≤ k₀ 1, seeFigure 1b, then integrating2.8directly fromt1tot2, we also get a contradiction.

Next we consider the case thatytis eventually positive. LetT₁be the constant such thatyt>0 fort≥T₁. It follows from1.1and2.7that,

r₂t

r₁txt

yt−r₂tytr1txt

≤ −ytqtφxk, 2.12

wheret∈k, k1withk ∈N andk≥T₁. Suppose to the contrary thatxthas arbitrarily large zeros, then there exist consecutive zeros ofxt,t3, andt4, such thatt4 > t3 > T1and xt3≥ 0, xt4 ≤0. Using the argument as above, we also arrive at a contradiction. Thus the proof is complete.

Remark 2.5. Lemma 2.4shows that, the oscillatory or asymptotic behavior of1.1is linked to nonoscillation of the second-order homogeneous equation2.7. The source of this interesting phenomenon can be explained as follows.

In1.1, if we letyr1x, thenyverifies the following equation:

r1t

r2tyt

ptyt r1tft, xt 0. 2.13

Let xt be any solution of 1.1. If xt is nonoscillatory, then r1tft, xt and thus r₁tr2tyt ptyt is eventually of one sign see 2.13. Hence, by the comparison methodas was shown by the proof ofLemma 2.4, the assumption that2.7 is nonoscillatory guarantees that

r₁t

r₂tyt

ptyt<0 or >0 2.14

is nonoscillatory. This fact means that, under the hypotheses ofLemma 2.4, any solutionxt of1.1is either oscillatory or is monotone.

Lemma 2.6. Suppose that assumption2.6is satisfied and thatxis a nonoscillatory solution of1.1 such thatxtL1xt>0 for everyt≥T ≥0. Thenxhas propertyV₂onT1,∞for someT₁.

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k1−1 t1 k1 k1+1 k1+m t2 k1+m+1 a

k0−1 k0 t1 t2 k0+1 k0+2

b

Figure 1: The relative position oft1andt2.

Proof. Assume without loss of generality thatxt>0, L₁xt>0 fort≥T. We assert firstly that for any integerk ≥ T,L2xk ≥ 0. If this is not true, then there exists an integer j ≥ T such thatL2xj:μ <0. By1.1, we obtain fort∈jn, jn1, n0,1,2, . . ., that

L2xt−ptxt−ft, xt<0. 2.15

This implies thatL₂xt≤L₂xj μ <0, t∈j, j1, and hence L2x

j1

L2x

j1₋

≤μ <0. 2.16

Using induction, we obtain thatL₂xjn≤L₂xj μ <0, n0,1,2, . . .. Thus L₂xt≤L₂x

jn

≤μ <0, t∈ jn, jn1

. 2.17

Integrating this inequality fromjntot, we find L1xt≤L1x

jn μR2

t, jn

, t∈ jn, jn1

. 2.18

Lettingt → jn1⁻, it follows from2.18and the continuity ofL₁xtthat L₁x

jn1

≤L₁x jn

μR₂

jn1, jn

. 2.19

Consequently,

L₁x

jm1 ^m

n0

L₁x

jn1

−L₁x jn

L₁x j

≤μ m n0

R₂

jn1, jn L₁x

j μR₂

jm1, j L₁x

j

−→ −∞, asm−→ ∞.

2.20

This contradicts thatL₁xt> 0, t ≥ T. Therefore,L₂xk ≥0 for any integerk ≥ T. By the continuity and monotonicity ofL2xt, we get thatL2xt≥0,t≥T. Finally, sinceL₂xt>0, we conclude thatL2xtis eventually positive.

This completes the proof.

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Lemma 2.7. Letxbe a solution of 1.1such thatxtxt<0, t≥T, whereTis a constant. If

tlim→ ∞xt λ /0, 2.21

then for any arbitrary constantδ >1, there exists an integerk0 ≥T such that for anyt > s≥k0, we have

λ < xs< δλ λ >0 or δλ < xs< λ λ <0,

−1^xt

L₂xt ptxt−L₂xs−psxs λ _t

s

Kqu−pu du

≥0. 2.22

Proof. Let us assume thatxtis eventually positive. The case whenxtis eventually negative can be similarly dealt with. For any constantδ >1, it follows from2.21andxt<0 that, there exists an integerk₀≥T such thatλ < xt< δλfort≥k₀. By1.1, we have

