Wen et al. Journal of Inequalities and Applications (2015) 2015:261 DOI 10.1186/s13660-015-0791-4

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Oscillations of even order half-linear impulsive delay differential equations with damping

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Kunwen Wen1*, Genqiang Wang2 and Lijun Pan1

Correspondence: wenkunwen@126.com 1 Department of Mathematics, Jiaying University, Meizhou, Guangdong 514015, P.R.China

Abstract

In this paper, a kind of half-linear impulsive delay differential equations with damping is studied. By employing a generalized Riccati technique and the impulsive differential inequality, we derive several oscillation criteria which are either new or improve several recent results in the literature. In addition, we provide several examples to illustrate the use of our results.

MSC: 34K06; 34K11

Keywords: even order; impulsive delay differential equation; half-linear; damping; oscillation

ft Spri

ringer

1 Introduction

Impulsive differential equations are used to simulate processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, etc., and therefore their qualitative properties are important. The phenomenon of oscillations is observed in ecology, physics, economic, etc. In [1], Chen and Feng showed a few examples and indicated that some of the oscillations did favor the stability of system, but some might destroy the balance of the system. Oscillatory properties are so important for the balance of the system that there are now quite a few results on oscillatory properties of their solutions since recent years [1-15]. In particular, Agarwal etal. in [14,15] discussed oscillation theory of differential equations and nonoscillation theory of functional differential equations with applications. Chen and Feng in [1] investigated oscillations of second order nonlinear impulsive differential equation by impulsive differential inequality. From then on, the authors in [3-7] generalized and improved the results of [1]. Furthermore, in [8-13], the delay effect to impulsive equations is considered and some interesting results of oscillations are obtained. Those papers have only considered first or second order differential equations (delay differential equations) with impulses. Recently, some scholars have been attracted by the problems of the oscillations of higher order differential equations and higher order impulsive differential equations and made relative advances therein in [16-26]. For example, Grace etal. in [22, 23] first studied oscillations of higher order nonlinear dynamic equations on time scales and got some interesting and exciting results. Pan et al. in [18] considered even order nonlinear differential equations with impulses of

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Wen et al. Journal of Inequalities and Applications (2015) 2015:261

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the form

x(2n) (t) +f (t, x) = 0, t > to, t = tk,

x(i) (t+) = $ (x(i)(tk)), i = 0, 1, ..., 2n - 1, k =1, 2, ...,

x(i)(tj)=x0°, i = 0,1,..., 2n -1,

where n is positive integer and 0 < t0 < ti < ••• < tk < ••• such that limk^TO tk = to. They obtained sufficient conditions which guaranteed oscillation of every solution of (i). Wen et al. in [19] considered even order nonlinear differential equations with impulses of the form

where n is a positive integer and 0 < t0 < t1 < ••• < tk < ••• such that limk^TO tk = to, p(t) >0. They generalized and improved the results in [16-18]. Pan in [20] considered nonlinear impulsive differential equations with damping of the form

where n is a positive integer and 0 < t0 < t1 < ••• < tk < ••• such that limk^TO tk = to. He obtained sufficient conditions which guaranteed the oscillation of every solution of (3).

References devoted to the study of the oscillations of higher order impulsive differential equations are [18-20]. Impulsive delay differential equations may be used for the mathematical simulation of processes which are characterized by the fact that their state changes by jumps and by the dependence of the process on its history at each moment of time. Those equations can more precisely describe the real processes of a system than impulsive differential equations. Therefore, it is necessary to consider both impulsive effect and delay effect on the oscillation of a differential equation. Many useful results on oscillation and nonoscillation of first order or second order impulsive delay differential equations have been obtained in [8-13], but references devoted to the study of the oscillations of higher order impulsive delay differential equations are relatively scarce.

This paper is motivated by several recent studies [14-26] of such higher order equations. Using impulsive differential inequality and the Riccati transformation, we study the oscillatory properties of even order half-linear impulsive delay differential equation with damping of the form

(r(t)x(2n-1)(t))' + f (t,x) = 0, t > to, t = tk,

x(i) (t+) = $ (x(i) (tk)), i = 0,1,..., 2n -1, k = 1,2,...,

x(i)(t+)=x0i), i = 0,1,..., 2n -1,

(r(t)x(2n-1)(t))' + q(t)x(2n-1)(t) +f (t,x(t)) = 0, t > t0, t = tk, x(i) (t+) = gki] (x(i)(tk)), i = 0,1,..., 2n - 1, k = 1,2,..., x(i)(t+)=x0i), i = 0,1,..., 2n -1,

(r (t) |x(2n-1) (t) | a-1x(2n-1) (t))' + q(t) |x(2n-1) (t) | a-1x(2n-1) (t)

+f (t, x(t), x(t - t)) = 0, t > t0, t = tk,

x(i) (t+) = gf (x(i)(tk)), i = 0,1,..., 2n - 1, k = 1,2,..., x(i)(t+)=x0i), i = 0,1,..., 2n -1,

x(t)=^(t), t0- t < t < t0,

{i) . x(i-1)(tk + h)-x(i-1)(tk) x(l)(tk)= lim ---,

h^o- h

x(i)(t++) = lim ^ + h)-x(i-1)(t+). v k' h^o+ h

$ :[to - t, to] ^ R has at most finite discontinuous points of the first kind and is left-continuous at these points. a > 0, t > 0,0 < to < ti < ••• < tk < ••• such that limk^TO tk = to, x(0)(t) = x(t), n is a positive integer.

