MIXED-TYPE DYNAMIC EQUATIONS ON TIME SCALES
Y. S¸AH˙INERReceived 31 January 2006; Revised 11 May 2006; Accepted 15 May 2006
We consider the equation (r(t)(yΔ(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
it by fΔ(t),
fΔ(t)= f
σ(t)−f(t)
σ(t)−t (1.2)
iftis right scattered. Whentis a right dense point, then the derivative is defined by
fΔ(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)fΔ(t), (f g)Δ(t)=fΔ(t)g(t) +fσ(t)gΔ(t). (1.4)
A functionFwithFΔ= 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, fΔ(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, fΔ=Δf, 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)yΔ(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)
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 equaoscilla-tion
zΔ(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].
Lemma2.1 [11, Corollary 2]. Assumeh(t)< t. Define
α:=lim sup t→∞ λ∈EsupQ
λe−λQh(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)ξμ(s)
−λQ(s)Δs,
ξl(z)=
⎧ ⎪ ⎨ ⎪ ⎩
log(1 +lz)
l ifl=0,
z ifl=0.
(2.4)
Theorem2.2 [2, Theorem 3.2]. Assume thatrΔ(t)≥0. Then every solution of (2.1) oscil-lates if
lim sup t→∞ λ∈EsupA
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)
Theorem2.3 [2, Theorem 3.3]. Assume thatrΔ(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.
Theorem3.1. LetE= {λ|λ >0, 1−λg(t)μ(t)>0}. Assume thatδ(t)< t. If
lim sup t→∞ supλ∈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 thatyΔ(t) has eventually a fixed sign. IfyΔhas a generalized zero on I=[t2,σ(t2)) for somet2> t1, then
which implies thatyΔ(t) cannot have another generalized zero after it vanishes or changes
Integration fromt3totyields
y(t)≤yt3
Substituting (3.10) into the last inequality, we obtain fort≥t∗,
zΔ(t) +1−pδ(t)γq(t)zδ(t)≤0, (3.12)
wherez(t)=r(t)(yΔ(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)
wherehn=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
Qi>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.
Lemma3.3 [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.
Theorem3.4. 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
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→∞ supλ∈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.
Theorem3.7. Assume thatδ(t)> tandτ(δ(t))< t. If
lim sup t→∞ supλ∈E
λe−λgτ(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) andyΔ(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
At the end, we obtain
Example 3.9. Consider the following mixed-type neutral dynamic equation:
⎛
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.
whereg(t)andEare as defined inTheorem 3.1, then (NE)dis oscillatory. Corollary3.12. Assume thatτ(t)< tandδ(t)< t. If
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Y. S¸ahiner: Department of Mathematics , Atilim University, 06836 Incek-Ankara, Turkey
