Thandapani et al. SpringerPlus (2016) 5:558 DOI 10.1186/S40064-016-2117-5

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RESEARCH

Open Access

Interval oscillation criteria

for second-order forced impulsive delay

differential equations with damping term

Ethiraju Thandapani1, Manju Kannan1 and Sandra Pinelas2*

CrossMark

'Correspondence: sandra.pinelas@gmail.com 2 Departamento De Ciencias Exactas E Naturais, Academia Militar, Av. Conde Castro Guimaraes, 2720-113 Amadora, Portugal Full list of author information is available at the end of the article

Abstract

In this paper, we present some sufficient conditions for the oscillation of all solutions of a second order forced impulsive delay differential equation with damping term. Three factors-impulse, delay and damping that affect the interval qualitative properties of solutions of equations are taken into account together. The results obtained in this paper extend and generalize some of the the known results for forced impulsive differential equations. An example is provided to illustrate the main result.

Keywords: Oscillation, Second-order, Impulse, Damping term, Differential equation

Mathematics Subject Classification: 34A37, 34C10

Background

In this paper, we consider the second-order impulsive differential equation with mixed nonlinearities of the form

(r(t)(x'(t))Y)' + p(t)(x'(t))Y + q(t)xY (t - 5)

+ En=i q<(t)lx(t - 5)|ai-1x(t - 5) = e(t), t = vk; (1)

x(T+) = akx(Tk), x (t+) = bkX(Tk)

where t > t0, k e N, {tk} is the impulse moments sequence with 0 < t0 = to < t1 ..., lim Tk = <x,

x(t£) = x(t— ) = lim x(t), x(t+) = lim x(t)

,, , — ,. x(xk + h) - x(xk)

x (Tk) = x (t- ) = lim ---,

k h^0- h

f, + .. x(Tk + h) - x(t+)

x (T+) = lim---.

k h^0+ h

Springer Open

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Let J c R be an interval and define PLC (J, R) = {x : J ^ R : x(t) is continuous on each interval (tk, tk+1), x(texist, and x(tk) = x(t—) for all k e N}.

For given t0 and <p e PLC([t0 — S, t0], R), we say x e PLC([t0 — S, to], R) is a solution of Eq. (1) with initial value < if x(t) satisfies (1) for t > t0 and x(t) = <(t) for all t e [t0 — S, t0]. A non-trivial solution is called oscillatory if it has infinitely many zeros;otherwise it is called non-oscillatory.

In recent years the theory of impulsive differential equations emerge as an important area of research, since such equations have applications in the control theory, physics, biology, population dynamics, economics, etc. For further applications and questions concerning existence and uniqueness of solutions of impulsive differential equation, see Bainov and Simenov (1993), Lakshmikantham et al. (1989). In the last decades, interval oscillation of impulsive differential equations was arousing the interest of many researchers, see Li and Cheung (2013), Liu and Xu (2007, 2009), Muthulakshmi and Thandapani (2011) and Ozbekler and Zafer (2009, 2011) considered the following equations

( (r(t)($a(x'))'+ p(t)$a(X) + q(t)^p(x) = e(t), t = Tk;

\ A(r(t)$a(x')) + qi^p(x) = ei, t = Tk, k e N.

As far as we know, it is the first article focusing on the interval oscillation for the impulsive differential equation with damping term. Their results well improved and extended the earlier one for the equations without impulse or damping. Recently Guo et al. (2014) considered a class of second order nonlinear impulsive delay differential equations with damping term and established some interval oscillation criteria for that equation.

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 (2010) gave the first research in this direction recently. They considered second order delay differential equations with impulses

' x''(t) + p(t)f (x(t - t)) = e(t), t > to, t = tk;

+ ' + (3)

x(t+) = akx(tk), x (t+) = bkx'(tk), k = 1,2,...

and established some interval oscillation criteria which developed some known results for the equations without delay or impulses (Liu and Xu 2007; El Sayed 1993). It is natural to ask if it is possible to research the interval oscillation of the impulsive delay equations with damping term. In this paper, motivated mainly by Huang and Feng (2010) and Ozbekler and Zafer (2009), we study the interval oscillation of second order nonlinear impulsive delay differential equations with damping term (1). We establish some interval oscillation criteria which generalize or improve some known results of Guo et al. (2012a, b, 2014), Liu and Xu (2007, 2009), Muthulakshmi and Thandapani (2011), Pandian and Purushothaman (2012), Ozbekler and Zafer (2009, 2011) and Li and Cheung (2013). Finally we give an example to illustrate our main result.

