doi:10.1155/2010/160761
Research Article
Comparison Theorems for the Third-Order Delay
Trinomial Differential Equations
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 J. Dˇzurina,[email protected]
Received 11 August 2010; Accepted 1 November 2010
Academic Editor: E. Thandapani
Copyrightq2010 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 study the asymptotic properties of third-order delay trinomial differential equationyt ptyt gtyτt 0. Employing new comparison theorems, we can deduce the oscillatory and asymptotic behavior of the above-mentioned equation from the oscillation of a couple of the first-order differential equations. Obtained comparison principles essentially simplify the examination of the studied equations.
1. Introduction
In this paper, we are concerned with the oscillation and the asymptotic behavior of the solution of the third-order delay trinomial differential equations of the form
yt ptyt gtyτt 0. E
In the sequel, we will assume that the following conditions are satisfied:
ipt≥0,gt>0,
iiτt≤t, limt→ ∞τt ∞.
By a solution ofE, we mean a functionyt∈C1Tx,∞,Tx ≥t
0that satisfiesEon
Remark 1.1. All functional inequalities considered in this paper are assumed to hold eventually, that is, they are satisfied for alltlarge enough.
In the recent years, great attention in the oscillation theory has been devoted to the oscillatory and asymptotic properties of the third-order differential equationssee1–20. Various techniques appeared for the investigation of such equations. Some of them1,19
make use of the methods developed for the second-order equations 16, 17, 20 like the Riccati transformation and the integral averaging method and extend them to the third-order equations. Our method is based on the suitable comparison theorems.
Lazer12has shown that the differential equation without delay
yt ptyt gtyt 0 E
1
has always a nonoscillatory solution satisfying the condition
ytyt<0. 1.1
We say thatEhas the property P0 if every nonoscillatory solutionyt satisfies
1.1. In6–8,12, the first criteria forE1to have propertyP0appeared. Those criteria have
been improved in18. Dˇzurina3has presented a set of comparison theorems that enable us to extend the results known for E1to the delay equation E. This method has been further elaborated by Parhi and Padhi13,14and Dˇzurina and Kotorov´a5. In this paper, we present a new comparison method for the studying properties ofE. We will compare
Ewith a couple of the first-order delay differential equations in the sense that the oscillation of these equations yields the studied properties ofE.
2. Main Results
It will be derived that the properties ofEare closely connected with the positive solutions of the corresponding second-order differential equation
vt ptvt 0, V
as the following lemma says.
Lemma 2.1. Ifvtis a positive solution of V, thenEcan be written as the binomial equation
v2t
1
vt y
vtgtyτt 0. EC
Proof. Straightforward computation shows that
1
vt
v2t
1
vt yt
yt−vt
vt yt yt ptyt. 2.1
For our next consideration, it is desirable forECto be in a canonical form, that is, we
require
∞
v−2tdt
∞
vtdt∞. 2.2
It is clear that ifvtis a positive solution ofV, then the second integral in2.2is divergent. So, at first we will investigate the properties of the positive solutions ofV, and then we will be able to study the oscillation of the trinomial equationEwith, the help of its binomial representationEC.
The following resultsee, e.g.,4,10or11is a consequence of Sturm’s comparison theorem and guarantees the existence of a nonoscillatory solution.
Lemma 2.2. If
t2pt≤ 1
4 or lim supt→ ∞ t
2pt< 1
4, 2.3
thenVpossesses a positive solution. If
lim inf t→ ∞ t
2pt> 1
4 or t
2pt≥ 1
4 ε, ε >0, 2.4
then all solutions of Vare oscillatory.
We present some properties ofVthat will be utilized later.
Lemma 2.3. Assume that 2.3 is fulfilled, then V always possesses a nonoscillatory solution satisfying2.2.
Proof. Letv1tbe a positive solution ofV. Ifv1tdoes not accomplish2.2, then another
solution ofVis given by
v2t v1t
∞
t
v−2
1 sds, 2.5
indeed, because
v
2 v1
∞
t v −2
1 sds−ptv1
∞
t v −2
Moreover,v1tmeets2.2by now. Really, if we denoteUt
Picking up all the previous results, we can conclude by the following.
Corollary 2.4. Assume that2.3is fulfilled, then the trinomial equationEcan be always written in its binomial formEC. Moreover,ECis in the canonical form.