Definition 1 A function x :[t0 - t, t0 + y ) ^ R (y > 0) is said to be a solution of (4) on [to - t, t0 + y) starting from (t, 0,x00),x01),.. .,x02n-1)) if

(i) x(i) (t) is continuous on [t0, t0 + y) \ {tk, k e N}, i = 0,1,..., 2n -1,

(ii) x(t)=0(t), t e [t0- t, t0], x(i)(t+)=x0i), i = 0,1,...,2n -1,

(iii) x(t) satisfies the first equality of (4) on [t0, t0 + y) \ {tk, k e N},

(iv) x(i) (t) has two-side limits and left-continuous at points tk, x(i) (tk) satisfies the second equality of (4), i = 0,1,..., 2n -1, k = 1,2,____

Remark 1 Let x0(t) = x(t),xi(t) = x'(t),...,x2„_i(t) = x(2n-1)(t). Then (4) can be changed into a differential system with impulses. By the same method in [21], one can get sufficient conditions that can guarantee the solution of (4) exists on [t0, to). In the following, we always assume the solution of (4) exists on [t0, to).

Definition 2 A solution of (4) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, it is said to be oscillatory.

In this paper, we investigate the oscillatory properties of (4). We first obtain two theorems to ensure every solution of (4) is oscillatory. The results extend and improve the earlier publications. Next, we obtain three corollaries by Theorem 1 and Theorem 2, and provide examples to show that although even order nonlinear delay differential equations without impulses may have nonoscillatory solutions, adding impulses may lead to oscillatory solutions. That is, impulses may change the oscillatory behavior of an equation.

2 Main results

We will establish oscillatory results based on combinations of the following conditions:

(A) r(t) > o and r(t), q(t) are both continuous on [to - t, to), f (t, u, v) is continuous on [to - t, to) x (-to, to) x (-to, to), uf (t, u, v)> o (uv > o), andf (t, u, v)/^(v) > p(t) (v = o), wherep(t) is positive and continuous on [to - t, to) and for any t > to,p(t) is not always equal to o on [t, to), y is differentiable on (-to, to) such that x^(x) > o (x =o), y'(x) > o.

(B) For k = 1,2,..., g|i] (x) are continuous on (-to, to) and there exist positive numbers ak, ¿ki] such that

4i] < Sk](x)/x < bf, i = 0,1,2,..., 2n -1.

(C) For i = 1,2,..., 2n -2,

t0<tk <s bk

^r ds = to

/•to i [2n-1]

— n ak_

1/ n 1 1 i [2n-2] ^0 ra (s) t0<tk<sbk

ds = to.

J2«-«

/to nt0<tk<s al2n-2] exP(/t, bk

The main results of the paper are as follows.

; dv) ds = to.

t0 ar(v) 7

Theorem 1 Assume that the conditions (A), (B), (C), and (D) hold. Suppose further that

ak°] > 1 and

i n p(s) exp(/ ~7T dv)ds /0 Vit0 r(v) /

&0,w =

t0,w = tk + r = tm (m > k),

(bk2n-1])a, t0,w = tk,

(b[mn-1]r, t0,w = tk + r = tm,

and t0,w = tk or tk + r (t1 = t0,1 < t0,2 < ••• < t0,w < t0,w+1 < •••), then every solution of (4) is oscillatory.

Theorem 2 Assume that the conditions (A), (B), (C), and (D) hold and that q>(ab) > y(a)y(b) for ab > 0. Furthermore suppose that

f n —p(s)exp( f dv\ds /0 t0<t0,W<t ^0,w \Jt0 r(v) J

l^0,w =

1 [2n-1]

^(ak )

, t0,w = tk + r = tm (m > k),

(bk2n 1])a, t0,w = tk and tk - r = tm (0 < m < k),

,,[2k-1],

p(ai„)

t0,w = tk + r = tm (m > k), , t0,w = tk and tk - r = tm (0 < m < k),

and t0,w = tk or tk + r (t1 = t0,1 < t0,2 < ••• < t0,w < t0,w+1 < •••), then every solution of (4) is oscillatory.

Remark 2 When a = 1 and not considering the delay effect, (4) reduces to (3). Our Theorem 1 and Theorem 2 generalize and contain results in [20]. When a = 1, q(t) = 0 and not

considering a delay effect, (4) reduces to (2). Our Theorem 1 and Theorem 2 are extensions of Theorem 1, Theorem 2 of [19], respectively.

3 Corollaries and examples

Corollary 1 Assume that the conditions (A), (B), (C), and (D) hold. Furthermore suppose thataf > 1, bk2n-1] < 1 and

f p(s)exp( f dv\ds = to, (9)

Jt0 \Jt0 r(v) /

then every solution of (4) is oscillatory.

