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New oscillation criteria are established for second-order mixed-nonlinear delay dynamic equations on time scales by utilizing an interval averaging technique.. A function x is called an

Volume 2010, Article ID 389109, 21 pages

doi:10.1155/2010/389109

Research Article

Oscillation of Second-Order Mixed-Nonlinear

Delay Dynamic Equations

M ¨ Unal1 and A Zafer2

1 Department of Software Engineering, Bahc¸es¸ehir University, Bes¸iktas¸, 34538 Istanbul, Turkey

2 Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey

Received 19 January 2010; Accepted 20 March 2010

Academic Editor: Josef Diblik

Creative Commons Attribution License, which permits unrestricted use, distribution, andreproduction 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 thecoefficient functions and the forcing term to be nonnegative

We assume that the time scaleT is unbounded above, that is, sup T  ∞ and definethe time scale interval t0, ∞Tby t0, ∞T: t0, ∞ ∩ T It is also assumed that the reader is

already familiar with the time scale calculus A comprehensive treatment of calculus on timescales can be found in 1 3

By a solution of1.1 we mean a nontrivial real valued function x : T → R such that

x ∈ C1

rd T, ∞T and rxΔ ∈ C1

rd T, ∞T for all T ∈ T with T ≥ t0, and that x satisfies 1.1

A function x is called an oscillatory solution of 1.1 if x is neither eventually positive nor

eventually negative, otherwise it is nonoscillatory Equation1.1 is said to be oscillatory if

and only if every solution x of 1.1 is oscillatory

Notice that when T  R, 1.1 is reduced to the second-order nonlinear delaydifferential equation

Another useful time scale isT  qN: {qm : m ∈ N and q > 1 is a real number}, which leads

to the quantum calculus In this case,1.1 is the q-difference equation

as we know when T  R, an interval oscillation criterion for forced second-order lineardifferential equations was first established by El-Sayed 4 5

nicely how the interval criteria method can be applied to delay differential equations of theform

xt  pt|xτt| α−1 x τt  et, α ≥ 1, 1.6

where the potential p and the forcing term e may oscillate Some of these interval oscillation

criteria were recently extended to second-order dynamic equations in 6 10

on oscillatory and nonoscillatory behavior of the second order nonlinear dynamic equations

on time scales can be found in 11–23

Therefore, motivated by Sun and Meng’s paper 24

introduced in 17

criteria for second-order nonlinear delay dynamic equations of the form1.1 Examples areconsidered to illustrate the results

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 {α i }, i  1, 2, , n be the n-tuple satisfying α1 > α2 > · · · > α m > 1 > α m1 > · · · >

α n > 0 Then, there exists an n-tuple {η1, η2, , η n } satisfying

Lemma 2.2 Let {α i }, i  1, 2, , n be the n-tuple satisfying α1 > α2 > · · · > α m > 1 > α m1 > · · · >

α n > 0 Then there exists an n-tuple {η1, η2, , η n } satisfying

The next two lemmas are quite elementary via differential calculus; see 23,25

Lemma 2.3 Let u, A, and B be nonnegative real numbers Then

The last important lemma that we need is a special case of the one given in 6

completeness, we provide a proof

Lemma 2.5 Let τ : T → T be a nondecreasing right-dense continuous function with τt ≤ t, and

a, b ∈ T with a < b If x ∈ C1rd Tis a positive function such that rtxΔt is nonincreasing

on Twith r > 0 nondecreasing, then

which completes the proof.

In what follows we say that a function Ht, s : T2 → R belongs to HTif and only if

H is right-dense continuous function on {t, s ∈ T2: t ≥ s ≥ t0} having continuous Δ-partialderivatives on{t, s ∈ T2: t > s ≥ t0}, with Ht, t  0 for all t and Ht, s / 0 for all t / s Note

that in caseHR, theΔ-partial derivatives become the usual partial derivatives of Ht, s The

partial derivatives for the casesHZandHNwill be explicitly given later

Denoting theΔ-partial derivatives HΔt t, s and HΔs t, s of Ht, s with respect to t and s by H1t, s and H2t, s, respectively, the theorems below extend the results obtained

as in 5 2t, s instead of Ut, s for simplicity.