Main results

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

(A1) r (t) e C 1([to, ro), (0, to)) and p(t), q(t), qi (t), e(t) e PLC ([to, to), R), i = 1,

2..., n, with r'(t) + p(t) > 0 for all t > t0; (A2) 5 > 0, Tk+i — Tk >5, k e N, ai > • • • > am > y > am+i > • • • > an > 0 are constants;

(A3) ak, bk are real constants satisfying bk > ak > 0, k = 1,2,____

We begin with the following notations: I(s) = max{i : t0 < t• < s}, r; = max{r(t) : t e [c,-, dj]}, j = 1,2 and

Ec.d = {u e C 1([cj, dj], R) : u(t) # 0, u(cj) = u(dj) = 0}.

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

Qdr ]= f 0 for I(c) = I(d),

c r^J I f(ri(C)+i)0(c) + Elided ^(TMr,-) for I(c) < I(d),

0(c) = (ai(cm)Y - (bi(c)+l)y and e(T(.) = - b

(ai(c)+i)y(ti(c)+i - c)Y af (t,- - t,-—)y

To prove our main results, we need the following lemmas.

Lemma 1 Lei (a1, a2,..., an) ¿e an n-tuple satisfying a1 > a2 > • • • > am > y > am+\ > ■ ■ ■ > an > 0. Then there exists an n-tuple (n1, n2, • • •, n«) satisfying

ni = Y (4)

and also either

Y^ ni < 1, 0 < n < 1 (5)

ni = 1, 0 < ni < 1. (6)

The proof of Lemma 1 can be found in Hassan et al. (2011) and Özbekler and Zafer (2011) which is the extension of (Lemma 1, Sun and Wong 2007).

Remark 1 For given constants a1 > a2 > ... am > y > am+1 > • • • > an > 0, Lemma 1 ensures the existence of n-tuple (n1, n2, • • •, n«) such that either (4) and (5) or (4) and (6) hold. Particularly when n = 2, and a1 > y > a2 > 0 in the first case we have

Y - a2(1 - no) ai(1 - no) - Y ni = -, n2 = -

ai - a2 ai - a2

where no be any positive number satisfying 0 < no < . This will ensure that 0 < n1, n2 < 1 and conditions (4) and (5) are satisfied. In the second case, we can solve (4) and (6) and obtain

Y - ai ai - y

ni =-, ni =-.

a1 - ai a1 - ai

The Lemma below can be found in Hardy et al. (1934).

Lemma 2 Let X and Y be non-negative real numbers. Then

2XY2-1 - X2 < (2 - 1) Y2, 2 > 1

where equality holds if and only if X = Y. 2

i / Y \ Y -Yl Let y > 0, A > 0, B > 0 and y > 0. Put 2 = 1 + Y-, X = By+1y, Y =1 y+xJ AyB y+1

in Lemma 2, we have

A - B TTl)^ B)'. (7)

Theorem 1 Suppose that for any T > 0, there exist Cj, dj / {rk}, j = 1,2 such that ci < d1 < d1 + S < c2 < d2 and q(t), qi(t) > 0, t e [c1 — S, d1] U [c2 — S, d2], i = 1,2,..., n and

( < 0 if t e [ci - S, Di],

e(t) = ^ (8) \> 0 if t e[c2 - S, ¿2],

and Uj e Ec.such that

t ^ va P(t)«(t) (Y + 1)u'(t) -

dt - c Q(t)Q(c)(t)|u(t)|Y+1 dt

■H (c)(

(Y + 1) Y + 1

I(dj) n . rdi

- Y, / Q(t)Qk(t) |u(t)|Y+1dt - / Q(t)Qj(d.)(t)|u(t)|Y+1dt

k=l (cj )+2 Tk-1 Ti (dj)

< r.fid.[|u(t)|Y+1], j = 1,2 (9)

Q(t) = q(t) + n0 II(nr1q«'(t))ni |e(i)r, no = 1 - ^ n«

(•=1 (=1

where ni > 0 are chosen according to given a\, a2,... an as in Lemma 1 satisfying (4) and (5), and

Qk(t) H ' J k = I(Cj),I(Cj) + 1,.. .,I(dj),

i < t e [Tk + 5, tk+i],

then every solution of Eq. (1) is oscillatory.

