Oscillatory behavior of solutions of third order delay and advanced dynamic equations

R E S E A R C H

Open Access

Oscillatory behavior of solutions of

third-order delay and advanced dynamic

equations

Murat Adıvar

_{, Elvan Akın}

_{and Raegan Higgins}

*_{Correspondence: [email protected]} 2_{Department of Mathematics and} Statistics, Missouri University of Science and Technology, Rolla, 65049, USA

Full list of author information is available at the end of the article

Abstract

In this paper, we consider oscillation criteria for certain third-order delay and advanced dynamic equations on unbounded time scales. A time scaleTis a nonempty closed subset of the real numbers. Examples will be given to illustrate some of the results.

MSC: 34N05; 39A10; 39A21

Keywords: oscillation; third-order; dynamic equations; time scales

1 Introduction

In this paper, we are concerned with oscillation criteria for solutions of the third-order delay and advanced dynamic equations



a(t)

x(t)α

+q(t)fxg(t)=  (.)

and



a(t)

x(t)α

=q(t)fxg(t)+p(t)hxk(t) (.)

on [t,∞)_Tsuch thatt∈Tandt≥, whereαis the ratio of two positive odd integers,

a,p,q∈Crd( [t,∞)_T, (,∞)) with

∞

aα_(s)s=∞_{, (.)}

andg,k∈Crd(T,T) are nondecreasing functions such thatg(t) <t<k(t) andlimt→∞g(t) = ∞. We also assume thatf,h∈C(R,R) such thatxf(x) > ,xh(x) > , f(x) and h(x) are nondecreasing forx=  satisfying

–f(–xy)≥f(xy)≥f(x)f(y) ifxy>  (.)

and

–h(–xy)≥h(xy)≥h(x)h(y) ifxy> . (.)

By a solution of (.) (or (.)) we mean a functionx∈C

rd( [Tx,∞)T,R),Tx≥t, which has the property that (/α)(x₎α_∈C

rd( [Tx,∞)T,R) and satisfies (.) (or (.)) for all large

t≥Tx. A nontrivial solution is said to be nonoscillatory if it is eventually positive or

even-tually negative and it is oscillatory otherwise. A dynamic equation is said to be oscillatory if all its solutions are oscillatory.

Since we are interested in the oscillatory behavior of solutions of (.) and (.) near infinity, we assume throughout this paper that our time scale is unbounded above. An excellent introduction of time scales calculus can be found in the books by Bohner and Peterson [] and [].

The purpose of this paper is to extend the oscillation results given in [] and []. Os-cillation criteria for third-order dynamic equations have recently been studied in [–]. Other papers related with oscillation of higher-order dynamic equations can be found in [, ]. Well-known books concerning the oscillation theory are [, ].

For simplification, we define the following operators:

Lx(t) =x(t), Lx(t) = 

a(t)

L_x(t)α, Lx(t) =Lx(t), Lx(t) =Lx(t).

Thus (.) and (.) can be written as

Lx(t) +q(t)f

xg(t)= 

and

Lx(t) =q(t)f

xg(t)+p(t)hxk(t),

respectively. In what follows we use the following notation. For (t,s,T)∈ [s,∞)_T× [T,∞)_T×[t,∞)T

A(t,s) =

t

s

aα_(u)(u–s)α_{u (.)}

and

B(t,s) =

t

s

aα_(u)(t–u)α_{u. (.)}

2 Oscillation criteria for (1.1)

In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (.).

Theorem . Let(.)and(.)hold and assume that

f(uα)

u ≥c>  (.)

for u= and a constant c.If for t∈[t,∞)T

lim sup t→∞

t

g(t)

q(s)fAg(s),t

s>

and

lim sup t→∞

t

g(t)

q(s)fBg(t),g(s)s>

c, (.)

where A and B are defined as in(.)and(.),respectively,then(.)is oscillatory.

Proof Letxbe a nonoscillatory solution of (.) and assume that without loss of generality

x(t) >  fort∈ [t,∞)Tand soLx(t)≤ eventually fort∈ [t,∞)T. Therefore, there exists at∈[t,∞)Tsuch thatLx(t) andLx(t) are of one sign for allt∈[t,∞)T. We now distinguish the following two cases:

(I) Lx(t) > andLx(t) > eventually; (II) Lx(t) < andLx(t) > eventually. We now start with the first case.

