On Properties of Third Order Differential Equations via Comparison Principles

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Volume 2012, Article ID 975298,10pages doi:10.1155/2012/975298

Research Article

On Properties of Third-Order Differential

Equations via Comparison Principles

B. Bacul´ıkov ´a and J. D ˇzurina

Department of Mathematics, Faculty of Electrical Engineering and Informatics, Technical University of Koˇsice, Letn´a 9, 042 00 Koˇsice, Slovakia

Correspondence should be addressed to B. Bacul´ıkov´a,[email protected]

Received 21 March 2012; Revised 18 June 2012; Accepted 19 June 2012

Academic Editor: Christian Corda

Copyrightq2012 B. Bacul´ıkov´a and J. Dˇzurina. 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.

The objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equationrtx_tγ_p_t_x_τ_t_{0, where studied equation}

is in a canonical form, that is,∞r−1/γsds ∞. Employing Trench theory of canonical operators, we deduce properties of the studied equations via new comparison theorems. The results obtained essentially improve and complement earlier ones.

1. Introduction

We are concerned with the oscillatory and asymptotic behavior of all solutions of the third-order functional diﬀerential equations:

rtx_tγ_ptxτ_t

0. E

In the sequel, we will assumer, τ, p∈Ct0,∞and

H1γis the ratio of two positive odd integers,

H2rt>0,pt>0, lim_{t→ ∞}τt ∞.

Moreover, we assume thatEis in a canonical form, that is,

_∞

t0

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By a solution of E we mean a function xt ∈ C1_Tx,_∞_,_Tx _≥ _t0_{, which has the}

property rtxtγ ∈ C2_T

x,∞ and satisfies E on Tx,∞. We consider only those

solutions xt ofEwhich satisfy sup{|xt| : t ≥ T} > 0 for all T ≥ Tx. We assume that

Epossesses such a solution. A solution ofEis called oscillatory if it has arbitrarily large zeros on Tx,∞and otherwise it is called to be nonoscillatory. Equation E is said to be

oscillatory if all its solutions are oscillatory.

Recently,Eand its particular casessee enclosed referenceshave been intensively studied. We establish new comparison theorems that permit to study properties ofEvia properties of the second-order diﬀerential equations, in the sense that the oscillation of the second-order equations yields desired properties ofE.

Our results complement and extend earlier ones presented in1–23.

Remark 1.1. All functional inequalities considered in this paper are assumed to hold eventually; that is, they are satisfied for alltlarge enough.

Remark 1.2. It is suﬃcient to deal only with positive solutions ofE.

2. Main Results

We begin with the classification of the possible nonoscillatory solutions ofE.

Lemma 2.1. Letxtbe a positive solution ofE. Thenxtsatisfies, eventually, one of the following

conditions:

Ixt<0,rtxtγ>0,rtxtγ<0;

IIxt>0,rtxtγ>0,rtxtγ<0.

Proof. The proof follows immediately from the canonical form ofE.

To simplify formulation of our main results, we recall the following definition:

Definition 2.2. We say thatEenjoys propertyAif all its positive solutions satisfy caseI

ofLemma 2.1.

PropertyAofEhas been studied by various authors; see enclosed references. We oﬀer new technique for investigation propertyAofEbased on comparison theorems and Trench theory of canonical operators.

Remark 2.3. It is known that condition

_∞

t0

psds ∞, 2.1

implies propertyAofE. Consequently, in the sequel, we may assume that the integral on the left side of2.1is convergent.

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Theorem 2.4. If the second-order diﬀerential inequality

1

ptzt

τt−t11/γ

r1/γ_τ_t τtz1/γτt≤0 E1

has not any solution satisfying

zt>0, zt<0,

1

ptzt

<0, P1

thenEhas property (A).

Proof. Assuming the contrary, let xt be a solution of E satisfying the Case II of

Lemma 2.1. Using the monotonicity ofrtxtγ, we see that

rtx_tγ_≥t

t1

rsx_sγ_d_s_≥_rt_x_tγ_t₋_t1. _2.2

Then evaluatingxtand then integrating fromt1tot, we are lead to

xt≥

_t

t1

s−t11/γ

r1/γ_s

rsx_sγ 1/γ _d_s. _2.3

Setting toE, we get

rtx_tγ_ptτt

t1

s−t11/γ

r1/γ_s

rsx_sγ 1/γ

ds≤0. 2.4

Integratingtto∞, we see thatyt rsxsγsatisfies

yt≥

_∞

t ps

_τ_s

t1

u−t11/γ

r1/γ_u y1/γududs. 2.5

Let us denote the right hand side of2.5byzt. ThenP1holds and moreover,

1

ptzt

τt−t11/γ

r1/γ_τt τty1/γτt 0. 2.6

Consequently, zt is a solution of the diﬀerential inequality E1, which contradicts our

assumption.

