Volume 2008, Article ID 458687,12pages doi:10.1155/2008/458687
Research Article
Iterated Oscillation Criteria for Delay Dynamic Equations of First Order
M. Bohner,1 B. Karpuz,2 and ¨O. ¨Ocalan2
1Department of Economics and Finance, Missouri University of Science and Technology, Rolla, MO 65409-0020, USA
2Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University, ANS Campus, 03200 Afyonkarahisar, Turkey
Correspondence should be addressed to B. Karpuz,[email protected] Received 9 June 2008; Accepted 4 December 2008
Recommended by John Graef
We obtain new sufficient conditions for the oscillation of all solutions of first-order delay dynamic equations on arbitrary time scales, hence combining and extending results for corresponding differential and difference equations. Examples, some of which coincide with well-known results on particular time scales, are provided to illustrate the applicability of our results.
Copyrightq2008 M. Bohner et al. 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.
1. Introduction
Oscillation theory onZ and R has drawn extensive attention in recent years. Most of the results onZ have corresponding results onRand vice versa because there is a very close relation between Zand R. This relation has been revealed by Hilger in1, which unifies discrete and continuous analysis by a new theory called time scale theory.
As is well known, a first-order delay differential equation of the form
xt ptxt−τ 0, 1.1
wheret∈Randτ ∈R: 0,∞, is oscillatory if
lim inf
t→ ∞
t
t−τpηdη > 1
e 1.2
holds2, Theorem 2.3.1. Also the corresponding result for the difference equation
Δxt ptxt−τ 0, 1.3
wheret∈Z,Δxt xt1−xtandτ∈N, is
lim inf
t→ ∞
t−1 η t−τ
pη>
τ τ1
τ1
1.4
2, Theorem 7.5.1. Li3and Shen and Tang4,5improved1.2for1.1to
lim inf
t→ ∞ pnt> 1
en, 1.5
where
pnt
⎧⎪
⎪⎨
⎪⎪
⎩
1, n 0,
t
t−τpηpn−1ηdη, n∈N. 1.6 Note that1.2is a particular case of1.5withn 1. Also a corresponding result of1.4for 1.3has been given in6, Corollary 1, which coincides in the discrete case with our main result as
lim inf
t→ ∞ pnt>
τ τ1
nτ1
, 1.7
wherepnis defined by a similar recursion in6, as
pnt
⎧⎪
⎪⎨
⎪⎪
⎩
1, n 0,
t−1
η t−τpηpn−1η, n∈N. 1.8
Our results improve and extend the known results in7,8to arbitrary time scales. We refer the readers to9,10for some new results on the oscillation of delay dynamic equations.
Now, we consider the first-order delay dynamic equation
xΔt ptxτt 0, 1.9
where t ∈ T, T is a time scale i.e., any nonempty closed subset ofR with supT ∞, p ∈ CrdT,R, the delay functionτ :T → Tsatisfies limt→ ∞τt ∞andτt ≤ tfor all t∈T. IfT R, thenxΔ xthe usual derivative, while ifT Z, thenxΔ Δxthe usual
forward difference. On a time scale, the forward jump operator and the graininess function are defined by
σt: inft,∞T, μt: σt−t, 1.10
wheret,∞T : t,∞∩Tandt ∈T. We refer the readers to11,12for further results on time scale calculus.
A functionf :T → Ris called positively regressive iff∈CrdT,Rand 1μtft>0 for allt ∈ T, and we writef ∈ RT. It is well known that iff ∈ Rt0,∞T, then there exists a positive functionxsatisfying the initial value problem
xΔt ftxt, xt0 1, 1.11
wheret0∈Tandt∈t0,∞T, and it is called the exponential function and denoted by ef·, t0. Some useful properties of the exponential function can be found in11, Theorem 2.36.
The setup of this paper is as follows: while we state and prove our main result in Section 2, we consider special cases of particular time scales inSection 3.
2. Main results
We state the following lemma, which is an extension of3, Lemma 2and improvement of 10, Lemma 2.
