A Survey on Oscillation of Impulsive Ordinary Differential Equations

doi:10.1155/2010/354841

Review Article

A Survey on Oscillation of Impulsive Ordinary

Differential Equations

Ravi P. Agarwal,

_{Fatma Karakoc¸,}

_{and A ˘gacık Zafer}

1_{Department of Mathematical Sciences, FL Institute of Technology, Melbourne, FL 32901, USA} 2_{Mathematics and Statistics Department, King Fahd University of Petroleum and Minerals,}

Dhahran 31261, Saudi Arabia

3_{Department of Mathematics, Faculty of Sciences, Ankara University, 06100 Ankara, Turkey} 4_{Department of Mathematics, Middle East Technical University, 06531 Ankara, Turkey}

Correspondence should be addressed to Ravi P. Agarwal,[email protected] Received 1 December 2009; Accepted 3 March 2010

Academic Editor: Leonid Berezansky

Copyrightq2010 Ravi P. Agarwal 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.

This paper summarizes a series of results on the oscillation of impulsive ordinary differential equations. We consider linear, half-linear, super-half-linear, and nonlinear equations. Several oscillation criteria are given. The Sturmian comparison theory for linear and half linear equations is also included.

1. Introduction

Impulsive differential equations, that is, differential equations involving impulse effect,

appear as a natural description of observed evolution phenomena of several real world

problems. There are many good monographs on the impulsive differential equations 1–

6. It is known that many biological phenomena, involving thresholds, bursting rhythm

models in medicine and biology, optimal control models in economics, pharmacokinetics, and

frequency modulates systems, do exhibit impulse effects. Let us describe the Kruger-Thiemer

model7for drug distribution to show how impulses occur naturally. It is assumed that the

drug, which is administered orally, is first dissolved into the gastrointestinal tract. The drug is then absorbed into the so-called apparent volume of distribution and finally eliminated

from the system by the kidneys. Letxtand ytdenote the amounts of drug at timetin

the gastrointestinal tract and apparent volume of distribution, respectively, and letk1andk2

be the relevant rate constants. For simplicity, assume thatk1/k2.The dynamic description of

this model is then given by

x−k1x,

y−k2yk1x.

In8, the authors postulate the following control problem. At discrete instants of time 0

t0 < t1 < t2 < · · · < tN, the drug is ingested in amountsδ0, δ1, δ2, . . . , δN.This imposes the

following boundary conditions:

xt_ixt−_iδi, i0,1,2, . . . , N,

yt_iyt−_i, x0−y0−0.

1.2

To achieve the desired therapeutic effect, it is required that the amount of drug in the apparent

volume of distribution never goes below a constant level or plateaum,say, during the time

intervalt1, tN1,wheretN1> tN. Thus, we have the constraint

yt≥m, t∈t1, tN1. 1.3

It is also assumed that only nonnegative amounts of the drug can be given. Then, a control

vector is a point u δ1, δ2, . . . , δN in the nonnegative orthant of Euclidean space of

dimension N 1. Hence, Ω {u ∈ RN1 _{: u ≥ θ}. Finally, the biological cost function}

fδ 1/2N_i0δ2

i minimizes both the side effects and the cost of the drug. The problem is

to find infu≥θfusubject to1.1–1.3.

The first investigation on the oscillation theory of impulsive differential equations

was published in 19899. In that paper Gopalsamy and Zhang consider impulsive delay

differential equations of the form

xt axt−τ

∞

j1

bjx

t−_jδt−tj

, t /tj, 1.4

xt ptxt−τ 0, t /ti,

xt_i−xt−_ibix

t−_i, i∈N{1,2, . . .},

1.5

where 0 < t1 < t2 < · · · < tj → ∞ asj → ∞,andτ is a positive real number. Sufficient

conditions are obtained for the asymptotic stability of the zero solution of1.4and existence

of oscillatory solutions of1.5. However, it seems that the problem of oscillation of ordinary

differential equations with impulses has received attention much later10. Although, the

theory of impulsive differential equations has been well established, the oscillation theory of

such equations has developed rather slowly. To the best of our knowledge, except one paper

11, all of the investigations have been on differential equations subject to fixed moments of

impulse effect. In11, second-order differential equations with random impulses were dealt

with, and there are no papers on the oscillation of differential equations with impulses at

variable times.

