doi:10.1155/2010/389109
Research Article
Oscillation of Second-Order Mixed-Nonlinear
Delay Dynamic Equations
M. ¨
Unal
1and A. Zafer
21Department of Software Engineering, Bahc¸es¸ehir University, Bes¸iktas¸, 34538 Istanbul, Turkey
2Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey
Correspondence should be addressed to M. ¨Unal,[email protected] Received 19 January 2010; Accepted 20 March 2010
Academic Editor: Josef Diblik
Copyrightq2010 M. ¨Unal and A. Zafer. 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.
New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique. No restriction is imposed on the coefficient functions and the forcing term to be nonnegative.
1. Introduction
In this paper we are concerned with oscillatory behavior of the second-order nonlinear delay dynamic equation of the form
rtxΔtΔp0txτ0t n
i1
pit|xτit|αi−1xτit et, t≥t0 1.1
on an arbitrary time scaleT, where
α1 > α2 >· · ·> αm>1> αm1>· · ·> αn >0, n > m≥1; 1.2
the functionsr,pi,e:T → Rare right-dense continuous withr >0 nondecreasing; the delay functionsτi:T → Tare nondecreasing right-dense continuous and satisfyτit≤tfort∈T withτit → ∞ast → ∞.
By a solution of1.1we mean a nontrivial real valued functionx:T → Rsuch that
x ∈ C1
rd T,∞T andrxΔ ∈ C1rd T,∞T for all T ∈ TwithT ≥ t0, and thatxsatisfies1.1. A functionxis called an oscillatory solution of1.1ifxis neither eventually positive nor eventually negative, otherwise it is nonoscillatory. Equation1.1is said to be oscillatory if and only if every solutionxof1.1is oscillatory.
Notice that when T R, 1.1 is reduced to the second-order nonlinear delay differential equation
rtxtp0txτ0t n
i1
pit|xτit|αi−1xτit et, t≥t0 1.3
while whenTZ, it becomes a delay difference equation
ΔrkΔxk p0kxτ0k n
i1
pik|xτik|αi−1xτik ek, k≥k0. 1.4
Another useful time scale isTqN:{qm:m∈Nand q >1 is a real number}, which leads to the quantum calculus. In this case,1.1is theq-difference equation
Δq
rtΔqxt
p0txτ0t n
i1
pit|xτit|αi−1xτit et, t≥t0, 1.5
whereΔqft fσt−ft/μt,σt qt, andμt q−1t.
Interval oscillation criteria are more natural in view of the Sturm comparison theory since it is stated on an interval rather than on infinite rays and hence it is necessary to establish more interval oscillation criteria for equations on arbitrary time scales as inT R. As far as we know when T R, an interval oscillation criterion for forced second-order linear differential equations was first established by El-Sayed 4. In 2003, Sun 5 demonstrated nicely how the interval criteria method can be applied to delay differential equations of the form
xt pt|xτt|α−1xτt et, α≥1, 1.6
where the potentialpand the forcing termemay oscillate. Some of these interval oscillation criteria were recently extended to second-order dynamic equations in 6–10. Further results on oscillatory and nonoscillatory behavior of the second order nonlinear dynamic equations on time scales can be found in 11–23, and the references cited therein.
2. Main Results
We need the following lemmas in proving our results. The first two lemmas can be found in
25, Lemma 1.
Lemma 2.1. Let{αi},i1,2, . . . , nbe then-tuple satisfyingα1 > α2 >· · ·> αm>1> αm1>· · ·>
αn>0. Then, there exists ann-tuple{η1, η2, . . . , ηn}satisfying
n
i1
αiηi1, n
i1
ηi<1, 0< ηi <1. 2.1
Lemma 2.2. Let{αi},i1,2, . . . , nbe then-tuple satisfyingα1 > α2 >· · ·> αm>1> αm1>· · ·>
αn>0. Then there exists ann-tuple{η1, η2, . . . , ηn}satisfying
n
i1
αiηi1, n
i1
ηi1, 0< ηi <1. 2.2
The next two lemmas are quite elementary via differential calculus; see 23,25.