Proof By a}-0 > 1, bk2n-1] < 1, we know that > 1. Therefore

f t 1 ifs q(v) \ f t ifs q(v) \

n P(s) exp / ——dv)ds > p(s) exp / ——dv)ds, (10)

Jt0 J^t g0w \Jt0 r(v) / Jt0 \Jt0 r(v) J

letting t —^ to, it follows from (9), (10) that (5) holds. By Theorem 1, we see that all solutions of (4) are oscillatory. □

Corollary 2 Assume that the conditions (A), (B), (C), and (D) hold and that there exists a constant S >0 such that

j[0U. 1 1 S

akl>1, H(11)

f to ( fs q(v) \

ssp(s) expl / ——dv)ds = to, (2)

J to \Jto r(v) /

then every solution of (4) is oscillatory.

Proof By alk0] > 1, -¿t^ > (tJrL)&, then for t e (tw, tw+i], we have

(bk ) k

CH ¿p(s) exp( d^dl

=£p(s) exp( £Sdv)ds+¡¿v £p(s) exp(£ Sdv)ds

1 i „/„\ „„„ I i q v d^ ds + •••

i p(s) exp^i Jt2 \Jt0

(b12n-1]b22n-1])a 4 \Jt0 r(v)

(b12n-1]b22n-1] • • • bWn-1])a Jt."\Jt0

L^f p(s) exp( ftSoqqv dv)ds

f p(s) exp f f q(v) d^ ds -

Jt2 \Jt0 r(v) J

Lp(s) exp( {fr

1 ' ' ' ' ' "v dv ) ds

> (b12n-1])^t1

(b12n-1]b22n-1])a

(¿[2«-l]¿[2»-l] ...b>-l])£

■ í p(s) expf Í qV) dV\ds

Jtw \Jt0 r(v) J

/*t2 / fs q(v) \ ft3 / fs q(v) \

/ p(s) expl / ——dv)ds + / t?p(s) expl / --—dvlds-./ti Vito r(v) / 4 Vito r(v) /

W, tW+iP(s)exp(ft dV)ds

í s?p(s) expf Í d^ds + í s?p(s) expf Í Jtl Vito r(v) J Jt, Vito

dv)ds + •

+ í s?p(s) exp( Í

>tw Vto r(v)

dv I ds

■ í s?p(s) expf Í q(V) d^ ds, ti Jti Vito r(v) /

letting t ^to, it follows from (12), (13) that (5) hold. By Theorem 1, we see that all solutions of (4) are oscillatory. □

Corollary 3 Assume that the conditions (A), (B), (C), and (D) hold and that q>(ab) > q>(a)q>(b) for ab > 0. If there exists a constant S > 0 such that

tk+i - tk > t ,

M01) (bk2n-i])«

tk+A ? tk )

fí fs q(V) \

I s?p(s) expl / ——dv)ds = to, ito Vito r(v) /

tften every solution of (4) is oscillatory.

Corollary 3 can be deduced from Theorem 2. The proof is similar to that of Corollary 2 and it is omitted.

Example 1 Consider the equation

(|x(2n1) (t) |x(2n1) (t)) - 1 |x(2n-1) (t) |x(2n1)(t) + t2x(t -2) = 0, t > 2, t = k, x(k+)=x(k), x(i)(k+) = k+Y x(i)(k), i = 1,2,...,2n -1;k =1,2,...,

x(t) = 0(t), t e [o, ,],

x(2)=xo,

where 4° = ¡k°] = i, «k] = bk'] = i+r, i = i, 2,..., 2n - i; q(t) = -, p(t) = r(t) = i, to = ,, tk = k, t = 2, a = 2, ç(x)= x. It is easy to see that the conditions (A), (B), (C), and (D) hold. Since _i_ = (M)2 =

= (k+^)2 = (kf)2, we may let S = 2, furthermore,

fTO ( fs q(v) \ fTO 1 ( fs 1 \

I sSp(s) expl / —— dv\ds = / s2 — expl - / - dv I ds ito Vito r(v) / J2 s V i<2 v /

, œ / ,s 1 x , œ 1

= y exp|^-y -dvjds = J 2~ds = to. By Corollary 2, every solution of (16) is oscillatory.

Example 2 Consider the equation

(t|x(2n1)(t)|a1x(2n1)(t))/ - x^^r^^t

+ t2x3(t -2) = 0, t > 2, t = k, x(k+) = k+ix(k), x(i)(//+) = x(i)(k), i = 1,2,...,2n -1;k =1,2,..., (17)

x(2) = x0, x()(22) = x0 , x(t) = 0(t), t e [0, 2],

where a-0] = bk°] = k-f, a/ = b-i] = 1, i = 1,2,..., 2n -1; q(t) = -1, p(t) = r(t) = t, t0 = 1, tk = k, t = i, tk+1 - tk = 1 > 2 = t, ^(x) = x3. It is easy to see that the conditions (A), (B), (C),

and (D) hold. Since a = (k+i)3 = (^/r1)3, we may let S = 3, furthermore

f to i fs q(v) \ f to 1 i ^s 1 \

I sSp(s)expl / —— dv\ds = / s3—■ expl - / -dv\ds

Jt0 VA, r(v) / J2, s2 \ ./±v /

to / ps

sexpl - I -dv\ds

1 \ /1 v

s exp^In Ins)ds

f s — ds

ds = to.

