Volume 2010, Article ID 796256,9pages doi:10.1155/2010/796256
Research Article
New Oscillation Criteria for Second-Order Delay Differential Equations with Mixed Nonlinearities
Yuzhen Bai and Lihua Liu
School of Mathematical Sciences, Qufu Normal University, Qufu 273165, China
Correspondence should be addressed to Yuzhen Bai,baiyu99@126.com Received 10 January 2010; Accepted 17 June 2010
Academic Editor: Binggen Zhang
Copyrightq2010 Y. Bai and L. Liu. 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 establish new oscillation criteria for second-order delay differential equations with mixed nonlinearities of the formptxtn
i1pitxt−τin
i1qit|xt−τi|αisgnxt−τi et, t≥ 0, wherept,pit,qit, andetare continuous functions defined on0,∞, andpt>0,pt≥0, andα1>· · ·> αm>1> αm1>· · ·> αn>0. No restriction is imposed on the potentialspit,qit, andetto be nonnegative. These oscillation criteria extend and improve the results given in the recent papers. An interesting example illustrating the sharpness of our results is also provided.
1. Introduction
We consider the second-order delay differential equations containing mixed nonlinearities of the form
ptxt n
i1
pitxt−τi
n i1
qit|xt−τi|αisgnxt−τi et, t≥0. 1.1
In what follows we assume thatτi ≥ 0,p ∈ C10,∞,pt > 0, pt ≥ 0, pi, qi, e ∈ C0,∞,α1>· · ·> αm>1> αm1>· · ·> αn>0 n > m≥1,and
∞
0
1
ptdt∞. 1.2
As usual, a solutionxtof1.1is called oscillatory if it is defined on some rayT,∞ withT ≥0 and has arbitrary large zeros, otherwise, it is called nonoscillatory. Equation1.1 is called oscillatory if all of its extendible solutions are oscillatory.
Recently, Mustafa1has studied the oscillatory solutions of certain forced Emden- Fowler like equations
xt at|xt|λsgnxt et, t≥t0≥1. 1.3
Sun and Wong2, as well as Sun and Meng3have established oscillation criteria for the second-order equation
ptxt
qtxt n
i1
qit|xt|αisgnxt et, t≥0. 1.4
Later in4, Li and Chen have extended1.4to the delay differential equation ptxt
qtxt−τ n
i1
qit|xt−τ|αisgnxt−τ et, t≥0. 1.5
As it is indicated in2,3, further research on the oscillation of equations of mixed type is necessary as such equations arise in mathematical modeling, for example, in the growth of bacteria population with competitive species. In this paper, we will continue in the direction to study the oscillatory properties of1.1. We will employ the method in study of Kong in 5and the arithmetic-geometric mean inequalitysee6to establish the interval oscillation criteria for the unforced1.1and forced1.1, which extend and improve the known results.
Our results are generalizations of the main results in 3, 4. We also give an example to illustrate the sharpness of our main results.
2. Main Results
We need the following lemma proved in2,3for our subsequent discussion.
Lemma 2.1. For any givenn-tuple{α1, α2, . . . , αn}satisfyingα1 > α2 > · · · > αm > 1 > αm1 >
· · ·> αn>0,there corresponds an n-tuple{η1, η2, . . . , ηn}such that n
i1
αiηi1, a
which also satisfies either
n i1
ηi<1, 0< ηi<1, b
or
n i1
ηi1, 0< ηi<1. c
For a given set of exponents{αi}satisfyingα1> α2>· · ·> αm>1> αm1>· · ·> αn>0, Lemma2.1ensures the existence of ann-tuple{η1, η2, . . . , ηn}such that eitheraandbhold oraandchold. Whenn2 andα1 >1> α2>0,in the first case, we have
η1 1−α2
1−η0
α1−α2 , η2 α1
1−η0
−1
α1−α2 , 2.1
whereη0 can be any positive number satisfying 0 < η0 < α1−1/α1.This will ensure that 0< η1,η2<1, and conditionsaandbare satisfied. In the second case, we simply solvea andcand obtain
η1 1−α2
α1−α2, η2 α1−1
α1−α2. 2.2
Following Philos7, we say that a continuous functionHt, sbelongs to a function classDa,b,denoted byH ∈ Da,b,if Hb, s > 0, Hs, a > 0 forb > s > a,andHt, shas continuous partial derivatives∂Ht, s/∂tand∂Ht, s/∂sina, b×a, b.Set
h1t, s ∂Ht, s/∂t 2
Ht, s , h2t, s −∂Ht, s/∂s 2
Ht, s . 2.3
Based on Lemma2.1, we have the following interval criterion for oscillation of1.1.