Lemma 2.3. Letu, A, andBbe nonnegative real numbers. Then
AuγB≥γγ−11/γ−1A1/γB1−1/γu, γ >1. 2.3
Lemma 2.4. Letu, A, andBbe nonnegative real numbers. Then
Cu−Duγ ≥γ−1γγ/1−γCγ/γ−1D1/1−γ, 0< γ <1. 2.4
The last important lemma that we need is a special case of the one given in 6. For completeness, we provide a proof.
Lemma 2.5. Letτ :T → Tbe a nondecreasing right-dense continuous function withτt≤t, and
a, b∈Twitha < b. Ifx∈C1rd τa, bTis a positive function such thatrtxΔtis nonincreasing on τa, bTwithr >0nondecreasing, then
xτt xσt ≥
τt−τa
σt−τa, t∈ a, bT. 2.5
Proof. By the Mean Value Theorem 2, Theorem 1.14
xt−xτa≥xΔηt−τa, 2.6
for some η ∈ τa, tT, for any t ∈ τa, bT. SincertxΔtis nonincreasing andrtis
nondecreasing, we have
and soxΔt≤xΔη,t≥η. Now
xt−xτa≥xΔtt−τa, t∈ τa, bT. 2.8
Define
μs:xs−s−τaxΔs, s∈ τt, σtT, t∈ a, bT. 2.9
It follows from2.8thatμs≥xτa>0 fors∈ τt, σtTandt∈ a, bT. Thus, we have
0<
σt
τt
μs
xsxσsΔs
σt
τt
s−τa xs
Δ
Δs σt−τa xσt −
τt−τa
xτt , 2.10
which completes the proof.
In what follows we say that a functionHt, s:T2 → Rbelongs toH
Tif and only if
His right-dense continuous function on{t, s∈T2:t≥s≥t
0}having continuousΔ-partial derivatives on{t, s∈T2:t > s≥t0}, withHt, t 0 for alltandHt, s/0 for allt /s. Note that in caseHR, theΔ-partial derivatives become the usual partial derivatives ofHt, s. The partial derivatives for the casesHZandHNwill be explicitly given later.
Denoting theΔ-partial derivativesHΔtt, sandHΔst, sofHt, swith respect tot
andsbyH1t, sandH2t, s, respectively, the theorems below extend the results obtained in 5to nonlinear delay dynamic equation on arbitrary time scales and coincide with them whenH2t, sis replaced byHt, s. Indeed, if we setHt, s Ut, s, then it follows that
H1t, s U1t, s
Uσt, s Ut, s, H2t, s
U2t, s
Ut, σs Ut, s. 2.11
WhenTR, they become
∂Ht, s
∂t
∂Ut, s/∂t
2Ut, s ,
∂Ht, s
∂s
∂Ut, s/∂s
2Ut, s 2.12
as in 5. However, we prefer usingH2t, sinstead ofUt, sfor simplicity.
Theorem 2.6. Suppose that for any given (arbitrarily large)T ∈Tthere exist subintervals a1, b1T
and a2, b2Tof T,∞T, wherea1< b1anda2< b2such that
pit≥0 for t∈ a1, b1T∪ a2, b2T, i0,1,2, . . . , n,
−1let>0 fort∈ al, blT, l1,2,
2.13
where
almin
τjal:j0,1,2, . . . , n
hold. Let{η1, η2, . . . ηn}be ann-tuple satisfying2.1ofLemma 2.1. If there exist a functionH∈ HT
and numberscν∈aν, bνTsuch that
1
H2cν, aν
cν
aν
QtH2σt, aν−rtH12t, aν
Δt
1
H2bν, cν
bν
cν
QtH2bν, σt−rtH22bν, t
Δt >0
2.15
forν1,2, where
Qt p0tτ0t−τ0aν
σt−τ0aν k0|et| η0
n
i1
pit
ηi
τit−τiaν
σt−τiaν αiηi
,
k0 n
i0
η−ηi
i , η01− n
i1
ηi,
2.16
then1.1is oscillatory.
Proof. Suppose on the contrary thatxis a nonoscillatory solution of1.1. First assume that
xtandxτjt j 0,1,2. . . , nare positive for allt≥t1for somet1 ∈ t0,∞T. Choosea1 sufficiently large so thatτjτja1≥t1. Lett∈ a1, b1T.