(I) Assume thatLx(t) >  andLx(t) >  fort∈[t,∞)T. Then we have

t

t

Lx(s)s=Lx(t) –Lx(t)≤Lx(t), t∈[t,∞)T.

SinceLxis nonincreasing, we have

(t–t)Lx(t)≤Lx(t), t∈[t,∞)T.

This implies that

x_(t)≥aα_{(t)(t–t)α}_Lα

x(t), t∈[t,∞)T.

Integrating the above inequality fromttot, we have

x(t)≥L



α

x(t)

t

t

aα_{(s)(s–t)}α_s=A(t,t)L



α

x(t), t∈[t,∞)T,

whereAis defined as in (.). Hence there existst∈[t,∞)Tsuch that

xg(t)≥Ag(t),t

L



α

x

g(t), t∈[t,∞)_T. (.)

From (.) and (.) in (.) we obtain

–Lx(t)≥q(t)f

Ag(t),t

fL



α

x

g(t), t∈[t,∞)_T.

Integrating the above inequality fromg(t) tot, we obtain

Lx

g(t)≥

t

g(t)

q(s)fAg(s),t

fL



α

x

g(s)s, t∈[t,∞)T

or

Lx

g(t)≥fL



α

x

g(t)

t

g(t)

q(s)fAg(s),t

Dividing both sides of the above inequality byf(L



α

x[g(t)]), taking thelim supof both sides ast→ ∞and using (.), we obtain a contradiction to (.).

(II) Assume thatLx(t) < ,Lx(t) >  fort∈[t,∞)T. Then we have

t

s

Lx(u)u=Lx(t) –Lx(s)≤–Lx(s), (t,s)∈[s,∞)T× [t,∞)T.

SinceLxis nonincreasing, we have

–Lx(s)≥(t–s)Lx(t), (t,s)∈ [s,∞)T×[t,∞)T

or

–x(s)≥aα_(s)(t–s)α_L



α

x(t), (t,s)∈[s,∞)T×[t,∞)T.

Integrating the above inequality fromstotwe obtain

x(s)≥B(t,s)L



α

x(t), (t,s)∈[s,∞)T×[t,∞)T, (.)

whereBis defined as in (.). Then there existst∈[t,∞)Tsuch that

xg(s)≥Bg(t),g(s)L



α

x

g(t), g(t),g(s)∈g(s),∞_T×[t,∞)_T. (.)

Integrating (.) fromg(t) totand using (.) along with the above inequality we have

Lx

g(t)≥

t

g(t)

q(s)fxg(s)s≥fL



α

x

g(t)

t

g(t)

q(s)fBg(t),g(s)s.

Dividing the above inequality byf(L



α

x[g(t)]), taking thelim supof both sides of the above inequality ast→ ∞and using (.), we obtain a contradiction to (.).

For the bounded solutions of (.) we have the following, which is immediate from The-orem ..

Corollary . In addition to(.)and(.),assume that(.)and(.)hold.Then all bounded solutions of (.)are oscillatory.

Now we prove the following result.

Theorem . Let(.)and(.)hold and assume that

lim u→

u

f(uα₎

= .

If

lim sup t→∞

t

g(t)

q(s)fBg(t),g(s)s> ,

Proof Letxbe a nonoscillatory bounded solution of (.). Without loss of generality as-sume that xis positive. Sincexsatisfies Case (II) in the proof of Theorem ., we have (.). Integrating (.) fromg(t) totand using (.) along with (.) yields

Lx

g(t)≥

t

g(t)

q(s)fxg(s)s≥fL



α

x

g(t)

t

g(t)

q(s)fBg(t),g(s)s.

By dividing the above inequality byf(L



α

x[g(t)]) and taking thelim supof both sides of the resulting inequality ast→ ∞, we obtain a contradiction. The proof is now complete.

We now consider a special case of (.) of the form



a(t)

x_(t)α

+q(t)xβ_{g(t)= , (.)}

whereβis a ratio of two odd positive integers, and we obtain some oscillatory criteria.

Theorem . Letα≥βand assume that ∞

q(s)Aβg(s),t

s=∞ (.)

and ∞

q(s)Bβg(s),t

s=∞, (.)

where A and B are defined as in(.)and(.),respectively.Then(.)is oscillatory.

