Asymptotic properties of solutions of third order nonlinear neutral dynamic equations

R E S E A R C H

Open Access

Asymptotic properties of solutions of

third-order nonlinear neutral dynamic

equations

Tuncay Candan

*

*_{Correspondence:}

[email protected] Department of Mathematics, Faculty of Art and Science, Ni ˘gde University, Ni ˘gde, 51200, Turkey

Abstract

The purpose of this article is to give oscillation criteria for the third-order neutral dynamic equation (r2(t)[(r1(t)[y(t) +p(t)y(

τ

(t))])]γ)+f(t,y(

δ

(t))) = 0, where

γ

≥1 is a ratio of odd positive integers withr1(t),r2(t), andp(t) are positive real-valued rd-continuous functions defined onT. We give new results for the third-order neutral dynamic equations and an example to illustrate the importance of our results.

Keywords: oscillation; dynamic equations; time scales; neutral equations

1 Introduction

In the present article, we are concerned with oscillations of the third-order nonlinear neu-tral dynamic equation

r(t)

r(t)

y(t) +p(t)yτ(t)γ+ft,yδ(t)=  ()

on a time scaleT. Throughout this paper it is assumed thatγ ≥ is a ratio of odd positive integers,τ(t) :T→Tandδ(t) :T→Rare rd-continuous functions such thatτ(t)≤t,

δ(t)≤t,limt→∞δ(t) =limt→∞τ(t) =∞andδ(t) >  is rd-continuous,r(t),r(t) andp(t) are positive real valued rd-continuous functions defined onT, ≤p(t)≤p<  is increas-ing. We define the time scale interval [t,∞)Tby [t,∞)T= [t,∞)∩T. Furthermore, f :T×R→Ris a continuous function such thatuf(t,u) >  for allu=  and there exists a rd-continuous positive functionq(t) defined onTsuch that|f(t,u)| ≥q(t)|uγ_|.

We use throughout this paper the following notations for convenience and for shorten-ing the equations:

x(t) =y(t) +p(t)yτ(t), x[]=rx

,

x[]=r

x[]γ, x[]=x[].

()

A nontrivial functiony(t) is said to be a solution of () ifx∈C_rd[ty,∞),rx∈Crd[ty,∞) andx[]∈C_rd[ty,∞) forty≥tandy(t) satisfies equation () forty≥t. A solution of () which is nontrivial for all largetis said to be oscillatory if it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory.

Recently, there has been many important research activity on the oscillatory behavior of dynamic equations. For example, on second-order dynamic equations, Saker [], and

Agarwalet al.[], Saker [], Hassan [] and Candan [, ] considered the following non-linear dynamic equations:

r(t)y(t) +p(t)y(t–τ)γ+ft,y(t–δ)= ,

r(t)y(t) +p(t)yτ(t)γ+ft,yδ(t)= ,

r(t)x(t)γ+p(t)xγ(t) = ,

and

r(t)y(t) +p(t)yτ(t)γ+ d

c

ft,yθ(t,ξ)ξ= ,

respectively, and they gave sufficient conditions which guarantee that every solution of the equation oscillates. Moreover, there are also some papers on third-order dynamic equa-tions. For instance, Erbeet al.[] considered the third-order nonlinear dynamic equation

c(t)a(t)x(t)+q(t)fx(t)= .

Later, Erbeet al.[] considered the third-order nonlinear dynamic equation

x(t) +p(t)x(t) = 

by giving Hille and Nehari type criteria. Then, Hassan [] studied the third-order nonlin-ear dynamic equation

a(t)r(t)x(t)γ+ft,xτ(t)= .

Lastly, Wang and Xu [] studied asymptotic properties of a certain third-order dynamic equation,

r(t)

r(t)x(t)

γ

+q(t)fx(t)= .

As we see from all the above, our equation, a neutral dynamic equation, is more general than other third-order dynamic equations and therefore it is very important. For some other important articles on oscillations of second-order nonlinear neutral delay dynamic equation on time scales and oscillations of third-order neutral differential equations, we refer the reader to the papers [, ], and [], respectively. We give [, ] as references for books on the time scale calculus.

2 Main results

Lemma  Assume that y is an eventually positive solution of()and

∞

t t r(t)

=∞,

∞

t

 r(t)



γ

t=∞. ()

Then,there is a t∈[t,∞)Tsuch that either

or

(ii) x(t) > , x(t) < , x[](t) > , t∈[t,∞)T.