By Corollary 3, every solution of (17) is oscillatory. Example 3 Consider the equation

(t2 |x"'(t) |x"'(t))' +64t-4(t - i)-7x7(t - 2) = o, t > 1, t = k, x(k+) = k+T x(k), x(i)(k+) = x(i)(k), 2 = 1,2,...,2n-1;k =1,2,...,

x(2) = xo, x()(22) = xo ,

x(t) = $(t), t e [o, 2],

where a-0] = bf] = -+i, a-i] = b-i] = 1, i = 1,2,3; r(t) = t2, q(t) = 0,p(t) = f t-4(t - , to = 1,

t- = k, t = 2, t-+1 -1- = 1 > 2 = t, a = 2, ^(x) = x7. It is easy to see that the conditions (A),

( [0])

(B), (C), and (D) hold. Since TnkJ = (/+i)7 = (^)7, we may let S = 7, furthermore

(b[ ])a k k

i sSp(s)exp| i q(v)dv\ds = i s7 —s 4| s- 1 | ds J to V J to r(v) J il 64 V V

27 f^s-IV5ds

64 J2 V V

27 /-to 1 > — -r ds

64 J2 (s- 2)2

By Corollary 3, every solution of (18) is oscillatory. But the delay differential equation

has a nonnegative solution x = \/t. This example shows that impulses play an important role in the oscillatory behavior of equations under perturbing impulses.

4 Preparatory lemmas

To prove Theorem 1 and Theorem 2, we need the following lemmas.

Lemma 1 (Lakshmikantham etal. [2]) Assume that

(H0) m e PC'(R+,R) and m(t) is left-continuous at tk, k = 1,2,____

(H1) For tk, k = 1,2,... and t > t0,

m'(t) < p(t)m(t) +q(t), t = tk, m(t+) < dkm(tk) + bk,

where p, q e PC(R+, R), dk > 0 and bk are real constants. Then for t > t0,

Lemma 2 Suppose that the conditions (A), (B), and (C) hold and x(t) is a solution of (4). We have the following statements:

(a) If there exists some T > to such that x(2n-1)(t) > 0 and (r(t)|x(2n-1)(t)|a-1x(2n-1)(t))' > 0 for t > T, then there exists some T1 > T such that x(2n-2)(t) > 0 for t > T1.

(b) If there exist i e {1,2,..., 2n - 2} and some T > t0 such that x(i) (t) > 0 and x(i+1)(t) > 0 for t > T, then there exists some Ti > T such that x(i-1)(t) > 0for t > Ti.

Proof (a) Without loss of generality, we may assume that T = t0, x(2n-1)(t) > 0 and (r(t) |x(2n-1) (t) |a-1x(2n-1) (t)) > 0 for t > t0. We first prove that there exists some j such that x(2n-2) (tj) > 0 for tj > t0. If this is not true, then for any tk > t0, we have x(2n-2)(tk) < 0. Since x(2n-2) (t) is increasing on intervals of the form (tk, tk+1], we see that x(2n-2) (t) < 0 for t > t0. Since r(t)|x(2n-1)(t)|a-1x(2n-1)(t) is increasing on intervals of the form (tk, tk+1], we see that for (t1, t2],

(t) |x(2n-1)(t)|a-1x(2n-1)(t) > r(t1) |x(2n-1^t+) |a-1x(2n-1^t+),

that is,

x(2n-1)(t) > ^^x(2n-1)(t+).

r a (t)

In particular,

ra fe)

Similarly, for (t2, t3], we have

x(2„-i)(i) > Mtix(2«-i)(i2+) > ^Mat-1]x{2"-1}(t2) >

r a (t) r a (t) r a (t)

By induction, we know that

x(2«-1)(t) > ^ a[2n-1]x(2"-1)(t1+), t = tk. (20)

r a(t) t1<tk <t

From the condition (B), we have

x(2n-2)(t+i) > èk2"-2]x(2-2)(tk), t > t1,k = 2,3,.... (21)

Set m(t) = -x(2n-2)(t). Then from (20) and (21), we see that

m'(t) < n ak2n-1]x(2n-1)(t1), t > t1, t = tk,

r a (t) t1<tk <t

m(t+) < b[2n-2]m(tk), k = 2,3,....

It follows from Lemma l that

m(t) < m(t+) n bk2n-2] -x(2n-l)(t+)rHti)f 0 bk2n-2] U

tl<tk<t ^l r a (s) s<tk<t tl<tk<t

{rt 1 «[2n-l] 1

m(t+) -x(2n-l)(t+)ra(tl) J- H Okn-T.

tl<tk< Jtl r a (s) tl<tk<s bk >

That is,

x(2n-2)(t) > U bk2n-2]ix(2n-2)(t+) + x(2n-l)(t+)r^(tlJ^^ U Ê4. (22)

tl<tk<t I Jtl ra (s) tl<tk<sbk J

Note that «k'] > 0, bk] > 0, and the second equality of the condition (B) holds. Thus we get x(2n-2)(t) > 0 for all sufficiently large t. The relation x(2n-2)(t) < 0 leads to a contradiction. So there exists some j such that tj > T and x(2n-2)(tj) > 0. Since x(2n-2)(t) is increasing on (tj+x-l, tj+x], X = l, 2,..., for (tj, tj+l], we have

x(2n-2)(t) > x(2n-2)(t+) > flj2n-2]x(2n-2)(tj) > 0.

