Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities

doi:10.1155/2009/938706

Research Article

Oscillation Criteria for Second-Order Forced Dynamic Equations with Mixed Nonlinearities

Ravi P. Agarwal

and A. Zafer

1Department of Mathematical Sciences, Florida Institute of Technology, Melbourne, FL 32901, USA

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

Correspondence should be addressed to A. Zafer,zafer@metu.edu.tr Received 21 January 2009; Accepted 19 April 2009

Recommended by Mariella Cecchi

We obtain new oscillation criteria for second-order forced dynamic equations on time scales containing mixed nonlinearities of the formrtΦαx^ΔΔ ft, x^σ et,t ∈ t0,∞_T with ft, x qtΦαx n

i1qitΦβix,Φ∗u |u|^∗−1u, where t0,∞_Tis a time scale interval with t0∈T, the functionsr, q, qi, e: t0,∞_T → Rare right-dense continuous withr >0,σis the forward jump operator,x^σt:xσt, andβ1>· · ·> βm> α > β_m1>· · ·βn >0. All results obtained are new even forTRandTZ. In the special case whenTRandα1 our theorems reduce toY. G. Sun and J. S. W. Wong, Journal of Mathematical Analysis and Applications. 3372007, 549–560. Therefore, our results in particular extend most of the related existing literature from the continuous case to arbitrary time scale.

Copyrightq2009 R. P. Agarwal 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.

1. Introduction

LetTbe a time scale which is unbounded above andt0 ∈Ta fixed point. For some basic facts on time scale calculus and dynamic equations on time scales, one may consult the excellent texts by Bohner and Peterson 1,2.

We consider the second-order forced nonlinear dynamic equations containing mixed nonlinearities of the form

rtΦαx^Δ_Δ

ft, x^σ et, t∈ t0,∞_T, 1.1

with

ft, x qtΦαx ⁿ

i1

qitΦβix, Φ∗u |u|^∗−1u, 1.2

where t0,∞_Tdenotes a time scale interval, the functionsr, q, qi, e: t0,∞_T → Rare right- dense continuous withr >0,σis the forward jump operator,x^σt:xσt, and

β₁>· · ·> β_m> α > β_m1>· · ·β_n>0. 1.3

By a proper solution of1.1on t0,∞_T we mean a functionx ∈ C¹_rd t0,∞_T which is defined and nontrivial in any neighborhood of infinity and which satisfies 1.1 for all t ∈ t0,∞_T, where C¹_rd t0,∞_T denotes the set of right-dense continuously differentiable functions from t0,∞_TtoR. As usual, such a solutionxtof1.1is said to be oscillatory if it is neither eventually positive nor eventually negative. The equation is called oscillatory if every proper solution is oscillatory.

In a special case,1.1becomes rtΦαx^Δ_Δ

ctΦβx^σ et, 1.4

which is called half-linear forβ α, super-half-linear forβ > α, and sub-half-linear for 0 <

β < α. IfTR,1.4takes the form

rtΦαx

ctΦβx et. 1.5

The oscillation of1.5has been studied by many authors, the interested reader is referred to the seminal books by Doˇsl ´y and ˇReh´ak 3and Agarwal et al. 4,5, where in addition to mainly oscillation theory, the existence, uniqueness, and continuation of solutions are also discussed. In 3, one may also find several results related to the oscillation of 1.4 when TZ, that is, for

ΔrkΦαΔxk ckΦβxk1 ek, 1.6

whereΔis the forward difference operator.

There are several methods in the literature for finding sufficient condition for oscillation of solutions in terms of the functions appearing in the corresponding equation, and almost all such conditions involve integrals or sums on infinite intervals 3–19. The interval oscillation method is different in a sense that the conditions make use of the information of the functions on a union of intervals rather than on an infinite interval. Following El-Sayed 20, many authors have employed this technique in various works 20–30. For instance, Sun et al. 26, Wong 28, and Nasr 25have studied1.5whenα1 andβ≥1, while the case α 1 and 0< β < 1 is taken into account by Sun and Wong in 16. The results in 25,28 have been extended by Sun 27to superlinear delay differential equations of the form

xt ct|xτt|^β−1xτt et. 1.7

Further extensions of these results can be found in 30,31, where the authors have studied some related super-half-linear differential equations with delay and advance arguments.