Proof Let x(t) be a non-oscillatory solution of Eq. (1). Without loss of generality, we may assume that x(t) > 0 and x(t — 5) > 0 for all t > t0 > 0. Define

r(t)(x'(t ))Y

= V, J , t e[ci - 5,di\. (10)

Then for all t = Tk, t > t0, we have

Va A (—y xY(t - S) e(t) p(t)a(t) Y Kt)| Y

a (t) = —q(t)-—--> qt (t)|x(t — 5)|a' Y--- + ——---—---—.

XY(t) XY(t) XY(t) r(t) (r(t))Y

By taking no := 1 - E¿=1 ni,

e(t)xY(t - 5)

Zo = no1

xY (t)

x-Y(t - 5)

Zi = n-1qi(t) ^ ° )5) xai-Y (t - 5), i = 1,2,..., n and using the the arithmetic-geometric mean inequality,

Zi >![ Zini < Zi > 0 i=0 i=0

we have

£ -+ief - n-*"* '-*«*(t' ""■Titr" '*-

" «^"--o'«- •)■ (12)

" xniY(t - 8) _ x(no +n1 )Y(t - 8) _ xY(t - 8)

xniY (t) _ x(n0 +n1 + -+nn)Y(t) _ xY (t)

Y[x(ai-Y)ni (t - S)x-n0Y(t - S) = 1, (=1

from (12), (11) becomes

VA^ r^Y(t - S) -n0 A -n, p(t)«(t) Y |«(t)| Y

« (t) <- q(t) --nn "H n, (t)|e(t)lnn -

(r(t))Y

_.xY(t - S) p(t)«(t) y l«(t)T Y _ = - Q(t)—^.---—---1—' t = Tk.

XY (t)

r(t) (r (t))l

For t = Tk, k = 1,2,..., we have

0J(rk ) = —).

Multiply both sides of (13) by |w(i) |Y+1 where u(t) s ECi d, and integrating from ci to d\, then using integration by parts on the left side, we have

I (di)

Y, |u(Tk)IY+Wk) - ®(T+)]

k=l(ci)+1

/"d1 /"d1 (t — 5)

< / (y + 1 )uY(t)u'(t)«(t)dt - Q(t)|u(t)|Y+1—---dt

Jc1 Jc1 xY (t)

» , Y+1

^ ^ |u(t)|Y + 1 dt -fd1 Y^ |u(t)|Y + 1dt

r(t) ./c, 1

'c1 (r(t ))Y

I (c1)+1

Y +! xY(t - 5) , 1 (d1-1) Tk+1 <-/ - Q(t)|u(t)|Y+1 dt - £ I

/c1 x (t) (c1

Y+! xY(t - 5), /"d1 r/ Q(t )|u(t )|Y+1 \ ;dt +

xY(t) ,/c1 A

Y + 1 XYit-il dt

" T1(d1)

|u(t)|Y + 1 (r (t))Y ■

Q(t)|u(t)| y

, - Jn XY(t)

d r/ ' P(t)u(t) \ Y

(Y + 1)u'(t) - P( 'J ) |«(t)||u(t)|>

Using (7) with

(Y + l)u'(t) -

p(t)u(t)

we have

(r (t ))Y

and y = |«(t)||u(t)|Y

Y+1 | U(t )| Y

/ , m /aai P(t)|u(t)|Y Y (Y + 1) |u (t) |--—- |«(t)||u(t)|Y-- j

r (t) / (r(t))Y

- r(t) (, . 1M p(t)l"(t)l\ Y + 1

- (^(y + 1)lu (t)|- 1

l«(t )l

(Y + 1)Y + 1

r (t) )

Now for t e [ci, d,] \ , k e N from (1) it is clear that

(r(t)(x'(t))Y)' + p(t)(x'(t))Y = e(t) - q(t)xY (t - 8) - ^ q,-(t)|x(t - 5)|ai-1x(t - 5) < 0.

That is

T '(t) + p(t h

((x'(t^tP) (x'(t))Y < 0

which implies that

f r'(s) + p(s) (x'(t))Y exp -—-ds

is non-increasing on [c1, d1] \ Tk.