Similarly, for (tj+i, i;+2],

x(2«-2)(i) > x(2n-2)(tj+1) > j^x^Oj+i) > aj2n-2]aj2n-2]x(2n-2)(tj) > 0.

We can easily prove that, for any positive integer X > 2 and t e (tj+x, tj+x+i],

x(2"-2)(t) > flj2"-2]flj2"-2] • • • aj+X-21x(2"-2)(tj) > 0.

Thus x(2n-2) (t) > 0 for t > tj. So there exists Ti > T such that x(2n-2)(t) > 0 for t > Ti. The proof of (a) is complete.

(b) Assume that for any tk > T, we have x{i-1)(tk) < 0. By x(i)(t) > 0, x(i+l)(t) > 0, t e (tk, tk+i], we see that x(,)(t) is nondecreasing on (tk, tk+i]. For t e (ti, t2], we have

x(i)(t) > x(i)(t+).

In particular,

x(i)(t2) > x(i)(t+).

Similarly, for t e (t2, t3], we have

x(i) (t) > x(i)(t+) > fl^fe) > fl2i]x(i)(t1+).

By induction, we know that

x(i)(t) > H ak]x(i)(t+), t > ti, t = tk. (3)

ti<tk <t

From the condition (ii), we have

x(i-1) (t+) > bl-1]x{i-1) (tk), k = 2,3,.... (24)

Set u(t) = -x(i-1)(t). Then from (23) and (24), we see that

U (t) < - Y[ afx(i\ t+), t > ti, t = tk,

ti<tk <t

u(t+) < bk-i]u(tk), k = 2,3,.... It follows from Lemma i that

u(t) < u(t+) n bk-i]-x(i)(t+)[t n bki-i] n ak] ds

ti<tk<t ti s<tk<t ti<tk<t

= n bk-i]iu(t+) -x»(fi)tt n Hh *}.

ti<tk<t l Jti ti<tk<s bk '

That is,

x(i-i)(t) > n bki-i](x(i-i) (ti+) + x(i)(t+)ft n A A. (25)

ti<tk <t ^ •"i ti<tk <s bk '

Note that aki] > 0, bk'] > 0, and the first equality of the condition (B) holds. Thus we get x(i-1)(t) > 0 for all sufficiently large t. The relation x(i-1)(t) < 0 leads to a contradiction. So there exists some j such that tj > T and x(i-1)(tj) > 0. Then

x(i-i)(t+) > aji-1]x(i-1)(tj) > 0.

Since x(i)(t) > 0, we see that x(i-1)(t) is increasing on (tj+m-1, tj+m], m = i, 2,____For (tj, tj+i],

we have

x(i-1)(t) > x(i-1)j >0.

In particular,

x(i-i)(tj+i) > x(i-1)(t;+) >0.

Similarly, for (tj+i, tj+2], we have

x(i-i)(t) > x(i-i)(t;+i) > aj+V^+i) > 0.

By induction, for (tj+m-i, tj+m], we have x(i-1)(t) > 0. So when t > tj+i, we have

x(i-1)(t) > 0.

Summing up the above discussion, we know that there exists some Ti > T such that x(i-1)(t) > 0, t > Ti.

The proof of Lemma 2 is complete. □

Remark 3 We may prove in a similar manner the following statements:

(a') If we replace the condition (a) in Lemma 2 'x(2n-1)(t) > 0 and (r(t)|x(2n-1)(t)r-1 x x(2n-i)(t))' > 0 for t > T with 'x(2n-i)(t) < 0 and (r(t)|x(2"-i)(t)|a-ix(2"-i)(t))' < 0 for t > Tunder the conditions (A), (B), and (C), then there exists some Ti > T such that x(2«-2)(t) < 0 for t > Ti. (b') If we replace the condition (b) in Lemma 2 'x(,)(t) > 0 and x('+i)(t) > 0 for t > T' with 'x(i) (t) < 0 and x(i+i) (t) < 0 for t > T' under the conditions (A), (B), and (C), then there exists some Ti > T such that x(,-i)(t) < 0 for t > Ti.

Lemma 3 Letx = x(t) be a solution of (4) and suppose that the conditions (A), (B), and (C) hold.

(a) If there exists some T > t0 such that x(t) > 0 and (r(t)|x(2n-1)(t)|a-1x(2n-1)(t))' < 0 for t > T, then x(2n-1)(t) > 0for all sufficiently large t.

(b) If there exist i e {1,2,..., 2n -1} and some T > t0 such that x(t) > 0 and x(i) (t) < Ofor t > T, then x(i-1) (t) > 0 for all sufficiently large t.

Proof (a) We first prove that x(2n-1) (t) > 0 for any tk > T. If this is not true, then there exists some tj > T such that x(2n-1)(tj) < 0. Since r(t) > 0 and r(t)|x(2n-1)(t)r-1x(2n-1)(t) is strictly decreasing on (tj+m-1, tj+m] for m = 1,2,... and for t e (tj, tj+1], we have

r(t)|x(2n-1)(t)r-1x(2n-1)(t) < r(tj)|x(2n-1)(tj+) T-1x(2n-1)(tj+)

< (flj2n-11)ar(tj)|x(2n-1)(tj)r-1x(2n-1)(tj) < 0.

