Asymptotic behavior of a third order nonlinear neutral delay differential equation

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R E S E A R C H

Open Access

Asymptotic behavior of a third-order

nonlinear neutral delay differential equation

Ying Jiang

1

_{and Tongxing Li}

2,3*

*_{Correspondence:}

[email protected]

2_{LinDa Institute of Shandong}

Provincial Key Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P.R. China

3_{School of Informatics, Linyi}

University, Linyi, Shandong 276005, P.R. China

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

Abstract

The objective of this paper is to study asymptotic nature of a class of third-order neutral delay diﬀerential equations. By using a generalized Riccati substitution and the integral averaging technique, a new Philos-type criterion is obtained which ensures that every solution of the studied equation is either oscillatory or converges to zero. An illustrative example is included.

MSC: 34K11

Keywords: asymptotic behavior; third-order neutral diﬀerential equation; oscillation; generalized Riccati substitution

1 Introduction

In this work, we study the oscillation and asymptotic behavior of a third-order nonlinear neutral diﬀerential equation with variable delay arguments

r(t)x(t) +P(t)xt–τ(t)+ m

i= Qi(t)fi

xt–σi(t)= , t≥t, ()

wherem≥ is an integer andt> . We assume that the following hypotheses are

satis-ﬁed.

(A) r∈C([t,∞), (,∞)),P,τ,Qi,σi∈C([t,∞), [,∞)),fi∈C(R,R), andufi(u) > for

u= ,i= , , . . . ,m; (A) r(t)≥,

∞

t r

–₍_t_{) d}_t₌_∞_{, and}__≤_P₍_t₎_≤_p < ;

(A) limt→∞(t–τ(t)) =limt→∞(t–σi(t)) =∞,i= , , . . . ,m;

(A) there exist constantsαi> such thatfi(u)/u≥αiforu= andi= , , . . . ,m. Throughout, we deﬁne

z(t) :=x(t) +P(t)xt–τ(t). ()

By a solution of equation (), we mean a functionx∈C([Tx,∞),R),Tx≥t, which has the

propertiesz∈C_([_T_x,_∞_),_R_),_rz_∈_C_([_T_x,_∞_),_R_{), and satisﬁes () on [}_T_x,_∞_{). We}

con-sider only those solutionsxof () which satisfy assumptionsup{|x(t)|:t≥T}>  for all

T≥Tx. We assume that () possesses such solutions. A solution of () is called oscillatory if it has arbitrarily large zeros on [Tx,∞); otherwise, it is termed nonoscillatory.

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As is well known, the third-order differential equations are derived from many different areas of applied mathematics and physics, for instance, deflection of buckling beam with a fixed or variable cross-section, three-layer beam, electromagnetic waves, gravity-driven flows,etc.In recent years, the oscillation theory of third-order differential equations has received a great deal of attention since it has been widely applied in research of physi-cal sciences, mechanics, radio technology, lossless high-speed computer network, control system, life sciences, and population growth.

Numerous research activities are concerned with the oscillation of solutions to different functional differential equations, for some related contributions, we refer the reader to [–] and the references cited therein. In the following, we provide some background details regarding the study of various classes of neutral differential equations. Baculíková and Džurina [] studied a second-order neutral differential equation

r(t)x(t) +p(t)xτ(t)+q(t)xσ(t)= .

Agarwalet al.[], Graceet al.[], and Zhanget al.[] considered a third-order nonlinear diﬀerential equation

a(t)b(t)x(t)+q(t)xγσ(t)= .

Baculíková and Džurina [], Candan [, ], Karpuz [], Li and Rogovchenko [], Li and Thandapani [], and Liet al.[, ] investigated a class of third-order neutral diﬀerential equations

r(t)x(t) +p(t)xτ(t)+q(t)xσ(t)= . () Deﬁneτ˜(t) :=t–τ(t) andσ˜i(t) :=t–σi(t),i= , , . . . ,m. It follows from conditions (A)

and (A) thatτ˜(t)≤t,σ˜i(t)≤t, andlimt→∞τ˜(t) =limt→∞σ˜i(t) =∞,i= , , . . . ,m. Hence,

equation () is a special case of (). As a matter of fact, equation () reduces to the form of () ifm=  andf(u) =u.

There are two techniques in the study of oscillation of third-order neutral differential equations. One of them is comparison method which is used to reduce the third-order neutral differential equations to the first-order differential equations or inequalities; see,

e.g., [–]. Another technique is the Riccati technique; see,e.g., [–, –]. In this pa-per, using ageneralizedRiccati substitution which diﬀers from those reported in [–, –], a new asymptotic criterion for () is presented. In what follows, all functional in-equalities are tacitly supposed to hold for all suﬃciently larget.

2 Some lemmas

Lemma  Assume that conditions(A)-(A)hold and x is a positive solution of().Then there are only the following two possible cases for z deﬁned by():

(I) z(t) > ,z(t) > ,z(t) > ,z(t)≤,and(r(t)z(t))≤; (II) z(t) > ,z(t) < ,z(t) > ,z(t)≤,and(r(t)z(t))≤,

for t≥T,where T≥tis suﬃciently large.