Define
wt −rtx
Δt
xt , t≥t1. 2.17
Using the delta quotient rule, we have
wΔt −
rtxΔtΔxt−rtxΔt2 xtxσt −
rtxΔtΔ xσt
rtxΔt2
xtxσt . 2.18
Notice that
xtxσt xtxt μtxΔtx2t
1−μtwt rt
x2t
rt
rt−μtwt 2.19
which implies
rt−μtwt rtx
σt
xt >0. 2.20
Hence, we obtain
wΔt −
rtxΔtΔ xσt
w2t
Substituting2.21into1.1yields
wΔt p0txτ0t xσt
w2t
rt−μtwt
n
i1
pit|xτit|αi−1xτit
xσt −
et
xσt. 2.22
By assumption, we can choosea1, b1 ≥ t1such that pit ≥ 0 i 1,2,3. . . , nandet ≤ 0 for allt∈ a1, b1T, wherea1 is defined as in2.14. Clearly, the conditions ofLemma 2.5are satisfied when,τ replaced withτjfor each fixedj 0,1,2, . . . , n. Therefore, from2.5, we have
xτjt
xσt ≥
τjt−τja1
σt−τja1, t∈ a1, b1T 2.23
and taking into account2.22yields
wΔt≥p0tτ0t−τ0a1
σt−τ0a1
w2t
rt−μtwt
n
i1
pkt
τit−τia1
σt−τia1 αi
xσtαi−1 |et| xσt.
2.24
Denote
Q∗0t:p0tτ0t−τ0a1
σt−τ0a1, Q
∗
it:pit
τit−τia1
σt−τia1 αi
. 2.25
From2.24, we have
wΔt≥Q∗0t w
2t
rt−μtwt
n
i1
Q∗itxσtαi−1 |et|
xσt. 2.26
Now recall the well-known arithmetic-geometric mean inequality, see 26,
n
i0
uiηi≥ n
i0
uηii , 2.27
whereη01−ni1ηiandηi>0,i1,2, . . . , n. Setting
u0η0 : |et|
xσt, uiηi:Q∗itxσtαi−1 2.28
in2.26yields
wΔt≥Q∗0t
w2t
rt−μtwt
n
i1
uiηiu0η0Q∗0t
w2t
rt−μtwt
n
i0
From2.29and taking into account2.27, we get
1and integrating both sides of the resulting inequality froma1toc1, a1< c1< b1yield
Fixsand note that
Therefore,
c1
a1
wΔtH2σt, a1Δt
c1
a1
wtH2t, a1
Δt
Δt
−
c1
a1
H1t, a1Hσt, a1wt Ht, a1H1t, a1wtΔt.
2.37
Notice that
c1
a1
wtH2t, a1
Δt
Δtwc1H2c1, a1−wa1H2a1, a1 wc1H2c1, a1 2.38
sinceHa1, a1 0 and hence, we obtain from2.34that
wc1H2c1, a1≥
c1
a1
QtH2σt, a1Δt
c1
a1
w2t
rt−μtwtH
2σt, a 1Δt
c1
a1
H1t, a1Hσt, a1wt Ht, a1H1t, a1wtΔt.
2.39
On the other hand,
w2tH2σt, s
rt−μtwt wtHσt, sH1t, s Ht, sH1t, swt
wtHσt, s
rt−μtwt
rt−μtwtH1t, s
2
−rt−μtwtH12t, s−wtHσt, sH1t, s Ht, sH1t, swt.
2.40
Taking into account thatHσt, s Ht, s μtH1t, s, we have
w2tH2σt, a 1
rt−μtwt wtHσt, a1H1t, a1 Ht, a1H1t, a1wt≥ −rtH
2 1t, a1.
2.41
Using this inequality in2.39, we have
wc1H2c1, a1≥
c1
a1
QtH2σt, a1−rtH12t, a1
Similarly, by following the above calculation step by step, that is, multiplying both
This contradiction completes the proof whenxtis eventually positive. The proof whenxt
is eventually negative is analogous by repeating the above arguments on the interval a2, b2T
instead of a1, b1T.