In the sequel, to be sure thatVpossesses a nonoscillatory solution, we will always assume that2.3holds.
Now, we are ready to study the properties ofEwith the help ofEC. Without loss
of generality, we can deal only with the positive solutions ofE. Since every solution ofE is also a solution ofEC, we are in view of a generalization of Kiguradze’s lemmasee4or 11in the following structure of the nonoscillatory solutions ofE.
Lemma 2.5. Assume that vt is a positive solution of V satisfying 2.2, then every positive
In the sequel, we will assume that the functionvtthat will be contained in our results is such solution ofVthat satisfies2.2. If we eliminate the solutions of degree 2ofE, we get the studied propertyP0ofE. The next theorem and its proof provide the details.
Theorem 2.6. If the first-order differential equation
Proof. Assume thatytis a positive solution ofE. It follows fromLemma 2.5thatytis either of degree 2or of degree 0. Ifytis of degree 2, then using thatzt v2t1/vtyt is decreasing, we are led to
1
Integrating fromt1tot, we obtain
yt≥
Or in other words,ztis a positive solution of differential inequality
zt
Remark 2.8. We note that ifEhas the propertyP0, then every positive solutionytsatisfies
D0, and then from the first two inequalities ofD0, we have the information only about the zero and the first derivative ofyt. We have no information about the second and the third derivatives, but on the other hand, we know the sign properties of the second and the third quasiderivatives ofyt.
Example 2.9. Consider the third-order trinomial equation of the form
yt α1−α
t2 y
t a
t3yλt 0, 2.13
with 0< λ <1, 0< α <1/2, anda >0. It is easy to see thatvt tαis the wanted solution of
V, and soE2reduces to
zt a
λ2−α
2−α1−2α 1
t O
t−22αzλt 0, 2.14
where in the function Ot−22α the terms unimportant for the oscillation of 2.14 are included. Applying the oscillation criterion from Corollary 2.7to 2.14, we see that2.13
has propertyP0provided that the parameterarealizes the following condition:
a λ2−α
2−α1−2αln
1
λ
> 1
e. 2.15
We note that for
aββ1β2βα1−αλβ, β >0, 2.16
one such solution isyt t−β.
Now, we turn our attention to oscillation ofE. We have known that oscillation ofE2 brings propertyP0ofE. If we eliminate also the caseD0ofLemma 2.5, we get oscillation ofE.
Theorem 2.10. Letτt>0. Assume that there exists a functionξt∈C1t
0,∞such that
ξt≥0, ξt> t, ηt τξξt< t. 2.17
If both the first-order delay equationsE2and
zt
vt ξt
t
v−2s
ξs
s
vxgxdxds
zηt0 E3
Proof. Assume thatytis a positive solution ofE. It follows fromLemma 2.5thatytis
Finally, integrating fromtto∞, one gets
yt≥ verify thatztis a solution of the differential inequality
zt
Then Theorem 1 in15shows that the corresponding differential equationE3has also a positive solution. This contradiction finishes the proof.
Applying the oscillation criterion from9toE2andE3, we obtain the sufficient condition forEto be oscillatory.
Corollary 2.11. Let τt > 0. Assume that there exists a function ξt ∈ C1t
Remark 2.12. There is an optional functionξtincluded inE3and conditionC2. There is no general rule for its choice. From the experience of the authors, we suggest to select such
ξtfor which the composite functionξ◦ξto be ”close to” the inverse functionτ−1tofτt.
In the next example, we provide the details.
Example 2.13. We consider2.13again. FollowingRemark 2.12, we setξt γt,1< γ <1/√λ, where these restrictions onγ result from2.17. Sincevt tα is a wanted solution ofV, thenE3reduces to
zt
1−γα−21−γ−α−1
2−α1α
a tz
λγ2t0. 2.23
Applying the oscillation criterion C2, we get in view of Corollary 2.11 that 2.13 is oscillatory provided thataverifies the following condition:
a
2−α1α
1−γα−21−γ−α−1ln
1
λγ2
> 1
e. 2.24
Obviously, we obtain the best oscillatory result if we choose suchγ ∈1,1/√λ, for which the function
fγ1−γα−21−γ−α−1ln
1
λγ2
2.25
attains its maximum. If we are not able to find the maximum value offγ, we simply put
γ 1√λ/2√λ, which is the middle point of the prescribed interval. In this case,2.24
takes the form
a
1−1√λ/2√λα−2
1−1√λ/2√λ−α−1
ln
4/1√λ2
2−α1α >
1 e.