L2xt ptxt

ptxt−ft, xt≤ptxt−qtφxt, t∈k, k1, 2.23

wherekk0, k01, k02, . . .. Sincext> xtandφxis nondecreasing, it follows from 2.23that

L₂xt ptxt

≤xt

pt−qtφxt

xt

≤λ

pt−Kqt

, t∈k, k1. 2.24

Integrating2.24fromξtoζ, wherek≤ξ < ζ < k1, we have

L₂xζ pζxζ≤L₂xξ pξxξ λ _ζ

ξ

pu−Kqu

du. 2.25

If there exists an integerk₁such thatk₁≤s < t < k₁1, then2.22follows from2.25 directly. Otherwise, there exist an integerk₂such that

k2≤s < k21< k22<· · ·< k2m < t≤k2m1. 2.26

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Using2.25and the continuity ofL2xt, we have that

L₂xt ptxt−L₂xs−psxs

L₂xk21 pk21xk21−L₂xs−psxs ^m−1

n1

L2xk2n1 pk2n1xk2n1−L2xk2n−pk2nxk2n L2xt ptxt−L2xk2m−pk2mxk2m

≤λ _k₂₁

s

pu−Kqu duλ

m−1

n1

_k₂_n1

k2n

pu−Kqu du

λ _t

k2m

pu−Kqu du

λ _t

s

pu−Kqu du.

2.27

Thus we conclude that2.22holds for anyk0≤s < t.

3. Main Results

In this section we present some suﬃcient conditions which guarantee that every solution of 1.1oscillates or converges to zero. For convenience, we let

Φt _∞

t

Kqτ−pτ

dτ, t≥0. 3.1

Theorem 3.1. Suppose that i 2.7is nonoscillatory,

ii 2.5and2.6are satisfied, and for everyT≥0,

tlim→ ∞

_t

T

qsexp _s

T

puR2u, T r1u du

ds∞, 3.2

iiione of the following two conditions holds:

ΦT ∞ for someT ≥0, 3.3

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or

for everyt≥0, Φt<∞, and there exists a constant δ >1 such that lim

t→ ∞

_t

T

1 r₁s

_t

s

Φu−δpu r₂u du

ds∞. 3.4

Then any solutionxof 1.1is oscillatory or satisfiesxt → 0 ast → ∞.

Proof. Letxtbe a nonoscillatory solution of1.1. Without loss of generality, we may assume thatxt>0 fort≥0. ByLemma 2.4, there exists a constantT such that one of the following cases holds:

Case 1 : xt>0, xt>0, t≥T, 3.5 Case 2 : xt>0, xt<0, t≥T. 3.6

We will firstly show that Case 1 is impossible. Indeed, if this case holds, then it follows fromLemma 2.6thatxthas propertyV2onT1,∞for someT1≥T. We define

wt L2xt

φxt. 3.7

Clearly,wt>0,t≥T1. It follows from1.1that

wt −ptxt ft, xk

φxk ≤ −qt−pt L2xt

φxk· xt

L₂xt, t∈k, k1, 3.8

wherek T1 1, T1 2, . . .. Noting that

L1xt L1xT1

_t

T1

L2xu

r₂u du≥L2xtR2t, T1, t≥T1, 3.9

we get from3.8that

wt≤ −qt−ptR2t, T1

r1t wt, t∈k, k1. 3.10

Denoted by

Ψt wtexp t

T1

psR2s, T1 r₁s ds

, 3.11

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then we get from3.10that

Ψt≤ −qtexp _t

T1

psR2s, T1 r₁s ds

, t∈k, k1. 3.12

Integrating3.12, we have

Ψ

k1⁻

−Ψk≤ − _k1

k

qtexp _t

T1

psR₂s, T1 r1s ds

dt. 3.13

Using the increasing property ofxt, it follows that

Ψk1 L₂xk1

φxk1exp

_k1

T1

≤ L₂xk1

φxk exp

_k1

T1

Ψ

k1⁻ .

3.14

Thus

Ψkm Ψk ^nkm

nk1

Ψn−Ψn−1

≤Ψk ^nkm

nk1

Ψ n⁻

−Ψn−1

≤Ψk− _km

k

qtexp _t

T1

psR2s, T1 r1s ds

dt,

3.15

wherem 1,2, . . .. For fixedk, by lettingm → ∞, we get from3.2thatΨj< 0 for large integerj, which is contrary to the fact thatΨt>0, t≥T₁.

