Báo cáo hóa học: " Research Article Iterated Oscillation Criteria for Delay Dynamic Equations of First Order" pdf

Karpuz,bkarpuz@aku.edu.tr 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 d

fference Equations

Volume 2008, Article ID 458687, 12 pages

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 ¨ Ocalan 2

1 Department of Economics and Finance, Missouri University of Science and Technology, Rolla,

MO 65409-0020, USA

2 Department of Mathematics, Faculty of Science and Arts, Afyon Kocatepe University,

ANS Campus, 03200 Afyonkarahisar, Turkey

Correspondence should be addressed to B Karpuz,bkarpuz@aku.edu.tr

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

Copyrightq 2008 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

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

where t ∈ R and τ ∈ R: 0, ∞, is oscillatory if

lim inf

t → ∞

t

t−τ

pηdη > 1

holds2, Theorem 2.3.1 Also the corresponding result for the difference equation

where t ∈ Z, Δxt xt  1 − xt and τ ∈ N, is

lim inf

t → ∞

t−1



η t−τ

pη >



τ

τ  1

τ1

1.4

lim inf

t → ∞ p n t > 1

where

p n t

⎧

⎪

⎪

t

t−τ

Note that1.2 is a particular case of 1.5 with n 1 Also a corresponding result of 1.4 for

1.3 has been given in 6, Corollary 1, which coincides in the discrete case with our main result as

lim inf

t → ∞ p n t >



τ

τ  1

nτ1

where p nis defined by a similar recursion in6, as

p n t

⎧

⎪

⎪

t−1



η t−τ

Now, we consider the first-order delay dynamic equation

where t ∈ T, T is a time scale i.e., any nonempty closed subset of R with sup T ∞,

p ∈ CrdT, R, the delay function τ : T → T satisfies lim t → ∞ τt ∞ and τt ≤ t for all

t ∈ T If T R, then xΔ xthe usual derivative, while if T Z, then xΔ Δx the usual

forward difference On a time scale, the forward jump operator and the graininess function are defined by

σt : inf t, ∞T, μt : σt − t, 1.10

wheret, ∞T : t, ∞ ∩ T and t ∈ T We refer the readers to 11,12 for further results on time scale calculus

A function f : T → R is called positively regressive if f ∈ CrdT, R and 1  μtft > 0 for all t ∈ T, and we write f ∈ RT It is well known that if f ∈ Rt0, ∞T, then there

exists a positive function x satisfying the initial value problem

where t0∈ T and t ∈ t0, ∞T, and it is called the exponential function and denoted by e f ·, t0

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

10, Lemma 2

Lemma 2.1 Let x be a nonoscillatory solution of 1.9 If

lim sup

t → ∞

t

τt

then

lim inf

where

y x t : xτt

Proof Since1.9 is linear, we may assume that x is an eventually positive solution Then, x is

eventually nonincreasing Let xt, xτt > 0 for all t ∈ t1, ∞T, where t1∈ t0, ∞T In view

of2.1, there exists ε > 0 and an increasing divergent sequence {ξn}n∈N ⊂ t1, ∞Tsuch that

σξ n

τξ npηΔη ≥

ξ n

τξ npηΔη ≥ ε ∀ n ∈ N0. 2.4

Now, consider the functionΓn:τξ n , σξ nT → R defined by

Γn t :

t

τξ npηΔη − ε

We see thatΓn τξ n  < 0 and Γ n ξ n  > 0 for all n ∈ N Therefore, there exists ζ n ∈ τξ n , ξ nT such that Γn ζ n ≤ 0 and Γn σζ n  ≥ 0 for all n ∈ N Clearly, {ζ n}n∈N ⊂ t1, ∞T is a

nondecreasing divergent sequence Then, for all n ∈ N, we have

σζ n

τξ npηΔη 2.5 ε

2 Γn σ ζ n

and

σξ n

ζ n pηΔη 2.5

σξ n

τξ npηΔη −



Γn ζ n

2



2− Γn ζ n

Thus, for all n ∈ N, we can calculate

x ζ n

≥ x ζ n

− x σ ξ n

1.9

σξ n

ζ n pηxτηΔη ≥ x τ ξ n

σξ n

ζ n pηΔη

2.7

2x τ ξ n

2

x τ ξ n

− x σ ζ n 1.9 ε

2

σζ n

τξ npηxτηΔη

2x τ ζ n

σζ n

τξ npηΔη 2.6≥



ε

2

2

x τ ζ n

,

2.8

y x ζ n

≤

 2

ε

2

Letting n tend to infinity, we see that 2.2 holds.