Because there are different integration intervals in (15), we will estimate x(t — S)/x(t) in each interval of t as follows. We first consider the situation where I(c1) < I(d1). In this case, all the impulsive moments in [c1, d{] are ti(ci)+i, ti(c2)+1, ... ti(d1).

Case 1 For t e (tk, t^+1] C [c1, d1] we have the following two sub cases:

(a) If Tk + S < t < Tk+1, then (t — S, t) c (Tk, Tk+1]. Thus there is no impulse moment in (t — S, t). For any s e (t — S, t), we have x(s) > x(s) — x(t+) = x'(f)(s — Tk), f e (Tk, s). Then

(x(s))Y > (x'(H))Y(s - Tj)Y. (17)

Since (x'(s))Y exp j'^ r (v;?++p(v) dv is non-increasing in [c1, t], we have

(x'OY exp f* dv > (x'(s))Y exp f dv. (18)

Jc1 r (v) Jc1 r(v)

From (17) and (18) we have

(X'(S))Y exP i's r'(v)+P(v) dv

)Y > (x (s)) expJci r(v) dv _ Y

(x(s))Y > -; (C1)+ )"-(S _ Tk)Y

~ exP r'(v)+P(v) dv

exp J c1 r(v) dv

> (x'(s))Y(s _ Tk)Y. (19)

Therefore < . Integrating both sides of the above inequality from t — S to t, we obtain

x(t — 5) t — Tk — S

-> - > 0.

x(t) t — Tk

(b) If Tk < t < Tk + S, then Tk — S < t — S < %k < t < Tk + S. There is an impulsive moment Tk in (t — S, t). Similar to (a), we have ^j) < s-T1 +s for any s e (tk — S, %k]. Upon integrating from t — S to Tk, we obtain

^ > ^ > o. (20)

x(tk) 5

For any t e (tk, tk + 5), we have

x(t) - x(T+) < x'(t+)(t - Tk), f e (Tk, t).

Using the impulsive conditions in Eq. (1) we get x(t) - akx(xk) < bkx'(xk)(t - Tk) x(t ) bkx'(Tk )

x(tk ) x(tk ) Using ^T) < I, we obtain

x(t) bk

—— < ak + — (t - tk). x(tk ) 5

That is

x(xk ) S

-(t - Tk) + ak.

x(t) akS + bk(t - Tk)

From (20) and (21), we have

x(t - 5) t - Tk

- > - > 0.

x(t) ak5 + bk(t - Tk)

Case 2 For t e [c,, t1( )+,) we have the following three sub-cases:

(a) If c1 < t < Tj( ) + S and tI( ) > c, — S, then t — S e [c, — S, t1( }) and there is an impulsive moment tI( ) in (t — S, t). Similar to Case 1(b), we have

x(t - 5) > t - V) > 0

x(t) aI(Ci)5 + bI(Ci)(t -

Tl(c l))

(b) If ti( ) + t < t < tI( )+1 and tI( ) > c, — S, then there are no impulsive moments in (t — S, t). Making a similar analysis of Case 1(a), we obtain

x(t-5) > t-5-Tl(ci) — 0

x(t) > t-Ti(ci) -

(c) If tI( ) > c, — S, then there are no impulsive moments in (t — S, t). So

x(t - 5) > t - 5 - ti(Ci) - °

x(t) t — TI(ci)

Case 3 For t e (t^ ), d,], there are three sub-cases:

(a) If t^j ) + S < d\, t e [ti(d ), t^ } + S), then there is an impulsive moment x[). Similar to Case 2(a), we have

x(t - S) > 1 - V) > 0

X(t) «I(d1)S + bI(d1)(t - TI(d1)) ~ '

(b) If TI(d^) + S < t < di then there are no impulsive moments in (t — S, t). Making a similar analysis of Case 2(b), we obtain

x(t - S) > t — S - t{dil ^ o

x(t) t — t

(c) If x1(j ) + S > d,, then there is an impulsive moment r^ ) in (t — S, t).

Similar to Case 3(a), we obtain

x(t - 5) > t - Ti(di) > ° x(t) a(di)5 + bi(A)(t- TKdi)) ~ .