Lemma 2.1. Letxbe a nonoscillatory solution of 1.9. If
lim sup
t→ ∞
t
τtpηΔη >0, 2.1
then
lim inf
t→ ∞ yxt<∞, 2.2
where
yxt: xτt
xt fort∈t0,∞T. 2.3
Proof. Since1.9is linear, we may assume thatxis an eventually positive solution. Then,xis eventually nonincreasing. Letxt, xτt>0 for allt∈t1,∞T, wheret1∈t0,∞T. In view of2.1, there existsε >0 and an increasing divergent sequence{ξn}n∈N⊂t1,∞Tsuch that
σξn
τξnpηΔη≥ ξn
τξnpηΔη≥ε ∀n∈N0. 2.4
Now, consider the functionΓn:τξn, σξnT → Rdefined by
Γnt: t
τξnpηΔη− ε
2. 2.5
We see thatΓnτξn<0 andΓnξn>0 for alln∈N. Therefore, there existsζn∈τξn, ξnT such that Γnζn ≤ 0 and Γnσζn ≥ 0 for all n ∈ N. Clearly, {ζn}n∈N ⊂ t1,∞T is a nondecreasing divergent sequence. Then, for alln∈N, we have
σζn
τξnpηΔη2.5 ε 2 Γn
σ ζn
≥ ε
2 2.6
and
σξn
ζn
pηΔη2.5 σξn
τξnpηΔη−
Γn
ζn
ε 2
≥ ε 2−Γn
ζn
≥ ε
2. 2.7
Thus, for alln∈N, we can calculate
x ζn
≥x ζn
−x σ
ξn
1.9σξn
ζn
pηxτηΔη≥x
τ ξn
σξn
ζn
pηΔη
2.7≥ ε 2x
τ ξn
≥ ε 2
x τ
ξn
−x σ
ζn
1.9 ε 2
σζn
τξnpηxτηΔη
≥ ε 2x
τ ζn
σζn
τξnpηΔη2.6≥ ε
2 2
x τ
ζn
,
2.8
and using2.3,
yx
ζn
≤ 2
ε 2
. 2.9
Lettingntend to infinity, we see that2.2holds.
For the statement of our main results, we introduce
αnt:
⎧⎪
⎪⎨
⎪⎪
⎩
1, n 0,
infλ>0
−λpαn−1∈Rτt,tT
1 λe−λpαn−1t, τt
, n∈N, 2.10
fort∈s,∞T, whereτns∈t0,∞T.
Lemma 2.2. Letxbe a nonoscillatory solution of 1.9. If there existsn0∈Nsuch that
lim inf
t→ ∞ αn0t>1, 2.11
then
tlim→ ∞yxt ∞, 2.12
whereyxis defined in2.3.
Proof. Since 1.9 is linear, we may assume thatxis an eventually positive solution. Then, xis eventually nonincreasing. There exists t1 ∈ t0,∞T such that xt, xτt > 0 for all t∈t1,∞T. Thus,yxt≥1 for allt∈t1,∞T. We rewrite1.9in the form
xΔt yxtptxt 0 2.13
fort∈t1,∞T. Integrating2.13fromttoσt, wheret∈t1,∞T, we get
0 xσt−xt μtyxtptxt>−xt
1−μtyxtpt
, 2.14
which implies−yxp∈ Rt1,∞T. From2.13, we see that xt x
t1
e−yxp t, t1
∀t∈ t1,∞
T, 2.15
and thus
yxt 1
e−yxpt, τt ∀t∈ t2,∞
T, 2.16
whereτt2 ∈ t1,∞T. NoteRt1,∞T ⊂ Rτt,∞T ⊂ Rτt, tTfort ∈ t2,∞T. Now define
znt:
⎧⎨
⎩
yxt, n 0,
inf
zn−1η:η∈τt, tT
, n∈N. 2.17
By the definition2.17, we haveyxη≥z1tfor allη∈τt, tTand allt∈t2,∞T, which yields−z1tp∈ Rτt, tTfor allt∈t2,∞T. Then, we see that
yxt2.16 1 e−yxpt, τt
2.17≥ 1 e−z1tpt, τt
z1t z1te−z1tpt, τt
2.10≥ α1tz1t 2.18
holds for allt∈t2,∞Tsee also13, Corollary 2.11. Therefore, from2.13, we have
xΔt z1tptα1txt≤0 2.19
fort∈t2,∞T. Integrating2.19fromttoσt, wheret∈t2,∞T, we get
0≥xσt−xt μtz1tptα1txt>−xt
1−μtz1tptα1t
, 2.20
which implies that−z1pα1∈ Rt2,∞T. Thus,−z2tpα1∈ Rτt, tTfor allt∈t3,∞T, whereτt3∈t2,∞T, and we see that
yxt2.16,≥2.17 1 e−z1pα1t, τt
2.17≥ 1 e−z2tpα1t, τt
z2t
z2te−z2tpα1t, τt
2.10≥ α2tz2t 2.21
for allt∈t3,∞T. By induction, there existstn01 ∈tn0,∞Twithτtn01∈tn0,∞Tand
yxt≥zn0tαn0t 2.22
for allt∈
tn01,∞
T. To prove now2.12, we assume on the contrary that lim inft→ ∞yxt<
∞. Taking lim inf on both sides of2.22, we get lim inf
t→ ∞ yxt≥lim inf
t→ ∞
zn0tαn0t
≥lim inf
t→ ∞ zn0tlim inf
t→ ∞ αn0t
2.17lim inf
t→ ∞ yxtlim inf
t→ ∞ αn0t,
2.23
which implies that lim inft→ ∞αn0t≤1, contradicting2.11. Therefore,2.12holds.