Recently, there have been also numerous papers on second-order forced dynamic equations on time scales, unifying particularly the discrete and continuous cases and

handling many other possibilities. For a sampling of the work done we refer in particular to 6,8,9,12,13,22,32,33and the references cited therein. In 22Anderson and Zafer have extended the above mentioned interval oscillation criteria to second-order forced super-half- linear dynamic equations with delay and advance arguments including

rtΦαx^Δt_Δ

ctΦβxτt et. 1.8

Our motivation in this study stems from the work contained in 34, where the authors have derived interval criteria for oscillation of second-order differential equations with mixed nonlinearities of the form

xft, x et, t≥t0, 1.9

with

ft, x qtx^m

i1

q_itΦβix 1.10

by using a Riccati substitution and an inequality of geometric-arithmetic mean type. As it is indicated in 34, 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. We aim to make a contribution in this direction for a class of more general equations on time scales of the form1.1by combining the techniques used in 22,34. Notice that whenα1,rt≡1, andT R,1.1coincides with1.9, and therefore our results provide new interval oscillation criteria even for T R when α /1.

Moreover, for the special case T Z we obtain interval oscillation criteria for difference equations with mixed nonlinearities of the form

ΔrkΦαΔxk qkΦαxk1 ⁿ

i1

q_ikΦβixk1 ek, 1.11

for which almost nothing is available in the literature.

2. Lemmas

We need the following preparatory lemmas. The first two lemmas are given by Wong and Sun as a single lemma 34, Lemma 1forα 1. The proof for the caseα /1 is exactly the same, in fact one only needs to replace the exponentsα_ibyβ_i/αin their proof.Lemma 2.3is the well-known Young inequality.

Lemma 2.1. For any givenn-tuple{β1, β₂, . . . , β_n}satisfying

β1>· · ·> βm> α > β_m1>· · ·> βn>0, 2.1

there corresponds ann-tuple{η1, η2, . . . , ηn}such that n

i1

β_iη_iα, n i1

η_i<1, 0< η_i <1. 2.2

Ifn2 andm1cf. 34for the caseα1one may take

η1 α−β₂ 1−η₀

β₁−β₂ , η2 β₁ 1−η₀

−α

β₁−β₂ , 2.3

whereη0is any positive number withβ1η0 < β1−α.

Lemma 2.2. For any givenn-tuple{β1, β2, . . . , βn}satisfying

β1>· · ·> βm> α > β_m1>· · ·> βn>0, 2.4

there corresponds ann-tuple{η1, η2, . . . , ηn}such that n

i1

β_iη_iα, n

i1

η_i1, 0< η_i<1. 2.5

Ifn2 andm1, it turns out that

η1 α−β₂

β₁−β₂, η2 β₁−α

β₁−β₂. 2.6

Lemma 2.3Young’s Inequality. Ifp >1 andq >1 are conjugate numbers1/p1/q1, then

|u|^p p |v|^q

q ≥ |uv|, ∀u, v∈R, 2.7

and equality holds if and only ifu|v|^q−2v.

Let γ > δ. Put u A^δ/γ, p γ/δ, andv Bα^1−δ/γγ −δ^δ/γ−1. It follows from Lemma 2.3that

Ax^γB≥γδ^−δ/γγ−δ^δ/γ−1A^δ/γB^1−δ/γx^δ 2.8

for allA, B, x≥0. Rewriting the above inequality we also have

Cx^δ−D≤δ^−γ/δδγ−δ^γ/δ−1C^γ/δD^1−γ/δx^γ 2.9

for allC, x≥0 andD >0.

3. The Main Results

Following 21,22,30, denote fora, b∈ t0,∞_Twitha < bthe admissible set Aa, b:

u∈C¹_rd a, b_T:ua 0ub, u /≡0 . 3.1

The main results of this paper are contained in the following three theorems. The arguments used in the proofs have common features with the ones developed in 22,30,34.