Therefore, we only need to consider Case 2:xt>0,xt<0,t≥T. We assert that

t→ ∞limxt λ0. 3.16

Suppose to the contrary thatλ >0. For any arbitrary constantδ >1, it follows fromLemma 2.7 that, there exists an integerk0 ≥Tsuch thatλ < xt< δλt≥k0and

L₂xt ptxt≤L₂xs psxs−λ _t

s

Kqu−pu

du, 3.17

wheret > s≥k₀.

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Now assume that3.3holds. Lettingsk0in3.17, we have

L2xt≤M−λ _t

k0

Kqu−pu

du, 3.18

whereMis a constant. By this inequality and3.3, there exists a negative constantμand a positive constantT1 ≥ k0 such thatL2xt < μfort ≥ T1. Integratingr1txt < μr₂⁻¹t fromT₁tottwice, we have

xt≤xT1 L₁xT1R1t, T1 μ _t

T1

R₂s, T1

r1s ds. 3.19

Thus we get from2.5thatxt<0 for larget, a contradiction.

Next, assume that3.4holds. We distinguish the following three subcases:iL₂xt≤ 0 for all larget,iiL2xt≥0 for all larget,iiiL2xtchanges sign for arbitrarily larget.

Caseiis equivalent to thatxt≤0 for all larget. Sincext<0, we conclude that xtis eventually negative, a contradiction.

In caseii, we get from2.22that

L2xs psxs≥λ _t

s

Kqu−pu

du, t > s≥T2≥k0, 3.20

whereT2is a constant such thatL2xt≥0 onT2,∞. Lettingt → ∞, we obtain that

L2xs δλps≥λΦs. 3.21

Integrating3.21fromstot, usingL1xt≤0, it turns out that

L1xs≤δλ _t

s

pu r₂udu−λ

_t

s

Φu

r₂udu. 3.22

Integrating again fromT₂tot, we find

xt≤xT2

_t

T2

δλ r₁s

_t

s

pu r₂udu

ds−

_t

T2

λ r₁s

_t

s

Φu r₂udu

ds. 3.23

By the condition3.4, we have thatxt<0 for all larget. This is a contradiction.

Finally, in caseiii, we let{tn}be the sequence of zeros ofL2xtsuch that limn→ ∞tn

∞. By choosingtt_nin2.22, we get

L₂xs psxs≥λ _t_n

s

Kqu−pu

du. 3.24

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Letn → ∞, it follows that

L₂xs δλps≥λΦs. 3.25

Integrating this inequality twice yields that

xtn≤xT0

_t_n

T0

δλ r₁s

_t_n

s

pu r₂udu

ds−

_t_n

T0

λ r₁s

_t_n

s

Φu r₂udu

ds, 3.26

which also leads to a contradiction.

Therefore, in Case 2 we conclude that3.16holds. This completes the proof of the theorem.

From the conclusion ofTheorem 3.1, we get the following corollary.

Corollary 3.2. Suppose that2.5,2.6hold and that2.7is nonoscillatory. Assume further that pt≤0 for larget. If_∞

T qtdt∞, then any solutionxof1.1is oscillatory or satisfiesxt → 0 ast → ∞.

Proof. Sincept≤0, it follows that

ΦT _∞

T

Kqτ−pτ dτ≥K

_∞

T

qsds∞. 3.27

Thus3.3is satisfied.

On the other hand, with the condition _∞

T qtdt ∞ we get 3.2 immediately.

Consequently, the expected conclusion follows fromTheorem 3.1directly.

We next consider the following equation which is diﬀerent from1.1:

r2t

r1txt

ptxt qtx^αt 0, t≥0, 3.28

or the more general equation

r₂t

r₁txt

ptxt ft, xt 0, t≥0, 3.29

with

ft, x≥qtφx, φx

x^α ≥K >0, qt>0, 3.30

whereα∈0,1∪1,∞is a quotient of odd integers. We note that the result ofTheorem 3.1 is not applicable. In what follows we give an oscillation criteria for3.29.

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Let

Φ1t _∞

t

K1qτ−pτ

dτ, t≥0. 3.31

Suppose thatK1qt−pt≥ 0 and not identically zero in any subinterval of0,∞for any K₁>0. Moreover, we need the following conditions:

Φ1T ∞, for someT ≥0, 3.32

or

for everyt≥0, Φ1t<∞, and there exists aδ >1, such that lim

t→ ∞

_t

T

1 r₁s

_t

s

Φ1u−δpu

r₂u du

ds∞. 3.33

Theorem 3.3. Suppose that i 2.7is nonoscillatory,

ii 2.5and2.6are satisfied, and for everyT≥0,

tlim→ ∞

_t

T

qsexp _s

T

puR2u, T r1u du

ds∞, 3.34

iiifor anyK1>0, one of 3.32and3.33is satisfied.

