OSCILLATION OF SECOND-ORDER NEUTRAL DELAY AND MIXED-TYPE DYNAMIC EQUATIONS ON TIME SCALES

MIXED-TYPE DYNAMIC EQUATIONS ON TIME SCALES

Y. S¸AH˙INER

Received 31 January 2006; Revised 11 May 2006; Accepted 15 May 2006

We consider the equation (r(t)(y^Δ(t))^γ)^Δ+ f(t,x(δ(t)))=0,t∈T, where y(t)=x(t) + p(t)x(τ(t)) andγis a quotient of positive odd integers. We present some sufficient con- ditions for neutral delay and mixed-type dynamic equations to be oscillatory, depending on deviating argumentsτ(t) andδ(t),t∈T.

Copyright © 2006 Y. S¸ahiner. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and repro- duction in any medium, provided the original work is properly cited.

1. Some preliminaries on time scales

A time scaleTis an arbitrary nonempty closed subset of the real numbers. The theory of time scales was introduced by Hilger [6] in his Ph.D. thesis in 1988 in order to unify continuous and discrete analysis. Several authors have expounded on various aspects of this new theory, see [7] and the monographs by Bohner and Peterson [3,4], and the references cited therein.

First, we give a short review of the time scales calculus extracted from [3]. For any t∈T, we define the forward and backward jump operators by

σ(t) :=inf{s∈T:s > t}, ρ(t) :=sup{s∈T:s < t}, (1.1) respectively. The graininess functionμ:T→[0,∞) is defined byμ(t) :=σ(t)−t.

A pointt∈Tis said to be right dense ift <supTandσ(t)=t, left dense ift >infTand ρ(t)=t. Also,tis said to be right scattered ifσ(t)> t, left scattered ift > ρ(t). A function f :T→Ris called rd-continuous if it is continuous at right dense points inTand its left-sided limit exists (finite) at left dense points inT.

For a function f :T→R, if there exists a number α∈R such that for all ε >0 there exists a neighborhoodU oftwith|f(σ(t))−f(s)−α(σ(t)−s)| ≤ε|σ(t)−s|, for alls∈U, then f isΔ-differentiable att, and we callαthe derivative of f attand denote

Hindawi Publishing Corporation Advances in Difference Equations Volume 2006, Article ID 65626, Pages1–9 DOI10.1155/ADE/2006/65626

it by f^Δ(t),

f^Δ(t)= fσ(t)−f(t)

σ(t)−t (1.2)

iftis right scattered. Whentis a right dense point, then the derivative is defined by f^Δ(t)=lim

s→t

f(t)−f(s)

t−s , (1.3)

provided this limit exists.

If f :T→RisΔ-differentiable att∈T, then f is continuous att. Furthermore, we assume thatg:T→RisΔ-differentiable. The following formulas are useful:

fσ(t)= f(t) +μ(t)f^Δ(t), (f g)^Δ(t)=f^Δ(t)g(t) +fσ(t)g^Δ(t). (1.4) A functionFwithF^Δ= f is called an antiderivative of f, and then we define

_b

a f(t)Δt=F(b)−F(a), (1.5)

wherea,b∈T. It is well known that rd-continuous functions possess antiderivatives.

Note that ifT=R, we haveσ(t)=t,μ(t)=0, f^Δ(t)=f(t), and _b

a f(t)Δt= _b

a f(t)dt, (1.6)

and ifT=Z, we haveσ(t)=t+ 1,μ(t)=1, f^Δ=Δ f, and _b

a f(t)Δt=

b−1 t=a

f(t). (1.7)

Iff is rd-continuous, then

_σ(t)

t f(s)Δs=μ(t)f(t). (1.8)

2. Introduction

In this paper, we are concerned with the oscillatory behavior of the second-order neutral dynamic equation with deviating arguments

r(t)y^Δ(t)^γΔ+ft,xδ(t)=0, t∈T, (NE)

wherey(t)=x(t) +p(t)x(τ(t)),γis a quotient of positive odd integers,r,p∈Crd(T,R) are positive functions,τ,δ∈Crd(T,T),τ(t)≤t, limt→∞τ(t)= ∞, limt→∞δ(t)= ∞, and

f :T×R→Ris continuous function such thatu f(t,u)>0 for allu=0.