Theorem 2.2. If, for anyT ≥ 0,there exista1,b1,c1,a2, b2 andc2 such thatT ≤ a1 < c1 < b1 ≤ a2< c2< b2,
pit≥0, t∈a1−τi, b1∪a2−τi, b2, i1,2, . . . , n, qit≥0, t∈a1−τi, b1∪a2−τi, b2, i1,2, . . . , n, et≤0, t∈a1−τi, b1, et≥0, t∈a2−τi, b2,
2.4
and there existHj∈ Daj,bj such that 1
Hj
cj, aj
cj
aj
QjsHj
s, aj
−psh2j1
s, aj ds
1 Hj
bj, cj
bj
cj
QjsHj
bj, s
−psh2j2
bj, s ds >0,
2.5
forj 1,2,wherehj1,hj2 are defined as in2.3,η1, η2, . . . , ηn are positive constants satisfyinga andbin Lemma2.1,η01−n
i1ηi,and
Qjt n
i1
pit
t−aj
t−ajτi
η0−1|et| η0n
i1
η−1i qit ηi
t−aj
t−ajτi
αiηi
, 2.6
then1.1is oscillatory.
Proof. Letxtbe a nonoscillatory solution of1.1. Without loss of generality, we may assume thatxt> 0 for allt ≥ t1−τ ≥ 0,wheret1depends on the solutionxtandτ max{τi}, i1, . . . , n.Whenxtis eventually negative, the proof follows the same argument by using the intervala2, b2instead ofa1, b1.Choosea1, b1 ≥t1such thatpit, qit≥0,et≤0 for t∈a1−τi, b1,andi1,2, . . . , n.
From1.1, we have thatxt≥0 fort∈a1−τi, b1.If not, there existst2∈a1−τi, b1 such thatxt2<0.Because
ptxt
≤0, 2.7
we haveptxt≤pt2xt2.Integrating fromt2tot, we obtain
xt≤xt2 pt2xt2 τ
t2
1
psds. 2.8
Noting the assumption1.2, we havext ≤ 0 for sufficient larget.This is a contradiction with xt > 0.From 2.7and the conditionspt > 0, pt ≥ 0, we obtainxt ≤ 0 for t∈a1−τi, b1.
Employing the convexity ofxt, we obtain
xt−τi
xt ≥ t−a1
t−a1τi, t∈a1, b1. 2.9
Define
ωt −ptxt
xt . 2.10
Recall the arithmetic-geometric mean inequality
n i0
ηiui≥n
i0
uηii, ui≥0, 2.11
whereη01−n
i1ηiandηi >0,i1,2, . . . , n, are chosen according to the givenα1, α2, . . . , αn
as in Lemma2.1satisfyingaandb. Let
u0t η−10 |et|, uit η−1i qitxt−τiαi. 2.12
We have
ωt −
ptxt
xt ω2t
pt
n
i1pitxt−τi
n
i1qitxt−τiαi−et
xt ω2t
pt
≥n
i1
pit
t−a1
t−a1τi
η−10 |et| η0 n
i1
η−1i qitηi
xαiηit−τi
xt ω2t
pt
n
i1
pit
t−a1
t−a1τi
η−10 |et| η0 n
i1
η−1i qitηi
xαiηit−τi n
i1xαiηit ω2t
pt
≥n
i1
pit
t−a1
t−a1τi
η−10 |et| η0n
i1
ηi−1qit ηi
t−a1
t−a1τi
αiηi
ω2t pt Q1t ω2t
pt .