Proof Letxbe a nonoscillatory solution of (.). Without loss of generality, assume thatx

is positive. We consider two cases as we did in the proof of Theorem .. (I) Assume thatLx(t) >  andLx(t) >  fort≥t. Using (.) in (.), we have

–L_x(t) =q(t)xβg(t)≥q(t)Aβg(t),t

L

β α

x

g(t), g(t)∈[t,∞)_T.

Integrating the above inequality fromg(t) touand lettingu→ ∞, we obtain

L–

β α

 x

g(t)≥

∞

g(t)

q(s)Aβ_g(s),t 

s.

Sinceα≥βand the right hand side of the above inequality is infinity by (.), we obtain a contradiction to the facts thatLxis positive and nondecreasing.

(II) Assume thatLx(t) < ,Lx(t) >  fort≥t. Then we have (.). Using (.) in (.), for (g(t),g(s))∈[g(s),∞)_T×[t,∞)T, we have

–L_xg(t)=q(t)xβg(t)≥q(t)Bβg(t),g(s)L



α

x

Integrating the above inequality fromg(t) touand lettingu→ ∞, we obtain

L–

β α

 x

g(t)≥

∞

g(t)

q(s)Bβ_g(s),t 

s.

Sinceα≥βand the right hand side of the above inequality is infinity by (.), we obtain a contradiction to the facts thatLxis positive and nondecreasing.

3 Oscillation criteria for (1.2)

In this section, we investigate some oscillation criteria for solutions of the third-order delay equation (.).

Theorem . Let(.)-(.)and(.)hold.Also assume that

h(uα)

u ≥b>  (.)

for u= and a constant b.If for t∈[t,∞)T

lim sup t→∞

k(t)

t

q(s)hAk(s),k(t)s>

b, (.)

and

lim sup t→∞

t

g(t)

q(s)fBg(t),t

s>

c, (.)

where A and B are defined as in(.)and(.),respectively,then(.)is oscillatory.

Proof Letxbe an eventually positive solution of (.). ThenLx(t)≥ eventually and so

Lx(t) andLx(t) are eventually of one sign. We now distinguish the following two cases: (I) Lx(t) > andLx(t) > eventually;

(II) Lx(t) > andLx(t) < eventually.

(I) Assume thatLx(t) >  andLx(t) >  fort∈[t,∞)T. Then we have

t

s

Lx(τ)τ=Lx(t) –Lx(s)≤Lx(t), (t,s)∈[s,∞)T× [t,∞)T.

SinceLxis nondecreasing, we have

Lx(t)≥(t–s)Lx(s), (t,s)∈[s,∞)T×[t,∞)T.

This implies that

x_(t)≥aα_(t)(t–s)α_Lα

x(s), (t,s)∈[s,∞)T×[t,∞)T.

Integrating both sides of the above inequality fromstot, we have

x(t)≥

t

s

aα_(u)(u–s)α_L



α

or

x(t)≥A(t,s)L



α

x(s), (t,s)∈[s,∞)T×[t,∞)T,

whereAis defined as in (.). Then

xk(τ)≥Ak(τ),k(t)L



α

x

k(t), k(t),τ,t∈[τ,∞)T×[t,∞)T×[t,∞)T.

Using the above inequality and (.) in (.), we have

Lx(τ)≥p(τ)h

xk(τ)≥p(τ)hAk(τ),k(t)hL



α

x

k(t).

Integrating both sides of the above inequality fromttok(t), we get

Lx[k(t)]

h(L



α

x[k(t)])

≥ k(t)

t

p(τ)hAk(τ),k(t)τ.

Taking thelim supof both sides of the above inequality ast→ ∞, we obtain a contradic-tion to (.).

(II) Assume thatLx(t) >  andLx(t) <  fort∈[t,∞)T. Then we have

t

s

Lx(u)u=Lx(t) –Lx(s), (t,s)∈[s,∞)T× [t,∞)T.

SinceLxis nondecreasing, we have

x(s)≥–aα_(s)(t–s)α_L



α

x(t), (t,s)∈[s,∞)T×[t,∞)T.

Integrating both sides of the above inequality fromttot, we have

x(t)≥–B(t,t)L



α

x(t), t∈[t,∞)T,

whereBis defined as in (.). This implies that there exists at∈[t,∞)Tsuch that

xg(t)≥–Bg(t),t

L



α

x

g(t), g(t)∈[t,∞)_T. (.)