Theorem 2.6 Suppose that for any given (arbitrarily large) T ∈ T there exist subintervals a1, b1 T

and a2, b2 Tof T, ∞T, where a1< b1and a2< b2such that

hold Let {η1, η2, η n } be an n-tuple satisfying 2.1 of Lemma 2.1 If there exist a function H ∈ HT

and numbers c ν ∈ a ν , b νTsuch that

Proof Suppose on the contrary that x is a nonoscillatory solution of 1.1 First assume that

xt and xτ j t j  0, 1, 2 , n are positive for all t ≥ t1for some t1 ∈ t0, ∞T Choose a1

sufficiently large so that τj τ j a1 ≥ t1 Let t ∈ a1, b1 T

Substituting2.21 into 1.1 yields

From2.29 and taking into account 2.27, we get

Multiplying both sides of2.32 by H2σt, a1 and integrating both sides of the resulting

inequality from a1to c1, a1< c1< b1yield

Similarly, by following the above calculation step by step, that is, multiplying bothsides of2.32 this time by H2b1, σs after taking into account that

This contradiction completes the proof when xt is eventually positive The proof when xt

is eventually negative is analogous by repeating the above arguments on the interval a2, b2 T

where a l  min{τ j a l  : j  0, 1, 2, , n} holds Let {η1, η2, , η n } be an n-tuple satisfying 2.1

of Lemma 2.1 If there exist a function H ∈ HRand numbers c ν ∈ a ν , b ν  such that

Corollary 2.8 Suppose that for any given (arbitrarily large) T ≥ t0there exist a1, b1, a2, b2∈ Z with

T ≤ a1< b1and T ≤ a2< b2such that for each i  0, 1, 2, , n,

p i t ≥ 0 for t ∈ {a1, a1 1, a1 2, , b1} ∪ {a2, a2 1, a2 2, , b2},

−1l

e t ≥ 0 for t ∈ {a l , a l  1, a l  2, , b l } l  1, 2, 2.49

where a l  min{τ j a l  : j  0, 1, 2, , n} holds Let {η1, η2, , η n } be an n-tuple satisfying 2.1 of

Lemma 2.1 If there exist a function H ∈ HZand numbers c ν ∈ {a ν  1, a ν  2, , b ν − 1} such that

Corollary 2.9 Suppose that for any given (arbitrarily large) T ≥ t0there exist a1, b1, a2, b2∈ N with

T ≤ a1< b1and T ≤ a2< b2such that for each i  0, 1, 2, , n,

where q a l  min{τ j q a l  : j  0, 1, 2, , n} holds Let {η1, η2, , η n } be an n-tuple satisfying 2.1

of Lemma 2.1 If there exist a function H ∈ H q and numbers q c ν ∈ {q a ν1, q a ν2, , q b ν−1} such that

Notice thatTheorem 2.6does not apply if there is no forcing term, that is, et ≡ 0 In

this case we have the following theorem

T

of T, ∞T, where a < b such that

p i T, i  0, 1, 2, , n, 2.55

where a  min{τ j a : j  0, 1, 2, , n} holds Let {η1, η2, , η n } be an n-tuple satisfying 2.2 in

Lemma 2.2 If there exist a function H ∈ HTand a number c ∈ a, bTsuch that

then1.1 with et ≡ 0 is oscillatory.