Proof The proof is similar to that of Baculíková and Džurina [, Lemma ], and hence is

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Lemma  Assume that conditions(A)-(A)hold and let x be a positive solution of()and corresponding z satisfy case(II)in Lemma.If

∞

t

∞

v 

r(u) m

i=

∞

u

Qi(s) ds

dudv=∞, ()

thenlim_t→∞x(t) =lim_t→∞z(t) = .

Proof Suppose thatxis a positive solution of (). Sincez(t) >  andz(t) < , there exists a ﬁnite constantl≥ such thatlim_t→∞z(t) =l≥. We shall prove thatl= . Assume now thatl> . Then for anyε> , there exists at≥Tsuch thatl+ε>z(t) >lfort≥t. Choose

 <ε<l( –p)/p. It is not hard to ﬁnd that

x(t) =z(t) –P(t)xt–τ(t)>l–P(t)xt–τ(t)>l–pz

t–τ(t)

>l–p(l+ε) :=N(l+ε) >Nz(t), ()

whereN:= (l–p(l+ε))/(l+ε) > . Using () and (), we conclude that

 =r(t)z(t)+ m

i= Qi(t)fi

xt–σi(t)

≥r(t)z(t)+ m

i=

αiQi(t)x

t–σi(t)

≥r(t)z(t)+N

m

i=

αiQi(t)z

t–σi(t)

≥r(t)z(t)+N

m

i=

αiQi(t)z(t). ()

Integrating () fromtto∞, we obtain

≥–r(t)z(t) +N

m

i=

αi

∞

t

Qi(s)z(s) ds.

Noting thatz(t) >l, we get

≥–z(t) + lN

r(t) m

i=

αi

_∞

t

Qi(s) ds. ()

Integrating () fromtto∞, we have

≥z(t) +lN

∞

t 

r(u) m

i=

αi

∞

u

Qi(s) ds

du. ()

Integrating () fromtto∞, we deduce that

_∞

t

_∞

v 

r(u) m

i=

αi

_∞

u

Qi(s) ds

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which contradicts (). Hence,l=  andlim_t→∞z(t) = . Then it follows from ≤x(t)≤

z(t) thatlim_t→∞x(t) = . The proof is complete.

Lemma (See [, Lemma ]) Assume that u(t) > ,u(t) > ,and u(t)≤for t≥t.If

σ∈C([t,∞), [,∞)),σ(t)≤t,andlimt→∞σ(t) =∞,then for everyα∈(, ),there exists a Tα≥tsuch that u(σ(t))/σ(t)≥αu(t)/t for t≥Tα.

Remark  Ifusatisﬁes conditions of Lemma , thenu(t–σi(t))/u(t)≥α(t–σi(t))/tfor

i= , , . . . ,mwhen using conditions (A) and (A).

Lemma (See [, Lemma ]) Assume that u(t) > ,u(t) > ,u(t) > ,and u(t)≤for t≥t.Then for eachβ∈(, ),there exists a Tβ≥tsuch that u(t)≥βtu(t)/for t≥Tβ. Remark  If u satisﬁes conditions of Lemma , then u(t–σi(t))/u(t–σi(t))≥β(t–

σi(t))/ fori= , , . . . ,mwhen using condition (A). 3 Main results

We use the integral averaging technique to establish a Philos-type (see Philos []) crite-rion for (). Let

D:=(t,s) :t≥s≥t

and D:=(t,s) :t>s≥t

.

We say that a functionH∈C(D,R) belongs to the classXif (i) H(t,t) = ,t≥t,H(t,s) > ,(t,s)∈D;

(ii) Hhas a nonpositive continuous partial derivative∂H/∂sonDwith respect to the second variable, and there exist functionsρ∈C_([_t

,∞), (,∞)),δ∈C([t,∞),R),

andh∈C(D,R)such that

∂H(t,s)

∂s +

δ(s) +ρ ₍_s₎

ρ(s)

H(t,s) = –h(t,s)H(t,s). ()

Theorem  Assume that conditions(A)-(A)and()are satisﬁed.If

lim sup t→∞



H(t,t)

t

t

H(t,s)G(s) – 

ρ(s)r(s)h

₍_t_,_s₎

ds=∞ ()

holds for someα∈(, ),β∈(, ),and for some H∈X,where

G(t) :=ρ(t)

αβ( –p)

 m

i=

αiQi(t)

(t–σi(t))

t +r(t)δ

₍_t_{) –}_r₍_t₎_δ₍_t₎

, ()

then every solution x of()is either oscillatory or satisﬁeslim_t→∞x(t) = .