Corollary 2.7. Suppose that for any given (arbitrarily large)T ≥t0there exist subintervals a1, b1 and a2, b2of T,∞such that
Corollary 2.8. Suppose that for any given (arbitrarily large)T ≥t0there exista1, b1, a2, b2∈Zwith
T ≤a1< b1andT ≤a2< b2such that for eachi0,1,2, . . . , n,
pit≥0 fort∈ {a1, a11, a12, . . . , b1} ∪ {a2, a21, a22, . . . , b2},
−1let≥0 fort∈ {al, al1, al2, . . . , bl}l1,2,
2.49
wherealmin{τjal:j0,1,2, . . . , n}holds. Let{η1, η2, . . . , ηn}be ann-tuple satisfying2.1of Lemma 2.1. If there exist a functionH∈ HZand numberscν∈ {aν1, aν2, . . . , bν−1}such that
1
H2c ν, aν
cν−1
taν
QtH2t1, a
ν−rtH12t, aν
1
H2b ν, cν
bν−1
tcν
QtH2bν, t1−rtH22bν, t
>0
2.50
forν1,2, where
H1t, aν:Ht1, aν−Ht, aν, H2bν, t:Hbν, t1−Hbν, t,
Qt p0tτ0t−τ0aν
t1−τ0aν k0|et| η0
n
i1
pit
ηi
τit−τiaν
t1−τiaν αiηi
,
k0 n
i0
η−ηi
i , η01− n
i1
ηi,
2.51
then1.4is oscillatory.
Corollary 2.9. Suppose that for any given (arbitrarily large)T ≥t0there exista1, b1, a2, b2∈Nwith
T ≤a1< b1andT ≤a2< b2such that for eachi0,1,2, . . . , n,
pit≥0 fort∈
qa1, qa11, . . . , qb1∪qa2, qa21, . . . , qb2,
−1let≥0 fort∈qal, qal1, . . . , qbl
, l1,2
2.52
whereqal min{τjqal:j0,1,2, . . . , n}holds. Let{η1, η2, . . . , ηn}be ann-tuple satisfying2.1 ofLemma 2.1. If there exist a functionH∈ Hqand numbersqcν ∈ {qaν1, qaν2, . . . , qbν−1}such that
1
H2qcν, qaν
cν−1
maν
qmQqmH2qm1, qaν
−rqmH12qm,qaν
1
H2qbν, qcν
bν−1
mcν
qmQqmH2qbν, qm1
−rqmH22qbν, qm
>0
forν1,2, where
then1.5is oscillatory.
Notice thatTheorem 2.6does not apply if there is no forcing term, that is,et≡0. In this case we have the following theorem.
Theorem 2.10. Suppose that for any given (arbitrarily large)T ∈Tthere exists a subinterval a, bT
of T,∞T, wherea < bsuch that
The arithmetic-geometric mean inequality we now need is
n
i1
uiηi≥ n
i1
uηi
i , 2.59
where 1ni1ηiandηi >0,i1,2, . . . , nare as inLemma 2.2.
Corollary 2.11. Suppose that for any given (arbitrarily large)T ≥t0there exists a subinterval a, b of T,∞, whereT ≤a < bwitha, b∈Rsuch that
pit≥0 fort∈ a, b, i0,1,2, . . . , n, 2.60
whereamin{τja:j 0,1,2, . . . , n}holds. Let{η1, η2, . . . , ηn}be ann-tuple satisfying2.2in Lemma 2.2. If there exist a functionH∈ HRand a numberc∈a, bsuch that
1
H2c, a
c
a
QtH2t, a−rtH12t, adt
1
H2b, c
b
c
QsH2b, t−rtH22b, tdt >0,
2.61
where
Qt p0tτ0t−τ0a
t−τ0a k0 n
i1
pit
ηi
τit−τia
t−τia αiηi
, k0 n
i1
η−ηi
i , 2.62
then1.3withet≡0is oscillatory.