Using the above inequality and (.) in (.), we have

Lx(t)≥q(t)f

xg(t)≥q(t)fBg(t),t

f–L



α

x

g(t).

Integrating both sides of the above inequality fromg(t) tot, we find

–Lx[g(t)]

f(–L



α

x[g(t)])

≥ t

g(t)

q(s)fBg(s),t

s.

Taking thelim supof both sides of the above inequality ast→ ∞, we obtain a

Theorem . Assume that

hα(u)

u ≥b> , (.)

and

fα(u)

u ≥c> , (.)

for u= and constants band c.If

lim sup t→∞

k(t)

t

a(s)

s

t u

t

p(r)ru 

α s> 

b

(.)

and

lim sup t→∞ B

g(t),t

∞

t

q(s)s 

α

> 

c

, (.)

where B is defined as in(.),respectively,then(.)is oscillatory.

Proof Letxbe an eventually positive solution of (.). As in the proof of Theorem . we have two cases to consider.

(I) Assume thatLx(t) >  andLx(t) >  fort∈ [t,∞)T. Integrating (.) fromttos yields

Lx(s)≥h

xk(t)

s

t

p(s)s.

Integrating the above inequality fromttos∈[s,∞)Tgives

x_(s )≥h



α_xk(t)a(s

)



α s

t s

t

p(s)ss



α

.

Again integrating the above inequality fromttok(t) we find

xk(t)≥hα_xk(t) k(t)

t

a(s)

s

t s

t

p(s)ss



α s.

Finally, dividing the above inequality byhα_{(x(k(t))) and taking thelim supof both sides of}

the above inequality ast→ ∞, we obtain a contradiction to (.). (II) Assume thatLx(t) >  andLx(t) <  fort∈[t,∞)T. Integrating

L_x(t)≥q(t)fxg(t)

fromtto∞we have

–Lx(t)≥f

xg(t)

∞

t

Using (.) along with the above inequality, we have

xg(t)≥Bg(t),t

fxg(t)

∞

t

q(s)s 

α

.

Dividing the above inequality byfα(x[g(t)]) and taking the lim supof both sides of the

resulting inequality ast→ ∞, we obtain a contradiction to (.). The proof is complete.

4 Examples

In this section we give examples to illustrate two of our main results. Recall

Theorem .([, Theorem .]) If f ∈Crdand t∈Tκ,then

σ(t)

t

f(τ)τ=μ(t)f(t).

And

Theorem .([, Theorem . (ii)]) If[a,b]consists of only isolated points and a<b,

then b

a

f(t)t=

t∈[a,b)

μ(t)f(t).

Our first example illustrates Theorem ..

Example . Consider the third-order delay dynamic equation



a(t)

x(t)α

+q(t)xα

t



= , (.)

wheret∈T= N. Hereα=

,q(t) =  + (–) lnt

ln,g(t) = t

, anda(u) =f(u) =u

α_{. Observe}

that ift∈T, then

q(t) =  + (–)n=

⎧ ⎨ ⎩

, nodd,

, neven.

First we show that (.) holds. Ifs= m_{andt= }n_{,m,n∈N, we have}

∞



aα_{(s)s= lim} t→∞

t



aα_{(s)s= lim} n→∞

ρ(n)

s=

s(s–s) = lim n→∞

n–

s=

s=∞.

It is clear thatf belongs toC(R,R), is nondecreasing forx= , and satisfiesxf(x) >  for

x=  and (.). Also, (.) holds since

fα(u)

u =

u

Observe that ifu= k_{ands= }m_{fork,m∈N, then}

Ag(s), =  ρ(m–)

u=

u(u– )= 

m–

k=

kk– = 

m–

k=

kk– .

Note k–  >  fork∈N. Hence

Ag(s), > 

m–

k=

k> ·(m–),

and sincef is nondecreasing andq(t)≥ onT, we obtain

q(s)fAg(s), ≥_(m–)_.

It follows that ift∈[,∞)_T

lim sup t→∞

t

g(t)

q(s)fAg(s),t

s≥lim sup n→∞

n–

s=_t

_(n–)_s

= _{lim sup}

n→∞

n–

m=n–

(n–)_m

> ,

and so (.) holds. It remains to show that (.) holds fort∈[,∞)_T. This requires that we determineB(g(t),g(s)). Using the above representations ofu,s,t, we have

Bg(t),g(s)= 

n–

k=m–

kn–– k> ·(n–)n–– n–.

