0 Advances in Difference Equations

a SpringerOpen Journal

RESEARCH

Open Access

Asymptotic behavior for third-order quasi-linear differential equations

Guixiang Qin1, Chuangxia Huang\Yongqin Xie1 and Fenghua Wen2*

Correspondence: wfh@amss.ac.cn 2Schoolof Business, CentralSouth University, Changsha, Hunan 410083, China

Fulllist of author information is available at the end of the article

Abstract

In this paper, a class of third-order quasi-linear differential equations with continuously distributed delay is studied. Applying the generalized Riccati transformation, integral averaging technique of Philos type and Young's inequality, a set of new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of this equations is given. Our results essentially improve and complement some earlier publications.

Keywords: third-order quasi-linear differential equations; oscillation; nonoscillation

1 Introduction

Consider the following third-order quasi-linear differential equation:

p b Y n' p d

x(t) + J p(t, i)x[r (t, i)] di \ +J q(t, f )f(x[a(t, f )]) df = 0. (1)

We build up the following hypotheses firstly: (HI) a(t) e C([t0, to), (0, to)) and f( a(s)-Y ds = to; (H2) p(t, f) e C([to, () x [a, b], [0, to)) and 0 <p(t) = /abp(t, f) df <p <1; (H3) t(t, f) e C([t0, to) x [a, b], R) is not a decreasing function for f and such that

t (t, i) < t and lim min t (t, i) = to;

t^œ ie[a,b]

(H4) q(t, f) e C([to, (),(0, ());

(H5) a(t, f) e C([t0, to) x [a, b], R) is not a decreasing function for f and such that

a (t, f ) < t and lim min a (t, f ) = to;

t >to f g[c,d]

(H6) f (x) e C(R, R) and SfX > 5 >0; (H7) y is a quotient of odd positive integers. Define the function by

ft Spri

z(t) = x(t) + / p(t, i)x[t(t, i)] di.

ringer

©#CPRQin et al.; licensee Springer. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A function x(t) is a solution of (1) means that x(t) e C2[Tx, to), Tx > t0, a{t){z"{t))v e C1[Tx, to) and satisfies (1) on [Tx, to). In this paper, we restrict our attention to those solutions of Eq. (1) which satisfy sup{|x(t)|: t > T} > 0 forall T > Tx. We assume that Eq. (1) possesses such a solution. A solution of Eq. (1) is called oscillatory on [Tx, to) if it is eventually positive or eventually negative; otherwise, it is called nonoscillatory.

In recent years, there has been much research activity concerning the oscillation theory and applications of differential equations; see [1-4] and the reference contained therein. Especially, the study content of oscillatory criteria of second-order differential equations is very rich. In contrast, the study of oscillatory criteria of third-order differential equations is relatively less, but most of works are about delay equations. Some interesting results have been obtained concerning the asymptotic behavior of solutions of Eq. (1) in the particular case. For example, [5] consider the third-order functional differential equations of the form

[a(t)(x"(t)) Y]' + q(t)fX a (t)]) = 0.

Zhang etal. [6] focus on the following the third-order neutral differential equations with continuously distributed delay:

pb ~|''~|' pd x(t)+ / p(t, f)x[t(t, f)] dfx + / q(t, fx[a (t, f)]) df = 0.

J a A A J c

Baculikova and Dzurina [7] are concerned with the couple of the third-order neutral differential equations of the form

[a(t)([x(t) + p(t)x[t(t)]]")Y]' + q(t)xv [a(t)] = 0. (7)

However, as we know, oscillatory behaviors of solutions of Eq. (1) have not been considered up to now. In this paper, we try to discuss the problem of oscillatory criteria of Philos type of Eq. (1). Applying the generalized Riccati transformation, integral averaging technique of Philos type, Young's inequality, etc., we obtain some new criteria for oscillation or certain asymptotic behavior of nonoscillatory solutions of this equations. We should point out that y is any quotient of odd positive integers in this paper, but it is required that y =1 in [6].

2 Several lemmas

We start our work with the classification of possible nonoscillatory solutions of Eq. (1).

Lemma 2.1 Letx(t) be a positive solution of (1), and z(t) is defined as in (4). Then z(t) has only one of the following two properties eventually:

(I) z(t) > 0, zZ(t) > 0, z"(t) > 0;

(II) z(t) > 0, z'(t) < 0, z"(t) > 0.