Theorem 3.1. Suppose that for any given T ∈ t0,∞_T there exist subintervals a1, b_1T and a2, b_2Tof T,∞_Tsuch that

q_it≥0 fort∈ a1, b_1T∪ a2, b_2T,i1,2, . . . , n,

−1^ket≥0/≡0 fort∈ ak, bk_T,k1,2. 3.2

Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.2 in Lemma 2.1. If there exists a function u ∈ Aak, b_k,k1,2, such that

_bk

ak

|u^σt|^α1

qt η|et|^η0n

i1

q^η_iⁱt

− |u^Δt|^α1rt

Δt≥0 3.3

fork1,2, where

η₀1−ⁿ

i1

η_i, ηⁿ

i0

η^−η_i ⁱ, 3.4

then1.1is oscillatory.

Proof. To arrive at a contradiction, let us suppose thatxis a nonoscillatory solution of1.1.

First, we assume thatxtis positive for allt∈ t1,∞_T, for somet₁ ∈ t0,∞_T. Lett∈ a1, b1_T, wherea1∈ t1,∞_Tis sufficiently large. Define

wt −rtΦα

x^Δt

Φαxt . 3.5

It follows that

w^Δt ft, x^σ

Φαx^σt− et

Φαx^σtrtΦ x^Δt

Φαxt^Δ

ΦαxtΦαx^σt , 3.6

and hence

w^Δt qt ⁿ

i1

qitΦβi−αx^σt |et|

Φαx^σtrtΦ x^Δt

Φαxt^Δ

ΦαxtΦαx^σt . 3.7

By our assumptions3.2we haveqit≥0 andet≤0 fort∈ a1, b1_T. Set

ui 1

ηiqitΦβi−αx^σt, u0 1 η0

|et|

Φαx^σt. 3.8

Then3.7becomes

w^Δt qt ⁿ

i0

ηiuirtΦ x^Δt

Φαxt^Δ

ΦαxtΦαx^σt . 3.9

In view of the arithmetic-geometric mean inequality, see 35, n

i0

η_iu_i≥ⁿ

i0

u^η_iⁱ, 3.10

and equality3.9we obtain

w^Δt≥qt η|et|^η0n

i1

q_i^ηit rtΦ x^Δt

Φαxt^Δ

ΦαxtΦαx^σt . 3.11

Multiplying both sides of inequality3.11by|u^σ|^α1and then using the identity

uΦαuw^Δu^σΦαu^σw^Δ |u|^α1Δw 3.12

result in

uΦαuw^Δ≥ |u^σ|^α1Q− |u^Δ|^α1rGu, w, 3.13

where

Qt qt η|et|^η0n

i1

q^η_iⁱt,

Gu, w |u^Δ|^α1r |u|^α1Δ w|u^σ|^α1r Φα

x^Δ

Φαx^Δ ΦαxΦαx^σ .

3.14

As demonstrated in 7,12, we know thatGu, w≥0, and thatGu, w 0 if and only if

u^Δ Φ⁻¹_α

−w r

u, 3.15

whereΦ⁻¹_α stands for the inverse function. In our case, since 1μΦ⁻¹_α −w/r x^σ/x > 0, dynamic equation 3.15 has a unique solution satisfying ua1 0. Clearly, the unique solution isu≡0. Therefore,Gu, w>0 on a1, b_1T.

For the benefit of the reader we sketch a proof of the fact thatGu, w≥0. Note that if tis a right-dense point, then we may write

Gu, w α1

Φ⁻¹α r

|Φ⁻¹_α ru^Δ|^α1

α1 wΦαuΦ⁻¹_α ru^Δ |wΦαu|^α1/α α1/α

. 3.16

Applying Young’s inequalityLemma 2.3with

pα1, u Φ⁻¹_α ru^Δ, vwΦαu, 3.17

we easily see thatGu, w≥0 holds. Iftis a right-scattered point, thenGcan be written as a function ofuμtu^Δandvuas

Gu, v 1 μ

r

μ^α|u|^α1 wr

Φα

Φ⁻¹α r μΦ⁻¹α w|uv|^α1−w |v|^α1

. 3.18

Using differential calculus, see 7, the result follows.