Theorem 2.3. Assume2.1. If there existsn0 ∈ Nsuch that2.11holds, then every solution of 1.9oscillates ont0,∞T.
Proof. The proof is an immediate consequence of Lemmas2.1and2.2.
We need the following lemmas in the sequel.
Lemma 2.4see7, Lemma 2. For nonnegativepwith−p∈ Rs, tT, one has
1− t
s
pηΔη≤e−pt, s≤exp
− t
s
pηΔη
. 2.24
Now, we introduce
βnt: sup
αn−1η:η∈τt, tT
2.25
forn∈Nandt∈s,∞T, whereτns∈t0,∞T. Lemma 2.5. If there existsn0∈Nsuch that
lim sup
t→ ∞
1 βn0t
1− 1
αn0t
>0 2.26
holds, then2.1is true.
Proof. There existst1∈t0,∞Tsuch that−pαn0−1∈ Rt1,∞T see the proof ofLemma 2.2.
Then,Lemma 2.4implies
αn0t 2.10≤ 1 e−pαn
0−1t, τt ≤ 1
1−t
τtpηαn0−1ηΔη
2.25≤ 1 1−βn0tt
τtpηΔη, 2.27
which yields
t
τtpηΔη≥ 1
βn0t
1− 1 αn0t
∀t∈ t1,∞
T. 2.28
In view of2.26, taking lim sup on both sides of the above inequality, we see that2.1holds.
Hence, the proof is done.
Theorem 2.6. Assume that there existsn0∈Nsuch that2.26and2.11hold. Then, every solution of 1.9is oscillatory ont0,∞T.
Proof. The proof follows from Lemmas2.1,2.2, and2.5.
Remark 2.7. We obtain the main results of7,8by lettingn0 1 inTheorem 2.6. In this case, we haveβ1t≡1 for allt∈t0,∞T. Note that2.1and2.26, respectively, reduce tos
lim inf
t→ ∞ α1t>1, lim sup
t→ ∞ α1t>1, 2.29
which indicates that2.26is implied by2.1.
3. Particular time scales
This section is dedicated to the calculation of αn on some particular time scales. For convenience, we set
pnt:
⎧⎪
⎪⎨
⎪⎪
⎩
1, n 0,
t
τtpn−1ηpηΔη, n∈N. 3.1 Example 3.1. Clearly, ifT Randτt t−τ, then3.1reduces to1.6and thus we have
α1t inf
λ>0
1 λexp
−λp1t
ep1t,
α2t inf
λ>0
1 λexp
−eλp2t
e2p2t
3.2
by evaluating2.10. For the general case, it is easy to see that
αnt enpnt 3.3
forn∈N. Thus if there existsn0 ∈Nsuch that
lim inf
t→ ∞ pn0t> 1
en0, 3.4
then every solution of 1.1 is oscillatory on t0,∞R. Note that 3.4 implies lim supt→ ∞p1t ≥1/e> 0. Otherwise, we have lim supt→ ∞pnt< 1/en forn 2,3, . . . , n0. This result for the differential equation1.1is a special case ofTheorem 2.3given inSection 2, and it is presented in3, Theorem 1,4, Corollary 1, and5, Corollary 1.
Example 3.2. LetT Zandτt t−τ, whereτ∈N. Then3.1reduces to1.8. From2.10, we have
α1t inf
λ>0 1−λpη>0 η∈t−τ,t−1Z
1 λ
t−1
η t−τ
1−λpη −1
≥ inf
λ>0 1−λpη>0 η∈t−τ,t−1Z
1 λ
1 τ
t−1 η t−τ
1−λpη −τ
≥inf
λ>0
1 λ
1−λ
τp1t
−τ τ1
τ τ1
p1t.