Proof Suppose to the contrary and assume that () has a nonoscillatory solutionx. With-out loss of generality, we can assume that there exists at≥tsuch thatx(t) > ,x(t–τ(t)) >

, andx(t–σi(t)) >  fort≥tandi= , , . . . ,m. By Lemma , we observe thatzsatisﬁes

either (I) or (II) fort≥T, whereT≥tis large enough. We consider each of the two cases

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Assume ﬁrst that case (I) holds. It follows fromz(t) >  that

x(t) =z(t) –P(t)xt–τ(t)≥z(t) –px

t–τ(t)

≥z(t) –pz

t–τ(t)≥( –p)z(t). ()

Using () and (), we deduce that

r(t)z(t)= – m

i= Qi(t)fi

xt–σi(t)

≤– m

i=

αiQi(t)x

t–σi(t)

≤–( –p)

m

i=

αiQi(t)z

t–σi(t). ()

Deﬁne a generalized Riccati substitution by

ω(t) :=ρ(t)

r(t)z(t)

z(t) +r(t)δ(t)

. ()

Then we have

ω=ρ

rz

z +rδ

+ρ

rz

z +rδ

=ρ

ρω+ρ(rδ)

₊_ρrz

z

=ρ

ρω+ρ(rδ)

₊_ρ(rz)

z –ρr

z z



. ()

By virtue of (), we conclude that

z z  = ω ρr–δ

 = ω ρr 

+δ– ωδ

ρr. ()

Substituting () and () into (), we obtain

ω=ρ(rz

₎

z +

ρ

ρω+ρ(rδ)

_–_ρ_r ω

ρ_r +δ _{– }ωδ

ρr

=ρ(rz

₎

z –ρ

rδ– (rδ)+

ρ ρ + δ

ω–ω



rρ

≤–( –p)ρ

m

i=

αiQi

z(t–σi(t))

z(t) –ρ

rδ– (rδ)+

ρ ρ + δ

ω–ω



rρ. ()

It follows from Remarks  and  that, for anyα∈(, ) andβ∈(, ),

z(t–σi(t))

z(t) =

z(t–σi(t))

z(t) ≥

αβ



(t–σi(t))

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i= , , . . . ,m. Combining () and (), we get

ω(t)≤–αβ( –p)  ρ(t)

m

i=

αiQi(t)

(t–σi(t))

t

–ρ(t)r(t)δ(t) –r(t)δ(t)+

ρ(t)

ρ(t) + δ(t)

ω(t) – ω

₍_t₎

r(t)ρ(t) = –G(t) +A(t)ω(t) –B(t)ω(t),

whereGis deﬁned as in (),A(t) := (ρ(t)/ρ(t)) + δ(t), andB(t) := /(r(t)ρ(t)). Replacing in the latter inequalitytwiths, multiplying both sides byH(t,s) and integrating with respect tosfrom someT(T≥T) tot, we derive fromH(t,t) =  and () that

t

T

H(t,s)G(s) ds

≤ t

T

H(t,s)–ω(s) +A(s)ω(s) –B(s)ω(s)ds

=H(t,T)ω(T) +

t

T

∂H(t,s)

∂s +A(s)H(t,s)

ω(s) –H(t,s)B(s)ω(s)

ds

=H(t,T)ω(T) – t

T

h(t,s)H(t,s)ω(s) +H(t,s)B(s)ω(s)ds

=H(t,T)ω(T) – t

T

H(t,s)B(s)ω(s) + h(t,s) √B(s)



ds+

t

T

h₍_t_,_s₎

B(s) ds

≤H(t,T)ω(T) + t

T

h₍_t_,_s₎

B(s) ds,

and hence

lim sup t→∞



H(t,T)

t

T

H(t,s)G(s) – 

ρ(s)r(s)h

₍_t_,_s₎

ds≤ω(T),

which contradicts condition ().

Assume now that case (II) holds. By virtue of Lemma ,lim_t→∞x(t) = . This completes

the proof.

Corollary  The conclusion of Theoremremains intact if condition()is replaced by the assumptions

lim sup t→∞



H(t,t)

t

t

H(t,s)G(s) ds=∞

and

lim sup t→∞



H(t,t)

t

t

ρ(s)r(s)h(t,s) ds<∞.

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Example  Fort≥, consider a third-order neutral delay diﬀerential equation

x(t) + 

x

t



+t–x

t



+ t–x

t



= . ()

Letρ(t) =t,δ(t) = , andH(t,s) = (t–s)_{. It is not diﬃcult to verify that all assumptions}

of Theorem  are satisﬁed. Hence, every solutionxof () is either oscillatory or satisﬁes limt→∞x(t) = . As a matter of fact, one such solution isx(t) =t–.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

Both authors contributed equally to this work. They both read and approved the ﬁnal version of the manuscript.

Author details

1_{Qingdao Technological University, Feixian, Shandong 273400, P.R. China.}2_{LinDa Institute of Shandong Provincial Key}

Laboratory of Network Based Intelligent Computing, Linyi University, Linyi, Shandong 276005, P.R. China.3_{School of}

Informatics, Linyi University, Linyi, Shandong 276005, P.R. China.

Acknowledgements

The authors are grateful to the editors and two anonymous referees for a very thorough reading of the manuscript and for pointing out several inaccuracies. This research is supported by NNSF of P.R. China (Grant No. 61403061) and the AMEP of Linyi University, P.R. China.

Received: 22 September 2014 Accepted: 8 December 2014 Published:23 Dec 2014

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Cite this article as:Jiang and Li:Asymptotic behavior of a third-order nonlinear neutral delay differential equation.