Now integrating the inequality3.13froma₁tob₁and usingGu, w>0 on a1, b_1T we obtain

_b1

a1

|u^σt|^α1Qt− |u^Δt|^α1rt Δt <0, 3.19

which of course contradicts3.3. This completes the proof whenxtis eventually positive.

The proof whenxtis eventually negative is analogous by repeating the arguments on the interval a2, b_2Tinstead of a1, b_1T.

A close look at the proof of Theorem 3.1reveals that one cannot takeet ≡ 0. The following theorem is a substitute in that case.

Theorem 3.2. Suppose that for any givenT ∈ t0,∞_Tthere exists a subinterval a1, b_1Tof T,∞_T such that

qit≥0 fort∈ a1, b1_T,i1,2, . . . , n. 3.20

Let {η1, η₂, . . . , η_n} be an n-tuple satisfying 2.5 in Lemma 2.2. If there exists a function u ∈ Aa1, b1such that

_b1

a1

|u^σt|^α1

qt η

n i1

q^η_iⁱt

− |u^Δt|^α1rt

Δt≥0, 3.21

where

ηⁿ

i1

η^−η_i ⁱ, 3.22

then1.1withet≡0 is oscillatory.

Proof. We proceed as in the proof ofTheorem 3.1to arrive at3.7withet≡0, that is,

w^Δt qt ⁿ

i1

q_itΦβi−αx^σt rtΦ x^Δt

Φαxt^Δ

ΦαxtΦαx^σt . 3.23

Setting

u_i 1

η_iq_itΦβi−αx^σt, 3.24

and using again the arithmetic-geometric mean inequality n

i1

ηiui≥ⁿ

i1

u^η_iⁱ, 3.25

we have

w^Δt≥qt η n

i1

q^η_iⁱt rtΦ x^Δt

Φαxt^Δ

ΦαxtΦαx^σt . 3.26

The remainder of the proof is the same as that ofTheorem 3.1.

As it is shown in 34 for the sublinear terms case, we can also remove the sign condition imposed on the coefficients of the sub-half-linear terms to obtain interval criterion which is applicable for the case when some or all of the functionsqit,i m1, . . . , n, are nonpositive. We should note that the sign condition on the coefficients of super-half-linear terms cannot be removed alternatively by the same approach. Furthermore, the functionet cannot take the value zero on intervals of interest in this case. We have the following theorem.

Theorem 3.3. Suppose that for any given T ∈ t0,∞_T there exist subintervals a1, b_1T and a2, b2_Tof T,∞_Tsuch that

qit≥0 fort∈ a1, b1_T∪ a2, b2_T,i1,2, . . . , m,

−1^ket>0 fort∈ ak, bk_T,k1,2. 3.27

If there exist a functionu∈ Aak, bk,k1,2, and positive numbersλiandμiwith m

i1

λ_i ⁿ

im1

μ_i1, 3.28

such that

_bk

ak

|u^σt|^α1

qt ^m

i1

Pit− ⁿ

im1

Rit

− |u^Δt|^α1rt

Δt≥0 3.29

fork1,2, where

Pit βiβi−α^α/βi−1α^−α/βiλ^1−α/β_i ⁱq_i^α/βit|et|^1−α/βi,

Rit βiα−βi^α/βi−1α^−α/βiμ^1−α/β_i ⁱ−q_i^α/βit|et|^1−α/βi, 3.30

with

−qi

t max

−qit,0

, 3.31

then1.1is oscillatory.