Unless otherwise is stated, throughout the paper, we assume the following conditions:

(H1) 0≤p(t)<1,

(H2)^∞(1/r(t))^1/γΔt= ∞,

(H3) there exists a nonnegative functionqdefined onTsuch that|f(t,u)| ≥q(t)|u|^γ. By a solution of (NE), we mean a nontrivial real-valued functionxsuch thatx(t) + p(t)x(τ(t)) andr(t)[(x(t) +p(t)x(τ(t)))^Δ]^γ are defined andΔ-differentiable fort∈T, and satisfy (NE) fort≥t0∈T. A solutionxhas a generalized zero attin casex(t)=0.

We say x has a generalized zero on [a,b] in case x(t)x(σ(t))<0 or x(t)=0 for some t∈[a,b), wherea,b∈Tanda≤b(xhas a generalized zero atb, in casex(ρ(b))x(b)<0 orx(b)=0). A nontrivial solution of (NE) is said to be oscillatory on [tx,∞) if it has infinitely many generalized zeros whent≥tx; otherwise it is called nonoscillatory. Finally, (NE) is called oscillatory if all its solutions are oscillatory.

In recent years, there has been a great deal of work on the oscillatory behavior of so- lutions of some second-order dynamic equations. To the best of our knowledge, there is very little known about the oscillatory behavior of (NE). Indeed, there are not many results about nonneutral second-order equation in the form of (NE) whenp(t)≡0. For some oscillation criteria, we refer the reader to the papers [1,2,9,12] and references cited therein.

Subject to our corresponding conditions, Agarwal et al. [2] considered the second- order neutral delay dynamic equation

r(t)x(t) +p(t)x(t−τ)^Δγ Δ

+ ft,x(t−δ)=0, (2.1)

whereτandδare positive constants. A part of this study contains two main theorems proven by the technique of reduction of order. Previously obtained result about oscilla- tion of first-order delay dynamic equation

z^Δ(t) +Q(t)zh(t)=0 (2.2)

is used to be compared with (2.1). One of them is the following which is auxiliary for the proof of the first theorem in [2].

Lemma 2.1 [11, Corollary 2]. Assumeh(t)< t. Define α:=lim sup

t→∞ sup

λ∈EQ

λe₋λQ

h(t),t, (2.3)

whereEQ= {λ|λ >0, 1−λQ(t)μ(t)>0,t∈T}, and e_−λQh(t),t=exp

_t

h(t)ξ_μ(s)−λQ(s)Δs, ξ_l(z)=

⎧⎪

⎨

⎪⎩

log(1 +lz)

l ifl=0,

z ifl=0.

(2.4)

Ifα <1, then every solution of (2.2) is oscillatory.

Theorem 2.2 [2, Theorem 3.2]. Assume thatr^Δ(t)≥0. Then every solution of (2.1) oscil- lates if

lim sup

t→∞ sup

λ∈EA

λe₋λA(t−δ,t)<1, (2.5)

where

A(t)=q(t)1−p(t−δ)^γ r(t−δ)

t−δ 2

γ

. (2.6)

Theorem 2.3 [2, Theorem 3.3]. Assume thatr^Δ(t)≥0. Then every solution of (2.1) oscil- lates if

lim sup

t→∞

_t

t−δA(s)Δs >1. (2.7)

Note that the monotonicity condition imposed on r is quite restrictive and there- foreTheorem 2.3applies only to a special class of neutral-type dynamic equations. Also, τ(t)=t−τandδ(t)=t−δbeing just linear functions cause further restrictions.

The above results are of special importance for us and in fact they motivate our study in this paper. Our purpose here, first of all, is to show that the conclusions of Theorems 2.2and2.3are valid without the monotonicity condition onrand requirementsτ(t)= t−τandδ(t)=t−δ. In the next section, we present some new oscillation criteria under very mild conditions and more general assumptions to extend the above results for the neutral delay and mixed dynamic equations.