In this survey paper, our aim is to present the resultswithin our reachobtained so far

on the oscillation theory of impulsive ordinary differential equations. The paper is organized

as follows. Section 2 includes notations, definitions, and some well-known oscillation

theorems needed in later sections. In Section 3, we are concerned with linear impulsive

2. Preliminaries

In this section, we introduce notations, definitions, and some well-known results which will be used in this survey paper.

LetJ t0,∞for some fixedt0 ≥0 and{tk}∞k1be a sequence inJ such thatt0 ≤t1 <

t2<· · ·< tk< tk1<· · · and limk→ ∞tk∞.

By PLCX, Y,we denote the set of all functionsψ:X → Y which are continuous for

t /tk and continuous from the left with discontinuities of the first kind att tk.Similarly,

PLCrX, Yis the set of functionsψ :X → Y having derivativeψr ∈PLCX, Y. One has

X ⊂J,Y Rn_{, orY R. In caseY R, we simply write PLCXfor PLCX,R. As usual,}

CX, Ydenotes the set of continuous functions fromXtoY.

Consider the system of first-order impulsive ordinary differential equations having

impulses at fixed moments of the form

xft, x, t /tk,

Δx|ttk fkx, k∈N,

2.1

whereft, x∈CJ×Rn_,Rn_,f

k∈CRn,Rn, and

Δx|ttk x

t_k−xt−_k 2.2

withxt±_k limt→t±_kxt. The notationΔxtkin place ofΔx|ttkis also used. For simplicity, it

is usually assumed thatxtk xt−_k.

The qualitative theory of impulsive ordinary differential equations of the form2.1

can be found in1–6,12.

Definition 2.1. A functionϕ∈PLC1J,Rnis said to be a solution of2.1in an intervalα, β

ifϕtsatisfies2.1fort∈α, β.

Fort0∈α, β, we may impose the initial condition

xt₀x0. 2.3

Each solution ϕtof 2.1which is defined in the interval α, β and satisfying the

conditionϕt₀ x0is said to be a solution of the initial value problem2.1-2.3.

Note that if t0 ∈ tk, tk1, then the solution of the initial value problem2.1-2.3

coincides with the solution of

xft, x, xt0 x0 2.4

ontk, tk1.

Definition 2.2. A real-valued functionxt, not necessarily a solution, is said to beoscillatory, if

it is neither eventually positive nor eventually negative. Otherwise, it is called nonoscillatory.

For our purpose we now state some well-known results on oscillation of second-order

ordinary differential equations without impulses.

Theorem 2.3see13. Letpt≥0.Then, the equation

ypty0 2.5

is oscillatory if

lim inf

t→ ∞ t _∞

t

psds > 1

4 2.6

and nonoscillatory if

t _∞

t

psds≤ 1

4 for larget. 2.7

Theorem 2.4see14. Letrandpbe continuous functions andrt, pt>0. If

lim

t→ ∞ _t

1

ds

rs ∞, tlim→ ∞ _t

1

psds∞, 2.8

then the equation

rty_pty

0 2.9

is oscillatory.

Theorem 2.5see15. Letf be a positive and continuously differentiable function fort≥ 0,and letft≤0.If

_∞

0

t2n−1ftdt <∞, 2.10

then the equation

yfty2n−10 2.11

has nonoscillatory solutions, wheren >1is an integer.

Theorem 2.6see15. Let ftbe a positive and continuous function fort ≥ 0,and n > 1 an integer. Then every solution of 2.11is oscillatory if and only if

_∞

0

3. Linear Equations

In this section, we consider the oscillation problem for first-, second-, and higher-order linear

impulsive differential equations. Moreover, the Sturm type comparison theorems for

second-order linear impulsive differential equations are included.