The monotonicity off and the fact thatq(t)≥ onTyield

q(s)fBg(t),g(s)>  ·_(n–)_n–_{– }n–_{= } ·_(n–)_.

Therefore

lim sup t→∞

t

t



q(s)fBg(t),g(s)s>lim sup n→∞

·_(n–) n–

s=n–

μ(s)

=lim sup n→∞ 



 ·_(n–)

n–

m=n– m

> ,

which shows that (.) holds. By Theorem ., (.) is oscillatory.

Example . Consider the third-order advanced dynamic equation



a(t)

x(t)α

=q(t)xα

t q

+p(t)xαqt, (.)

whereαis the ratio of two positive odd integers andt∈T=qN,q> . Herea(t) = (_(q–)q _t)α_,

q(t) =

q q–++(–)

lnt

lnq

(q–)t ,p(t) =



t_qα_(q–),g(t) =_qt,k(t) =qt, andh(u) =f(u) =uα. Then

hα(u)

u =

fα(u)

u =

u

u=  =b=c> , u= ,

and so (.) and (.) hold. Next we show that (.) holds. Fort∈[,∞)_T, we have

lim sup t→∞

k(t)

t

a(s)

s

t u

t

p(r)ru 

α s

=lim sup t→∞

σ(t)

t

a(s)

s

t u

t

p(r)ru 

α s

+

σ(t)

σ(t)

a(s)

s

t u

t

p(r)ru 

α s

+

σ(t)

σ_(t)

a(s)

s

t u

t

p(r)ru 

α s

=lim sup t→∞

μ(t)

a(t)

t

t u

t

p(r)ru 

α

+μσ(t)aσ(t) σ(t)

t u

t

p(r)ru 

α

+μσ(t)aσ(t) σ(t)

t u

t

p(r)ru 

α

=lim sup t→∞

μσ(t)aσ(t)μ(t)

t

t

p(r)r 

α

+μσ(t)aσ(t) σ(t)

t u

t

p(r)ru 

α

=lim sup t→∞

μσ(t)aσ(t) 

α

μ(t)

t

t

p(r)r+μσ(t) σ(t)

t

p(r)r 

α

=lim sup t→∞

μσ(t)aσ(t) 

α_{μσ(t)μ(t)p(t)}α

=lim sup t→∞

(q– )qt(q– )α_qt



α q

(q– )q_t 

tα_{q(q– )}α

=lim sup t→∞

qα

Sinceq(t) >  for allt∈T, we have

∞

t

q(s)s≥ σ(t)

t

q(s)s=μ(t)q(t) = q

q– +  + (–) lnt

lnq _> q q– .

This implies

lim sup t→∞

Bg(t),t

∞

t

q(s)s 

α

≥

q q– 

 α lim sup t→∞ t q 

aα_(u)

t q–u

 α u = q q– 

 α lim sup n→∞ n– k= q

(q– )qk

qn q –q

k 

α

qk(q– )

=q

q q– 

 α lim sup n→∞ n– k= qn

q –q k



α

> .

Thus (.) holds. By Theorem ., (.) is oscillatory.

5 Discussion

While we were able to unify most results for (.) given in [] and [], the comparison result

TheoremLet(.)-(.)hold. If the first-order delay dynamic equations

y(t) +q(t)fAg(t),t

fyα_{g(t)=  (.)}

and

z_{(t) +q(t)f}

B

t+g(t)

 ,g(t)

f

zα

t+g(t)



=  (.)

are oscillatory, then(.)is oscillatory.

cannot be extended since t+g_(t) ∈Tis satisfied for few time scales. While the result holds forT=RandT=Z, this condition is not satisfied onqN,q> . Being aware that time scales are not generally closed under addition, we were able to prove the following.

Theorem . Let(.)-(.)hold.Furthermore,assume the delay function g:T→Tis a bijection.If the first-order delay dynamic equations

y(t) +q(t)fAg(t),t

fyα_{g(t)=  (.)}

and

z_{(t) +q(t)f}

B

t+g(t)

 ,g(t)

f

zα

t+g(t)



=  (.)