Proof. Suppose that1.1has a nonoscillatory solution. We may assume thatxtis eventually positive on a1, b1_Twhena1is sufficiently large. Ifxtis eventually negative, then one can repeat the proof on the interval a2, b2_T. Rewrite1.1as follows:

rtΦαx^Δ_Δ

qtΦαx^σ gt, x^σ 0, t∈ a1, b_1T, 3.32

with

gt, x ^m

i1

qitx^βiλi|et|

− ⁿ

im1

−qitx^βix−μi|et|

. 3.33

Clearly,

gt, x≥^m

i1

qitx^βiλi|et|

− ⁿ

im1

−qitx^βi−μi|et|

, 3.34

where

−qit max

−qit,0

. 3.35

Applying2.8and2.9to each summation on the right side with

Aq_it, Bλ_i|et|, γβ_i, δα,

C −qit, Dμ_i|et|, δβ_i, γα, 3.36

we see that

gt, x≥ _m

i1

Pit− ⁿ

im1

Rit

x^α, 3.37

where

P_it β_iβi−α^α/βi−1α^−α/βiλ^1−α/β_i ⁱq_i^α/βit|et|^1−α/βi, R_it

β_i α

1−β_i α

α/βi−1

μ^1−α/β_i ⁱ−q_i^α/βit|et|^1−α/βi. 3.38

From3.32and inequality3.37we obtain rtΦαx^Δ_Δ

QtΦαx^σ≤0, t∈ a1, b_1T, 3.39

where

Qt qt ^m

i1

P_it− ⁿ

im1

R_it. 3.40

Set

wt −rtΦα

x^Δt

Φαxt . 3.41

In view of inequality3.39it follows that

w^Δt≥Qt rtΦ

x^Δt

Φαxt^Δ

ΦαxtΦαx^σt . 3.42

The remainder of the proof is the same as that ofTheorem 3.1, hence it is omitted.

4. Applications

To illustrate the usefulness of the results we state the corresponding theorems for the special casesT R,T Z, andT q^N,q > 1. One can easily provide similar results for other specific time scales of interest.

4.1. Differential Equations

LetTR, then we havef^Δf,σt t, and rtΦαx

qtΦαx ⁿ

i1

q_itΦβix et, t∈ t0,∞, 4.1

wherer, q, qi, e: t0,∞ → Rare continuous functions withr > 0, andβ1 >· · · > βm > α >

β_m1>· · ·β_n>0. LetA1a, b:{u∈C¹ a, b:ua 0ub, u /≡0}.

Theorem 4.1. Suppose that for any givenT∈ t0,∞there exist subintervals a1, b₁and a2, b₂of T,∞such that

qit≥0 fort∈ a1, b1∪ a2, b2,i1,2, . . . , n,

−1^ket≥0/≡0 fort∈ ak, bk,k1,2. 4.2

Let {η1, η₂, . . . , η_n} be an n-tuple satisfying 2.2 in Lemma 2.1. If there exists a function u ∈ A1ak, b_k,k1,2, such that

_bk

ak

|ut|^α1

qt η|et|^η0n

i1

q^η_iⁱt

− |ut|^α1rt

dt≥0 4.3

fork1,2, where

η₀ 1−ⁿ

i1

η_i, ηⁿ

i0

η^−η_i ⁱ, 4.4

then4.1is oscillatory.

Theorem 4.2. Suppose that for any givenT ∈ t0,∞there exists a subinterval a1, b1of T,∞ such that

q_it≥0 fort∈ a1, b₁,i1,2, . . . , n. 4.5 Let {η1, η₂, . . . , η_n} be an n-tuple satisfying 2.5 in Lemma 2.2. If there exists a function u ∈ A1a1, b₁such that

_b1

a1

|ut|^α1

qt η

n i1

q^η_iⁱt

− |ut|^α1rt

dt≥0, 4.6

where

ηⁿ

i1

η^−η_i ⁱ, 4.7

then4.1withet≡0 is oscillatory.

Theorem 4.3. Suppose that for any givenT∈ t0,∞there exist subintervals a1, b1and a2, b2of T,∞such that

q_it≥0 fort∈ a1, b₁∪ a2, b₂,i1,2, . . . , m,

−1^ket>0 fort∈ ak, bk,k1,2. 4.8

If there exist a functionu∈ Aak, b_k,k1,2, and positive numbersλ_iandμ_iwith m

i1

λi ⁿ

im1

μi1, 4.9

such that

_bk

ak

|u^σt|^α1

qt ^m

i1

P_it− ⁿ

im1

R_it

− |ut|^α1rt

dt≥0 4.10

fork1,2, where

Pit βiβi−α^α/βi−1α^−α/βiλ^1−α/β_i ⁱq_i^α/βit|et|^1−α/βi,

Rit βiα−βi^α/βi−1α^−α/βiμ^1−α/β_i ⁱ−q_i^α/βit|et|^1−α/βi, 4.11 with

−qi

t max

−qit,0

, 4.12

then4.1is oscillatory.