2.13
Multiplying both sides of2.13byH1b1, t∈ Da1,b1and integrating by parts, we find that
−ωc1H1b1, c1≥ b1
c1
Q1sH1b1, s−psh212b1, s ds. 2.14
That is,
−ωc1≥ 1 H1b1, c1
b1
c1
Q1sH1b1, s−psh212b1, s ds. 2.15
On the other hand, multiplying both sides of 2.13 by H1t, a1 ∈ Da1,b1 and integrating by parts, we can easily obtain
ωc1≥ 1
H1c1, a1 c1
a1
Q1sH1s, a1−psh211s, a1 ds. 2.16
Equations2.15and2.16yield 1
H1c1, a1 c1
a1
Q1sH1s, a1−psh211s, a1 ds
1 H1b1, c1
b1
c1
Q1sH1b1, s−psh212b1, s ds≤0,
2.17
which contradicts2.5forj1.The proof of Theorem2.2is complete.
Remark 2.3. Whenτ1 · · · τn 0,Σni1pit qt,the conditionsqt≥0 fort ∈a1, b1∪ a2, b2,pt≥0 and1.2can be removed. Therefore, Theorem2.2reduces to Theorem 1 in 3.
Remark 2.4. Whenτ1· · ·τn τ,Σni1pit qt,Theorem2.2reduces to Theorem 1 in4 for which the conditionsqt≥0 fort∈a1−τ, b1∪a2−τ, b2, pt≥0 and1.2are needed.
There are some mistakes in the proof of Theorem 1 in4.
The following theorem gives an oscillation criterion for the unforced1.1.
Theorem 2.5. If, for anyT ≥ 0,there exista,b,andcsuch thatT ≤ a < c < b,pit ≥ 0,and qit≥0 fort∈a−τi, b, i1,2, . . . , n,and there existsH∈ Da,b,such that
1 Hc, a
c
a
Hs, aQs−psh21s, a ds 1 Hb, c
b
c
Hb, sQs−psh22b, s ds >0, 2.18 where
Qt n
i1
pit
t−a t−aτi
n
i1
ηi−1qit ηi
t−a t−aτi
αiηi
, 2.19
η1, η2, . . . , ηnare positive constants satisfyingaandcin Lemma2.1, andh1,h2are defined as in 2.3, then the unforced1.1is oscillatory.
Proof. Letxtbe a nonoscillatory solution of1.1. Without loss of generality, we may assume thatxt> 0 for allt ≥ t1−τ ≥ 0,wheret1depends on the solutionxtandτ max{τi}, i1, . . . , n.Similar to the proof in Theorem2.2, we can obtain
xt−τi
xt ≥ t−a
t−aτi, t∈a, b. 2.20 Define
ωt −ptxt
xt . 2.21
Recall the arithmetic-geometric mean inequality n
i1
ηiui≥n
i1
uηii, ui≥0, 2.22
whereηi >0,i1,2, . . . , n,are chosen according to the givenα1, α2, . . . , αnas in Lemma2.1 satisfyingaandc. Let
uiη−1i qitxt−τiαi. 2.23
We can obtain
ωt −
ptxt
xt ω2t
pt
n
i1pitxt−τi n
i1qitxt−τiαi
xt ω2t
pt
≥n
i1
pit
t−a t−aτi
n
i1
ηi−1qitηi
xαiηit−τi
xt ω2t
pt
n
i1
pit
t−a t−aτi
n
i1
ηi−1qitηi
xαiηit−τi n
i1xαiηit ω2t
pt
≥n
i1
pit
t−a t−aτi
n
i1
η−1i qit ηi
t−a t−aτi
αiηi
ω2t pt Qt ω2t
pt .