SinceE1 is in noncanonical form, we apply Trench theory 24to transform it to

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Denote

t

_∞

t psds. 2.7

Theorem 2.5. If the diﬀerential equation

2_t

pt yt

τt−t11/γ

r1/γ_τt τtt1/γτty1/γτt 0 E2

is oscillatory, thenE1has not any solution satisfyingP1.

Proof. Letztbe a positive solution ofE1, such thatP1holds. By direct computation, we

can verify Trench result that the operator

Lz

1

pt zt

2.8

is equivalent to

L z _t1

2_t

pt

_zt

t

. 2.9

Therefore the diﬀerential inequalityE1can be written in the form

2_t

pt

_zt

t

τt−t11/γ

r1/γ_τt τttz1/γτt≤0. 2.10

Applying the substitutiony z/, we can see thatyis a positive solution of the diﬀerential inequality

2_t

pt yt

τt−t11/γ

r1/γ_τt τtt1/γτty1/γτt≤0. 2.11

Moreover, since

_∞ _ps

2_sds ∞, 2.12

our inequality is in canonical form, but according to Theorem 2 of 18, we get that the corresponding diﬀerential equationE2 has also a positive solution. A contradiction. The

proof is complete.

Combining Theorems2.4and 2.5, we get the following criterion for propertyAof

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Theorem 2.6. If the second-order diﬀerential equationE2is oscillatory, thenEhas property (A).

Remark 2.7. We do not stipulate whether or notτ is a delayed or advanced argument. Using any oscillatory condition forE2, we obtain criteria for propertyAof third-order equation

E. We oﬀer several such results.

Theorem 2.8. Letγ >1 andτt≤t. If

_∞

t0

τ1/γ_s

r1/γ_τ_sτssds ∞, 2.13

Proof. ByTheorem 2.6, it is suﬃcient to prove thatE2is oscillatory. Assume the contrary,

that is, letytbe a positive solution ofE2. Then

y_t_>₀

2_t

pt yt

<0. 2.14

An integration ofE2fromtto∞leads to

2_ty_t

pt ≥

_∞

t

τs−t11/γ

r1/γ_τs τss1/γτsy1/γτsds

≥c

_∞

t

τ1/γ_s

r1/γ_τ_s τss1/γτsy1/γτsds,

2.15

wherec∈0,1. Integrating again fromt1toτt, we obtain

yτt≥c

_τ_t

t1

pv

2_v

_∞

v

τ1/γ_s

r1/γ_τ_s τss1/γτsy1/γτsdsdv

≥c

_τ_t

t1

pv

2_v

_∞

t

τ1/γ_s

r1/γ_τ_s τss1/γτsy1/γτsdsdv

c

_τ_t

t1

ps

2_s ds

_∞

t

τ1/γ_s

r1/γ_τs τss1/γτsy1/γτsds.

2.16

Let us denote

Ft

_∞

t

τ1/γ_s

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Then

y1/γ_τ_t

F1/γ_t ≥

c

_τ_t

t1

ps

2_s ds

_1/γ

≥ c11/γ

1/γ_τt. 2.18

Multiplying the previous inequality with τ1/γtr−1/γτtτtt1/γτt and then integrating fromt1tot, we have

c11/γt

t1

τ1/γ_s

r1/γ_τs τssds≤

_t

t1

−F_s

F1/γ_s ds≤

F1−1/γ_t1

1−1/γ . 2.19

Lettingtbe∞, we get a contradiction with2.13.

Example 2.9. Consider the third-order nonlinear delay diﬀerential equation

tx_t3 a

t2 xλt 0, Ex1

with a > 0, and 0 < λ < 1. It is easy to check that condition 2.13 is fulfilled and then

Theorem 2.8implies thatEx1enjoys propertyA.

Our results are new even forγ 1. Employing a generalization of Hille’s criterion10 for oscillation ofE2withγ 1, we get in view ofTheorem 2.6.