Then any solutionxof3.29is oscillatory or satisfiesxt → 0 ast → ∞.

Proof. From the proof ofTheorem 3.1, it suﬃces to verify in Case 2 that there exists a positive constantK₁such that

L₂xt ptxt≤L₂xs psxs−λ _t

s

K₁qu−pu

du, 3.35

for anyt > s≥k₀, wherek₀≥Tis the integer such thatλ < xt< δλ,t≥k₀. By3.29, we obtain that

L2xt ptxt

ptxt−ft, xt

≤xt

pt−qtφxt

xt

≤xt

pt−Kqtx^α−1t ,

3.36

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t∈k, k1, wherekk0, k01, k02, . . .. Ifα >1, then it follows from this inequality that L₂xt ptxt

≤λ

pt−Kλ^α−1qt

, 3.37

whileα <1, we have

L2xt ptxt

≤λ

pt−Kδλ^α−1qt

, 3.38

LetK₁Kλ^α−1ifα >1 andK₁Kδλ^α−1ifα <1, it turns out for the both cases that L₂xt ptxt

≤λ

pt−K₁qt

, t∈k, k1. 3.39

The rest of the proof is exactly the same as inTheorem 3.1and hence is omitted.

The following result follows fromTheorem 3.3directly.

Corollary 3.4. Suppose that2.5,2.6hold and that2.7is nonoscillatory. Assume further that pt ≤ 0 for large t. If_∞

T qtdt ∞, then any solutionxtof 3.29is oscillatory or satisfies xt → 0 ast → ∞.

Remark 3.5. In the literature dealing with the third-order delay diﬀerential equation, the Riccati transformation wt ρtL2xt/xgt, where gt is the delay and ρt is a diﬀerentiable positive function, is used widely, see3,4and the references therein. However, in our paper we find that the transformationwt L₂xt/φxt see3.7plays the same role as the more general one

wt ρtL2xt

φxt . 3.40

In fact, if we replace3.7by3.40in the proof ofTheorem 3.1, then it yields that

wt≤ −ρtqt ρt

ρt −ptR2t, T r₁t

wt. 3.41

Similarly toTheorem 3.1, we need the following conditioninstead of3.2

tlim→ ∞

_t

T

ρsqsexp

s T

−ρu

ρu puR2u, T r₁t du

ds∞ 3.42

to obtain a contradiction. Noting that

ρsexp

− _s

T

ρu

ρudu

ρsexp

− _ρs

ρT

dρ ρ

ρT, 3.43

we conclude that3.42is equivalent to3.2.

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Therefore, we cannot get a more general result by using the “more general”

transformation3.40. This is diﬀerent from the theory of function diﬀerential equation with the continuous delay.

4. Examples

In this section, we give several examples to illustrate our main results. For the convenience of readers, let us firstly recall the famous lemma of Kneser13. Consider the following second- order ordinary diﬀerential equation:

yt atyt 0, 4.1

whereatis a locally integrable function oft. Kneser13shows that4.1is nonoscillatory ifat≤ 1/4t²and is oscillatory ifat≥ 1ε/4t², whereεis an any arbitrary positive constant.

Example 4.1. Consider the third-order delay diﬀerential equation with piecewise constant argument

xt 3√ t

e^t17t²xt 1t²

x^αte^1txt0, t≥0, 4.2

whereα >0 is a quotient of odd integers. Clearly,pt 3√

t/e^t17t², ft, x t²x^αe^1tx. Letqt 1t²,φx x^α, then|ft, x| ≥qt|φx|. By the results of13,yt ptyt 0 is nonoscillatory. A simple calculation shows thatpt < 0 fort > 1. Therefore, it is easy to see fromCorollary 3.4that any solution of4.2is either oscillatory or converges to zero.

t1xt 1

5t5xt t1^−6/5xt

1e^xt

0, t≥0, 4.3

wherer1t t1, r2t 1, pt 5t5⁻¹, andft, x t1^−6/5x1e^x. Letqt t 1^−6/5, φx x, then|ft, x| ≥qt|φx|. It is easy to see that the Euler equation 5τ²zτ zτ 0 is nonoscillatory, hence 5t1²ytyt 0 is nonoscillatory. A simple calculation leads to

Φu _∞

u

qτ−pτ

dτ 5

u1^1/5 1 5u5, Φu−2pu≥ 4

5u1^1/5, and hence _t

s

Φu−2pu

r2u du≥1t^4/5−1s^4/5. 4.4

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Thus _t

T

1 r1s

_t

s

Φu−2pu

r2u du ds≥

ln1t−5

4 −ln1T

1t^4/5 5

41T^4/5−→ ∞, 4.5

ast → ∞.

