Behavior of solutions of a third order dynamic equation on time scales

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R E S E A R C H

Open Access

Behavior of solutions of a third-order

dynamic equation on time scales

M Tamer ¸Senel

*

*_{Correspondence:} [email protected] Department of Mathematics, Faculty of Science, Erciyes University, Kayseri, 38039, Turkey

Abstract

In this paper, we will establish some suﬃcient conditions which guarantee that every solution of the third-order nonlinear dynamic equation

(

r1(t)(r2(t)x(t))

)

+P

(

t,x(t),x(t))+F

(

t,x(t))= 0 oscillates or converges to zero on an arbitrary time scaleT.

1 Introduction

The theory of time scales, which has recently received a lot of attention, was introduced by Stefan Hilger in his Ph.D. Thesis [] in order to unify continuous and discrete analysis. A time scaleTis an arbitrary nonempty closed subset of the reals, and the cases when this time scale is equal to the reals or to the integers represent the classical theories of diﬀerential and of diﬀerence equations. A book on the subject of time scales by Bohner and Peterson [] summarizes and organizes much of the time scale calculus. Many other interesting time scales exist and they give rise to plenty of applications, among them the study of population dynamic models (see []). In the last few years, there has been much research activity concerning the oscillation and nonoscillation of solutions of some dy-namic equations on time scales, and we refer the reader to the papers [–].

Following this trend, in this paper, we will consider the third-order nonlinear dynamic equation

r(t)

r(t)x(t)

+Pt,x(t),x(t)+Ft,x(t)= , t≥t, ()

wherer(t),r(t) are positive, real-valued, rd-continuous functions deﬁned on the time

scale interval [a,b] (throughouta,b∈Twitha<b). We assume throughout that:

(i) F:T×R→Ris a function such thatuF(t,u) > , F(t,u)_u ≥f(t)foru= , (ii) P:T×R_→_R_{is a function such that} P(t,u,v)

u ≥q(t),uP(t,u,v) > foru= , (iii) q,f :T→Rare rd-continuous functions such thatq(t) +f(t) > for allt∈T.

Since we are interested in oscillatory behavior, we suppose that the time scale under con-sideration is not bounded above,i.e., it is a time scale interval of the form [a,∞). By a solution of (), we mean a nontrivial real-valued function satisfying equation () fort≥a. A solutionx(t) of () is said to be oscillatory if it is neither eventually positive nor eventu-ally negative; otherwise, it is called nonoscillatory. Equation () is said to be oscillatory if

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all its solutions are oscillatory. Our attention is restricted to those solutions of () which exist on some half-line [tx,∞) and satisfysup{|x(t)|:t>t}>  for anyt≥tx.

2 Some lemmas

In this section, we state and prove some lemmas which we will need in the proofs of our main results.

Lemma . Suppose that x is an eventually positive solution of()and

∞

t  r(t)

t=∞,

∞

t  r(t)

t=∞. ()

Then there is a t∈[t,∞)such that either (i) x(t) > ,x₍_t_{) > }_,₍_r

(t)x(t))> ,t∈[t,∞),or (ii) x(t) > ,x₍_t_{) < }_,₍_r

(t)x(t))> ,t∈[t,∞).

Proof Letxbe an eventually positive solution of (). Then there existst∈[to,∞) such

thatx(t) >  fort∈[t,∞). From () we have

r(t)

r(t)x(t)

= –Pt,x(t),x₍_t₎_–_F_t_,_x₍_t₎

≤–x(t)q(t) –x(t)f(t)

< 

fort∈[t,∞). Hence,r(t)(r(t)x(t)) is strictly decreasing on [t,∞). We claim that

r(t)(r(t)x(t)) >  on [t,∞). Assume not, then there is a t ∈ [t,∞) such that

r(t)(r(t)x(t))< ,t∈[t,∞). Then we can choose a negative constantcandt∈[t,∞)

such thatr(t)(r(t)x(t))≤c<  fort∈[t,∞). Dividing byr(t) and integrating fromt

tot, we obtain

r(t)x(t)≤r(t)x(t) +c

t

t s r(s)

.

Lettingt→ ∞, thenr(t)x(t)→–∞by (). Thus, there is at∈[t,∞) such that for

t∈[t,∞),r(t)x(t)≤r(t)x(t) < . Dividing byr(t) and integrating fromttot, we

obtain

x(t) –x(t)≤r(t)x(t)

t

t s r(s)

,

which implies thatx(t)→–∞ast→ ∞by (), a contradiction with the fact thatx(t) > . Hence, we have

r(t)x(t)

> , t∈[t,∞).