Theorem 2.10. Letτt≤tandτt>0. If

lim inf

t→ ∞

1

τt

_∞

t

τsτ_ssτs

rτs ds>

1

4, 2.20

Now, we are prepared to provide another criterion for property A based on the Riccati transformation.

Let us denote

Qt τ1/γt

r1/γ_τ_t τtt1/γτt. 2.21

Theorem 2.11. Letγ≥1 andτt≤t. If

_∞

t0

Qs

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and for somec >0

_∞

t0

Qs

s −c

p2_s11/γ_τ_s

3_spτ_sτ_s

ds ∞, 2.23

Proof. ByTheorem 2.6, it is suﬃcient to prove thatE2is oscillatory. Assume the contrary,

that is, letytbe a positive solution ofE2. Thenytsatisfies2.14. Since2t/ptyt

is decreasing, then there exists that

lim

t→ ∞

2_t

ptyt ≥0. 2.24

We claim that 0. If not, then it is easy to see that

yt≥

_t

t1

2_s

ps ys

ps

2s ds≥

_t

t1

ps

2sds >

2 1

t. 2.25

Setting the last inequality toE, we get

0≥

2_t

pt yt

τt−t11/γ

r1/γ_τt τtt1/γτty1/γτt

≥

2_t

pt yt

k

2

1/γ _Qt

1/γ_τt,

2.26

wherek∈0,1is arbitrary. Integrating the previous inequality fromt1tot, one gets

2_t1

pt1 yt1≥k

2

1/γt

t1

Qs

1/γ_τs ds. 2.27

Lettingtbe∞, we get a contradiction with2.22and we conclude that

lim

t→ ∞

2_t

pt yt 0. 2.28

On the other hand, it follows fromE2that for any constantk∈0,1, we have

2_t

pt yt

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We choose a positive constantc1, such thatc γc1−1/γ₁ /4k. It follows from2.28that

2_t

pt yt≤

c1

2, 2.30

eventually. Integrating fromt1tot, we obtain

yt≤yt1 c1

2

_t

t1

2_s

ps ds≤

c1

t, 2.31

or equivalently

y1/γ−1_τt_≥_c1/γ−1

1 1−1/γτt. 2.32

We set

wt

1/t2_t/pt_y_t

y1/γ_τt . 2.33

Thenwt>0 and

w_t pt

t wt

1/t2_t/pt_y_t

y1/γ_τt −

1

γ

y_τtτ_t

yτt wt, 2.34

which in view of2.29implies

w_t_≤ pt

t wt−k

Qt

t −

1

γ

y_τ_tτ_t

yτt wt. 2.35

It follows from2.14that

2_τt

pτt yτt>

2_t

pt yt. 2.36

Therefore

w_t_{≤ −k}Qt

t

pt

t wt−

τ_t

γ

2_t

2_τt

pτt

pt

y_t

yτt wt

−kQt_t pt_t wt−τ_γt t

2_τtpτty

1/γ−1_τtw2_t

≤ −kQt

t

pt

t wt−

c1/γ−1₁ τ_t

γ

t

2_τtpτt1−1/γτtw

2_t,

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where we have used2.32. Applying the inequalityAw−Bw2 _≤_A2_/₄_B_{, we are led to}

w_t_{≤ −k}Qt

t ck

p2_s11/γ_τs

3_spτsτ_s. 2.38

Integrating fromt1tot, we obtain in view of2.23

wt≤wt1−k

_t

t0

Qs

s −c

p2_s11/γ_τs

3_spτ_sτ_s

ds−→ −∞ 2.39

ast → ∞. A contradiction. The proof is complete now.

Example 2.12. Consider once more the third-order nonlinear delay diﬀerential equation

tx_t3 a

t2 xλt 0, Ex2

witha >0, and 0< λ <1. It is easy to check that condition2.22is fulfilled and the condition

2.22reduces to

_∞

t0

1

s1/3

a1/3_λ2/3₋_c 1

a2/3_λ1/3

ds ∞. 2.40

Choosingc aλ/2 the condition2.40holds true and thenTheorem 2.11implies thatEx2

enjoys propertyA.

In this paper, we have presented new comparison theorems for deducing propertyA ofEfrom the oscillation of the suitable second-order delay diﬀerential equation. Our results here generalize those presented for linear diﬀerential equations19,25.

Acknowledgment

Research is in this paper supported by S.G.A. KEGA 019-025TUKE-4/2010.

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