Combining all these cases, we have xY(t - S)

Q^i) for t 6[Cl) r/(ci)+1 ],

XY (t)

> { Ql(t) for t e (vk, vk+i], k = I(ci) + 1 ,...,I(Di) - 1 , (22) Q1^)(t) for t e (TI(Di)+1, D1]

Using (16) and (22) in (15) we get

Hdt) k=I (c1)+1

Y, I "(Tk )IY+1[®(Tk) - ®(*k+)]

fd1 r(t) ( , p(t)|u(t)|\Y+1 r TI(C1)+1

((y + 1)|u'(t)|- p( )("( )y dt - C1 Q(t)|u(t)|Y+1Q/1(C1)(t)dt

Jc1 (Y + 1)Y+1V r(t) , JC1

I(d^1) rk+1 rd1

_ Y' ■ Q(t)|u(t)riQi(d1 r

k=I (C1)+^ Tk JTl(d1)

' ' / Tk+1 1^1 - £ / Q(t )|u(t)|Y+1Q1(t)dt - Q(t )|u(t)|Y+1Qi1(d)(t)dt. (23)

For any t e (c,, t1( )+, ], we have x(t) — x(c1) = x'(f)(t — c1), f e (c1, t). Since x(c1) > 0, we have x(t) > x'(f)(t — c1). Then

(x(t))Y > (x'(f))Y(t — ci)Y. (24)

Using the monotonicity of (x'(t))Y exp ^ fci r (sl'+Jp(s) dSj, and (24) we have

(x'(t))Y exp (fct ^^ds)

(x(t))Y > --(\ 1 (t - ci)Y

exp (H ds)

> (x'(t))Y(t - ci)Y

for some f s (c1, t). It follows

(x'(t))Y < 1 (x(t))Y < (t - ci)Y .

Letting t ^ r/( )+,, from (9), we have / \ ri

0(V)+1) " (Ti(ci)+i - C1)Y . (25)

Making a similar analysis on (xk-i, xk ], k = I (ci) + 2,..., I (di), we can prove that

0(Tk) - (T^-b^. (26)

From (24), (25) and (A3), we obtain

I (d) _ bY

Y \u(rk)\y+l^(rk)

k=I (ci) + l -k

_ I(di) v bY

> Y-I(Cl) + 1 bl(Cl) + 1 \"(TI(c )+1 )\Y + 1ri + V y -k _ bk \u(Tk)\Y + 1ri " «Y(C1) + 1(TI(c1) + 1 _ C1)Y ' ' I(C1)+^ 1 k=f^.}+2 -1 (Tk _ Tk_ 1)Y ' KkM 1

= nadu [\u(t )\Y+1]. (27)

I(di) I(di) ^y _ y/

Y \u(Tk)\Y+Wk) _ ®(r+)]= Y, -^-YT^- !"(Tk )IY),

k=I(ci)+2 k=I(ci) + 1 -k

from (23) we have r(t)

Id (Y + 1)Y+1 ^ - w ' Jci

(Y + 1)|U(t)| - «)Y+ * - ^ Q(t)|u(t)|Y+QWt)dt

I(d1)-1 № z-di

- Y, / Q(t)|u(t)lY+1Ql(t)dt - Q(t)Kt)|Y+1Ql(dl)(t)dt > ri^fl[|u(t)|Y+1]

k=1(ci )+2 JTk- 1 jTl(dl) 1

which contradicts (9).

If I (ci) = I (di), then ftd [|w(i )|Y+1] = 0 and there is no impulsive moments in [c1, d1]. Similar to the proof of (22), we obtain

£ <^+1+1((Y + Dlu'Ct)| - +1 dt - J^)+1 Q(t)|u(t)|Y+1Q/(Clm > 0.

It is again a contraction with our assumption. The proof when x(t) is eventually negative is analogous by repeating a similar argument on the interval [c2, d2 ]. □

Following Kong (1999) and Philos (1989), we introduce a class of functions: D = {(t, s) : to < s < t}, Hi,H2 e C 1(D, R). A pair of functions (Hi,H2) is said to belong to a function class H, if Hi(t, t) = Hi(t, t) = 0, Hi(t, s) > 0,H2(t,s) > 0 for t > s and there exist h1, h2 e Lioc (D, R) such that

= hi (t, s)Hi(t, s), ^H-M = -h2(t, s)H2(t, s). (28)

For I e (cj, dj), j = 1,2,

i (d )-i

fT' (Cj )+i i -A tXk+1 1

Fij = Hi(t, Cj )Q(t)Q}{c. )(t)dt + Y / Hi(t, Cj )Q(t)Ql (t)dt

^cj k =I (cj )+i Tk

+ [' Hi(t,Cj)Q(t)Q*(t)dt

'TI (dj)

j )Q(t)QI(di)(

(y + i)Y + iJc,

r(t)Hi(t, Cj)