On the other hand,

exp t

T

puR2u, T r₁u du

1

5 _t

T

s−T s1²ds 1

5 1T

1t ln1t−1−ln1T

, 4.6

which yields that

exp _t

T

puR2u, T r₁u du

ae^b1t⁻¹1t^1/5≥a1t^1/5, 4.7

wherea, bare positive constants. Hence _t

T

qsexp _s

T

puR2u, T r1u du

ds≥a

_t

T

1

1sds−→ ∞ ast−→ ∞. 4.8

Obviously, the other conditions ofTheorem 3.1are also satisfied. Hence we conclude that any solution of4.3is either oscillatory or converges to zero.

e^−t

e^−tx e^−2t

4 xt 3t 2sin 2tln

t1

√2

x³t 0, t≥0. 4.9

Herept e^−2t/4, ft, x 3t/2sin 2tlnt1/√

2x³. Letqt 3t/2sin 2tlnt 1/√

2, φx x³, then|ft, x|qt|φx|. Since the equatione^−te^−typtyt 0 can be reduced toy−y 1/4y0, we conclude that the former is nonoscillatory. It is easy to see thatpt<0 and_∞

0 qtdt∞. Therefore, ByCorollary 3.4, any solution of4.9is either oscillatory or converges to zero.

1 ln1t

2t3x 1

t²2t²xt

gt− 41t t³2t³

x^αt 0, t≥0, 4.10

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whereα >0 is a quotient of odd integersgtis a integrable function such that _∞

gtdt∞, gt> 41t

t³2t³ t >0. 4.11

Letr₁t 2t3, r2t 1/ln1t,pt 1/t²2t², andft, x gt−41t/t³2t³x^α. We will show that

r2tyt pt

r1tyt 0 4.12

is nonoscillatory. We introduce the change of variables:

sst

_t

0

ln1τdτ ₋₁

, yt s⁻¹zs, 4.13

which transforms4.12into

zs

ptr2t s⁴r₁t

tts

zs 0. 4.14

Note that

ptr2t s⁴r1t

sst _t

0ln1τdτ₄

2t3t²2t²ln1t, 4.15

and that

t²2t²4 t²

2 t 2

4 _t

0

1τdτ 2

≥4 _t

0

ln1τdτ 2

, 2t3ln1t>1, t≥2,

4.16

we obtain that

ptr2t s⁴r₁t

sst ≤ _t

0ln1τdτ₄ 4_t

0ln1τdτ2 1

4s². 4.17

Thus by Kneser13, it follows that4.14 and hence4.12is nonoscillatory.

Letqt gt−41t/t³2t³, then it is easy to find that_∞

qtdt∞. Therefore, byCorollary 3.4, any solution of4.10is either oscillatory or converges to zero.

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Acknowledgments

The authors would like to thank the referee very much for his valuable comments and suggestions. Liang was supported by the NSF of China no. 11201086 and Foundation for Distinguished Young Talents in Higher Education of Guangdong, China Grant 2012LYM 0087.

References

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in Delay Diﬀerential Equations and Dynamical Systems, vol. 1475 of Lecture Notes in Mathematics, pp. 1–15, Springer, Berlin, Germany, 1991.

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7 A. R. Aftabizadeh, J. Wiener, and J.-M. Xu, “Oscillatory and periodic solutions of delay diﬀerential equations with piecewise constant argument,” Proceedings of the American Mathematical Society, vol. 99, no. 4, pp. 673–679, 1987.

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Third-Order Differential Equation with Piecewise Constant Argument

Research Article

Oscillation Criteria of Certain

Third-Order Differential Equation with Piecewise Constant Argument

Haihua Liang and Gen-qiang Wang

1. Introduction

2. Definitions and Preliminary Lemmas

3. Main Results

4. Examples

Acknowledgments

References

Submit your manuscripts at http://www.hindawi.com

Mathematics

Complex Analysis

Optimization

Combinatorics

Function Spaces

The Scientific World Journal

Algebra

Decision Sciences

Discrete Mathematics

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