Let p = r(tj)|x(2n-1)(tj)|a-1 x(2n-1)(t;-) < 0, we have r(t)|x(2n-1)(t)r-1x(2n-1)(t) < (aj2n-11)ap < 0. Similarly, for t e (tj+1, tj+21, we have

r(t)|x(2n-1)(t)|a-1x(2n-1)(t) < r(tj+i)|x(2n-1)(tj+1) r-1x(2n-1)(t+4)

< (aj2n-11)a (aj2n-11)ap < 0.

We can easily prove that, for any positive integer n > 1 and t e (tj+n, tj+n+11, we have r(t)|x(2n-1)(t)rV2n-1)(t) < (afn-11aj+n-11 • ••aj+rTp < 0.

Hence, x(2n-1)(t) < 0 for t > tj+1. By the result (a') of Remark 2, for sufficiently large t, we have x(2n-2)(t) < 0. Using the result (b') ofRemark 2 repeatedly, for all sufficiently large t, we get x(t) < 0. This is contrary with x(t) > 0 for t > T. Hence, we have x(2n-1)(tk) > 0 for any tk > T. So we get x(2n-1)(t) > 0 for all sufficiently large t.

(b) We first prove that x(i-1)(tk) > 0 for any tk > T. If this is not true, then there exists some tj > T such that x(i-1)(tj) < 0. Since x(i-1)(t) is strictly monotony decreasing on (tj+n, tj+n+11 for n = 0,1,2,... and for t e (tj, tj+11, we have

x(i-1)(t) <x(i-1)(t;) < aji-11x(i-1)t) < 0.

Similarly, for t e (tj+i, tj+21, we have

x(i-1)(t) <x(i-1)(tj+1) < aji-11aj+-111x(i-1)(tj) < 0.

We can easily prove that, for any positive integer n > 2 and t e (tj+n, tj+n+11, we have

x(i-1)(t) < flji-11flj+-111 • • • aj+-1Ix(i-1)(tj) < 0.

Hence, x(i-1)(t) < 0 for t > tj+1. By the result (b') of Remark 2, for sufficiently large t, we have x(i-2)(t) < 0. Similarly, by using the result (b') of Remark 2 again, we can conclude that for all sufficiently large t, x(t) < 0. That is contrary with x(t) > 0 for t > T. Hence, we have x(i-1)(tk) > 0 for any tk > T. So we get x(i-1) (t) > 0 for all sufficiently large t. The proof of Lemma 3 is complete. □

Lemma 4 Letx = x(t) be a solution of (4). Suppose that T > t0 andx(t) > 0for t > T. f the conditions (A), (B), (C), and (D) hold, then there exist some T' > T and l e{1,3,..., 2n -1} such that for t > T',

jx(i)(t)>0, i = 0,1,..., l;

|(-1)(i-1)x(i)(t)>0, i = l +1,..., 2n -1. ( )

Proof Let x(t) > 0 for t > T. We first prove that x(2n-1)(tk) > 0 for any tk > T. If this is not true, then there exists some tj > T such that x(2n-i)(tj) < 0. By (4) and the condition (A), for t e (tj+m-i, tj+m], m = 1,2,..., we have

(|x(2n-1)(t)r-1x(2n-1)(t))' + r/(t) + g(t)|x(2n-1)(t)r-1x(2n-1)(t) = 0,

r(t) r(t)

that is,

(|x(2n-i)(t)r-ix(2n-i)(t) exp j ds

f (t,x(t),x(t - t)) ft r'(s) + q(s) ,

=--71-exp -—-ds

r(t) tj r(s)

p(t)p(x(t - t)) ft r'(s) + q(s) ,

<--—-exp -—— ds < 0. (27)

r(t) tj r(s)

Let s(t) = |x(2n-i)(t) |a-1x(2n-1) (t) exp ds, we have s'(t) < 0, s(t) is monotonically

decreasing on (tj+m-i, tj+m], m = 1,2,____

For t e (tj, tj+i], we have

s(t) < s(tj+) < (aj2n-1])as(tj) < 0,

particularly, we have

s(tj+i) < (aj2n-1])as(tj) < 0. Similarly, for t e (tj+i, tj+2], we have

s(t) < s(tj+i) < (aj+n-1])as(tj+i) < (aj+n-1])a(aj2n-1])as(tj) < 0. By induction, for t e (tj+m-i, tj+m], m = 1,2,..., we obtain

s(t) < s(tj+m-i) < (aj+m-1^ • • • (aj+n-1])a(aj2n-1])as(j)

= n (ak2n-1])as(tj) < 0. (28)

tj<tk<t

Since s(t) < 0, s'(t) < 0, s(t) is not always equal to 0 on any interval [t, to), we have s(t) < 0 for sufficiently large t, therefore, we get x(2"-1)(t) < 0 for sufficiently large t, without loss of

generality, we may let x(2n 1)(i) < 0 for t > tj. Let s(tj) = — ya (y > 0), using (28), we have |x(2n-1)(i)r-1x(2n-1)(i) expf V(s) + q(s) ds < n («i2n-1])as(tj).