U U N P(t)

hi(t,C) - r(t)

I (dj )-1

H2(dj, t)Q(t)Q}(, )(t)dt + Y

H2 (dj, t)Q(t)Q1 (t)dt

H2(dj, t )Q(t )Ql(d )(t )dt

'TI (dj)

(Y + 1)Y+1 A

r(t)H2(t, Cj)

k=I (1,- HI-7Tk

7 , , A P(t)

hl(dj,t) - r(t)

Theorem 2 Suppose that for any T > 0, there exist Cj, dj, j = 1,2,1 / {rk} such that ci < 11 < d1 < c2 < 12 < d2 and (8) ho/ds. If there exists (Hi, H2) e H such that

Hi(1i, ci) where

ri,i +

H2(di, li)

r2,i > A(Hi,H2; cj, dj)

A(Hi, H2; Cj, dj) = -

^¿[Hi(., Cj)] + j fyj [H2(dj,.)] H2(dj > 1j ) j

[Hi(1j, Cj)

then every so/ution of Eq. (1) is osci//atory.

Proof Let x(t) be a non-oscillatory solution of Eq. (1). Proceeding as in proof of Theorem 1, we get (13) and (14). Noticing whether or not there are impulsive moments in [c1,11] and [11, d1], we should consider the following four cases, namely: I (ci) < I (li) < I (di); I (ci) = I (li) < I (di); I (ci) < I (li) = I (di) and I (c1) = I (11) = I (d1). Moreover, in the discussion of the impulse moments of x(t — 5), it is necessary to consider the following two cases: rI{1 +g > 1j and rI(1.< 1j. Here we only consider the case I(ci) < I(1i) < I(di); with rI(1 > 1j. For the other cases, similar conclusions can be obtained.

For this case there are impulsive moments rJ(ci) + i, rJ(ci) + 2,..., r/(1} in [c1, d1] and TI(h)+i, TI(h)+2,..., TI(di) in [11, d{]. Multiplying both sides of (13) by H1(t, c1) and integrating it from ci to 11, we have

fAl xY(t - S) fAl ,

Hi(t, ci)Q(t) \ J dt <-/ Hi(t, aW(t)dt Jd XY (t) Jcl

[1l p(t)w(t)

Hi (t, ci)dt

r11 YKt)| Y

'ci (r (t))?

Hi(t, ci)dt.

Applying integration by parts on first integral of R.H.S of last inequality, we get

f*1 xY (t - 5)

c1 Hi(t' ci)Q(t) ^Tdt

^ (4 - bY \

<- ^ Hi (Tk, ciW k y k «(Tk) - Hi(2i, ci )«(*1)

rxI (c1)+1 /(d^"1 fTk+1 r* 1

p(t) y l«(t)l y

--Ta «(t)--T"

r (t) (r(t))Y

k=I(c1)+^Tk Vo,

H1(t, C1)dt

h1(t, C1)«(t)

I (d1)

k=I (C1) + 1

( dY bY \

<- ^ H1 (Tk, C1) ak ~Y k )«(Tk) - H1(h, C1 )«(*1)

+ ri«1)+1 + ^ T+1 + /*1

WC1 k=I(c1) + 1J Tk •^TI(d1)

h1(t, C1)«(t)

p(t) r(t)

l«(t )|-

Yl«(t)l Y 1

(r (t ))Y

H1 (t, C1)dt.

Using (7) with A = have

hi (t, ci) - f§

, B = , y = |rn(t)| in the last inequality, we

f*1 xY (t - 5)

IC1 Hi(t' ci)Q(t) ^rdt

1 (dl) / nY hv'

ak - hk

<- ^ Hi (Tk, ci)|

k=l (ci) + 1

«(Tk) - Hi(*i, ci)«(*i)

/•*i

(Y + 1)Y + 1

r(t)Hi(t, ci)

u u ^ p(t)

hi(t, ci)--—

Similar to the proof of Theorem 1, we need to divide the integration interval [ci, *i] into several subintervals for estimating the function xX(—)S). Now,

C*1 xY(t - 5) ftKc1)+1 1

H1(t,C1)Q(t) 'Dt > / H1(t,C1)Q(t)Q1(C1)(t)Dt

xy (t)

1(D1)"1 /tK+1 rd1

+ Y H1(t, C1)Q(t)Q1 (t )dt + H1(t, C1)Q(t ^^(t )dt.