Corollary 2.12. Suppose that for any given (arbitrarily large) T ≥ t0 there exists a, b ∈ Zwith
T ≤a < bsuch that
pit≥0 fort∈ {a, a1, . . . , b}, i0,1,2, . . . , n, 2.63
whereamin{τja:j 0,1,2, . . . , n}holds. Let{η1, η2, . . . , ηn}be ann-tuple satisfying2.2in Lemma 2.2. If there exist a functionH∈ HZand a numberc∈ {a1, a2, . . . , b−1}such that
1
H2c, a c−1
ta
QtH2t1, a−rtH2 1t, a
1
H2b, c b−1
tc
QtH2b, t1−rtH22b, t>0,
where
Corollary 2.13. Suppose that for any given (arbitrarily large) T ≥ t0 there exista, b ∈ Nwith
T ≤a < bsuch that
It is obvious thatTheorem 2.6is not applicable if the functionspitare nonpositive forim1, m2, . . . , n. In this case the theorem below is valid.
Theorem 2.14. Suppose that for any given (arbitrarily large)T ∈Tthere exist subintervals a1, b1T
and a2, b2Tof T,∞T, wherea1< b1anda2< b2such that
pit≥0 fort∈ a1, b1T∪ a2, b2T, i0,1,2, . . . , n,
−1let>0 fort∈ al, blT, l1,2,
wherealmin{τjal:j0,1,2, . . . , n}holds. If there exist a functionH∈ HT, positive numbers
then1.1is oscillatory.
Proof. Suppose that1.1has a nonoscillatory solution. Without losss of generality, we may assume thatxtandxτit i0,1,2, . . . , nare eventually positive on a1, b1Twhena1is sufficiently large. Ifxtis eventually negative, one may repeat the same proof step by step on the interval a2, b2T.
and applyingLemma 2.3to each term in the first sum, we obtain
whereμiαiαi−11/αi−1fori1,2, . . . , m. Setting
Substituting the above last equality into2.75, we have
wΔt≥p0txτ0t
Notice that the second sum in2.78can be written as
1
and hence applying theLemma 2.4yields
whereβi αi1−αi1/αi−1 andpi max{−pit,0}fori m1, m2, . . . , n.Using2.79,
2.80, and2.78into2.78, we obtain
wΔt≥p0tτ0t−τ0a1
σt−τ0a1 m
i1
τit−τia1
σt−τia1
μiλi|et|1−1/αip1/αi it
− n
im1
τit−τia1
σt−τia1
βiνi|et|1−1/αipi1/αit w 2t
rt−μtwt.
2.84
Setting
Qt p0tτ0t−τ0a1
σt−τ0a1 m
i1
μiλi|et|1−1/αip1/αi it
τit−τia1
σt−τia1
− n
im1
βiνi|et|1−1/αipi1/αit
τit−τia1
σt−τia1 ,
2.85
we have
wΔt≥Qt w
2t
rt−μtwt. 2.86
The rest of the proof is the same as that ofTheorem 2.6and hence it is omitted.
Corollary 2.15. Suppose that for any given (arbitrarily large)T ≥t0there exist subintervals a1, b1 and a2, b2of T,∞, whereT ≤a1< b1andT ≤a2< b2such that
pit≥0 fort∈ a1, b1∪ a2, b2, i0,1,2, . . . , n,
−1let>0 fort∈ al, bl, l1,2,
2.87
wherealmin{τjal:j0,1,2, . . . , n}holds. If there exist a functionH∈ HR, positive numbers
λiandνisatisfying
m
i1
λi n
im1
νi1, 2.88
and numberscν∈aν, bνsuch that
1
H2cν, aν
cν
aν
QtH2t, aν−rtH12t, aν
dt
1
H2bν, cν
bν
cν
QtH2bν, t−rtH22bν, t
dt >0
forν1,2,where
Qt p0tτ0t−τ0aν
t−τ0aν m
i1
μiλi|et|1−1/αip1/αi it
τit−τiaν
t−τiaν
− n
im1
βiνi|et|1−1/αipi1/αit
τit−τiaν
t−τiaν
2.90
with
μiαiαi−11/αi−1, βiαi1−αi1/αi−1, pimax
−pit,0
, 2.91
then1.3is oscillatory.
