Hindawi Publishing Corporation Abstract and Applied Analysis Volume 2012, Article ID 351709,23 pages doi:10.1155/2012/351709

Research Article

Interval Oscillation Criteria of

Second Order Mixed Nonlinear Impulsive

Differential Equations with Delay

Zhonghai Guo,1 Xiaoliang Zhou,2 and Wu-Sheng Wang3

1 Department of Mathematics, Xinzhou Teachers University, Shanxi, Xinzhou 034000, China

2 Department of Mathematics, Guangdong Ocean University, Guangdong, Zhanjiang 524088, China

3 Department of Mathematics, Hechi University, Guangxi, Yizhou 546300, China

Correspondence should be addressed to Xiaoliang Zhou, zxlmath@yahoo.cn Received 19 December 2011; Revised 10 April 2012; Accepted 11 April 2012 Academic Editor: Agacik Zafer

Copyright © 2012 Zhonghai Guo et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

We study the following second order mixed nonlinear impulsive differential equations with delay (r(t)Q«(x'(t)))'+P0(t)®«(x(t)) + Zh pi(t)®fi (x(t-a)) = e(t), t > t0,t = tux(t+) = akx(rk),x'(t+) = bkx'(rk), k = 1,2,..., where <bt(u) = |u|*-1u, a is a nonnegative constant, [rk} denotes the impulsive moments sequence, and Tk+1 - Tk > a. Some sufficient conditions for the interval oscillation criteria of the equations are obtained. The results obtained generalize and improve earlier ones. Two examples are considered to illustrate the main results.

1. Introduction

We consider the following second order impulsive differential equations with delay

(r(t)O^x'(t)))' + p0(t)Oa(x(t)) + ^pi(t)Oßi(x(t - a)) = eit), t > t0, t //Tk,

i=1 (1.1)

x(r£) = akx(rk ), x'(r£) = bkx'(rk ), k = 1,2,...,

where O^(u) = |u|* 1 u, a is a nonnegative constant, [rk} denotes the impulsive moments sequence, and Tk+i - Tk> a, for all k e N.

Let J c R be an interval, and we define

PLC(J,R) := {y : J —> R | y is continuous everywhere except each Tk at which y(r£) and y(rk) exist and y(rk) = y(rk), k e N•

For given t0 and <p e PLC([t0 - o,t0],R), we say x e PLC([t0 - o, to),R) is a solution of (1.1) with initial value $ if x(t) satisfies (1.1) for t > t0 and x(t) = $(t) for t e [t0 - o, t0].

A solution of (1.1) is said to be nonoscillatory if it is eventually positive or eventually negative. Otherwise, this solution is said to be oscillatory.

Impulsive differential equation is an adequate mathematical apparatus for the simulation of processes and phenomena observed in control theory, physics, chemistry, population dynamics, biotechnologies, industrial robotics, economics, and so forth. Because it has more richer theory than its corresponding without impulsive differential equation, much research has been done on the qualitative behavior of certain impulsive differential equations (see [1,2]).

In the last decades, there is constant interest in obtaining new sufficient conditions for oscillation or nonoscillation of the solutions of various impulsive differential equations, see, for example, [1-9] and the references cited therein.

In recent years, interval oscillation of impulsive differential equations was also arousing the interest of many researchers. In 2007, Ozbekler and Zafer [10] investigated the following equations:

(m(t)<^(y'))' + q(0<My) = f (t), t /di, A(m(%«(y')) + qmiy) = fi, t = 9i, (i e N),

(m(t)y')' + s(t)y' + q(t)q>ß(y) = f (t), tf dt, A(m(t)y') + W(y) = fi, t = 6i, (ß > 1),

where y*(u) = |u|* 1u, ¡5 > a, [qi] and [fi] are sequences of real numbers. In 2009, they further gave a research [11] for equations of the form

(r(t)^(x'))' + p(t)ya(x') + q(t)q>ß(x)= e(t), tf di,

M r(t)<Mx')) + q^ß(x) = ei, t = 0i,

and obtained some interval oscillation results which improved and extended the earlier ones for the equations without impulses.

For the mixed type Emden-Fowler equations

(r (f)x'(0) ' + p(i)x(i)^pI(i)|x(i)|tt*-1x(i)= e(t), t frk,

¿=1 (1.6)

^Tfc) = akx(Tk)> X\Tk) = bkx'(rk), k e N,

Liu and Xu [12] established some interval oscillation results. Recently, Ozbekler and Zafer [13] investigated the more general cases

(r (t)®a(x'(t))) ' k q(t)Oa(x(t)) (t)Opk (x(t)) = e(t), t = dit

k=1 (1.7)

x(9k) = Uix(di), x'(9k ) = bix'(0i),

where¡¡1 > ■■■ > ¡¡m> a> ¡¡mk1 > ■■■ > ¡¡n> 0.

However, for the impulsive equations, almost all of interval oscillation results in the existing literature were established only for the case of "without delay." In other words, for the case of "with delay" the study on the interval oscillation is very scarce. To the best of our knowledge, Huang and Feng [14] gave the first research in this direction recently. They considered second order delay differential equations with impulses

x"(t) k p(t) f (x(t - t )) = e(t), t > to, t = tk,k = 1,2,..., x(t+k) = akx(tk), x'( t+k) = bkx'(tk), k = 1,2,...

and established some interval oscillation criteria which developed some known results for the equations without delay or impulses [15-17].

Motivated mainly by [13, 14], in this paper, we study the interval oscillation of the delay impulsive (1.1). By using some inequalities, Riccati transformation and H functions (introduced first by Philos [18]), we establish some interval oscillation criteria which generalize and improve some known results. Moreover, examples are considered to illustrate the main results.