From (31) and (32),we obtain

fT'(c 1)+1 , ^ pk+1 1

/ Hi(t, ci)Q(t)Q1I(ci)(t)dt + J2 / Hi(t, ci)Q(t)Ql(t)dt

JC1 k=I(ci)+1J Tk

rd i 1 rh

+ / H1(t, c1)Q(t)Q (t)dt - v+1 / r(t)H1(t, c1)

Jrl(dl) I(d1) (Y + 1)Y+1 Jc1

/ Y i Y

'4 - K

h1(t, c1) - pM

«D ( aY - hY\

< Y, H1 (Tk, c^ a^-Y^ U(Tk) - H1(h,c1)MA). (33)

k=I(d)+1 \ -k '

On the other hand multiplying both sides of (13) by H2(d1, t) and then integrating from 11 to d1 and using similar analysis to the above, we can obtain

rTKh)+i , rTk+1 ,

/ H2(di, t)Q(t)Q}ai)(t)dt + Y, / H2(di, t)Q(t)Ql (t)dt

hl(dl, t) - ^ r(t)

k=I (h)+1 Tk

/di 1 rd 1

^) H2(di, t)Q(t)Q)idi> (t)dt - (Y + 1)y+1 J, r(t)H2(t, C1)

1 (d1) i aY - bY \

<- Y, H2(du Tk) -t-y-t U(Tk) - H2(du h)MA)- (34)

k=I (A1)+1 \ -k '

Dividing (33) and (34) by H1(l1, c1) and H2(d1,l\) respectively and adding them, we get

"rU + ^—TT r2,i

H1 (Xu c{) ^ H2(DI, )

( i ^ (al - K \

VHi(^i' C1) k=f)+i V ak )

1 '(Di) . (al - blx

+ WITT: E H2(D1, rk)[ Utk). (35)

H2(d1'X1) k=ife+1 V aY / 1

Using the same method as in (27), we obtain

v^ f - bl \ ,

- Y Hi(tk, ci) U(tk) <- ri^1 [Hi(., Cl)]

k=/(Cl)+l \ ak )

1 (di) f aY - bY \

- Y H2(di, Tk) k Y k «(Tk) <- ^J [H2(dl,.)]. (36) k=l(h)+1 \ ak )

From (33) and (36), we obtain

Huh*TlA+mhz)r* -K1 [Hi("ci)]+[H2(di")])

< A(Hi, H2; c;-, dj)

which is a contradiction to the condition (29). When x(t) < 0, we choose interval [c2, d2] to study Eq. (1). The proof is similar and hence omitted. Now the proof is complete. □

Remark 2 When p(t) = 0, Eq. (1) reduces to the equation studied by Guo et. al (2012b). Therefore our Theorem 1 provides an extension of Theorem 2.3 with p(t) = 1 to damped impulsive differential equation.

Remark 3 When S = 0, that is, the delay disappears and our results reduces to that of Theorem 2.1 and Theorem 1 with p(t) = 1 in Pandian and Purushothaman (2012).

Remark 4 When p(t) = 0 and y = 1 our Theorem 1 is a generalization of Theorem 2.2 in Li and Cheung (2013).

Remark 5 When the impulse is disappear, i.e., ak = = 1 for all k = 1,2,..., the delay term S = 0 and p(t) = 0 Eq. (1) reduces to the situation studied in Hassan et al. (2011). Therefore our Theorem 1 extends Theorem 2.1 of Hassan et al. (2011).