3.5
In the second line above, the well-known inequality between the arithmetic and the geometric mean is used. In the next step, we see that
α2t inf
λ>0 1−λpηα1η>0
η∈t−τ,t−1Z
1 λ
t−1
η t−τ
1−λα1ηpη −1
≥ inf
λ>0
1−λτ1/ττ1p1ηpη>0 η∈t−τ,t−1Z
1 λ
t−1
η t−τ
1−λ
τ1 τ
τ1
p1ηpη −1
≥ inf
λ>0
1−λτ1/ττ1p1ηpη>0 η∈t−τ,t−1Z
1 λ
1 τ
t−1 η t−τ
1−λ
τ1 τ
τ1
p1ηpη −τ
≥inf
λ>0
1 λ
1−λ
τ τ1
τ τ1
p2t
−τ τ1
τ
2τ1 p2t.
3.6
By induction, we get
αnt≥ τ1
τ
nτ1
pnt 3.7
forn∈N. Therefore, every solution of1.3is oscillatory ont0,∞Zprovided that there exists n0∈Nsatisfying
lim inf
t→ ∞ pn0t>
τ τ1
n0τ1
. 3.8
Note that3.8implies that lim supt→ ∞p1t≥τ/τ1τ1>0. Otherwise, we would have lim supt→ ∞pnt<τ/τ1nτ1forn 2,3, . . . , n0. This result for the difference equation 1.3is a special case ofTheorem 2.3given inSection 2, and a similar result has been presented in6, Corollary 1.
Example 3.3. LetT qN0 : {qn :n∈N0}andτt t/qτ, whereq >1 andτ ∈N. This time scale is different than the well-known time scalesRandZsincets/∈Tfort, s ∈ T. In the present case,3.1reduces to
pnt
⎧⎪
⎪⎨
⎪⎪
⎩
1, n 0,
q−1τ
η 1
t qηp
t qη
pn−1
t qη
, n∈N, 3.9
and the exponential function takes the form
e−p
t, q−τt τ
η 1
1−q−1p t
qη t
qη
. 3.10
Therefore, one can show
λe−λp t, q−τt
λ τ
η 1
1−λq−1p t qη
t qη
≤λ
1−λq−1 τ
τ η 1
p t
qη t
qη τ
≤ τ
τ1 τ1
1 p1t
3.11
and
α1t≥ τ1
τ τ1
p1t. 3.12
For the general case, forn∈N, it is easy to see that
αnt≥ τ1
τ
nτ1
pnt. 3.13
Therefore, if there existsn0∈Nsuch that
lim inf
t→ ∞ pn0t>
τ τ1
n0τ1
, 3.14
then every solution of
xΔt ptx t
qτ
0, wherexΔt xqt−xt
q−1t , 3.15
is oscillatory ont0,∞qN0. Clearly,3.14ensures lim supt→ ∞p1t≥τ/τ1τ1 > 0. This result for theq-difference equation3.15is a special case ofTheorem 2.3given inSection 2, and it has not been presented in the literature thus far.
Example 3.4. LetT {ξm:m∈N}andτξm ξm−τ, where{ξm}m∈Nis an increasing divergent sequence andτ ∈N. Then, the exponential function takes the form
λe−λp
ξm, ξm−τ λ
m−1
η m−τ
1−λ
ξη1−ξη
p ξη
. 3.16
One can show that2.10satisfies
αn
ξm
≥ τ
τ1 nτ1
pn
ξm
, 3.17
where3.1has the form
pn
ξm
⎧⎪
⎪⎨
⎪⎪
⎩
1, n 0,
m−1
η m−τ
ξη1−ξη
p ξη
pn−1 ξη
, n∈N. 3.18
Therefore, existence ofn0∈Nsatisfying
lim inf
m→ ∞ pn0
ξm
>
τ τ1
n0τ1
3.19
ensures byTheorem 2.3that every solution of
xΔ ξm
p ξm
x ξm−τ
0, where xΔ ξm
x ξm1
−x ξm
ξm1−ξm , 3.20
is oscillatory onξτ,∞T. We note again that lim supm→ ∞p1ξm≥τ/τ1τ1>0 follows from3.19.
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