Proof Let x(t) be a positive solution of (1), eventually (if it is eventually negative, the proof is similar). Then [a(t)(z"(t))Y]' < 0. Thus, a(t)(z"(t))Y is decreasing and of one sign and it follows hypotheses (H2)-(H7) that there exists ti > t0 such that z"(t) is of fixed sign for t > t1. If we admit z"(t) < 0, then there exists a constant M >0 such that

z (t) <--p t > t1.

a(t) 1

Integrating from t1 to t, we get

z'(t) < z'(ti) -M i a(s) y ds. (9)

Let t — to and using (Hi), we have Z(t) — -to. Thus z'(t) < 0 eventually, which together with z"(t) < 0 implies z(t) < 0, which contradicts our assumption z(t) > 0. This contradiction shows that z"(t) > 0, eventually. Therefore z'(t) is increasing and thus (I) or (II) holds for z(t), eventually. □

Lemma 2.2 Let x(t) be a positive solution of (i), and correspondingly z(t) has the property (II). Assume that

pTO /"TOT i pTO ç d

Jta Jv _a(u) Ju Jc

dudv = to. (10)

lim x(t) = 0. (11)

t—>TO

Proof Let x(t) be a positive solution of Eq. (1). Since z(t) satisfies the property (II), it is obvious that there exists a finite limit

lim z(t) = l. (2)

t—>to

Next, we claim that l = 0. Assume that l >0, then we have l < z(t) < l + e for all e >0 and t enough large. Choosing e < l(1 -p)/p, we obtain

/> b r- b

x(t)=z(t)- I p(t, f)x [t (t, f)] df > l - I p(t, f)z [t (t, f)] df

> l -p(t)z[t(t, a)] > l -p(l + e)

= K (l + e)>Kz(t), (13)

where K = tPeel>0. □

Combining (H6), (13) with (1), one can get

q(t, f)(z[a(t, f )])Y df

q(t, f) df

< -SKY(z[aa(t)])Yq1(t), (14)

where q1(t) = ff q(t, f) df and a0(t) = a (t, d). Integrating inequality (14) from t to to, we get immediately

qi(s)(z[o0(s)])Y ds. (15)

Using z(o0(s)) > l, we have

z" (t) > si/yjt qi (s) ^^7 >sllYa<j) I I q(s, %)d% 7;

f to / i /. to /. d

—zZ (t) > 5l/YKl — q(s, %) df ds I du; (16)

Jt \a(u) Ju Jc )

n to p to / i /> co /> d \ y

z(ti) > 51/Y Kl q(s, %) d% ds) dudv.

Jti Jv \a(u) Ju Jc )

We have a contradiction with (iO) and so it follows that limt—TO z(t) = 0, which implies that lim x(t) = 0. (i7)

t—>TO

Lemma 2.3 [7] Assume that u(t) > 0, u'(t) > 0, u"(t) < 0 on [t0, to). Then, for each a e (0, i), there exists Ta > t0 such that

u(o (t)) u(t) „^^ ,14A

-—— > a- for all t > Ta. (8)

o (t) t

Lemma 2.4 [8] Let z(t) > 0, z'(t) > 0, z"(t) > 0, z"'(t) < 0 on [Ta, to). Then there exist ß e (0, i) and Tß > Ta such that

z(t) > ßtz'(t) for all t > Tß. (9)

3 Main results

For simplicity, we introduce the following notations:

D ={(t,s): t > s > fe}; D0 = {(t,s): t > s > fe}. (20)

A function H e C1(D, R) is said to belong to X class (H e X) if it satisfies

(i) H(t, t) = 0, t > t0; H(t, s) > 0, (t, s) e D0;

(ii) < 0, there exist p e Ci([t0, to), (0, to)) and h e C(D0,R) such that

+ P|H(t, s) = -h(t, s)(H(t, s)) ^. (2i)

Theorem 3.1 Assume that (i0) holds, there exist p e Ci([t0, to), (0, to)) and H e X such that

it limsup H7tT)

t—TO H (t, t0) Jt0

■ a(s)p (s)hY+i(t, s) H(t, s)Q(s)--(y + i) +i

'aßo2(s, c)x Y 'd

ds = to, (22)

Q(s) = 5(i-p)Yp(s)(^" q(t,%) d%. (23)

Suppose, further, that a'(t) > 0. Then every solution x(t) ofEq. (1) is either oscillatory or converges to zero.