Example 1 Consider the following impulsive differential equation

(((2 + cost)x'(t)5))' + (1 + sin t)(x'(t))5 + mi(cos t)|x(t - §)l2x(t - |) +M2(cos t)|x(t - f)l2x(t - 8) = sin2t, t = 2kn ± f;

x(t+) = 1x(tk),x' (t+) = 3x'(tk), tk = 2kn ± 4, k = 1,2,

Here r(t) = 2 + cost, p(t) = 1 + sin t, q1(t) = mi cos t, q2(t) = m2 cost, e(t) = sin2t, Y = |, a1 = |, a2 = 3 and m1, m2 are positive constants. Also S = j, tk+1 — tk = n/2 > n/8. For any T > 0, we can choose k large enough such that T < ci = 4kn — n < di = 4kn and C2 = 4kn + § < d2 = 4kn + §, k = 1,2.... Then there is an impulsive moment Tk = 4kn — \ in [c1, d1] and an impulsive moment tk+1 = 4kn + f in [c2, d2]. Now choose no = 1/5, n1 = 3/5, n2 = 1/5, therefore

25 3 1 1

Q(t) = 5— (mi) 5 (m2) 51 cos 11| sin 115

If we take u1(t) = u2(t) = sin4t, t^ ) = 4kn — 7n, rI(d^) = 4kn — 4, then by a simple calculation, the left side of Eq. (9) is the following:

''dl r(t) ( , p(t )|u(t)|\ y+1 Wi)+! Y+, ,

() '(y + 1)|u'(t)| - dt -/ Q(t)|u(t)|y+1 Ql(Cl)(t)dt

(Y + 1)y+1

I(d1-1) /Tk+1 pdi

-Y Q(t)lu(t)IY+1Ql(t)dt - / Q(t)'u(t)'Y+1Qi1(d )(t)dt

k=I (ci) + 1J Tk Jt' (d1) 1

(14)14

f4kn /56

(2 + cos t) — | cos4t |-

hkn - 2 v 5

(1 + sin t)| sin4t 14 (2 + cos t)

¡4k n - f r4kn

Q(t)| sin4t| -5

Q(t)| sin4t| -5

14k n - §

Q(t)| sin4t | -5

t - 4kn + f

t - 4kn + f

Aci)+i(t + n - 4k n + f),

t - f - 4kn + f t - 4kn + f

(mi) 5 (^2) 5 (1.5196) - 0.6739.

4kn - n

4k n - §

4kn - n

Since 1(c1) = k — 1,1(d1) = k, r1 = 3, we have ri[|u(t)|Y+1 ] = 3| sin4(Tk)|14 *k - bk )

V ak J

The condition (9) is satisfied in [ci, d1 ] if

(mi) 5 (m2) 1 (1.5196) < 0.6739 (38)

Similarly, we can show that for t e [c2, d2], the condition (9) is satisfied if

(mi) 5 (m2) 1 (0.7553) < 0.5233 (39)

Since the condition (38) is weaker than (39) we can choose the constants mi, m2 small enough such that (39) holds. Hence by Theorem 1 every solution of Eq. (37) is oscillatory. In fact for m1 = 1/5, m2 = 1/6, every solution of Eq. (37) is oscillatory.

Remark 6 The result obtained in Guo et al. (2012a, b, 2014) and Erbe et al. (2010) cannot be applied to Example 1, since the results in Guo et al. (2012a) can be applicable only to equations having only one nonlinear term and the results in Guo et al. (2012b), Guo et al. (2014), Erbe et al. (2010) can be applied to equations without damping term. Therefore our results extent and compliment to Guo et al. (2012a, b, 2014), Hassan et al. (2011), Li and Cheung (2013), Pandian and Purushothaman (2012) and Erbe et al. (2010).

Conclusion

In this paper we have obtained interval oscillation criteria for Eq. (1) which extend and generalize some known results in Guo et al. (2012a), Li and Cheung (2013), Hassan et al. (2011) and Ozbekler and Zafer (2011), Pandian and Purushothaman (2012).

Authors' contributions

All authors contributed equally to this paper. All authors read and approved the final manuscript. Author details

1 Ramanujan Institute for Advanced Study in Mathematics, University of Madras, Chennai 600 005, India. 2 Departamento De Ciencias Exactas E Naturais, Academia Militar, Av. Conde Castro Guimaraes, 2720-113 Amadora, Portugal.

Acknowledgements

The author E. Thandapani thanks University Grants Commission of India for awarding EMERITUS FELLOWSHIP [No. 6-6/2013-14/EMERITUS/-2013-14-GEN-2747/(SA-II)] to carry out this research. The author K. Manju gratefully acknowledges the Research Fellowship granted by the University Grants Commission (India) for Meritorious students in Sciences. Further the authors thank the referees for their constructive and useful suggestions which improved the content of the paper.

Competing interests

The authors declare that they have no competing interests.

Received: 25 August 2015 Accepted: 6 April 2016 Published online: 04 May 2016

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