For the statement of our main results, we introduce

α n t :

⎧

⎪

⎪

inf

λ>0

−λpα n−1∈R τt,tT 



1

λe −λpα n−1 t, τt



for t ∈ s, ∞T, where τ n s ∈ t0, ∞T

Lemma 2.2 Let x be a nonoscillatory solution of 1.9 If there exists n0∈ N such that

lim inf

then

lim

where y x is defined in2.3

Proof Since 1.9 is linear, we may assume that x is an eventually positive solution Then,

x is eventually nonincreasing There exists t1 ∈ t0, ∞T such that xt, xτt > 0 for all

t ∈ t1, ∞T Thus, y x t ≥ 1 for all t ∈ t1, ∞T We rewrite1.9 in the form

for t ∈ t1, ∞T Integrating2.13 from t to σt, where t ∈ t1, ∞T, we get

0 xσt − xt  μty x tptxt > −xt1− μty x tpt , 2.14

which implies−y x p ∈ Rt1, ∞T From 2.13, we see that

xt x t1

e−y x p t, t1

∀t ∈t1, ∞

and thus

e−y x p t, τt ∀t ∈

t2, ∞

where τt2 ∈ t1, ∞T NoteRt1, ∞T ⊂ Rτt, ∞T ⊂ Rτt, tT for t ∈ t2, ∞T Now define

z n t :

⎧

⎨

⎩

inf

By the definition2.17, we have yx η ≥ z1t for all η ∈ τt, tTand all t ∈ t2, ∞T, which yields−z1tp ∈ Rτt, tT for all t ∈ t2, ∞T Then, we see that

y x t 2.16 1

e−y p t, τt

2.17

e−z tp t, τt

z1t

z1te −z tp t, τt

2.10

≥ α1tz1t 2.18

holds for all t ∈ t2, ∞Tsee also 13, Corollary 2.11 Therefore, from 2.13, we have

for t ∈ t2, ∞T Integrating2.19 from t to σt, where t ∈ t2, ∞T, we get

0≥ xσt − xt  μtz1tptα1txt > −xt1− μtz1tptα1t , 2.20

which implies that−z1pα1∈ Rt2, ∞T Thus, −z2tpα1∈ Rτt, tT for all t ∈ t3, ∞T,

where τt3 ∈ t2, ∞T, and we see that

y x t 2.16, 2.17≥ 1

e−z1pα1t, τt

2.17

e−z2tpα1t, τt

z2t

z2te −z2tpα1t, τt

2.10

≥ α2tz2t

2.21

for all t ∈ t3, ∞T By induction, there exists t n0 1∈ t n0, ∞Twith τt n0 1 ∈ t n0, ∞Tand

for all t ∈

t n0 1, ∞

T To prove now2.12, we assume on the contrary that lim inft → ∞ y x t <

∞ Taking lim inf on both sides of 2.22, we get

lim inf

t → ∞ y x t ≥ lim inf

t → ∞

z n0tα n0t

≥ lim inf

t → ∞ z n0tlim inf

t → ∞ α n0t

2.17

lim inf

t → ∞ y x tlim inf

t → ∞ α n0t,

2.23

which implies that lim inft → ∞ α n0t ≤ 1, contradicting 2.11 Therefore, 2.12 holds.

Theorem 2.3 Assume 2.1 If there exists n0 ∈ N such that 2.11 holds, then every solution of

1.9 oscillates on t0, ∞T.

Proof The proof is an immediate consequence of Lemmas2.1and2.2

We need the following lemmas in the sequel

Lemma 2.4 see 7, Lemma 2 For nonnegative p with −p ∈ Rs, tT, one has

1−

t

s

pηΔη ≤ e −p t, s ≤ exp



−

t

s

pηΔη



Now, we introduce

for n ∈ N and t ∈ s, ∞T, where τ n s ∈ t0, ∞T

Lemma 2.5 If there exists n0∈ N such that

lim sup

t → ∞

1

β n0t



α n0t



holds, then2.1 is true

Proof There exists t1∈ t0, ∞Tsuch that−pα n0 −1∈ Rt1, ∞T see the proof ofLemma 2.2

α n0t 2.10≤ 1

e−pα n0−1 t, τt ≤

1

1−t τt pηα n0 −1ηΔη

2.25

1− β n0tt

τt pηΔη , 2.27

which yields

t

τt

pηΔη ≥ 1

β n0t



α n0t



∀t ∈t1, ∞

In view of2.26, taking lim sup on both sides of the above inequality, we see that 2.1 holds Hence, the proof is done

Theorem 2.6 Assume that there exists n0∈ N such that 2.26 and 2.11 hold Then, every solution

of 1.9 is oscillatory on t0, ∞T.