Proof We will just highlight the proof since it is the same as the proof ofTheorem 2.6 We

should remark here that taking et ≡ 0 and η0 0 in proof ofTheorem 2.6, we arrive at

The arithmetic-geometric mean inequality we now need is

i1 η i and η i > 0, i  1, 2, , n are as inLemma 2.2

Corollary 2.11 Suppose that for any given (arbitrarily large) T ≥ t0there exists a subinterval

of T, ∞, where T ≤ a < b with a, b ∈ R such that

where a  min{τ j a : j  0, 1, 2, , n} holds Let {η1, η2, , η n } be an n-tuple satisfying 2.2 in

Lemma 2.2 If there exist a function H ∈ HRand a number c ∈ a, b such that

i1

η −η i

i , 2.62

then1.3 with et ≡ 0 is oscillatory.

Corollary 2.12 Suppose that for any given (arbitrarily large) T ≥ t0 there exists a, b ∈ Z with

T ≤ a < b such that

p i t ≥ 0 for t ∈ {a, a  1, , b}, i  0, 1, 2, , n, 2.63

where a  min{τ j a : j  0, 1, 2, , n} holds Let {η1, η2, , η n } be an n-tuple satisfying 2.2 in

Lemma 2.2 If there exist a function H ∈ HZand a number c ∈ {a  1, a  2, , b − 1} such that

η −η i

i , 2.65

then1.4 with et ≡ 0 is oscillatory.

Corollary 2.13 Suppose that for any given (arbitrarily large) T ≥ t0 there exist a, b ∈ N with

T ≤ a < b such that

p i t ≥ 0 for t ∈

q a , q a1 , , q b

, i  0, 1, 2, , n 2.66

where q a  min{τ j q a  : j  0, 1, 2, , n} holds Let {η1, η2, , η n } be an n-tuple satisfying 2.2

in Lemma 2.2 If there exist a function H ∈ H qN and a number q c ∈ {q a , q a1 , , q b } such that

i1

η −η i

i ,

2.68

then1.5 with et ≡ 0 is oscillatory.

It is obvious thatTheorem 2.6is not applicable if the functions p i t are nonpositive for i  m  1, m  2, , n In this case the theorem below is valid.

Theorem 2.14 Suppose that for any given (arbitrarily large) T ∈ T there exist subintervals a1, b1 T

and a2, b2 Tof T, ∞T, where a1< b1and a2< b2such that

p i t ≥ 0 for t ∈ a1, b1 T∪ a2, b2 T, i  0, 1, 2, , n,

−1l

e t > 0 for t ∈ a l , b l T, l  1, 2, 2.69

where a l  min{τ j a l  : j  0, 1, 2, , n} holds If there exist a function H ∈ HT, positive numbers

Proof Suppose that1.1 has a nonoscillatory solution Without losss of generality, we may

assume that xt and xτ i t i  0, 1, 2, , n are eventually positive on a1, b1 Twhen a1issufficiently large If xt is eventually negative, one may repeat the same proof step by step

where μ i  α i α i− 11/α i−1for i  1, 2, , m Setting

where β i  α i 1 − α i1/α i−1 and p i  max{−p i t, 0} for i  m  1, m  2, , n Using 2.79,

2.80, and 2.78 into 2.78, we obtain

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, b2 1< b1and T ≤ a2< b2such that

Corollary 2.16 Suppose that for any given (arbitrarily large) T ≥ t0 there exist a1, b1, a2, b2 ∈ Z

with T ≤ a1< b1and T ≤ a2< b2such that for each i  0, 1, 2, , n,

μ i  α i α i− 11/α i−1, β i  α i 1 − α i1/α i−1, p i max−p i t, 0, 2.96

then1.4 is oscillatory.

Corollary 2.17 Suppose that for any given (arbitrarily large) T ≥ t0there exist a1, b1, a2, b2 ∈ N

with T ≤ a1< b1and T ≤ a2< b2such that for each i  0, 1, 2, , n,

Example 3.3 Let A ≥ 0 and B, C > 0 be constants Define p0t  A, p1t  B and p2t  C for t  2 10jk , k  −3, −2, −1, 0, 1, 2, 3, j ≥ 1; otherwise, the functions are defined arbitrarily Consider the q-difference equation, q  2,

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