This implies thatr(t)x(t) is strictly increasing on [t,∞). It follows from this that either

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Lemma . Assume that

∞

t

ϕ(t)t=∞, ()

whereϕ(t) =q(t) +f(t)and x(t)is a solution of()that satisﬁes Case(ii)in Lemma.. Thenlim_t_→∞x(t) = .

Proof Letxbe a solution of () satisfying Case (ii) in Lemma ., that is,

x(t) > , x(t) < , r(t)x(t)

> , t∈[t,∞).

Thenlimt→∞x(t) =b≥. Assumeb>  and we now show that this leads to a contradic-tion. From () andx≥b,

r(t)

r(t)x(t)

= –Pt,x(t),x(t)–Ft,x(t)

≤–x(t)q(t) –x(t)f(t)

= –x(t)q(t) +f(t)

≤–bϕ(t),

whereϕ(t) =q(t) +f(t), fort∈[t,∞). Now let

u(t) =r(t)

r(t)x(t)

, t∈[t,∞),

then we have

u₍_t₎_≤_–_bϕ₍_t_), _t_∈_[_t ,∞).

Integrating the last inequality fromttot, we have

u(t)≤u(t) –b

t

t ϕ(s)s.

Using () it is possible to choose at∈[t,∞), suﬃciently large, such thatu(t) <  for all

t∈[t,∞), which is a contradiction, and this completes the proof.

Lemma . Assume that x(t)is a solution of()satisfying Case(i)of Lemma..Then there exists t∈[t,∞)such that

x(t)≥δ(t,t)r(t) r(t)

r(t)x(t)

for t≥t, ()

whereδ(t,t) = t

t

s r(s).

Proof From Case (i) of Lemma ., we havex(t) as a solution of () satisfying

x(t) > , x(t) > , r(t)x(t)

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for t≥ t. Using x(t) is a solution of (), we get (r(t)(r(t)x(t))) <  and hence

r(t)(r(t)x(t))is decreasing on [t,∞). Hence,

r(t)x(t) =r(t)x(t) +

t

t

r(s)(r(s)x(s))

r(s)

s

≥r(t)δ(t,t)

r(t)x(t)

, t≥t,

and this leads to () and the proof is complete.

3 Main results

In this section, we establish some suﬃcient conditions which guarantee that every solution x(t) of () oscillates on [t,∞) or converges ast→ ∞.

Theorem . Assume that()holds.Furthermore,assume that there exists a positive func-tion z(t)such that z₍_t₎_{is rd-continuous on}_[_t

,∞)and if

lim sup

t→∞

t

t

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s=∞, ()

for all suﬃciently large t,where Q(t,t) = z(t)_r_δ_(t)(t,t),ϕ(t) =q(t) +f(t),then every solution x(t)

of()is oscillatory orlim_t_→∞x(t)exists(ﬁnite).

Proof Letx(t) be a nonoscillatory solution of (). We only consider the case whenx(t) is eventually positive, since the case whenx(t) is eventually negative is similar. By Lemma . either Case (i) or Case (ii) in Lemma . holds. Assumex(t) satisﬁes Case (i) in Lemma .. Deﬁne the Riccati-type functionw(t) by

w(t) =z(t)r(t)(r(t)x

₍_t₎₎

x(t) , t≥t. ()

By the product rule,

w₍_t_{) =}_r 

σ(t)rx(t)

σ(t)z(t) x(t)

+z(t) x(t)

r(t)

r(t)x(t)

=r

σ(t)rx(t)

σ(t)z(t) x(t)

+z(t) x(t)

–Pt,x(t),x(t)–Ft,x(t).

Using () we have

w₍_t₎_≤_–_z₍_t₎_q₍_t_{) –}_z₍_t₎_f₍_t_{) +}_z₍_t₎w σ₍_t₎

zσ₍_t₎ –z(t)

x₍_t₎

x(t)xσ₍_t₎

r(t)

r(t)x(t) σ

.