Corollary 2.16. Suppose that for any given (arbitrarily large)T ≥ t0 there exista1, b1, a2, b2 ∈ Z withT ≤a1< b1andT ≤a2< b2such that for eachi0,1,2, . . . , n,
pit≥0 fort∈ {a1, a11, . . . , b1} ∪ {a2, a21, . . . , b2}
−1let>0 fort∈ {al, al1, . . . , bl}, l1,2,
2.92
wherealmin{τjal:j0,1,2, . . . , n}holds. If there exist a functionH∈ HZ, positive numbers
λiandνisatisfying
m
i1
λi n
im1
νi1, 2.93
and numberscν∈ {aν1, aν2, . . . , bν−1}such that
1
H2cν, aν cν−1
taν
QtH2t1, aν−rtH12t, aν
1
H2b ν, cν
bν−1
tcν
QtH2bν, t1−rtH22bν, t
>0
2.94
forν1,2, where
H1t, aν:Ht1, aν−Ht, aν, H2bν, t:Hbν, t1−Hbν, t,
Qt p0tτ0t−τ0aν
t1−τ0aν m
i1
μiλi|et|1−1/αip1/αi it
τit−τiaν
t1−τiaν
− n
im1
βiνi|et|1−1/αipi1/αit
τit−τiaν
t1−τiaν
with
then1.4is oscillatory.
Corollary 2.17. Suppose that for any given (arbitrarily large)T ≥ t0there exista1, b1, a2, b2 ∈ N
3. Examples
In this section we give three examples whenn2, andα1 2,α2 1/2 in1.1. That is, we consider
xΔΔt p0txτ0t p1t|xτ1t|xτ1t p2t|xτ1t|−1/2xτ2t 0. 3.1
For simplicity we takeHt, s t−s, thusH1t, s −H2t, s 1. Note thatη1 1/3 and
η22/3 byLemma 2.2.
Example 3.1. LetA≥0 andB, C >0 be constants. Consider the differential equation
xt Axt−1 B|xt−2|xt−2 C|xt−1|−1/2xt−1 0. 3.2
Letaj,bj2, andcj1,j∈N. We calculate
Qt A
t−j t−j1
3 3
√
4B 1/3
C2/3
t−j
t−j22/3t−j11/3 3.3
and see that2.61holds if
4A9BC21/3>27. 3.4
Since all conditions ofCorollary 2.11are satisfied, we conclude that3.2is oscillatory when
3.4holds.
Example 3.2. LetA≥ 0 andB, C >0 be constants. Definep0t A,p1t B, andp2t C fort 10jk,k −3,−2,−1,0,1,2,3,j ≥1; otherwise, the functions are defined arbitrarily. Consider the difference equation
Δ2xt p
0txt−1 p1t|xt−2|xt−2 p2t|xt−1|−1/2xt−1 0. 3.5
Leta10j,b10j3, andc10j1. We derive
Qt A t−10j t−10j2
3 3
√
4
BC21/3 t−10j
t−3j32/3t−10j41/3 3.6
and see that positivity in2.64satisfies if
A 9
BC21/3 4√3
5 > 48
5 . 3.7
Since all conditions ofCorollary 2.12are satisfied, we conclude that3.5is oscillatory if3.7
Example 3.3. LetA ≥0 andB, C >0 be constants. Definep0t A,p1t Bandp2t C fort 210jk,k −3,−2,−1,0,1,2,3,j ≥ 1; otherwise, the functions are defined arbitrarily. Consider theq-difference equation,q2,
Δ2
qxt p0tx
t
2 p1t
x
t
4 x
t
4 p2t
x
t
8
−1/2
x
t
8 0. 3.8
Leta10j,b10j3, andc10j1. We have
Qt At−2
10j
4t−210j 3 3
√
4
BC21/3 t−2
10j
8t−210j2/316t−210j1/3. 3.9
We see that2.67holds for allA≥ 0 andB, C >0. Since all conditions ofCorollary 2.12are satisfied, we conclude that3.8is oscillatory ifA≥0 andB, C >0 are positive.
Acknowledgments
The paper is supported in part by the Scientific and Research Council of TurkeyTUBITAK under Contract 108T688. The authors would like to thank the referees for their valuable comments and suggestions.
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