Proof Assume thaty(t) >  fort≥tand thereforey(τ(t)) >  andy(δ(t)) >  fort≥t>t. Consequently,x(t) > , eventually. Using () in () and the fact that|f(t,u)| ≥q(t)|uγ_{|, we} obtain

x[](t) +q(t)yδ(t)γ ≤, t∈[t,∞)T. ()

Hence, we conclude that x[]_{(t) is a strictly decreasing function on [t}

,∞)T. We claim

thatx[]_{(t) >  on [t}

,∞)T. If not, then there exists at∈[t,∞)Tsuch thatx[](t) <  on

[t,∞)T. Then, there exist a negative constantcandt∈[t,∞)Tsuch that

x[](t)≤c< , t∈[t,∞)T

and it follows that

x[](t)≤

c r(t)



γ

. ()

Integrating () fromttotand using (), we obtain

r(t)x(t)≤r(t)x(t) +c



γ t

t

 r(s)



γ

s,

which implies thatr(t)x(t)→–∞ast→ ∞. Therefore, there exists at∈[t,∞)Tsuch

that

r(t)x(t)≤r(t)x(t) < , t∈[t,∞)T. ()

Dividing both sides of () byr(t) and integrating fromttot, we obtain

x(t) –x(t)≤r(t)x(t) t

t

 r(s)

s.

Hence, we see from () thatx(t)→–∞ast→ ∞, which contradicts the fact thatx(t) > , and thereforex[]_{(t) >  fort∈[t}

,∞)T. As a result ofx[](t) >  fort∈[t,∞)Tit follows

thatr(t)x(t) <  on [t,∞)T orr(t)x(t) >  on [t,∞)T, which completes the proof.

Lemma  Let y be an eventually positive solution of().Assume that Case(i)of Lemma holds.Then,there exists a t∈[t,∞)Tsuch that

x(t)≥r(t,t) r(t)

x[](t)γ_{, t}∈_[t

where r(t,t) = _tt_ s (r(s))



γ and

x(t)≥r(t,t)

x[](t)γ_{, t}∈_[t

,∞)T,

where r(t,t) = _tt_r_r(_s₍,_st₎)s.

Proof Sincex[]_{(t) is strictly decreasing on [t}

,∞)T, we have

r(t)x(t)≥r(t)x(t) –r(t)x(t)

= t

t

[x[]_(s)]γ

(r(s))



γ

s,

it follows that

x(t)≥[x []_(t)]γ

r(t) t

t s (r(s))



γ

or

x(t)≥r(t,t) r(t)

x[](t)



γ_{, t}∈_[t

,∞)T. ()

Similarly, integrating () fromttot, we obtain

x(t)≥x[](t)γ t

t

r(s,t) r(s)

s

or

x(t)≥r(t,t)

x[]_(t)γ_{, t}∈_[t

,∞)T.

This completes the proof.

Lemma  Let y be an eventually positive solution of().Assume that Case(ii)of Lemma holds.If

∞

t

 r(t)

∞

t

 r(s)

∞

s

q(u)u /γ

st=∞, ()

thenlimt→∞y(t) = .

Proof Since Case (ii) of Lemma  is satisfied,

lim

t→∞x(t) =l≥.

We claim thatlim_t→∞x(t) = . Assume thatl> . Then for any> , we havel<x(t) <l+

for sufficiently larget≥t. Choose  <<l(–_pp). On the other hand, since

we have

y(t)≥x(t) –pxτ(t)

>l–p(l+)

=k(l+)

>kx(t), t≥t≥t,

wherek=l–p_l+(l+) > . Then,

yδ(t)γ ≥kγxδ(t)γ, t≥t≥t. ()

Substituting () into (), we obtain

x[](t)≤–q(t)kγxδ(t)γ, t≥t. ()

Integrating () fromtto∞, we get

x[](t)≥kγ

_∞

t

q(s)xδ(s)γs, t≥t

or usingx(δ(t)) >l,

x[](t)≥kl

 r(t)

∞

t

q(s)s /γ

, t≥t. ()

Integrating () fromtto∞and dividing both sides byr(t), we have

–x(t)≥ kl r(t)

∞

t

 r(u)

∞

u

q(s)s /γ

u, t≥t. ()

Integrating () fromtto∞, we obtain

x(t)≥kl

∞

t

 r(t)

∞

t

 r(s)

∞

s

q(u)u /γ

st,

which contradicts () and thereforel= . By making use of ≤y(t)≤x(t), we conclude

thatlimt→∞y(t) = .