Proof The proof follows from Lemmas2.1,2.2, and2.5

Remark 2.7 We obtain the main results of7,8 by letting n0 1 inTheorem 2.6 In this case,

we have β1t ≡ 1 for all t ∈ t0, ∞T Note that2.1 and 2.26, respectively, reduce tos

lim inf

t → ∞

α1t > 1, lim sup

t → ∞

which indicates that2.26 is implied by 2.1

3 Particular time scales

convenience, we set

p n t :

⎧

⎪

⎪

t

τt

Example 3.1 Clearly, if T R and τt t − τ, then 3.1 reduces to 1.6 and thus we have

α1t inf

λ>0



1

λ exp

− λp1t



ep1t,

α2t inf

λ>0



1

λ exp

− eλp2t



e2p2t

3.2

by evaluating2.10 For the general case, it is easy to see that

for n ∈ N Thus if there exists n0 ∈ N such that

lim inf

t → ∞ p n0t > 1

lim supt → ∞ p1t ≥ 1/e > 0 Otherwise, we have lim sup t → ∞ p n t < 1/e n for n 2, 3, , n0 This result for the differential equation 1.1 is a special case ofTheorem 2.3given inSection 2, and it is presented in3, Theorem 1, 4, Corollary 1, and 5, Corollary 1

Example 3.2 Let T Z and τt t − τ, where τ ∈ N Then 3.1 reduces to 1.8 From 2.10,

we have

λ>0

1−λpη>0

η∈t−τ,t−1Z

 1

λ

t−1

η t−τ

1 − λpη

−1

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

τ p1t

−τ



τ  1 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

λ>0

1−λpηα 1η>0

η∈t−τ,t−1Z

 1

λ



η t−τ

1 − λα1ηpη

−1

λ>0

1−λτ1/ττ1 p1ηpη>0

η∈t−τ,t−1Z

 1

λ

t−1

η t−τ



1− λ



τ  1 τ

τ1

p1ηpη

λ>0

1−λτ1/ττ1 p1ηpη>0

η∈t−τ,t−1Z

 1

λ

 1

τ

t−1



η t−τ



1− λ



τ  1 τ

τ1

p1ηpη

≥ inf

λ>0

 1

λ



1−λ

τ



τ  1 τ

τ1

p2t

−τ



τ  1 τ

2τ1

p2t.

3.6

By induction, we get

α n t ≥



τ  1 τ

nτ1

for n ∈ N Therefore, every solution of 1.3 is oscillatory on t0, ∞Zprovided that there exists

n0∈ N satisfying

lim inf

t → ∞ p n0t >



τ

τ  1

n0τ1

Note that3.8 implies that lim supt → ∞ p1t ≥ τ/τ  1 τ1

> 0 Otherwise, we would have

lim supt → ∞ p n t < τ/τ  1 nτ1 for n 2, 3, , n0 This result for the difference equation

1.3 is a special case ofTheorem 2.3given inSection 2, and a similar result has been presented

in6, Corollary 1

Example 3.3 Let T qN 0 : {qn : n ∈ N0} and τt t/q τ , where q > 1 and τ ∈ N This time

present case,3.1 reduces to

p n t

⎧

⎪

⎪

q − 1τ

η 1

t

q η p



t

q η



p n−1



t

q η



and the exponential function takes the form

e−p t, q −τ t

η 1



1− q − 1p



t

q η



t

q η



Therefore, one can show

λe −λp t, q −τ t

λτ

η 1



1− λq − 1p t

q η



t

q η



≤ λ



1−λq − 1

τ

τ



η 1

p



t

q η



t

q η

τ

≤



τ

τ  1

τ1 1

p1t

3.11

and

α1t ≥



τ  1 τ

τ1

For the general case, for n ∈ N, it is easy to see that

α n t ≥



τ  1 τ

nτ1

Therefore, if there exists n0∈ N such that

lim inf

t → ∞ p n0t >



τ

τ  1

n0τ1

then every solution of

xΔt  ptx



t

q τ



is oscillatory ont0, ∞ qN0 Clearly,3.14 ensures lim supt → ∞ p1t ≥ τ/τ  1 τ1

> 0 This

result for the q-difference equation 3.15 is a special case ofTheorem 2.3given inSection 2, and it has not been presented in the literature thus far

Example 3.4 Let T {ξ m : m ∈ N} and τξ m  ξ m−τ, where{ξ m}m∈Nis an increasing divergent

sequence and τ ∈ N Then, the exponential function takes the form

λe −λp ξ m , ξ m−τ

λ m−1

η m−τ

1− λ ξ η1 − ξ η

One can show that2.10 satisfies

α n ξ m

≥



τ

τ  1

nτ1

p n ξ m

p n ξ m

⎧

⎪

⎪

m−1

η m−τ

ξ η1 − ξ η

p ξ η

p n−1 ξ η

Therefore, existence of n0∈ N satisfying

lim inf

m → ∞ p n0 ξ m

>



τ

τ  1

n0τ1

3.19

xΔ ξ m

 p ξ m

x ξ m−τ

x ξ m1

− x ξ m

is oscillatory onξ τ , ∞T We note again that lim supm → ∞ p1ξ m  ≥ τ/τ  1 τ1 > 0 follows

from3.19

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holds for all t ∈ t2, ∞Tsee also 13, Corollary 2.11 Therefore,