2. Main Results

Throughout the paper, we always assume that the following conditions hold:

(Ai) the exponents satisfy that ¡i > ■■■ > ¡m> a> ¡¡mk\ > ■■■ > ¡¡n> 0;

(A2) r(t) e C([t0, œ), (0, œ)) is nondecreasing, e(t), pi(t) e PLC([t0, œ),R), i = 0,1,...,n;

(A3) {ak} and {bk} are real constant sequences such that bk > ak> 0, k e N.

It is clear that all solutions of (1.1) are oscillatory if there exists a subsequence {ki} of {k} such that aki < 0 for all i e N. So, we assume ak> 0 for all k e N in condition (A3).

In this section, intervals [c1,d1] and [c2,d2] are considered to establish oscillation criteria. For convenience, we introduce the following notations (see [12]). Let

k(s) = max{i : t0 <Ti < s}, rj = max{r(t) : t e [Cj,dj]},

Q(cj,dj) = | Wj e C1 [Cj,dj] : Wj (t) / 0, w^j = Wj(dj) = 0}, j = 1,2.

For two constants c,d / {rk} with c < d and k(c) < k(d) and a function y e C([c, d],R), we define an operator Q : C([c, d], R) ^ R by

K ^ - at )1 k(d) ba_ aa

Qdw] = y(Tk(c)+1) /7 k(c)+\« + : y(Tt) Jt ai)', (2-2)

afc(c)+^Tk(c)+1 - c) i=k(c)+2 ai (Ti - Ti-1)

where ^ts = 0 if s > t.

In the discussion of the impulse moments of x(t) and x(t - a), we need to consider the following cases for k(cj) < k(dj)

(51) Tk(cj) + a <cj and Tk(d,) + a > dj,

(52) Tk(ci) + a <cj and Tk(dj) + a < dj,

(53) Tk(cj) + a>cj and Tk(A}) + a > dj,

(54) Tk(cj) + a>cj and Tk(d;) + a < dj,

and the cases for k(cj) = k(dj)

(51) Tk(cj) + a < cj,

(52) cj < Tk(ci) + a < dj,

(53) Tk(cj) + a > dj.

Combining (S*) with (S*), we can get 12 cases. In order to save space, throughout the paper, we study (1.1) under the case of combination of (S1) with (S1) only. The discussions for other cases are similar and omitted.

The following preparatory lemmas will be useful to prove our theorems. The first is derived from [19] and second is from [20].

Lemma 2.1. For any given n-tuple {p\,p2,...,pn} satisfying (A1)/ then there exists an n-tuple {n ,n2,...,nn} such that

= a, & = L 0 <nl< 1, (2.3)

i=1 i=1

where 1 e (0,1].

Lemma 2.2. Suppose X and Y are nonnegative, then

1XY1-1 - X1 < (1 - 1)Y1, 1 > 1, (2.4)

where equality holds if and if X = Y.

Let a > 0, B > 0, A > 0, and y > 0. Put

1 = 1 + 1, X = Aa/(a+1)y, Y =( J^)aBaA-al/(a+i). (2.5)

a y \a +1/ v ;

Abstract and Applied Analysis It follows from Lemma 2.2 that

aa Ba+1

By - Ay(a+^/a <-a-r B-. (2.6)

y y ~ (a + 1)a+1 Aa K !

Theorem 2.3. Assume that for any T > t0, there exist cj,dj / {Tk}, j = 1,2, such that T < c1 < d1 < c2 < d2 and for j = 1,2

pi(t) > 0, t e [cj - a,dj ] \{Tk}, i = 0,1,2,...,n;

(-1)je(t) > 0, t e [cj - o,dj] \{Tk}. {2'?)

Ifthereexist wj (t) e Q.(cj,dj) and p(t) e C1 ([cj,dj ], (0, <x>)) suchthat,for k(cj) <k(dj), j = 1,2,

Ç Tk(cj )k1

(t - Tk(cj) - 0)

Wj (t) /-Y^dt

j [t - Tk(Cj))

k(dj )-1 EE

i=k(cj )k1

bâ^ k\> (^dt

(t - Tk(d,)) bHd, )(t k 0 - Tk(j ))C

\ Wj (t)-^- J/ s*dt

JTk(d>) K(d,)(t k 0 - Tk(dj)

kj\(t)(v0(t)\Wj (t)\ak1 - r (t)(\w> (t)|k a+) > j [|wj|ak1],

j (2.8)

where pj is maximum value of p(t) on [cj,dj] and, for k(cj) = k(dj), j = 1,2,

fa (»(&+P^Wj (or1 - r (o(\wj (0|k ipM) " 1)dt > 0,

where Wj (t) = nom\e(t)\n°YXn=in—ni (Pi(t))ni Wj (t)\a+1 with n = 1 -Y2=\ m and m,m,...,rin are positive constants satisfying conditions of Lemma 2.1, then (1.1) is oscillatory.

Proof. Assume, to the contrary, that x(t) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that x(t) > 0 and x(t — o) > 0 for t > t0. In this case the interval of t selected for the following discussion is [ci,di]. Define

"(')=p(')nsX^, ' e [c^- (2-10)

It follows, for t = Tk, that

u'(t) = -p(t)po(t) - p(t)

P(t)u(t) -- „

P(t) (p(t)r (t))1/a

+ PÜu(t) ---|u(t)|(a+1)/a.

Qa(x(t - a)) Ca(x(t))

(2.11)

Now, let

vo = n-'^ethj-y Vl = n-Vi(0%-«(*(f - a)), i = 1,2.....n, (2.12)

where ni,n2,---,nn are positive constants satisfying conditions of Lemma 2.1 and no = 1 -2i=i ni. Employing in (2.11) the arithmetic-geometric mean inequality (see [20])

Znv > 11 vn (2.13)

¿=0 1=0 "

and in view of (2.3), we have that

u'(t) < -p(t)po(t) -p(t*p(t)CCx(X-a1 -^^|u(t)|(a+1)/a + Pt)u(t), (2.14)

Ca(x(t)) (p(t)r(t)) /a P(t)

y(t) = n0no Ktffl V^fa (t))ni. (2.15)

First, we consider the case k(c\) < k(di).