3. Main results

Since we deal with the oscillatory behavior of (NE) on time scales, throughout the paper, we assume that the time scaleTunder consideration satisfies supT= ∞. We label (NE) as (NE)dor (NE)m that refers to neutral delay or mixed dynamic equation ifδ(t)< tor δ(t)> t, respectively.

Theorem 3.1. LetE= {λ|λ >0, 1−λg(t)μ(t)>0}. Assume thatδ(t)< t. If lim sup

t→∞ sup

λ∈E

λe_−λgδ(t),t<1, (3.1)

whereg(t)=[1−p(δ(t))]^γq(t), then (NE)_dis oscillatory.

Proof. Assume, for the sake of contradiction, that (NE)d has a nonoscillatory solution x(t). We may assume thatx(t) is eventually positive, since the proof whenx(t) is even- tually negative is similar. Becauseδ(t),τ(t)→ ∞ast→ ∞, there exists a positive num- bert1≥t0, such thatx(δ(t))>0 andx(τ(t))>0 fort≥t1. We also see thaty(t)>0 for t≥t1. We may claim thaty^Δ(t) has eventually a fixed sign. Ify^Δhas a generalized zero on I=[t2,σ(t2)) for somet2> t1, then

r(t)y^Δ(t)^γΔ_I= −ft,xδ(t)<0, (3.2)

which implies thaty^Δ(t) cannot have another generalized zero after it vanishes or changes sign once on the intervalI. Suppose thaty^Δ(t)<0 fort≥t3≥σ(t2). It is easy to see from (NE)dthatr(t)(y^Δ(t))^γis nonincreasing. So we have

r(t)y^Δ(t)^γ≤rt3

y^Δt3

γ

=d <0, t≥t3. (3.3)

Integration fromt3totyields

y(t)_≤yt3

+d^1/γ _t

t3

1

r(s)^1/γΔs. (3.4)

In view of (H2), it follows from (3.4) that the function y(t) takes on negative values for sufficiently large values oft. This contradicts the fact that y(t) is eventually positive, we must have y^Δ(t)>0 fort≥t3. Using this fact together withτ(t)≤tandx(t)< y(t), we see that

y(t)=x(t) +p(t)xτ(t)≤x(t) +p(t)yτ(t)≤x(t) +p(t)y(t) (3.5) or

x(t)≥

1−p(t)y(t), t≥t3. (3.6)

Because of (H2), we have for sufficiently larget_≥t3, _t

t3

1

r^1/γ(s)Δs >1. (3.7)

By the nonincreasing property ofr^1/γy^Δ, y(t)=yt3

+ _t

t3

y^Δ(s)Δs

≥ _t

t3

1 r^1/γ(s)

r^1/γ(s)y^Δ(s)Δs≥r^1/γ(t)y^Δ(t) _t

t3

1 r^1/γ(s)

(3.8)

and using (3.7), we get

y(t)≥r^1/γ(t)y^Δ(t), t≥t3. (3.9) There exists a numbert_∗=δ(t3)< t3≤tsuch that the following holds from inequalities (3.6) and (3.9):

xδ(t)≥

1−pδ(t)r^1/γδ(t)y^Δδ(t), t≥t_∗. (3.10) In view of (NE)dand (H3), we have

r(t)y^Δ(t)^γΔ+q(t)x^γδ(t)≤0. (3.11)

Substituting (3.10) into the last inequality, we obtain fort≥t_∗,

z^Δ(t) +1−pδ(t)^γq(t)zδ(t)_≤0, (3.12) wherez(t)=r(t)(y^Δ(t))^γ is an eventually positive solution. This contradicts condition

(3.1), the proof is complete.

Remark 3.2. In case thatT=N, (2.2) reduces to the first-order delay difference equation zn+1−zn+Qnzn−h=0, (3.13) whereh_n=n−h,h∈Nandn > h≥1. Erbe and Zhang [5] proved that (3.13) is oscilla- tory provided that

lim sup

n→∞

n i=n−h

Qi>1. (3.14)

In the proof ofTheorem 2.3, first (2.1) is reduced to a first-order delay dynamic equation in the form of (2.2) and then, by similar steps of the proof of well-known oscillation criterion given by Ladas et al. [8] for (2.2) whenT=R, a contradiction is obtained in view of condition (2.7). But whenT=N, considering definition (1.7), condition (2.7) is derived as

lim sup

n→∞

n−1 i=n−h

Q_i>1 (3.15)

which is not the same as condition (3.14).