3.1. Oscillation of First-Order Linear Equations

Let us consider the linear impulsive differential equation

xatx0, t /tk,

Δx|ttkakx0, k∈N,

3.1

together with the corresponding inequalities:

xatx≤0, t /tk,

Δx|ttkakx≤0, k∈N,

3.2

xatx≥0, t /tk,

Δx|_ttkakx≥0, k∈N.

3.3

The following theorems are proved in1.

Theorem 3.1. Leta∈PLCJand1−ak/0, k ∈N.Then the following assertions are equivalent.

1The sequence{1−ak}has infinitely many negative terms.

2The inequality3.2has no eventually positive solution.

3The inequality3.3has no eventually negative solution.

4Each nonzero solution of 3.1is oscillatory.

Proof. 1⇒2.Let the sequence{1−ak}have infinitely many negative terms. Let us suppose

that the assertion2is not true; that is, the inequality3.2has an eventually positive solution

xt, t≥T0.Lettk≥T0be such that 1−ak<0.Then, it follows from3.2that

xt_k≤1−akxtk<0, 3.4

which is a contradiction.

2⇔3. The validity of this relation follows from the fact that ifxtis a solution of

the inequality3.2, then−xtis a solution of the inequality3.3and vice versa.

2and3⇒4.In fact, if3.1has neither an eventually positive nor an eventually

negative solution, then each nonzero solution of3.1is oscillatory.

4⇒1.Ifxtis an oscillatory solution of3.3, then it follows from the equality

xt xt0exp −

_t

t0 asds

t0≤tk<t

1−ak 3.5

The following theorem can be proved similarly.

Theorem 3.2. Leta∈PLCJ,and1−ak/0, k∈N.Then the following assertions are equivalent.

1The sequence{1−ak}has finitely many negative terms.

2The inequality3.2has an eventually positive solution.

3The inequality3.3has an eventually negative solution.

4Each nonzero solution of 3.1is nonoscillatory.

It is known that3.1without impulses has no oscillatory solutions. But 3.1 with

impulses can have oscillatory solutions. So, impulse actions determine the oscillatory

properties of first-order linear differential equations.

3.2. Sturmian Theory for Second-Order Linear Equations

It is well-known that Sturm comparison theory plays an important role in the study of qualitative properties of the solutions of both linear and nonlinear equations. The first paper

on the Sturm theory of impulsive differential equations was published in 1996. In 10,

Bainov et al. derived a Sturmian type comparison theorem, a zeros-separation theorem,

and a dichotomy theorem for second-order linear impulsive differential equations. Recently,

the theory has been extended in various directions in 16–18, with emphasis on Picone’s

formulas, Wirtinger type inequalities, and Leighton type comparison theorems.

We begin with a series of results contained in 1, 10. The second-order linear

impulsive differential equations considered are

xptx0, t /tk,

Δxtk 0, Δxtk pkxtk 0, k∈N,

3.6

yqty0, t /tk,

Δytk 0, Δytk qkytk 0, k∈N,

3.7

wherepandqare continuous fort∈J, t /tk,and they have a discontinuity of the first kind

at the pointstk∈J,where they are continuous from the left.

The main result is the following theorem, which is also valid for differential

inequalities.

Theorem 3.3see1,10. Suppose the following.

1Equation3.7has a solutionytsuch that

yt>0, t∈a, b, ya yb−0. 3.8

2The following inequalities are valid:

pt≥qt, pk≥qk, t∈a, b, tk∈a, b. 3.9

3pt> qtin a subinterval ofa, borpk> qkfor sometk∈a, b.

Proof. Assume that3.6has a solution xtsuch that xt > 0, t ∈ a, b.Then from the relation

xtyt−xtytxtyt−xtyt, t∈a, b, t /tk, 3.10

an integration yields

xb−yb−−xb−yb−−xaya xaya

_b

a

xtyt−xtytdt

a<tk<b

Δx_t

kytk−xtkΔytk

. 3.11

From3.6,3.7, condition1, and the above inequality, we conclude that

0≤

_b

a

qt−ptxtytdt

a<tk<b

qk−pk

xtkytk. 3.12

But, from conditions 2 and 3, it follows that the right side of the above inequality is

negative, which leads to a contradiction. This completes the proof.