4.2. Difference Equations

LetTZ, then we havef^Δk Δfk fk1−fk,σk k1, and

ΔrkΦαΔxk qkΦαxk1 ⁿ

i1

q_ikΦβixk1 ek, k∈ k0,∞_N, 4.13

where k0,∞_N{k0, k₀1, k₀2, . . .},r, q, q_i, e: k0,∞_N → Rwithrk>0, andβ₁>· · ·>

βm> α > β_m1>· · ·βn>0. Let a, b_N{a, a1, a2, . . . , b}, andA2a, b:{u: a, b_N → R, ua 0ub, u /≡0}.

Theorem 4.4. Suppose that for any given K ∈ k0,∞_N there exist subintervals a1, b1_N and a2, b_2Nof K,∞_Nsuch that

q_i j

≥0 forj ∈ a1, b_1N∪ a2, b_2N,i1,2, . . . , n,

−1^ke j

≥0/≡0 forj∈ ak, bk_N,k1,2. 4.14

Let {η1, η₂, . . . , η_n} be an n-tuple satisfying 2.2 in Lemma 2.1. If there exists a function u ∈ A2ak, b_k,k1,2, such that

bk−1 jak

|uj1|^α1

q j

η|ej|^η0n

i1

q_i^ηi j

− |Δuj|^α1r j

≥0 4.15

fork1,2, where

η0 1−ⁿ

i1

ηi, ηⁿ

i0

η^−η_i ⁱ, 4.16

then4.13is oscillatory.

Theorem 4.5. Suppose that for any given K ∈ k0,∞_N there exists a subinterval a1, b_1N of K,∞_Nsuch that

qi

j

≥0 forj∈ a1, b1_N,i1,2, . . . , n. 4.17

Let {η1, η₂, . . . , η_n} be an n-tuple satisfying 2.5 in Lemma 2.2. If there exists a function u ∈ A2a1, b₁such that

b1−1 ja1

|uj1|^α1

q j

η n

i1

q^η_iⁱ j

− |Δuj|^α1r j

≥0, 4.18

where

ηⁿ

i0

η^−η_i ⁱ, 4.19

then4.13withek≡0 is oscillatory.

Theorem 4.6. Suppose that for any given K ∈ k0,∞_N there exist subintervals a1, b_1N and a2, b2_Nof K,∞_Nsuch that

qi

j

≥0 forj∈ a1, b1_N∪ a2, b2_N,i1,2, . . . , m,

−1^ke j

>0 forj∈ ak, bk_N,k1,2. 4.20

If there exist a functionu∈ A2ak, bk,k1,2, and positive numbersλiandμiwith m

i1

λ_i ⁿ

im1

μ_i1, 4.21

such that

bk−1 jak

|uj1|^α1

q j

^m

i1

P_i j

− ⁿ

im1

R_i j

− |Δuj|^α1r j

≥0 4.22

fork1,2, where

P_it β_iβi−α^α/βi−1α^−α/βiλ^1−α/β_i ⁱq_i^α/βit|et|^1−α/βi,

R_it β_iα−β_i^α/βi−1α^−α/βiμ^1−α/β_i ⁱ−q_i^α/βit|et|^1−α/βi, 4.23 with

−qi

t max

−qit,0

, 4.24

then4.13is oscillatory.

4.3.q-Difference Equations

LetTq^Nwithq >1, then we haveσt qt,f^Δt Δqft fqt−ft/qt−t,and

Δq

rtΦα

Δqxt

ptΦα

x qt

ⁿ

i1

p_itΦβi

x qt

et, t∈ t0,∞_q, 4.25

where t0,∞_q :{q^t0, q^t01, q^t02, . . .}witht₀ ∈N,r, p, p_i, e: t0,∞_q → Rwithrt> 0, and β₁ >· · · > β_m > α > β_m1 > · · ·β_n > 0. Let a, b_q {q^a, q^a1, q^a2, . . . , q^b}witha, b ∈N, and A3a, b:{u: a, b_q → R, uq^a 0uq^b, u /≡0}.