2.24
Multiplying both sides of2.24byHb, t∈ Da,band integrating by parts, we obtain b
c
Hb, tωtdt≥ b
c
Hb, tQtdt
b
c
Hb, tω2t pt dt,
−Hb, cωc≥ b
c
Hb, tQtdt
b
c
Hb, tω2t
pt −2ωth2b, t Hb, t
dt
b
c
Hb, tQt−pth22b, t dt b
c
Hb, t
pt ωt−
pth2b, t 2
dt
≥ b
c
Hb, tQt−pth22b, t dt.
2.25
It follows that
−ωc≥ 1
Hb, c b
c
Hb, tQt−pth22b, t dt. 2.26
On the other hand, multiplying both sides of2.24byHt, a∈ Da,band integrating by parts, we have
ωc≥ 1
Hc, a c
a
Ht, aQt−pth21t, a dt. 2.27
Equations2.26and2.27yield
1 Hc, a
c
a
Ht, aQt−pth21t, a dt 1 Hb, c
b
c
Hb, tQt−pth22b, t dt <0, 2.28
which contradicts2.24. The proof of Theorem2.5is complete.
Remark 2.6. Whenτ1 · · · τn 0,Σni1pit qt,the conditionsqt ≥ 0 fort ∈ a, b, pt≥0 and1.2can be removed. Therefore, Theorem2.5reduces to Theorem 2 in3.
Remark 2.7. Whenτ1· · ·τn τ,Σni1pit qt,Theorem2.5reduces to Theorem 2 in4 for which the conditionsqt≥0 fort∈a−τ, b,pt≥0 and1.2are needed.
3. Example
In this section, we provide an example to illustrate our results.
Consider the following equation:
xtksint x
t−π
8
α1sgnx
t−π 8
lcostx t−π
4 α2sgnx t−π
4 −mcos 2t, t≥0, 3.1
wherek,l,mare positive constants,α1>1, and 0< α2<1.Here
pt 1, p1t p2t 0, q1t ksint, q2t lcost, τ1 π
8, τ2 π
4, et −mcos 2t. 3.2
According to the direct computation, we have
Qjt k0|cos 2t|η0sintη1costη2
t−aj
t−ajτ1
α1η1 t−aj
t−ajτ2
α2η2
, j 1,2, 3.3
wherek0 η−10 /mη0η1−1/kη1η−12 /lη2, η0 can be any positive number satisfying 0< η0 <
α1−1/α1,andη1,η2satisfy2.1. For anyT ≥0,we can choose
a12iπ, a2 b12iππ
4, b22iππ
2, c12iππ
8, c22iπ3π 8 ,
3.4
fori0,1, . . . ,andH1t, s H2t, s t−s2.By simple computation, we obtainhj1t, s hj2t, s 1,j1,2.From Theorem2.2, we have that3.1is oscillatory if
2iππ/8
2iπ
Q1ss−2iπ2ds
2iππ/4
2iππ/8Q1s 2iππ
4 −s 2ds > π 4, 2iπ3π/8
2iππ/4 Q2s
s−2iπ−π 4
2ds
2iππ/2
2iπ3π/8Q2s
2iπ π
2 −s 2ds > π 4.
3.5
IfH1t, s H2t, s sin2t−s,by simple computation, we obtain hj1t, s hj2t, s cost−sforj 1,2.From Theorem2.2, we have that3.1is oscillatory if
2iππ/8
2iπ
Q1ss−2iπ2ds
2iππ/4
2iππ/8Q1s 2iππ
4 −s 2ds > π 16
√2 8 , 2iπ3π/8
2iππ/4 Q2s
s−2iπ−π 4
2ds
2iππ/2
2iπ3π/8Q2s
2iππ
2 −s 2ds > π 16
√2 8 .
3.6
Acknowledgments
The authors would like to thank the referees for their valuable comments which have led to an improvement of the presentation of this paper. This project is supported by the National Natural Science Foundation of China 10771118 and STPF of University in Shandong Province of ChinaJ09LA04.
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