In this case, we assume impulsive moments in [c1,d1] are t^)^,^^)^,...,^^). Choosing w1(t) e Q(c1,d1), multiplying both sides of (2.14) by |w1 (t) |a+1 and then integrating it from c1 to d1, we obtain

aTk(c1 )+1 k(d1)-1 fTi+1 f d1 \

+ X + y^wm^dt

C1 i=KC, )+W Ti J Tk(d, ) I

\Jc1 i=k(C1)+1J Ti jTk(d1)/

pTfe(cj)+1 k(d1)-1 ^Ti+1 fd^ \ /p'(t)

.P(t) "" (p(t)r(t)) \xa(t - a)

< fr1" + k|-1 fTi+1 + fd1 Vptfu(t) - a 1/a |u(t)|(1+a)/^|w1(t)|a+1dt

\-'C1 i=k(c1) + ^ Ti JTk(d1 )J \pltl t-tt\~ttW

lTk(c1)+1 k(d1)-l

C1 i=k(C1)+1

f1 p(t)po(t)|W1 (t)|a+1dt,

r +a ^—

(2.16)

where W\(t) = p(t)f (t)\wi(t)\a+1. Using the integration by parts formula in the left-hand side of above inequality and noting the condition w\(c\) = w\(d\) = 0, we obtain

£ \w1(Tl)\a+1 [u(Ti) - u«)]

i=k(ct)+l

I iTk(c1)+1 k(d1)-1 fTi+t fdt \

< M + 2 \ +{ )

y-'c i=k(a) + lJ Ti Tk(di) J

(a + 1)Oa(wi(t))w'1 (t)u(t)

P'(t) P(t)

u(t)\wi(t)\

(p(t)r (t))

|u(t)|

(1+a)/a

\Wl(t)\

Tk(c1)+1 k(d1)-1

\) + ^

\^c1 i=k(ci) + 1

' fTi+v r Ti+1"1 s d1 \

_ Ti JTi+OJ JTk(d1)/

\ xa(t - O) ' xa(t)

W1(t)dt

P0 (t)\W1(t)\a+1dt,

- P(t)P0

I fTk(c1)+1 k(d1)-1 fTi+1 M \

Ü + ^ \ + )

y-'c1 i=k(c1) + 1J Ti Tk(d1) J

a + mwMnumn w<0| + if+WJ))

(p(l)r (I))

(1+a)/a

\ W1(t) \

Tk(c1)+1 k(d1)-1

C1 i=k(c1)+1

sTi+o * Ti+11 r d1

J Ti J Ti+O J Tk(d

'Tk(d1),

xa(t - o) xa(t)

W1(t)dt

-o (t) \ W1(t) \ a+1dt.

(2.17)

Letting y = \W1(t)\a\u(t)\, B = (a + 1)( \w1 (t)\ + \p'(t)\\w^t)\/(a + 1)\ p(t)\), A = a/(p(t)r(t))1/a and using (2.6), we have for the integrand function in above inequality that

(a + 1)\ W1(t)\a|w1 (t)|\ u(t)\ -

r1/a(t)

(1+a)/a

\W1(t) \

^ m (J1 ' IP'(t)l \ w1(t)\ \

< p(t)r(t)( |w1(t)| + pa + 1)p(t) )

(2.18)

In view of the impulse condition in (1.1) and the definition of u we have, for t = Tk, k = 1,2,..., that

u«) = -au(Tk). (2.19)

From (2.19), we have

k(d1) k(d1) / ba

X \w1(Ti)\a+1 [u(Ti) -u«)]= £ (1 - -L )\w1(Tl)\a+1u(rl). (2.20)

t=k(c1)+1 i=k(c1 )+1 \

Therefore, we get

k№) / ba\ E (1 - -¿a )\w1(Ti)\a+1u(Ti) i=k(c1)+A ai /

<Jd1 Ww (t)i + r dt

P (A (a/ I ' (Al IP (t)|\W1(t)\\

Jc1 P(t)r(t\|Wl(t)| + (* + 1)p(t) )

~ fTi+O fTi+1

rd1 \xa(t - a),

fTk(c1)+1 k(d1)-1 fTi+O fTi+1 fd

+ Z + + )^7tr- W1(t)dt

JC1 i=k(c1) + 1 L Ti JTi+oJ JTk(d1) / x w

[d1 p(t)po(t)\w1(t)\a+1dt.

On the other hand, for t e [ci,di] \ {Ti},

(2.21)

(r(t)Oa(x'(t)))' = e(t) - po(t)Oa(x(t)) - (x(t - a)) < 0. (2.22)

Hence r(t)Oa(x'(t)) is nonincreasing on [c1,d1] \ {Tk}.

Because there are different integration intervals in (2.21), we will estimate x(t-a)/x(t) in each interval of t as follows.

Case 1. t e (Ti,Ti+1 ] c [c1,d1], for i = k(c1) + 1,...,k(d1) - 1.

Subcase 1. If Ti + a < t < Ti+1, then (t - a, t) c (Ti,Ti+1]. Thus there is no impulsive moment in (t - a, t). For any s e (t - a, t), we have

x(s) - x(T+) = x'(&)(s - Ti), ¿1 e (Ti,s).

(2.23)

Since x(t+) > 0, r(s) is nondecreasing, function is an increasing function and

r(t)Oa(x'(t)) is nonincreasing on (Ti,Ti+1), we have

*«(x(s)) > ^ (x(s)) > ^«(^(s - *)) - ri^ll^M^lllcs - ^ > r^^,(s)) (s - T)a = ®«(x'(s)(s - Ti)), h e (rus).