Theorem 4.7. Suppose that for any givenT ∈ t0,∞_qthere exist subintervals a1, b_1qand a2, b_2q of T,∞_qsuch that

pit≥0 fort∈ a1, b1_q∪ a2, b2_q,i1,2, . . . , n,

−1^ket≥0/≡0 fort∈ ak, b_kq,k1,2. 4.26

Let {η1, η₂, . . . , η_n} be an n-tuple satisfying 2.2 in Lemma 2.1. If there exists a function u ∈ A3ak, bk,k1,2, such that

bk−1 jak

q^j

|uq^j1|^α1

p q^j

η|eq^j|^η0n

i1

p^η_iⁱ q^j

− |Δquq^j|^α1r q^j

≥0 4.27

fork1,2, where

η01−ⁿ

i1

ηi, ηⁿ

i0

η^−η_i ⁱ, 4.28

then4.25is oscillatory.

Theorem 4.8. Suppose that for any givenT ∈ t0,∞_qthere exists a subinterval a1, b_1qof T,∞_q such that

p_it≥0 fort∈ a1, b_1q,i1,2, . . . , n. 4.29

Let {η1, η2, . . . , ηn} be an n-tuple satisfying 2.5 in Lemma 2.2. If there exists a function u ∈ A3a1, b₁such that

b1−1 ja1

q^j

|uq^j1|^α1

p q^j

η n

i1

p^η_iⁱ q^j

− |Δquq^j|^α1r q^j

≥0, 4.30

where

ηⁿ

i0

η^−η_i ⁱ, 4.31

then4.25withet≡0 is oscillatory.

Theorem 4.9. Suppose that for any givenT ∈ t0,∞_qthere exist subintervals a1, b_1qand a2, b_2q of T,∞_qsuch that

pit≥0 fort∈ a1, b1_q∪ a2, b2_q,i1,2, . . . , m,

−1^ket>0 fort∈ ak, b_kq,k1,2. 4.32

If there exist a functionu∈ A3ak, b_k,k1,2, and positive numbersλ_iandμ_iwith m

i1

λ_i ⁿ

im1

μ_i1 4.33

such that

bk−1 jak

q^j

|uq^j1|^α1

p q^j

^m

i1

P_i q^j

− ⁿ

im1

R_i q^j

− |Δquq^j|^α1r q^j

≥0 4.34

fork1,2, where

P_it β_iβi−α^α/βi−1α^−α/βiλ^1−α/β_i ⁱq_i^α/βit|et|^1−α/βi,

Rit βiα−βi^α/βi−1α^−α/βiμ^1−α/β_i ⁱ−q_i^α/βit|et|^1−α/βi 4.35

with

−pi

t max

−pit,0

, 4.36

then4.25is oscillatory.

5. Examples

We give three simple examples to illustrate the importance of our results. For clarity, we have takenn2 andet≡0. Then,

η₁ α−β₂

β₁−β₂, η₂ β₁−α

β₁−β₂, β₁> α > β₂>0. 5.1

Example 5.1. Consider the constant coefficient differential equation |x|^α−1x

a|x|^α−1xb|x|^β1−1xc|x|^β2−1x0, t≥0, 5.2

whereb, c >0 andaare real numbers.

Letut sint−a1,a1 mand b1 mπ,m ∈ Nis arbitrarily large. Applying Theorem 4.2we see that every solution of5.2is oscillatory if

a b

η1

_η1 c η2

_η2

≥1. 5.3

Example 5.2. Consider the constant coefficient difference equation

Δ

|Δxk|^α−1Δxk

a|xk1|^α−1xk1 b|xk1|^β1−1xk1 c|xk1|^β2−1xk1 0, k≥1,

5.4

whereb, c >0 andaare real numbers.