To overcome this difficulty, we intend to use the following sufficient condition estab- lished by S¸ahiner and Stavroulakis [10] for (2.2) to be oscillatory on any time scaleT. Lemma 3.3 [9, Theorem 2.4]. Assume thath(t)< t. If

lim sup

t→∞

_σ(t)

h(t)Q(s)Δs >1, (3.16)

then (2.2) is oscillatory.

Theorem 3.4. Assume thatδ(t)< t. If lim sup

t→∞

_σ(t)

δ(t)

1−pδ(s)^γq(s)Δs >1, (3.17)

then (NE)dis oscillatory.

Proof. Suppose the contrary thatxis a nonoscillatory solution of (NE)d and following the same steps as inTheorem 3.1, we obtain (3.12). The rest of the proof is exactly the same as that ofLemma 3.3, see [10]. The proof is complete.

Remark 3.5. The above theorems are applicable even ifris not monotone and deviating argumentsτ(t) andδ(t) are variable functions oft. Moreover, in caser(t)>(t/2)^γ for sufficiently larget, Theorems3.1and3.4are stronger than Theorems2.2and2.3.

Example 3.6. Consider the following neutral delay dynamic equation:

⎛

⎝1 t

x(t) +p(t)x t 2

Δ3⎞

⎠

Δ

+q(t)x³(^√t)=0. (3.18)

r(t) satisfies (H2) but it is not increasing. Moreover, delay termsτ(t)=t/2 andδ(t)=√ t are not in the form oft−τandt−δ for any constantsτ,δ >0, respectively. Therefore, Theorems2.2and2.3cannot be applied to (3.18). On the other hand, if

lim sup

t→∞ sup

λ∈E

λe₋λg(^√t,t)<1, (3.19)

or

lim sup

t→∞

_σ(t)

√t

1−p(^√t)³q(s)Δs >1 (3.20)

is satisfied, then by Theorem3.1or3.4, respectively, (3.18) is oscillatory.

Remember that (NE) is a mixed-type neutral dynamic equation whenδ(t)> t, because of that the equation contains both delay and advanced arguments. Now, we state some sufficient conditions for mixed-type neutral dynamic equations (NE)mto be oscillatory.

We just give an outline for the proof of next theorem.

Theorem 3.7. Assume thatδ(t)> tandτ(δ(t))< t. If

lim sup

t→∞ sup

λ∈E

λe₋λg

τ(δ(t),t<1, (3.21)

whereg(t) andEare as defined inTheorem 3.1, then (NE)mis oscillatory.

Proof. Assume that (NE)mhas a nonoscillatory solutionx(t). Without loss of generality, we assume thatx(t) is eventually positive. Proceeding as in the proof ofTheorem 3.1, it is known thatx(t)< y(t) andy^Δ(t)>0. Therefore, for sufficiently larget4, we obtain instead of (3.6),

yτ(t)≤y(t)=x(t) +p(t)xτ(t)≤x(t) +p(t)yτ(t) (3.22) or

x(t)≥

1−p(t)yτ(t), t≥t4. (3.23) Using this with inequality (3.9), we get

xδ(t)≥

1−pδ(t)r^1/γτδ(t)y^Δτδ(t). (3.24)

At the end, we obtain

z^Δ(t) +1−pδ(t)^γq(t)zτδ(t)≤0, (3.25) wherez(t)=r(t)(y^Δ(t))^γ is an eventually positive solution. This contradicts condition

(3.29), the proof is complete.

Theorem 3.8. Assume thatδ(t)> tandτ(δ(t))< t. If

lim sup

t→∞

_σ(t)

τ(δ(t))

1−pδ(s)^γq(s)Δs >1, (3.26)

then (NE)dis oscillatory.