(2.24)

Therefore,

x'(s) l

v ' <-. (2.25)

x(s) s - Ti

Integrating both sides of the above inequality from t - a to t, we obtain

x(t - a) t - Ti - a

x(t) t - Ti

, t e (Ti + a,ri+i]. (2.26)

Subcase 2. If Ti <t <Ti + a, then Ti - a < t - a <Ti <t <Ti + a. There is an impulsive moment Ti in (t - a, t). For any t e (Ti,Ti + a), we have

x(t) - x(t+) = x'(l2)(t - Ti ), ¿2 e (Ti,t). (2.27)

Using the impulsive condition of (l.l) and the monotone properties of r (t), ®a(-) and r(t)Oa(x'(t)), we get

(2.28)

*.(,(.) - «(T)) = - t,)« < (, - t,)«

= r(Ti)Oa(bix'(Ti)(t - Ti))

r(&) .

Since x(t,) > 0, we have

- -) < ^(b-K|('-T.))- (2.29)

In addition,

x(Ti) > x(Ti) - x(Ti - o) = x'(l3)a, ¿3 e (t, - o,t,). (2.30)

Using the same analysis as (2.24) and (2.25), we have

x'(ri) 1 x(ri) a'

From (2.29) and (2.31) and note that the monotone properties of and r(t), we get

- -^nKbi (t - *)) s < (t - *)

(2.32)

x(t) bi —M- <ai + — (t - Ti). x(ri) a

(2.33)

In view of (A3), we have

x(Ti) >____

x(t) aai + bi(t - Ti) ~ bi(t + a - Ti)

(2.34)

On the other hand, similar to the above analysis, we get

x'(s) 1

x(s) s - Ti + a'

s e (ri - a,Ti).

(2.35)

Integrating (2.35) from t - a to Ti, where t e (ri, Ti + a), we have

x(t - a) > t-T > 0. (2.36)

x(Ti) a

From (2.34) and (2.36), we obtain

x(t - a) t - Ti . .

V m ' > uTl-t e (Ti,Ti + a). (2.37)

x(t) bi(t + a - Ti)

Case2 (t e [ci,Tk(ci)+i]). SinceTk(ci)+ a<ci,thent-a e [ci-a,Tko+i-a] c (Tk(ci),Tk(ci)+i-a]. So, there is no impulsive moment in (t - a,t). Similar to (2.26) of Subcase i, we have

x(t - aK t - Tk(ci) - a . _ r ^ n,,a)

^JT > t -Tk{ci) , t e fa,^]. (Z38)

Case 3 (t e T(di),di]). Since Tk(di) + a > di, then t - a e (Tk(di) - a,di - a] c (Tk(di) - a,Tk(di)). Hence, there is an impulsive moment Tk(di) in (t - a, t). Making a similar analysis of Subcase 2, we obtain

x(t - a K t - Tk(di) ^^ A i oqca

WA >7-T.-r > 0, t e (jk(di),di\. (2.39)

x(t) bk(di){t + a - Tk(di))

From (2.21), (2.26), (2.37), (2.38), and (2.39) we get

k(d1) / ba \ E (1 - aa )\w1(Ti)\a+1u(Ti) i=k(c1)+A ai /

< I p(t)r(t)( K(t)| + ir(: ) dt- fk(c1)+1 W1(t)(t-Tk(c1) dt

( + ) ( ) (2.40)

^ (t - Ti - o)a

k(d1)-1

i=k(c1)+1

fTi+O (t _ T.)a fTi+1 It, W1(t)~LW1(,) (t - t,)

- f1 W«)^^dt - f1 p(t)p0(t)\w1(t)\a+1dt.

jTk(d1) b£(d1)(t + O - Tk(d1)) Jc1

On the other hand, for t e (Ti-1,Ti] c [c1,d1], i = k(c1) + 2,.. .,k(d1), we have

x(t) - x(Ti-1)= x'(¿)(t - Ti-1), £ e (Ti-1,t). (2.41)

In view of x(Ti-1) > 0 and the monotone properties of ®a(-), r(t)Oa(x'(t)) and r(t), we obtain

-a(x(t)) > ®a(x'(£))®a(t - Ti-1) > r^j-^x'(t))Oa(t - Ti-1). (2.42)

This is

r(t)Oa(x'(t)) < p1r(¿) (2 43)

P(t) O(x(t)) < (t^. (2.43)

Letting t ^ T- , we have

u(Ti) = P(t)< T^V, i = k(c1) + 2.....k(d1). (2.44)

-a(x(Ti)) (Ti - Ti-1)

Using similar analysis on (c1,Tk(c1)+1], we get

u(Tk(cO+1) < --——rf. (2.45)

\Tk(c1 ) + 1 - c1)

Then from (2.44), (2.45), and (A3), we have k(d1) / ba \

E [Of -1 )\w1(Ti)\a+1u(Ti) <p1 i=k(d)+A ai /

= pmQdl [w^1],

+ k(d1) w^Tfc(d)+0 |a+10(c1)+ E \W1(Ti)\a+1^(Ti)

i=k(c1)+2

where 0(d) = (bU(c1)+1 - a^)/a^^)+1 - c1)a and £(;) = (bf - af)^; - T,-1)a.

From (2.40) and (2.46), we obtain

wM^ÇIZSdt

(t - Tk(Cl))"

k(dx)-\ X

i=k(ci)+\

f" "'M - «pkidlf m (247)

'Tk(dt) K(dt)(.t + a - Tk(di))

.......J>« ^"^dt

f W'Vt^—^^^i dt

[ p(t)po(t)\wi(t)\a+1dt -Çd P(t)r(t)(\w[(t)\ + dt

<pmQdï [wr'].

This contradicts (2.8).