Letuj 1−−1^j, anda₁2mandb₁ 2m2,m∈Nis arbitrarily large. It follows fromTheorem 4.5that every solution of5.4is oscillatory if

a b

η₁ η1

c η₂

_η2

≥2. 5.5

Example 5.3. Consider the constant coefficientq-difference equation Δq

|Δqxt|^α−1Δqxt

a|xqt|^α−1x qt

b|xqt|^β1−1x qt c|xqt|^β2−1x

qt

0, t≥1, 5.6

whereq >1,b, c >0 andaare real numbers.

Letut q^b1 −tt−q^a1, anda₁ mandb₁ m2,m∈ Nis arbitrarily large. In view ofTheorem 4.8, we see that every solution of5.6is oscillatory if

a b

η₁ η1

c η₂

η2

>0. 5.7

6. Remarks

Equation 1.1 has been studied by Sun and Wong 34 for the case T R and α 1.

Our results in Section 4.1 coincide with theirs when α 1, and therefore the results can be considered as an extension fromα 1 to α > 0. Since the results in 34 are linked to many well-known oscillation criteria in the literature, the interval oscillation criteria we have obtained provide further extensions of these to time scales.

The results in Sections4.2and4.3are all new for all values of the parameters. Although there are some results for difference equations in the special case n 1, there is hardly any interval oscillation criteria for theq-difference equations case.

Moreover, since our main results inSection 4are valid for arbitrary time scales, similar interval oscillation criteria can be obtained by considering other particular time scales.

(2) Generalization

The results obtained in this paper remain valid for more general equations of the form rtΦαx^Δt_Δ

qtgx^{σ n}

i1

qitfix^σ et, t∈ t0,∞_T, 6.1

provided thatg, fi :R → Rare continuous and satisfy the growth conditions

xgx≥ |x|^α1, xfix≥ |x|^βi1 ∀x∈R. 6.2

To see this, we note that ifxtis eventually positive, then taking into account the intervals whereqandq_iare nonnegative, the above inequalities result in

rtΦαx^Δt_Δ

qtΦαx^{σ n}

i1

q_itqix^σ≤et, t∈ t0,∞_T. 6.3

The arguments afterward follow analogously.

(3) Forms Related to1.1

Related to1.1are the dynamic equations with mixed delta and nabla derivatives rtΦαx^Δ_∇

ft, x et, t∈ t0,∞_T, 6.4

rtΦαx^∇_Δ

ft, x et, t∈ t0,∞_T, 6.5

rtΦαx^∇_∇

ft, x^ρ et, t∈ t0,∞_T, 6.6

whereρdenotes the backward jump operator and

ft, x qtΦαx ⁿ

i1

qitΦβix. 6.7

It is not difficult to see that time scale modifications of the previous arguments give rise to completely parallel results for the above dynamic equations. For an illustrative example we provide below the version ofTheorem 3.1for6.4. The other theorems for6.4,6.5, and 6.6can be easily obtained by employing arguments developed for1.1in this paper.

Theorem 6.1. Suppose that for any given T ∈ t0,∞_T there exist subintervals a1, b_1T and a2, b2_Tof T,∞_Tsuch that

q_it≥0 fort∈ a1, b_1T∪ a2, b_2T,i1,2, . . . , n,

−1^ket≥0/≡0 fort∈ ak, b_kT,k1,2. 6.8

Let {η1, η₂, . . . , η_n} be an n-tuple satisfying 2.2 in Lemma 2.1. If there exists a function u ∈ Bak, bk:{u∈C¹_ld a, b_T:ua 0ub, u /≡0},k1,2, such that

_bk

ak

|ut|^α1

qt η|et|^η0n

i1

q^η_iⁱt

− |u^∇t|^α1r^ρt

Δt≥0 6.9

fork1,2, where

η0 1−ⁿ

i1

ηi, ηⁿ

i0

η^−η_i ⁱ, 6.10

then6.4is oscillatory.

(4) An Open Problem

It is of theoretical and practical interest to obtain interval oscillation criteria when there are only sub-half-linear terms in1.1, that is, whenβ_i < αholds for alli 1,2, . . . , n. Also, the open problems stated in 34for the special caseT Rwithα 1 naturally carry over for 1.1.