2.26
Thus, it follows fromTheorem 2.10that2.13is oscillatory provided that2.26holds. Applying MATLAB, we can draw the graph offγwithα0.3,λ0.5 and verify that the maximum value offγis reached forγ1.24. On the other hand, the middleγ1.20.
Therefore, Theorems2.6and2.10imply that ifα0.3,λ0.5, and
a >1.1726, then2.13has the propertyP0,
a >41.3856, then2.13is oscillatory. 2.27
On the other hand, if we apply the middleγ, we get a bit weaker result for oscillation of
Remark 2.14. The oscillation ofEis a new phenomena in the oscillation theory. The previous results3,5,13do not help to study this case, because they are based on transferring the properties of the ordinary equation E1 to the delay equation E, and sinceE1 is not oscillatory, we cannot deduce oscillation ofEfrom that ofE1.
Our comparison method is based on the canonical representation EC of E.
Although the condition2.3ofLemma 2.2guarantees the existence of the wanted solution
vtofVso that canonical representationECis possible, a natural question arises; what to
do if we are not able to findvtbecause it is needed in the crucialE2andE3? In the next considerations, we crack this problem. Employing the additional condition, we revise both
E2andE3into the form that instead ofvtrequires its asymptotic representation which essentially simplifies our calculations.
We say thatv∗tis an asymptotic representation ofvtif limt→ ∞vt/v∗t 1. We denote this fact byvt∼v∗t.
The following result is recalled from2.
Theorem 2.15. If
∞
spsds <∞, 2.28
thenVhas a solutionvtwith the propertyvt∼1.
CombiningTheorem 2.15together with Corollaries2.7and2.11, we get new oscillatory criterion forE.
Theorem 2.16. Assume that2.28holds and
lim inf t→ ∞
t
τtgu
τu−t12
2 du > 1
e, C
∗
1
thenEhas the property (P0).
If, moreover,τt >0 and there exists a functionξt ∈ C1t
0,∞such that2.17holds
and
lim inf t→ ∞
t
ηt ξu
u ξs
s gxdxdsdu > 1
e, C∗2
thenEis oscillatory.
Proof. It follows fromTheorem 2.15that for anyC∈0,1, we have
C < vt< 1
1 1.05 1.1 1.15 1.2 1.25 1.3 1.35 1.4 1.45 0
0.002 0.004 0.006 0.008 0.01 0.012 0.014 0.016 0.018 0.02
γ
Max[α=0.3, λ=0.5] =0.019645 at γ=1.24,middleγ=1.2071
Figure1
eventually. Moreover,C1∗implies that there existsC∈0,1such that
1
e <lim inft→ ∞ C
4
t
τtgu
τu−t12
2 du
lim inf t→ ∞
t
τtCgu τu
t1 C s
t1 1
C−2dxdsdu
≤lim inf t→ ∞
t
τtvugu τu
t1 vs s
t1v
−2xdxdsdu,
2.30
where we have used2.29. We see thatC1holds andCorollary 2.7guarantees the property
P0ofE.
The proof of the second part runs similarly, and so it can be omitted.
Example 2.17. Consider the third-order trinomial equation of the form
yt α1−α
t3 yt
a
t3yλt 0, 2.31
with 0< λ <1, 0< α <1/2, anda >0. It is easy to see that2.28holds. Now,C1∗reduces to
aλ2
2 ln
1
λ
> 1
e, 2.32
On the other hand, settingξt γt, where 1< γ <1/√λ, the conditionC∗2takes the
that in view ofTheorem 2.16yields the oscillation of2.31.
3. Summary
In this paper, we have presented a new comparison principle for studying the oscillatory and asymptotic behavior of the third-order delay trinomial equationE. Our method essentially makes use of its binomial representationEC, which is based on the existence of the suitable
positive solution of the corresponding second-order equation V, so that we can deduce propertyP0or even oscillation ofEfrom the oscillation of a couple of the first-order delay
equationsE2andE3. Moreover, in a partial case, we can examine the studied properties ofEwithout finding a positive solution ofV. Obtained comparison theorems are easily applicable.
Acknowledgment
This research was supported by S.G.A. KEGA no. 019-025TUKE-4/2010.
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