Next we consider the case k(ci) = k(di). From the condition (Si) we know that there is no impulsive moment in [ci, di]. Multiplying both sides of (2.47) by \wi(t)\a+i and integrating it from ci to di, we obtain

U(t)\wi(t)\a+'dt -\u(t)\(a+1)/a\wi(t)\a+'dt

JC1 Jci (p(t)r(t))1/a (2.48)

- Îd W'(t)dt -jc P(t)Po(t)\wi(t)\a+'dt.

Similar to the proof of (2.21), we have

Id' ^W,(0 + p(t)p0(t)\Wj wr - P(t)rW(|wj (()| + ip«- "'

dt < 0. (2.49)

Using same way as Subcase i, we get

x(t - a) > t - C'

x(t) t - C' + a

t e [C',d'].

(2.50)

From (2.49) and (2.50) we obtain

ç d' J C'

W'(t)/L(t - ^ + p(t)po(t)\wj (t)\'+' - p(t)r(t)( \wj (t)\ + lp'mwi (t)

(t - C' + a)0

(a + ')p(t)

dt < 0.

This contradicts condition (2.9).

When x(t) < 0, we can choose interval [c2,d2] to study (1.1). The proof is similar and will be omitted. Therefore, we complete the proof.

Remark 2.4. In article [14], the authors obtained the following inequalities:

x(t) tj + t - t

xx^ > aj ; >0 t e (tj, tj + t), (2.52)

See [14, equation (2.9)],

x(t) -T >tt-j > 0, t e (tj, tj + t). (2.53)

x(tj) t - \j j

See [14, equation (2.10)].

Dividing [14, equation (2.10)] by [14, equation (2.9)], they obtained

^^ > / - tj . > 0, t e (tj, tj + t). (2.54)

x(t) aj(tj + t - t)

See [14, equation (2.11)]

This is an error. Moreover, similar errors appeared many times in the later arguments, for example, in inequalities (2.15), (2.19), and (2.20) in [14]. Moreover, the above substitution can lead to some divergent integrals, for example, the integrals in (2.22), (2.24) in [14]. Therefore, the conditions of their Theorems 2.1-2.5 must be defective. In the proof of our Theorem 2.3, this error is remedied.

Remark 2.5. When a = 0, that is, the delay disappears, (1.1) reduces to (1.7) studied by Ozbekler and Zafer [13]. In this case, our result with p(t) = 1 is Theorem 2.1 of [13].

Remark 2.6. When a = 0, that is, the delay disappears in (1.1) and a = 1, our result reduces to Theorem 2.1 of [12].

Remark 2.7. When ak = bk = 1 for all k = 1,2,... and a = 0, that is, both impulses and delay disappear in (1.1), our result with a = 1 and p(t) = 1 reduces to Theorem 1 of [21].

In the following we will establish a Kong-type interval oscillation criteria for (1.1) by the ideas of Philos [18] and Kong [22].

Let D = {(t,s) : t0 < s < t}, H1,H2 e C1(D,R), then a pair function H1,H2 is said to belong to a function set H, defined by (H1,H2) e H, if there exist h1,h2 e Lloc(D,R) satisfying the following conditions:

(C1) H1 (t, t) = H2(t, t) = 0, H1 (t,s) > 0,H2(t,s) > 0 for t > s;

(C2) (d/dt)H1(t,s) = h1(t,s)H1(t,s), (d/ds)H2(t,s) = h2(t,s)H2(t,s).

We assume that there exist cj,dj,Sj / {;k,k = 1,2,—}(j = 1,2) such that T < c1 < 61 < d1 < c2 < 62 < d2 for any T > t0. Noticing whether or not there are impulsive

moments of x(t) in [cj,Sj] and [Sj,dj], we should consider the following four cases, namely, (S5) k(cj) < k(6j) < k(dj); (S6) k(cj) = k(Sj) < k(dj); (S7) k(cj) < k(Sj) = k(dj) and (S8) k(cj) = k(Sj) = k(dj). Moreover, in the discussion of the impulse moments of x(t - a), it is necessary to consider the following two cases: (S5) rk(Sj) + a > Sj and (S6) rk(Sj) + a < Sj .In the following theorem, we only consider the case of combination of (S5) with (S5). For the other cases, similar conclusions can be given and the proofs will be omitted here.

For convenience in the expression below, we define, for j = i, 2,

ni,j =■

Hi( 6jrcj)

fTHcj )+i _

Hi( t,cj)

k(Sj )-1

i=k(cj )+1

(t - Tk(Cj) - tf) (t - Tk(Cj))

raH (tc \ (t - Ti)

lJbf(t + a - Ti) t

a ¡-Ti+i

fTi+1 _

Hi( t,cj)

J Ti +CT

(t - T - O)a (t - T)a

C6' _ (t - Tk(Sj)) r6'

üi{ t,Cj)-±--- dt + p(t)po(t)Hi( t,Cj)dt

JTk(6>) K(6,)\t + ° - Tk(6')) Jci

ni,i =■

(a + 1) 1

p(t)r(t)Hi( t,cj)

hi( t,cj) +

H2( dj,6j)

fTk(6j)+0 _ t

H2( d'j)-^

j6, ba (

' bk(6,) V

" ' P(t)

(t - Tk(6j))

bt(6,){t + - Tk(6j)) (t - Tk(6j) - o)

rTk(6j)+1 __I t - Tk(6j) - C

' H2(dj,t\\---

jTk(6,)+° [t - Tk(6j )J

k(dj)-1

i=k(6j )+1

r+°H2(di,t\1 a(t Ti)\adt+ P H2(dj,t\)0

Jt 2Kj,Jb?(t + a - Ti)a Jt+o ^ ' ' (t-Ti

(t-Ti)a

HH2 dj,t

'Tk(d>)

(t - Tk(d'))

K(d,)(t + - Tk(dj))

(a + 1)a+1 J6j

f' p(t)r (t)H2 (dj, t\ h2(dj,t\ ■ p'(t) 6j

- p(t)po(t)H2{ d',t)dt 6j

whereHi(t,Cj)= Hi{i,c])f (f), ftidpt) = H2{d]/t)f (f) and f (f) = n—no|e(f)|non?=1n?(Mf))n' with no = 1 - X"=1 n and n1,n2,---,n" are positive constants satisfying conditions of Lemma 2.1.