The following corollaries follow easily fromTheorem 3.3.

Corollary 3.4Comparison Theorem. Suppose the following.

1Equation3.7has a solutionytsuch that

yt/0, t∈a, b, ya yb−0. 3.13

2The following inequalities are valid:

pt≥qt, pk≥qk, t∈a, b, tk∈a, b. 3.14

3pt> qtin some subinterval ofa, borpk> qkfor sometk∈a, b.

Then, each solutionxtof 3.6has at least one zero ina, b.

Corollary 3.5. If conditions (1) and (2) ofCorollary 3.4are satisfied, then one has the following.

1Each solutionxtof3.6for which|xa||xb−|>0has at least one zero ina, b.

2Each solutionxtof3.6has at least one zero ina, b.

Corollary 3.6Oscillation Theorem. Suppose the following.

1There exists a solutionytof 3.7and a sequence of disjoint intervalsan, bn⊂Jsuch

that

lim

n→ ∞an∞, ya

n y

b−_n0, andyt/0 _3.15

2The following inequalities are valid fort∈an, bn, tk∈an, bn,andn∈N;

pt≥qt, pk≥qk. 3.16

Then all solutions of 3.6are oscillatory, and moreover, they change sign in each intervalan, bn.

Corollary 3.7Comparison Theorem. Let the inequalitiespt≥qt, pk≥qkhold fort > T≥α

andtk> T.Then, all solutions of 3.7are nonoscillatory if3.6has a nonoscillatory solution.

Corollary 3.8 Separation Theorem. The zeros of two linearly independent solutions of 3.6 separate one another; that is, the two solutions have no common zeros, and ifa, b a < bare two consecutive zeros of one of the solutions, then the intervala, bcontains exactly one zero of the other solution.

Corollary 3.9Dichotomy Theorem. All solutions of 3.6are oscillatory or nonoscillatory.

We can useCorollary 3.7, to deduce the following oscillation result for the equation:

zft, z, z0, t /tk,

Δz|ttk 0, Δz

ttkfk

z, z0.

3.17

Theorem 3.10see1,10. Suppose the following.

1The function ft, u, v is such that if zt is a continuous function for t ≥ α having a piecewise continuous derivative zt for t ≥ α, then the function ft, zt, zt is piecewise continuous fort≥α.

2The following inequalities are valid:

uft, u, v≥qtu2_{, t≥α, u, v∈R}2_,

ufku, v≥qku2, tk≥α, u, v∈R2.

3.18

Then every solutionzt of 3.17 defined fort ≥ T ≥ αis oscillatory if 3.7 has an oscillatory solution.

Recently, by establishing a Picone’s formula and a Wirtinger type inequality, ¨Ozbekler

and Zafer 17 have obtained similar results for second-order linear impulsive differential

equations of the form

lx ktx_{rtxptx0, t /θ}

i,

l0x Δ

ktx_p

ix0, tθi,

3.19

Lymtystyqty0, t /θi,

L0

y Δmtyqiy0, tθi,

where{pi}and{qi}are real sequences,k, m, r, s, p, q∈PLCIwithkt>0 andmt>0 for

allt∈I⊂J.

LetI0be a nondegenerate subinterval ofI. In what follows we shall make use of the

following condition:

kt/mt wheneverrt/st∀t∈I0. H

It is well-known that conditionHis crucial in obtaining a Picone’s formula in the case when

impulses are absent. IfHfails to hold, then Wirtinger, Leighton, and Sturm-Picone type

results require employing a so-called “device of Picard.” We will show how this is possible

for impulsive differential equations as well.

LetHbe satisfied. Suppose thatx, y∈CI0such thatx, y∈PLCI0andkx, my∈

PLC1I0. These conditions simply mean that xand y are in the domain of l, l0 and L, L0,

respectively. Ifyt/0 for anyt∈I0, then we may define

wt _ytxtytktx_{t−xtmtyt fort∈I}

0. 3.21

For clarity, we suppress the variablet. Clearly,

Employing the identity

the following Picone’s formula is easily obtained.