Proof Assume that Eq. (1) has a nonoscillatory solution x(t). Without loss of generality, we may assume that x(t) > 0, t > t1, x(r (t, /)) > 0, (t, /) e [t0, to) x [a, b], x(a(t, f )) > 0, (t, f ) e [to, to) x [c,d], z(t) is defined as in (4). By Lemma 2.1, we have that z(t) has the property (I) or the property (II). If z(t) has the property (II). Since (10) holds, then the conditions in Lemma 2.2 are satisfied. Hence limt—TO x(t) = 0. When z(t) has the property (I), we obtain

/> b r- b

x(t) = z(t) - I p(t, /)x[t(t, /)] d/ > z(t) - I p(t, /)z[t(t, /)] d/

J a J a

> z(t) - z[T(t, b)] p(t, /) d/x > (1 -p)z(t). (24)

Using (H5) and (H6), we have

(a(t)[z"(t)]Y)' < -5(1 -p)Y(z[a1(t)])Yq1(t), (25)

where q1(t) = ff q(t, f ) df and a1(t) = a(t, c). Let iz"(t)\ Y

w(t) = p(t)a(t) , t > t1. (6)

,z'(t) Then

w (t)--— w(t)

<-S(1-p)Y,1W(zzf)Y-Y^'wn, (27)

Choosing u(t) = z'(t) in Lemma 2.2, we obtain

1 aa1(t) m . .

> , t > Ta > t1. (28)

z (t) tz'(a1(t))

Using Lemma 2.3, we get

z(a1(t)) > M(t)z'(a1(t))t > Tp > Ta. (9)

Combining with (27)-(29), we have

p '(t) / 1 \1/Y Y+1

w'(t) < -Q(t) + w(t) - y —^ w~ (t), (30)

p(t) \a(t)p (t)J

where Q(t) is defined by (23). Let

= Z B'« = Y ( ^ )1№-

For t > t2 > Tp ,we have

ft ft y+1

H(t,s)Q(s) ds < H(t,s)[-w'(s)+A(s)w(s)-B(s)w^+~(s)] ds Jt2 Jt2

ft Y+1

= H(t,t2)w(t2)^ [h(t,s)F(t,s)+B(s)(F(t,s)) Y ] ds, (31)

where F(t, s) = w(s)Hy+ (t, s). By Young's inequality

(B&(s)F(t,s))^ (yb ^(s)h+Dr1 y |h(t)|F(t) (32) -y+1-+-Y+1-> —i|h(t,s)|F(t,s), (32)

we obtain

,, y+i N ^ Ni , N a(s)p(s)hY+1(t, s) , N

B(s)F + (t,s) > |h(t,s)|F(t,s) - ()p () ^ ). (33)

11 ( y + 1)+1

Applying (33) to inequality (31), we obtain

et r t

ww r«(s)P(s)hY+1(t,s)

H (t, s)Q(s) ds < H (t, t2)wfe)+ / —:-—;-ds

Jt? Jt2 ( Y + 1)Y+1

■ i [h(t, s) + |h(t, s)|]F(t, s) ds. (34)

Therefore, we have

a(s)p (s)h Y+!(t, s)

1t w(t2) > H(^) i

H(t, s)Q(s) ds -■

ds. (35)

( y +1)y+!

The last inequality contradicts (22). □

Theorem 3.2 Assume that other conditions of Theorem 3.1 are satisfied except condition (22). Further, for every T, the following inequalities hold:

H (t, s) , N

0 < inf liminf ' < oo (36)

s>T t—oo H(t, T) - y '

ft a(s)p(s)hY+\t, s)

limsup - -ds < oo. (7)

t—x Jt H (t, t )

If there exists f e C([t0, (), R) such that

limsup I

t—( Jt

fl+\s) _p(s)a(s)_

ds = oo, (8)

lim—sup H(t,T) L

tv a(s)p(s)h Y+1(t, s)

H(t,s)Q(s) -

( Y + 1)y+!

ds > f (T), (39)

where ^+(s) = maxj^(s), 0}, then every solution x(t) ofEq. (1) is either oscillatory or converges to zero.

Proof As the proof of Theorem 3.1, we can see that (31) holds. It follows that 1 ft

limsup^— (H(t,s)Q(s) - G(t,s)) ds t^-ro H(t, t2) Jt2

1 ft Y+1

< w(t2) - liminf —--- [h(t,s)F(t,s) + B(s)(F(t,s)) Y + G(t,s)] ds,

t^TO H(t, t2) Jt2

where G(t, s) = By (45), we get

1 ft z±1

^(t2) < wfeMiminf —-- [h(t,s)F(t,s) + B(s)(F(t,s)) Y + G(t,s)]ds,

H (t, t2) Jt2

and hence

1 rt Y+1

0 < liminf —--- [h(t,s)F(t,s)+B(s)(F(t,s)) y + G(t,s)]ds

t^TO H(t, t2) Jt2

< W(t2) - ^(t2) < TO.