Theorem 2.8. Assume (2.7) holds. If there exists a pair of (H1 ,H2) e H such that

n«+ Hjj^ > ■1-2' (156)

then (1.1) is oscillatory.

Proof. Assume, to the contrary, that x(f) is a nonoscillatory solution of (1.1). Without loss of generality, we assume that x(f) > 0 and x(f — o) > 0 for f > fo. In this case the interval of f selected for the following discussion is [ci,di]. Similar to the proof of Theorem 2.3, we can get (2.14) and (2.19). Multiplying both sides of (2.14) by H1 (f,C1) and integrating it from C1 to 61, we have

f1 Hi (t, ci)u'(t)dt < f1 Hi (t, ci)(^u(t)--|w(f) |(i+a)/a ) dt

Jci Jci \ P(t) (p(t)r(t))1/a /

rfii_ xa (t-a ) f fii

Hi(t,ci) \ ) } dt - Hi(t,ci)p(t)po(t)dt, Jci x (t) Jci

Pt)r (t))i/a / (2.57)

where Hi(t,ci) ■ Hi(t,ci)p(t)y(t), f(t) ■ n°n°KOPE&inO'' (pt(t))n* with n° ■ i - £h m and ni, n2,...,n„ are positive constants satisfying conditions of Lemma 2.i.

Noticing impulsive moments Tk(ci)+i,Tk(ci)+2,...,Tk(si) are in [ci,6i] and using the integration by parts formula on the left-hand side of above inequality, we obtain

f5i / fTk(ci)+i fTk(ci)+2 f5i \

Hi(t,ci)u'(t)dt ■ ( + + ••• + )Hi(t,ci)du(t)

Jci X-'ci Jrk(ci)+i J Tk(Si) J

' Tk(ci)+i

k(5i) / ba \

■ E (i 0 -a Hi(TI,ci)u(TI) + Hi(6i,ci)u(6i) (2.58)

¿■k(ci)+A a'/

fTk(ci)+i fTk(ci)+2 f5i \

+ + ••• + )Hi(t,ci)hi(t,ci)u(t)dt.

Jci J Tk(ci)+i J Tk(6i) /

Substituting (2.58) into (2.57), we obtain

61 _ ya(t-n) k(61) /ba

H1 (tc) x-XLf dt < E -1

c1 x w i=k(c1)+1

Tk(c1)+1 fTk.(c1)+2

(61) ba

ll aaa- 0 H'(T"

CTk(c1)+1 CTk(c1)+2 C61

(J + + •••+ w,

c1 Tk(c1)+1 Tk(61)

c)u(Ti) - H1(Ö1,c1)u(Ö1)

h1(t, c1)

' Tk(c1)+1

\u(t)\-

(p(t)r(t))

\u(t)\

(a+1)/a

(2.59)

[ 1 p(t)po(t)H1(t,n)dt.

Letting A = a/(p(t)r(t))i/a, B = \hi(t,ci) + p'(t)/p(t)\, y = \u(t)\ and using (2.6) to the right-hand side of above inequality, we have

C61__x*(t-o) k(61) /ba

J H1(t,c1) -a{t) 'dt < £ [-L - 1 )H1(Ti,n)u(Ti) - H1(Ö1,c1)u(Ö1)

i=k(c1)+1 i

1 c61 + (071^1, p(,)r(,)H'(,,c')

h1(t,c1)

dt (2.60)

p(t)po

(t)H1(t,n)dt.

Similar to the proof of Theorem 2.3, we need to divide the integration interval [ci,Si] into several subintervals for estimating the function x(t - a)/x(t). Using the methods of (2.26), (2.37), (2.38), and (2.39) we estimate the left-hand side of above inequality as follows:

61 H1(t,n) -a([ - G) dt c1

(Tk(c1)+1 ~ (t - Tk(c1) - o)a

> H1(t,n)K k(c) / dt

Jc1 (t - Tk(c1))

k(61)-1

H1 (tc)

(t - Ti)a

i=k(c1)+1

T H1(t,c1)ha a ^

JTk(61) bl(61)(t + ° - Tk(61))

b0(t + O - Ti)

t - Tk(61) a

.f+'H^-t-p-fj,

J Ti +CT (t Ti)

(2.61)

Abstract and Applied Analysis From (2.60) and (2.61), we have

fc1"H,(,,e)T"-1' d,

C1 (f — Tk(c1))

k(S1)—1

i=k(d)+1

61 - (f — n(S1))a

H1(f,C1 df + fi+1 H1(f,C1)(f — Ti 0fdf

J Ti+o

bf(f + o — t)

(f — Ti)a

+ f R1(f,C1)-ua u V

"^(61) bak(61)(f + 0 — Tk(61))

(a + 1)a+1

r (f)H1(f,C1)

h1(f,C1)

a+1 f61

df + p(f)po(f)H1(f,C1)df

k(6{) / ba

< y — 1 )H1(Ti,C1 )u(t) — H1(61,C1)M(61).

k(61) /ha \ Z (ha—1)

i=k(C1)+1 \ hi /

(2.62)

On the other hand, multiplying both sides of (2.14) by H2 (d1, f) and using similar analysis to the above, we can obtain

nK+°H1(i„ f) — ^

k(d1)-1

i=k(61 )+1

bl(61)(f + 0 — Tk(61))

rTk(61) Tk(61)

a . g,(d„t)C—— *

(f — Tk(61))