Theorem . Assume thatδ(σ(t)) =σ(δ(t)).Furthermore,suppose that(), (),and

∞

t

Q(s)s=∞, ()

where Q(s) =q(s)( –p)γ_{, hold. Then, every solution y(t) of ()is either oscillatory on} [t,∞)Torlimt→∞y(t) = .

the case wheny(t) is negative the proof is similar. As we see from Lemma  we have two cases to consider. First we assume thatx(t) satisfies Case (i) in Lemma . Then, by using () we see that

y(t)≥x(t) –p(t)xτ(t)≥( –p)x(t), t≥t≥t

or

yδ(t)γ ≥( –p)γxδ(t)γ, t≥t≥t. ()

Substituting () into (), we obtain

x[](t)≤–q(t)( –p)γ_xδ(t)γ _{= –Q(t)xδ(t)}γ_{, t≥t}

. ()

Furthermore, using Pötzche’s chain rule, we find

xδ(t)γ

=γ





hxδ(t)σ+ ( –h)xδ(t)γ–xδ(t)dh

≥γ





hxδ(t)+ ( –h)xδ(t)γ–xδ(t)dh

=γxδ(t)γ–xδ(t)δ(t) > . ()

Define the function

z(t) = x []_(t)

(x(δ(t)))γ, t≥t. ()

It is easy to see thatz(t) > . Taking the derivative ofz(t), we see that

z(t) = x []_(t)

(x(δ(t)))γ +

x[](t)σ

 (x(δ(t)))γ

= x []_(t)

(x(δ(t)))γ –

x[](t)σ ((x(δ(t))) γ₎

(x(δσ_(t)))γ_(x(δ(t)))γ. ()

Substituting () into () and using (), respectively, we have

z(t)≤–Q(t) –x[](t)σ ((x(δ(t))) γ₎

(x(δσ_(t)))γ_(x(δ(t)))γ

≤–Q(t) –x[](t)σγ(x(δ(t)))

γ–_{x(δ(t))δ(t)}

(x(δσ_(t)))γ_(x(δ(t)))γ

= –Q(t) –γx[](t)σ x

_(δ(t))δ(t)

(x(δσ_(t)))γ_x(δ(t))

≤–Q(t) –γx[](t)σx

_(δ(t))δ(t)

Using () in () and the fact thatx[]_{(t) is strictly decreasing, we obtain from ()}

Finally, integrating () fromttot, we get

z(t) –z(t)≤

which contradicts (). When Case (ii) holds, we can conclude from Lemma  that

lim_t→∞y(t) = .

Theorem . Suppose that(), ()hold andδ(σ(t)) =σ(δ(t)).Furthermore,assume that there exists a positive rd-continuous-differentiable functionα(t)such that

lim sup

Proof Suppose to the contrary thaty(t) is nonoscillatory solution of (). We assume that y(t) >  fort≥t, theny(τ(t)) >  andy(δ(t)) >  fort≥t>t. We first consider thatx(t) satisfies Case (i) in Lemma . We proceed as in the proof of Theorem ., and we obtain (). Let us define the function

z(t) =α(t) x []_(t)

(x(δ(t)))γ, t≥t. ()

It is clear thatz(t) > . Taking the derivative ofz(t), we see that

Substituting () into (), we obtain

By making use of the inequality

λABλ––Aλ≤(λ– )Bλ, λ> ,A,B≥ ()

Integrating both sides of () fromttotthen yields

t continuous-partial derivativeHs_{(t,s) on}D

with respect to the second variable.

where C(t,s) =max{,Hs_{(t,s) +}H(t,s)(α(s))+

ασ_(s) }.Then every solution y(t)of()is either oscil-latory on [t,∞)Torlimt→∞y(t) = .

Proof Assume thaty(t) is a nonoscillatory solution of (). Definez(t) as in (). We proceed as in the proof of Theorem . to obtain (). Multiplying both sides of () byH(t,s), integrating with respect tosfromttot, we get

t

Then, using the inequality () in (), we have t

When Case (ii) holds, we can conclude from Lemma  thatlim_t→∞y(t) = .

Example . Consider the following third-order neutral nonlinear dynamic equation:

t

t

y(t) + 

y

t 



+ ty



t 

= , t∈[t,∞)T,t> , ()

where γ = ,r(t) =t, r(t) =r(t) =t p(t) = _,τ(t) =δ(t) = t_, andq(t) = t–. We can verify that all conditions of Theorem . are satisfied, therefore every solution of () is oscillatory orlim_t→∞y(t) = . In fact,y(t) =t–_{is a solution of ().}

Competing interests

The author declares that they have no competing interests.

Received: 11 September 2013 Accepted: 6 January 2014 Published:27 Jan 2014

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Cite this article as:Candan:Asymptotic properties of solutions of third-order nonlinear neutral dynamic equations.