Theorem 3.11Picone’s formula17. LetHbe satisfied. Suppose thatx, y ∈CI0such that

In a similar manner one may derive a Wirtinger type inequality.

Theorem 3.12Wirtinger type inequality 17. If there exists a solutionxof 3.19such that

x /0ona, b, then

Next, we give a Leighton type comparison theorem.

Theorem 3.15Leighton type comparison17. Suppose that there exists a solutionx∈Ωrkof

Proof. Letα a andβ b− ∈ I0. Sincexand y are solutions of 3.19and 3.20,

respectively, we havelx≡l0x≡Ly≡L0y≡0. Employing Picone’s formula3.25, we

see that

x y

ykx−xmy

b−

a

b−

a

q−p− s−r

2

4k−m−

s2

4m

x2 k−m

x s−r

2k−mx

2

m

y2

xy−xy− s

2mxy

2

dt

a≤θi<b−

qi−pi

x2.

3.29

The functions under integral sign are all integrable, and regardless of the values of

yaoryb, the left-hand side of3.29tends to zero as → 0. Clearly,3.29results in

Lx≤0, 3.30

which contradicts3.28.

Corollary 3.16 Sturm-Picone type comparison. Let x be a solution of 3.19 having two consecutive zerosa, b∈I0. Suppose thatHholds, and

k≥m, 3.31

q≥p s−r

2

4k−m

s2

4m 3.32

for allt∈a, b, and

qi≥pi 3.33

for alli∈Nfor whichθi∈a, b.

If either3.31or3.32is strict in a subinterval ofa, bor3.33is strict for somei∈N, then every solutionyof 3.20must have at least one zero ona, b.

Corollary 3.17. Suppose that conditions3.31-3.32are satisfied for allt∈t∗,∞for some integer

t_∗≥t0, and that3.33is satisfied for alli∈Nfor whichθi≥t∗. If one of the inequalities3.31–3.33

is strict, then3.20is oscillatory whenever any solutionxof3.19is oscillatory.

As a consequence of Theorem 3.15 and Corollary 3.16, we have the following

oscillation result.

Corollary 3.18. Suppose for a givent1 ≥ t0there exists an intervala, b ⊂t1,∞for which that

IfHdoes not hold, we introduce a setting, which is based on a device of Picard,

leading to different versions ofCorollary 3.16.

Indeed, for anyh∈PLC1I,we have

Then, we have the following result.

Theorem 3.19Device of Picard17. Letr, s∈PLC1Iand letxbe a solution of 3.19having two consecutive zerosaandbinI. Suppose that

Corollary 3.20. Suppose that3.38-3.39are satisfied for allt∈ t∗,∞for some integert∗ ≥ t0,

and that 3.40 is satisfied for alli ∈ Nfor which θi ≥ t∗. If r, s ∈ PLC1t∗,∞ and one of the

inequalities 3.38–3.40 is strict, then3.20is oscillatory whenever any solution xof 3.19 is oscillatory.

As a consequence ofTheorem 3.19, we have the following Leighton type comparison

result which is analogous toTheorem 3.15.

Theorem 3.21Leighton type comparison17. Letr, s∈PLC1a, b. If there exists a solution

x∈Ωrkof3.19such that

Lx:

b

a

q−p−1

2

s−r− s

2

4m

x2dt k−mx2

dt

a≤θi<b

qi−pi−

1

2Δs−Δr

x2 _>0,

3.41

then every solutionyof 3.20must have at least one zero ina, b.

As a consequence of Theorems3.19and3.21, we have the following oscillation result.

Corollary 3.22. Suppose that for a givent1 ≥t0 there exists an intervala, b ⊂ t1,∞for which

conditions of eitherTheorem 3.19orTheorem 3.21are satisfied. Then3.20is oscillatory.

Moreover, it is possible to obtain results for 3.20 analogous to Theorem 3.12 and

Corollary 3.13.