H2(d1,f),J/, ^df + f H2(d1,f) ———— df

T H2(d1 ,f)

bf(f + o — Ti) ./Ti+o

(f — Tk(d1))

(f — Ti)fl

—df + f 1 p(f)po(f)H2(d1,f)df

(a + 1)a+1 J 61

r (f)H2 (d1,f)

h2(d1, f)

k(d1) / b«

Z ( -a — 1 )H2(d1,Ti)u(Ti) + H2(d1,61)u(61).

i=k(61)+1 \ hi

(2.63)

Dividing (2.62) and (2.63) by H1(61,C1), and H2(d1,61) respectively, and adding them, we

nu + n2,1 <

1 k(61) / ba \

Z (ha—0 H1

i=k(C1)+1\ hi /

H1(61,C1) 1

C1)u(Ti)

H2(d1, 61) i=k(61) + A h

k(d1) / ba

Z lb? — 1 )H2(d1,Ti)u(Ti).

Using the same methods as (2.46), we have

fc(5i) /-a \

X ( -a - 1 )Hi(Tl,ci)u(Tl) < piriQ^ [Hi(-,ci)],

(2.65)

k(di) / r—a \

Z r-f - 1 H2(di,Tt)u(Tt) < piriQdl [H2(di, •)].

i=k(Si ) + 1\ ai /

From (2.64), (2.65), we can obtain a contradiction to the condition (2.56).

When x(t) < 0, we choose interval [c2,d2] to study (1.1). The proof is similar and will be omitted. Therefore, we complete the proof. □

Remark 2.9. When a = 0, that is, the delay disappears and a = 1 in (1.1), our result Theorem 2.8 reduces to Theorem 2.2 of [12].

3. Examples

In this section, we give two examples to illustrate the effectiveness and nonemptiness of our results.

Example 3.1. Consider the following delay differential equation with impulse:

I / n \ 13/2 / n \ I / n \ 1/2 / n \ x" (0+ - —) I x\t - J^J + | xyt - —) I x\t - n) = - sin(2i), t/n,

x(t+) = akx(Tk), x'(r+) = bkx'(Tk), t = Tk,

where Tk : T„r1 = 2nn + 5n/18, t„/2 = 2nn + 11n/18,n e N and pY,p2 are positive constants.

For any T > 0, we can choose large n0 such that T < c1 = 2nn + n/6, d1 = 2nn + n/3, c2 = 2nn + n/2, d2 = 2n + 2n/3, n = n0,n0 + 1,.... There are impulsive moments Tn/1 in [c1,d1] and Tn,2 in [c2,d2]. From Tn,2 - Tn/1 = n/3 > n/12 and Tn+1,1 - Tn,2 = 5n/3 > n/12 for all n > n0, we know that condition Tk+1 - Tk > a is satisfied. Moreover, we also see the conditions (S1) and (2.7) are satisfied.

We can choose no = n1 = n2 = 1/3 such that Lemma 2.1 holds. Let w1(t) = w2(t) = sin(6t) and p(t) = 1. It is easy to verify that Wi(t) = 3(^2)1/3| sin(2t)|1/3sin2(6t). Byasimple calculation, the left side of (2.8) is the following:

wm (' ; T"->; ? dt

- (' ; Tfc(-l))

(' ; Ti)a ^ f™Tir/ A' ; T ; a)a

/x__ \a A

J, Wi(t)+LWi(t)"(^d'

f1 Wi(') a ^ Nad' + f p(')po(')\wi(')\a+1d'

JTk(di) »a(d1A' + a ; Tk(di)) J-1

fd1 m ,J I ' ml № I|Wl(t)i^\

Lip(t)r (t\ lw(t)l+lanwr)

C2nn+5n/18 W t - 2(n - 1)n - 11n/18 - n/12

Jiun+n/e 1() t - 2(n - 1)n - 11n/18

r2nn+n/3 t - 2nn - 5n/18 J (2nn+n/30, 2, , Wx{t)-—7-=-r ,'a-ttttt dt - 36cos26tdt

'2nn+5n/18 bn,1(t - 2nn - 5n/18 + n/12) J2nn +n/6

t + 25n/18

j5n/18^f J + 47 n/36 jn/3 t - 5n/18 „

W,{t)t , ^J m 0 dt H W^—-—„ dt - 3n

/5n/18

bn^t - 7n/36)

0 71 RN

: 0.199 + 0- ) - 3n.

On the other hand, we have

Qd1 H =

27 bn,1 - fln,1

4n an 1

Thus if

3^2(0.199 + > 3n + 27 bn1 - ^

4n an 1

the condition (2.8) is satisfied in [ci,di]. Similarly, we can show that for t e [c2,d2] the condition (2.8) is satisfied if

3^2(0.057 + ^ > 3n + 27 bn,2 - an,2

4n an2

Hence, by Theorem 2.3, (3.1) is oscillatory, if (3.4) and (3.5) hold. Particularly, let ak = bk, for all k e N, condition (3.4) and (3.5) become

0 715N 0.199 + 0.— ) > n,

,_0.003 \

0.057 + -- ) > n.

Example 3.2. Consider the following equation:

x"(t) + ^(t)

xlt - 3

3/2 , 2

Xl t - 3 ) + ^2p2 (t)

xt - 3

-1/2 / 2 Xt- 3

= e(t), t /Tk,

x(t+) = a kX(Tk ),

'(r+)= bkx'(Tk ), k = 1,2.....

where p1,p2 are positive constants; Tk : Tn/1 = 9n + 3/2, Tn,2 = 9n + 5/2, Tn,3 = 9n + 15/2, Tn,4 = 9n + 17/2 (n = 0,1,2,...) and Tk+1 - Tk> a = 2/3. In addition, let

'(t - 9n)3, t e [9n, 9n + 3],

P1(t)= p2(t) ^ 33, t e [9n + 3,9n + 6],

3 (3.8)

^ (9n + 9 - t)3, t e [9n + 6,9n + 9],

e(t)=(t - 9n - 3)3, t e [9n, 9n + 9].