Jtj r(s) t^V

By the above equality, we obtain

(2n-1)(t)Ia-1x(2n-1)(t) < -va n (al2n-1])a expf— /t r(s) +q(s) ds1

(t)|a-1x(2n-1)(t) < -y^ (ak2n-1])a exp(—jT^)^ ds) Noting that x(2n—1) (t) < 0 for t > tj, we can get

-|x(2n-1)(t)|a < n (ak2n-1])a .

tj <tk <t That is,

x(2n-1)(t) < —y n a[2n-1] expf— f * ds) < 0, (29)

tj<t/<t ' V Jtj ar(s) J

by Lemma 3, we have x(2n—2)(t) > 0 for sufficiently large t, without loss of generality, let x(2n—2)(t) > 0, t > tj. In view of the condition (B), we have

x(2n—2)(t+) < 42n—2V2n—2)fe), k = j + 1,j + 2,.... (30)

By (29) and (30), applying Lemma 1, we obtain

x(2n—2)(t) < x(2n—2)(t+) n bfn—2]

tj<tk <t

— y iCn .^n^M — ^

= [] bk2n—2] x(2n—2)(j tj<tk <t

l0jjn<sbh eXP( —l ar(v)

— y/ n^k^ eJ—frr±^ dv\ds

letting t ^to, applying (31) and the condition (D), we get x(2n 2)(t) < 0, which is contracted with x(2n—2) (t) > 0, t > tj. Sowe have x(2n—1) (tk) > 0 for any tk > T. Since x(2n—1) (t) > 0 for t > tj, here, without loss of generality, we may let x(2n—1) (t)> 0 for t > t0. Then x(2n—2) (t) is strictly increasing on (tk, tk+1]. If for any tk, x(2n—2)(tk) < 0, then x(2n—2) (t) <0fort > T1.If there exists some tj such that x(2n—2)(tj) > 0, since x(2n—2)(t) is strictly monotony increasing and 42n—2] > 0, then x(2n—2) (t)> 0 for t > tj. Thus there exists T2 > T1 such that x(2n—2) (t) > 0 for t > T2. So one of the following statements holds:

(A1) x(2n—1)(t) > 0, x(2n—2)(t) >0, t > T2; (B1) x(2n—1)(t) > 0, x(2n—2)(t) <0, t > T2.

If (A1) holds, by the result (b) of Lemma 2, x(2n 3)(t) > 0 for all sufficiently large t. Using the result (b) of Lemma 2 repeatedly, for all sufficiently large t, we can conclude that

x(2n-1)(t) > 0, x(2n-2)(t) > 0, ..., x'(t) > 0, x(t) > 0.

If (B1) holds, by Lemma 3, we have for all sufficiently large t. Similarly, there exists some T3 > T2 such that one of the following statements holds:

(A2) x(2n-3)(t) > 0, x(2n-4)(t) >0, t > T3; (B2) x(2n-3)(t) > 0, x(2n-4)(t) <0, t > T3.

Repeating the discussion above, we can see eventually that there exist some T' > T and l e{1,3,...,2n-1} such that for t > T',

|x(i)(t) > 0, i = 0,1,..., l; I (-i)(i-i)x(i)(t)> 0, i = l + 1,..., 2n - 1.

The proofis complete. □

Remark 4 We may prove in a similar manner the following statements.

If we replace the condition in Lemma 4 'x(t) > 0 for t > T ' with 'x(t) < 0 for t > T', and under the conditions (A), (B), (C), and (D), then there exist some T' > T and l e {1,3,..., 2n -1} such that, for t > T,

jx(i)(t)<0, i = 0,1.....l; (32)

|(-1)(i-i)x(i)(t)<0, i = l + 1,..., 2n -1. ( )

5 Proofs of main theorems

We now turn to the proofs of Theorem 1 and Theorem 2.

Proof of Theorem 1 If (4) has a nonoscillatory solution x = x(t), without loss of generality, let x(t) > 0 (t > to). By Lemma 4, there exist T > t0 and an integer l e{1,3,..., 2n -1} such that for t > T,

x(t) > 0, x'(t) > 0, x(2n-1)(t)>0. (33)

= r(t)|x(2n-1)(t)|a-1x(2n-1)(t) (34)

p(x(t - T))

We see that u(t+) > 0 (k = 1,2,...), u(t) > 0 for t > T. By (4), (33), and the condition (A), we get

u (t) = -q(t)|x(2n-i)(t)rix(2n-i)(t) -f (t, x(t), x(t - t)) p(x(t - T))

r(t)|x(2n-1)(t)|a-1x(2n-1)(t)p'(x(t- T))x'(t- T) p2(x(t - T))

-q(t)\x(2n-1)(t)\a-1x(2n-1)(t) -f (t,x(t),x(t - r)) _ y(x(t - r))

—--TT u(t)-p(t), t = to,w

It follows from the conditions (B), a^ > 1, and y'(x) > 0 that

r(t+)\x(2"-1)(t+)\a-1x(2"-1)(t+) u[ k)= v(x(tk - r)+)

42"-11)" r(tk )\x<2n-1>(tk )\a-1x<2n-1>(tk)

<p(x(tk -r))

= (b[2n-11)a u(tk), tk - r = tm (0 < m < k),

b?"-1])a r(tk )\x(2n-1)(tk )\a-1x(2n-1)(tk)

y(alnx(tk -r)) (bk2K-1])" r(tk )\x(2n-1)(tk )\a-1x(2n-1)(tk)