In order to prove Theorem ., it is necessary to define the function F: [t,∞)_T× [,∞)→[,∞) to be a nondecreasing function with respect to its second argument and with the property thatF(·,z(·))∈Crd( [t,∞)T, [,∞)) for any functionz∈Crd( [t,∞)T, [,∞)). It is also necessary to assume thatτ :T→Tis a bijection withτ(t) <tfor allt∈T

andlim_t→∞τ(t) =∞, and to use the following definition and theorem. Definition . Lett∈[t,∞)T. By a solution of the dynamic inequality

y(t) +Ft,yτ(t)≤ (.)

on an interval [t,∞)T, we mean a rd-continuous functiony defined on the interval [τ(t),∞)T, which is rd-continuously differentiable on [t,∞)Tand satisfies (.) for all

t∈[t,∞)T. A solutionyof (.) is said to be positive ify(t) >  for everyt∈[τ(t),∞)T.

Theorem . Let y be a positive solution on an interval [t,∞)T,t ≥t of the delay

dynamic inequality(.).Moreover,we assume that F is positive on any set of the form

[t,τ–(t)]_T×(,∞),t∈[t,∞)T.Then there exists a positive solution x on [τ–(t),∞)T

of the delay dynamic equation

x(t) +Ft,xτ(t)=  (.)

such that

lim t→∞x(t) = 

and

x(t)≤y(t) for every t∈[t,∞)T.

Proof Letybe a positive solution (.). From (.) we obtain for allt,t∈Twitht≥t≥t

y(t)≥y(˜t) +

t

t

Fs,yτ(s)s>

t

t

Fs,yτ(s)s. (.)

Hence, lettingt→ ∞we get

y(t)≥

∞

t

Fs,yτ(s)s for everyt∈[t,∞)T. (.)

LetXbe the set of all nonnegative continuous functionsxon the interval [t,∞)Twith

x(t)≤y(t) for everyt≥t. Then by using (.) we can easily verify that for any functionx

inXthe formula

(Sx)(t) =

⎧ ⎨ ⎩

∞

t F(s,x[τ(s)])s, t∈[τ

–_(t ),∞)T,

∞

τ–_(t)F(s,x[τ(s)])s+

τ–(t)

t F(s,y[τ(s)])s, t∈[t,τ–(t))T

argument. Thus, the operatorSis monotone. Next, we put

x=y, and xν=Sxν–, ν= , , . . .

and observe that{xν}ν=,,is a decreasing sequence of functions inX. Furthermore, define

x= lim

ν→∞xνpointwise on [t,∞)T.

Then by applying the Lebesgue dominated convergence theorem, we obtainx=Sx. That is

x(t) =

⎧ ⎨ ⎩

∞

t F(s,x[τ(s)])s ift∈[τ

–_(t ),∞)T,

_∞

τ–_(t)F(s,x[τ(s)])s+

τ–(t)

t F(s,y[τ(s)])s ift∈[t,τ–(t))T.

(.)

From (.) it follows that

x(t) +Ft,xτ(t)=  for allt∈τ–(t),∞_T

and hencexis a solution of (.) on the intervalt∈[τ–(t),∞)T. Also (.) yields

lim

t→∞x(t) = .

Moreover, it is clear thatx(t)≤y(t) fort∈[t,∞)T. It remains to prove thatxis positive on the interval [t,∞)T. Taking into account thatyis positive on the interval [τ(t),∞)T and the positivity ofFon [t,τ–(t)]T×(,∞) we have

x(t)≥

τ–(t)

t

Fs,yτ(s)s> 

for eacht∈[t,τ–_(t))

T. So,xis positive on an interval [t,τ–(t))T. Next we will show thatxis also positive on [τ–(t),∞)T. Assume thattis the first zero ofxin [τ–(t),∞)T. Thenx(t) >  fort∈[τ–(t),t)Tandx(t) = . Then (.) yields

 =x(t) =

∞

t

Fs,xτ(s)s

and consequently

Fs,xτ(s)=  for alls∈[t,∞)_T.

That is, we can choose at∗∈[t,τ–_(t))

Tsuch that

Ft∗,xτt∗= .

On the other hand, taking into account thatx(t) >  fort∈[τ–_(t

),t)Tand the positivity ofFon [t,τ–(t)]T×(,∞) we get

Ft∗,xτt∗> .

Note that Theorem . holds for any unbounded time scaleT. Now we present the proof of Theorem ..

Proof Letxbe an eventually positive solution of (.). We consider two cases as we did in the proof of Theorem ..