For any t0 > 0, we choose n large enough such that t0 < 9n, and let [c1,d1] = [9n + 1,9n + 3], [c2,d2] = [9n + 7,9n + 9], 61 = 9n + 2 and 62 = 9n + 8. It is easy to see that condition (2.7) in Theorem 2.8 is satisfied. Letting H1(t,s) = H2(t,s) = (t - s)3, we get h1(t,s) = -h2(t,s) = 3/(t - s). By simple calculation, we have

'9n+3/^„ n-_\2 t - 9n - V6,

n1,1 = 3^1^ (t - 9n - 1)3(9n + 3 - t)(t - 9n)2 —-— dt

9n+1 t - 9n + 1/2

(t - 9n - 1)3(9n + 3 - t)(t - 9n)2-—\ - 9- 3/2.. dH - -

J 9n+3/2V" ^ v vv' bn,1(t - 9n - 5/6)"'I 8

3^Q3/2 u> - 1)3M(3+-;)(M - 1/6)du + f U(U -1)3(3 - U,)(U- 3/2) du\ - 9

u +1/2 J3/2 bn,1(u - 5/6)

0.29^\ 9

9n+13/6

„ ,_0.290 \ 9

+ mr)- 9,

4/. n x2 t - 9n - 3/2

n,1 = 3^1^ (9n + 3 - t)4(t - 9n)2 -dt

' J<>n+2 bn,1(t - 9n + 5/6)

r'2 (9n + 3 - ,)4(, - 9n)2id*-™dt

J 9n + 13/6 t - 9n - 3/2

(9n + 3 - t)4(t - 4 - 9

f _ _ _

J 9n+5/2 bn,2(t - 9n - 11/6^ J 8

13/6 u2(3 - u)4(u - 3/2) Jx f5/2 u2(3 - u)4(u - 13/6)

_ f13/6 u2(3 - u) (u - 3/2) f

= ^nL ^ 13/6 u - 3/2

2 bn 1 u / 13/6

3 u2(3 - u)4(u - 5/2) | 9 at>-

5/2 bn,2 (u - 11/6) [ 8

„ ._/0.102 nni_, 0.005 \ 9

+ °.056 + br)- 9.

Then the left-hand side of the inequality (2.56) is

^ ^ „ ,_/ or,, 0.392 0.005X 9

nu + n2,i « 0.376 + -— + -— ) - -. (3.10)

\ bn,1 bn,2 / 4

Because n = r2 = 1, Tk(ci)+i = Tk(Si) = Tn, 1 = 9n + 3/2 e (ci,6\) and rk(6i)+i = rk(di) = Tn,2 = 9n + 5/2 e (61, d1), it is easy to get that the right-hand side of the inequality (2.56) for j = 1 is

mht) )] + Hiidbu alt hm,)] = ^ + ^ - (311)

Thus (2.56) is satisfied with j = 1 if

-0.392 0.005 \ 9 bn,i - fln,i bn,2 - fln,2 /ai0,

3HiH2\ 0.376 + --+ -- > - + —^-- + —^--. (3.12)

vr r V bn,i bn,2 / 4 4fln,i 4an,2

When j = 2, with the same argument as above we get that the left-hand side of inequality (2.56) is

ni,2 + n2,2

3 3K-n-1 f15/2 (u - 9)2(u - 3)(u - 7)3(u - 19/6)

= Wi^ll -7-5/2-lu

(u - 9)2(u - 3)(u - 7)3(u - 15/2) J

(u - 3)(9 - u) (u - 15/2) ^ 1 f17/2 (u - 3)(9 - u)5(u - 19/6)lu

115/2 bn,3(u - 41/6)

+ T49/6 (u - 3)(9 - u)5(u - 15/2) lu + f

J8 bn,3(u - 41/6) J49/6 u - 5/2

+ f9 (u - 3)(9 - u)5(u - 17/2) 9

J17/2 bnA(u - 47/6) „ ,_0.724 0.001 \ 9

0.724 0.001 \ 9

3 bn,4

(3.13)

and the right-hand side of the inequality (2.56) is

r2 r^Si r t w m r2 r-\li<TT t j \ i bn,3 - an,3 bn,4 - an,4 /0 i a\

m&C) qs2 [Hi( ;c2)] + mas Qi2 iH2(l2,' )] = -ramr + -ar. (3A4)

Therefore, (2.51) is satisfied with j = 2 if

-/ 0.724 0.001 \ 9 bn,3 - fln,s bn,4 - fln,4

3^nii2( 0.400 + --+ -- > - + —^-- + —^--. (3.15)

bn,3 bn,4 / 4 4an,s 4an,4

Hence, by Theorem 2.8, (3.7) is oscillatory if

0.392 0.005 N 9 bn,1 - an,1 ¿V - an,2

„ ,_0.392 0.005 \ 9

^^(0.376 + — + _) > 9

„ ,_0.724 0.001 \ 9

bn,i b n,2 / 4 4an, 1 4an,2

0.724 0.001 \ 9 bn 3 - an 3 bn 4 - fln 4

■ 1 > - + —-- + —--.

4an 3 4an

3 bn 4 4 4an 3 4an 4

Particularly, when ak = bk, for all k e N, condition (3.16) becomes

„ ,_/„^ 0.392 0.005 \ 9

^^(0376+—+mr) > 9

n __/ 0.724 0.001 \ 9

(3.16)

(3.17)

Acknowledgments

The authors thank the anonymous reviewers for their detailed and insightful comments and suggestions for improvement of the paper. They were supported by the NNSF of China (11161018), the NSF of Guangdong Province (10452408801004217).

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