= (b[2n-1])a u(tk), tk - r = tm (0 < m < k),

y(x(tk-r))

i((tk k r)+) =

r((tk k r)+)\x(2n-1)((tk k r)+)\a-1x(2"-1)((tk k r)+) y(x(t+))

r(tk+r) \x(2n-1> (tk+r) \"-1x<2«-1) (tk+r) <p(a^x(tk))

< r(tk+r) \x<2"-1) (tk+r) r^2«-1) (tk+r)

— y(x(tk))

= u(tk k r), tk k r = tm (k < m), r(tm)\x<2n-1>(tm)\g-1x(2n-1)(tm)

y(ak°]x(tk))

< (fc!m«-1Tr(tm) \x(2«-1> (tm) \a-1x(2n-1) (tm)

_ y(ak01x(tk))

< (bm-1] )a r(tk+r) \x(2n-1> (tk+r) \a-1x(2«-1) (tk+r)

— y(x(tk))

= (bjm«-11)au(tk k r), tk k r = tm (k < m).

So we get

u'(t) — - q(t) u(t)-p(t), t = to,w r(t)

where to,w = tk or tk k r (t1 = to,1 < to,2 < • •• < to>w < to,w+1 < • • • ) and is defined by (6). Applying Lemma 1, we obtain

(f t q(s) \ f t if t q(v) \

—ttds\ - n 0o,wp(s)expl / —— dvlds

Tok <t JT r(s) / JT s<tk <t \Js r(v) )

— Yl 0o,w exp( / T

q(s) r(s)

<T ')-/t n<s ^ Af^ dVh

It is easy to see from (5) and (38) that u(t) < 0 for sufficiently large t. This is contrary to u(t) > 0 for t > T. Thus every solution of (4) is oscillatory. The proof of Theorem 1 is complete. □

Proof of Theorem 2 If (4) has a nonoscillatory solution x = x(t), without loss of generality, let x(t) > 0 (t > t0). By Lemma 4, there exists T > t0 and an integer l e {1,3,..., 2n - 1} such that for t > T,

x(t)>0, x'(t) > 0, x(2n-1)(t)>0.

Let u(t) be defined by (34), then u(t+) > 0 (k = 1,2,...), u(t) > 0 for t > T. By (4), and the condition (A), we also can get

U (t) < - ^ U(t) -p(t), t = t0,w.

It follows from the conditions (B), q>(ab) > y(a)y(b) (ab > 0), and q>'(x) > 0 that

r(t+)|x(2"-1)(t+)|a-1x(2"-1)(t+) U[ k)= v(x(tk - T)+)

bk2g-V r(tk )|x(2n-1)(tk )|g-1x(2n-1>(tk ) p(x(tk -T ))

= (bk2n-1])au(tk), tk - T = tm (0 < m < k),

i [2n-1]

)a r(tk )|x<2n-1>(tk )|a-1x<2n-1>(tk )

^(ajm1x(tk -T ))

(b'^V r(tk )|x<2n-1)(tk )|a-1x<2n-1)(tk )

,[2k-1]v

<p(àmm°)<p(x(tk -t ))

l((tk k T)k) =

k, [0K u(tk), tk - T = tm (0 < m < k),

r((tk k T)+)|x(2n-1)((tk k T)+)|a-1x(2n-1)((tk k T)+) <p(x(t+))

r(tk+t ) |x(2n-1> (tk+t ) |g-1x(2n-1> (tk+t ) ^(ak']x(tk )) r(tk+t ) |x(2n-1> (tk+t ) |a-1x<2n-1> (tk+t )

V^MMtk ))

^(ak )

u(tk k t), tk k t = tm (k < m),

r(tm)|x<2n-1>(tm)|g-1x(2n-1)(tm)

P(a|0]x(tk ))

^ (b|mn-1T r(tm) |x(2n-1) (tm) |g-1x(2n-1> (tm)

< -IÔ1-

V(ak x(tk))

< (b|mn-1] )a r(tk+t) |x(2n-1) (tk+t) |a-1x(2n-1> (tk+t)

_ ^(ak0I)^(x(tk ))

^(ak )

u(tk k t), tk k t = tm (k < m).

So we have

u (t) < - ^ u(t)-p(t), t = t0w, r(t)

u(tk,w) < l¿0,wu(t0,w),

where tQ>w = tk or tk + t (ti = % < to,2 < • • • < to,w < t0>w+i < • • • ) and ^o,w is defined by (8). Applying Lemma I, we obtain

It is easy to see from (7) and (42) that u(t) < 0 for sufficiently large t. This is contrary to u(t) > 0 for t > T. Thus every solution of (4) is oscillatory. The proof of Theorem 2 is

Competing interests

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

All authors contributed to each part of this work equally and read and approved the final manuscript. Author details

department of Mathematics, Jiaying University, Meizhou, Guangdong 514015, P.R. China. 2Department ofComputer Science, Guangdong Polytechnic Normal University, Guangzhou, Guangdong 510665, P.R. China.

Acknowledgements

The authors wish to express their sincere thanks to the anonymous referees and the handling editor for many constructive comments leading to the improved version of this paper.

Received: 31 December 2014 Accepted: 26 June 2015 Published online: 28 August 2015

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