(I) Assume thatLx(t) >  andLx(t) >  fort∈ [t,∞)T. Then (.) holds. Using this and (.) in (.), we obtain

–L_x(t) =q(t)fxg(t)≥q(t)fAg(t),t

fL



α

x

g(t), t∈[t,∞)T

or

y(t) +q(t)fAg(t),t

fyα_g(t)≤_{, t}∈_[t,∞_)T,

where y(t) :=Lx(t) for t∈ [t,∞)T. SinceLx(t) >  for allt∈[t,∞) andA(g(t),t) >  for allt∈[t,∞), by Theorem ., there exists a positive solutionzof (.) such that

limt→∞z(t) = , which contradicts the hypothesis that (.) is oscillatory.

(II) Assume thatLx(t) <  andLx(t) >  fort≥t. As in the proof of Theorem ., we have (.). Then fort∈[t,∞)_T, we have

xg(t)≥B

t+g(t)  ,g(t)

L



α

x

t+g(t)



.

Using this inequality and (.) in (.) fort∈[t,∞)Tyields

–L_x(t) =q(t)fxg(t)≥q(t)f

B

t+g(t)  ,g(t)

f

L



α

x

t+g(t)



or

–z(t)≥q(t)f

B

t+g(t)  ,g(t)

f

zα

t+g(t)



,

wherez(t) :=Lx(t) fort∈[t,∞)T. Similar to Case (I) above, by Theorem ., there exists a positive solutionyof (.) such thatlimt→∞y(t) = , which contradicts the fact that (.)

is oscillatory.

In [] and [], the authors prove a comparison result for (.) similar to the one given at the beginning of this section. That result involved t+k_(t). Again, since time scales are not generally closed under addition, this result cannot be extended to a general time scaleT.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

All authors read and approved this submitted manuscript.

Author details

1_{Department of Mathematics, Izmir University of Economics, Izmir, Turkey.}2_{Department of Mathematics and Statistics,} Missouri University of Science and Technology, Rolla, 65049, USA.3_{Department of Mathematics and Statistics, Texas Tech} University, Lubbock, 79423, USA.

References

1. Bohner, M, Peterson, A: Dynamic Equations on Time Scales: An Introduction with Applications. Birkhäuser Boston, Boston (2001)

2. Bohner, M, Peterson, A (eds.): Advances in Dynamic Equations on Time Scales. Birkhäuser Boston, Boston (2003) 3. Agarwal, RP, Grace, SR, Wong, PJY: On the oscillation of third order nonlinear difference equations. J. Appl. Math.

Comput.32(1), 189-203 (2010)

4. Agarwal, RP, Grace, SR, Wong, PJY: Oscillation of certain third order nonlinear functional differential equations. Adv. Dyn. Syst. Appl.2(1), 13-30 (2007)

5. Erbe, L, Peterson, A, Saker, SH: Oscillation and asymptotic behavior of a third-order nonlinear dynamic equation. Can. Appl. Math. Q.14(2), 129-147 (2006)

6. Erbe, L, Hassan, TS, Peterson, A: Oscillation of third order functional dynamic equations with mixed arguments on time scales. J. Appl. Math. Comput.34(1-2), 353-371 (2010)

7. Hassan, TS: Oscillation of third order nonlinear delay dynamic equations on time scales. Math. Comput. Model. 49(7-8), 1573-1586 (2009)

8. Han, Z, Li, T, Sun, S, Cao, F: Oscillation criteria for third order nonlinear delay dynamic equations on time scales. Ann. Pol. Math.99(2), 143-156 (2010)

9. Agarwal, R, Akın-Bohner, E, Sun, S: Oscillation criteria for fourth-order nonlinear dynamic equations. Commun. Appl. Nonlinear Anal.18(3), 1-16 (2011)

10. Grace, SR, Bohner, M, Sun, S: Oscillation of fourth-order dynamic equations. Hacet. J. Math. Stat.39(4), 545-553 (2010) 11. Agarwal, RP, Bohner, M, Grace, SR, O’Regan, D: Discrete Oscillation Theory, xiv+961 pp. Hindawi Publishing

Corporation, New York (2005)

12. Agarwal, RP, Grace, SR, O’Regan, D: Oscillation Theory for Difference and Functional Differential Equations, viii+337 pp. Kluwer Academic, Dordrecht (2000)

10.1186/1029-242X-2014-95