Define the functions a(t) and j(t) as follows:

a(t) =

H(t, t2) ,/t2

s)F(t, 5) ds,

P(t) = Hfh)Jt B(5)(f(t,s)) Y ds. From (37) and (42), we obtain liminf[a(t) + P(t)] < to.

The remainder of the proof is similar to the theorem given in [9-11] and hence is omitted. If z(t) has the property (II), since (10) holds, by Lemma 2.2, we have limt^TO x(t) = 0. □

Theorem 3.3 If we replace (37) by 1 ft

limsup —-- I H(t,s)Q(s) ds < to,

H(t, t0) Jt0

and assume that the other assumptions of Theorem 3.2 hold, then every solution ofEq. (1) is either oscillatory or converges to zero.

Proof The proof is similar to Theorem 3.2 and hence is omitted.

Remark 3.4 When y = 1, Theorems 3.1-3.3 with condition (37) reduce to Theorems 3.13.3 of Zhang [6], respectively.

Competing interests

The authors declare that they have no competing interests. Authors' contributions

The work presented here was carried out in collaboration between allauthors. GQ carried out the design of the study and drafted the manuscript. CH and YX conceived, instructed the design of the study and polished the manuscript. FW participated in discussion and completed the revision of the manuscript. Allauthors read and approved the final manuscript.

Author details

'College of Mathematics and Computing Science, Changsha University of Science and Technology, Changsha, Hunan 410004, China. 2Schoolof Business, CentralSouth University, Changsha, Hunan 410083, China.

Acknowledgements

The authors would like to thank Prof Yuanhong Yu and the anonymous reviewer for their constructive and valuable comments, which have contributed much to the improved presentation of this paper. This work was supported by the NationalScienceand Technology Major Projects of China (Grant No. 2012ZX10001001-006), the NationalNaturalScience Foundation of China (Grant No. 11101053, 71171024 71371195), the Scientific Research Fund of Hunan Provincial Education Department of China (Grant No. 11A008) and the Planned Science and Technology Project of Hunan Province of China (Grant No. 2012SK3098, 2013SK3143).

Received:2July2013 Accepted: 14October2013 Published:#PUBLICATION_DATE References

1. Agarwal, RP, Grace, SR, O'Regan, D: Oscillation Theory for Difference and FunctionalDifferentialEquations. Kluwer Academic, Dordrecht (2000)

2. Agarwal, RP, Grace, SR, O'Regan, D: Oscillation Theory for Second Order Dynamic Equations. Taylor & Francis, London (2003)

3. Erbe, LH, Kong, Q, Zhang, BG: Oscillation Theory for FunctionalDifferentialEquations. Dekker, New York (1995)

4. Philos, CG: Oscillation theorems for linear differentialequations of second order. Arch. Math. 53,482-492 (1989)

5. Baculikova, B, Dzurina, J: Oscillation theorems for higher order neutraldifferentialequations. Appl. Math. Comput. 219,3769-3778 (2012)

6. Zhang, Q, Gao, L, Yu, Y: Oscillation criteria for third-order neutraldifferentialequations with continuously distributed delay. Appl. Math. Lett. 25, 1514-1519(2010)

7. Baculikova, B, Dzurina, J: Oscillation of third-order neutraldifferentialequations. Math. Comput. Model. 52, 215-226 (2010)

8. Kiguradze, IT: On the oscillation of solutions of the equations m + a(t)\u\m sign u = 0. Sb. Math. 65(2), 172-187 (1964)

9. Rogovchenko, YV: Oscillation theorems for second-order equations with damping. Nonlinear Anal. 41,1005-1028 (2000)

10. Tiryaki, A, Yaman, S: Asymptotic behavior of a class of nonlinear functionaldifferentialequations of third order. Appl. Math. Lett. 14,327-332(2001)

11. Tiryaki, A, Aktas, MF: Oscillation criteria of a certain class of third order nonlinear delay differentialequations with damping. J. Math. Anal. Appl. 325, 54-68 (2007)

#DIGITAL_OBJECT_IDENTIFIER

Cite this article as: Qin et al.: Asymptotic behavior for third-order quasi-linear differential equations. Advances in Difference Equations #CITATION

Submit your manuscript to a SpringerOpen journal and benefit from:

► Convenient online submission

► Rigorous peer review

► Immediate publication on acceptance

► Open access: articles freely available online

► High visibility within the field

► Retaining the copyright to your article

Submit your next manuscript at ► springeropen.com