Using () andx₍_t_{) > , we obtain}

w₍_t₎_≤_–_z₍_t₎_q₍_t_{) –}_z₍_t₎_f₍_t_{) +}_z₍_t₎w σ₍_t₎

zσ₍_t₎

–z(t)r(t)(r(t)x

₍_t₎₎₍_r

(t)(r(t)x(t)))σδ(t,t)

r(t)(xσ(t))

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Sincer(t)(r(t)x(t))is decreasing, we have

w(t)≤–z(t)ϕ(t) +z(t)w

σ₍_t₎

zσ₍_t₎ –z(t)

δ(t,t)

r(t)

(wσ₍_t₎₎

(zσ₍_t₎₎

= –z(t)ϕ(t) +z₍_t₎w σ₍_t₎

zσ₍_t₎ –Q(t,t)

(wσ₍_t₎₎

(zσ₍_t₎₎, ()

whereϕ(t) =q(t) +f(t) andQ(t,t) =z(t)_rδ_(t,t_(t)). From () we have

w₍_t₎_≤_–_z₍_t₎_ϕ₍_t_{) +} (z ₍_t₎₎

Q(t,t)

–Q(t,t)

wσ₍_t₎

zσ₍_t₎ –

 

 Q(t,t)

z₍_t₎ 

≤–

z(t)ϕ(t) – (z

₍_t₎₎

Q(t,t)

. ()

Integrating () fromttot, we obtain

–w(t)≤w(t) –w(t)≤–

t

t

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s, ()

which yields

t

t

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s≤w(t) ()

for all larget. This is contrary to () and so Case (i) is not possible. If Case (ii) in Lemma . holds, then clearlylimt→∞x(t) exists (ﬁnite).

On the basis of Lemma . and Theorem ., we have the following results.

Corollary . If () and () hold, then every solution of () is either oscillatory or limt→∞x(t) = .

Corollary . Assume that()holds.If

lim sup

t→∞

t

t

sϕ(s) – r(s) sδ(s,t)

s=∞,

then every solution of()is either oscillatory orlimt→∞x(t)exists.

Example . We consider the third-order dynamic equation

x(t) +a( + (x

₍_t₎₎₎

t x(t) + b

tx(t) = , a> ,b> ,t≥, ()

wherer(t) =r(t) = ,P(t,x(t),x(t)) =a( + (x(t)))x(t)/t,F(t,x(t)) =bx(t)/t. Letf(t) =

b/tandq(t) =a/t. It is not difficult to verify that all conditions of Corollary . are satisfied. Hence, every solution of equation () is oscillatory or satisfieslimt→∞x(t) = .

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Theorem . Assume that()holds.Furthermore,assume that there is a positive function z(t)such that z₍_t₎_{is rd-continuous on}_[_t

,∞),and for all suﬃcient large t,

lim sup

t→∞  tm

t

t (t–s)m

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s=∞, ()

where m≥,Q(t,t)andϕ(t)are as in Theorem..Then every solution of()is either

oscillatory orlim_t_→∞x(t)exists.

Proof Proceeding as in Theorem ., we assume that () has a nonoscillatory solution, say x(t) >  for allt≥t, wheretis chosen so large that Lemma . and Lemma . hold. By

Lemma . there are two possible cases. First, if Case (i) holds, then by deﬁning againw(t) by () as in Theorem ., we havew(t) >  and (), that is,

z(t)ϕ(t) – (z

₍_t₎₎

Q(t,t) ≤

–w(t).

Therefore,

t

t (t–s)m

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s≤–

t

t

(t–s)mw₍_s₎_s_. _()

Integrating by parts of the right-hand side leads to

t

t

(t–s)mw(s)s= (t–s)mw(s)|t_t_–

t

t

h(t,s)wσ(s)s

= –(t–t)mw(t) –

t

t

h(t,s)wσ(s)s, ()

whereh(t,s) := ((t–s)m₎s_{. Note that}

h(t,s) =

⎧ ⎨ ⎩

–m(t–s)m–_, _μ₍_s_{) = ;} (t–σ(s))m–(t–s)m

μ(s) , μ(s) > ,

andm≥,h(t–s)≤ fort≥σ(s). It follows from () that

t

t

(t–s)mw₍_s₎_s_≥_–(_t_–_t

)mw(t).

Then from () we have

t

t (t–s)m

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s≤(t–t)mw(t).

Then

 tm

t

t (t–s)m

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s≤

t–t

t

m

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Hence,

lim sup

t→∞

 tm

t

t (t–s)m

z(s)ϕ(s) – (z

₍_s₎₎

Q(s,t)

s≤w(t),

which is a contradiction of (). If Case (ii) holds, then, as before,limt→∞x(t) exists and

the proof is complete.

Competing interests

The author declares that they have no competing interests.

Acknowledgements

This work was supported by the Research Fund of the Erciyes University. Project Number: FBA-11-3391. The author is grateful to the anonymous referee for a careful checking of the details and for helpful comments that improved this paper.

Received: 6 November 2012 Accepted: 4 February 2013 Published: 12 February 2013 References

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