Example 3.9. Consider the following mixed-type neutral dynamic equation:

⎛

⎝1 t

x(t) + 1−1 t

x(^√t)

Δ1/3⎞

⎠

Δ

+ t

σ(t)t^1/3x^1/3 t² 64

=0, t≥9. (3.27) r(t) satisfies (H2). Assumptions ofTheorem 3.8which areδ(t)=t²/64> tandτ(δ(t))= t/8< thold fort≥9. Since

_σ(t)

t/8 1− 1−64

s²

1/3 s

σ(s)s^1/3Δs≥ t 8

_σ(t)

t/8

4

sσ(s)Δs=1

2 8− t σ(t)

≥7

2, (3.28) condition (3.26) is satisfied. Therefore (3.27) is oscillatory.

Remark 3.10. Theorems3.7and3.8are also valid for (NE)d. If we assumeτ(t)< tinstead ofτ(t)≤t, assumptionτ(δ(t))< tis already satisfied whenδ(t)< tand the proofs do not change. Assumptionτ(t)< t implies the immediate resultτ(δ(t))< δ(t). Therefore, we conclude the following which are stronger conditions for neutral delay dynamic equation (NE)_d.

Corollary 3.11. Assume thatτ(t)< tandδ(t)< t. If

lim sup

t→∞ sup

λ∈E

λe_−λgτδ(t),t<1, (3.29)

whereg(t) andEare as defined inTheorem 3.1, then (NE)dis oscillatory.

Corollary 3.12. Assume thatτ(t)< tandδ(t)< t. If

lim sup

t→∞

_σ(t)

τ(δ(t))

1−pδ(s)^γq(s)Δs >1, (3.30)

then (NE)_dis oscillatory.

We note that obtained results in this section generalize and extend some sufficient conditions about oscillation previously established to neutral and nonneutral differential difference and dynamic equations.

References

[1] R. P. Agarwal, M. Bohner, and S. H. Saker, Oscillation of second order delay dynamic equations, to appear in The Canadian Applied Mathematics Quarterly.

[2] R. P. Agarwal, D. O’Regan, and S. H. Saker, Oscillation criteria for second-order nonlinear neutral delay dynamic equations, Journal of Mathematical Analysis and Applications 300 (2004), no. 1, 203–217.

[3] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applica- tions, Birkh¨auser Boston, Massachusetts, 2001.

[4] M. Bohner and A. Peterson (eds.), Advances in Dynamic Equations on Time Scales, Birkh¨auser Boston, Massachusetts, 2003.

[5] L. H. Erbe and B. G. Zhang, Oscillation of discrete analogues of delay equations, Differential and Integral Equations 2 (1989), no. 3, 300–309.

[6] S. Hilger, Ein Maßkettenkalk¨ul mit Anwendung auf Zentrumsmannigfaltigkeiten, Ph.D. thesis, Universit¨at W¨urzburg, W¨urzburg, 1988.

[7] , Analysis on measure chains—a unified approach to continuous and discrete calculus, Re- sults in Mathematics 18 (1990), no. 1-2, 18–56.

[8] G. Ladas, Ch. G. Philos, and Y. G. Sficas, Sharp conditions for the oscillation of delay difference equations, Journal of Applied Mathematics and Simulation 2 (1989), no. 2, 101–111.

[9] Y. S¸ahiner, Oscillation of second-order delay differential equations on time scales, Nonlinear Anal- ysis 63 (2005), no. 5–7, e1073–e1080.

[10] Y. S¸ahiner and I. P. Stavroulakis, Oscillations of first order delay dynamic equations, to appear in Dynamic Systems and Applications.

[11] B. G. Zhang and X. Deng, Oscillation of delay differential equations on time scales, Mathematical and Computer Modelling 36 (2002), no. 11–13, 1307–1318.

[12] B. G. Zhang and Z. Shanliang, Oscillation of second-order nonlinear delay dynamic equations on time scales, Computers & Mathematics with Applications 49 (2005), no. 4, 599–609.

Y. S¸ahiner: Department of Mathematics , Atilim University, 06836 Incek-Ankara, Turkey E-mail address:ysahiner@atilim.edu.tr

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