Theorem 3.23Wirtinger type inequality17. If there exists a solutiony of 3.20such that

y /0ona, b, then fors∈PLC1a, band for allη∈Ωsm,

Wη:

_b

a

q− s

2

2m−

s

2

η2−mη2

dt

a≤θi<b

qi−

1

2Δs

η2≤0. 3.42

Corollary 3.24. If there exists anη ∈ Ωsm withs ∈ PLC1a, bsuch thatWη > 0,then every

solutionyof3.20must have at least one zero ina, b.

As an immediate consequence of Corollary 3.24, we have the following oscillation

result.

Corollary 3.25. Suppose that for a given t1 ≥ t0 there exists an interval a, b ⊂ t1,∞ and a

functionη∈Ωsmwiths∈PLC1Ifor whichWη>0. Then3.20is oscillatory.

3.3. Oscillation of Second-Order Linear Equations

The oscillation theory of second-order impulsive differential equations has developed rapidly

Let us consider the second-order linear differential equation with impulses

xptx0, t /tk,

xt_kbkxtk, x

t_kckxtk, k∈N,

3.43

wherep∈CJ,R,{bk}and{ck}are two known sequences of real numbers, and

xtk lim h→0−

xtkh−xtk

h , x

_t

k

lim

h→0

xtkh−x

t_k

h . 3.44

For 3.43, it is clear that ifbk ≤ 0 for all large k, then 3.43is oscillatory. So, we assume

thatbk > 0, k 1,2, . . . .The following theorem gives the relation between the existence of

oscillatory solutions of3.43and the existence of oscillatory solutions of second-order linear

nonimpulsive differential equation:

ypty0. 3.45

Theorem 3.26see19. Assume thatbkck, k1,2, . . . .Then the oscillation of all solutions of

3.43is equivalent to the oscillation of all solutions of 3.45.

Proof. Letxtbe any solution of3.43. Setyt xt_t0≤tk<tb−1

k fort > t0.Then, for all

n≥1,we have

yt

n xtnbn

t0≤tk≤tn

b−_k1xtn

t0≤tk<tn

b_k−1ytn. _3.46

Thus,ytis continuous onJ.Furthermore, fort /tn, n∈N,we have

yt

t0≤tk<t b−_k1

lim

h→0

xth−xt h x

_t

t0≤tk<t

b−_k1. 3.47

Forttn,it can be shown that

yt−_nyt_n xtn

t0≤tk<tn

b−_k1. _3.48

Thus,ytis continuous if we define the value ofytatttnas

ytn xtn

t0≤tk<tn

b−_k1. _3.49

Now, we have fort /tn,

yt xt

t0≤tk<t

and forttn,

yt−_nyt_n −ptnytn. 3.51

Thus, we obtain

yt ptyt 0, t≥t0. 3.52

This shows thatytis the solution of3.45.

Conversely, ifytis the continuous solution of3.45, we setxt yt_t0≤tk<tbkfor

t > t0.Then,xt−n xtn,andxtn ytnt0≤tk≤tnbk bnxtn.Furthermore, fort /tn,we

have

xt yt

t0≤tk<t

bk, xt yt

t0≤tk<t

bk, _3.53

and so

xt_n bny

t−_n

t0≤tk<tn

bkcnxtn,

xt ptxt

t0≤tk<t bk

yt ptyt0, t /tn.

3.54

Thus,xtis the solution of3.43. This completes the proof.

By Theorems3.26and2.3, one may easily get the following corollary.

Corollary 3.27. Assume thatpt≥0, bkck, k∈N. Then,3.43is oscillatory if

lim inf

t→ ∞ t _∞

t

psds > 1

4 3.55

and nonoscillatory if

t _∞

t

psds≤ 1

4, for large t. 3.56

Whenbk/ck,andbk, ck > 0, k ∈ N,oscillation criteria for3.43can be obtained by

means of a Riccati technique as well. First, we need the following lemma.

Lemma 3.28. Assume thatpt≥0, pt/≡0on any intervaltk, tk1,and letxtbe an eventually

positive solution of3.43. If

∞

n0

n

i0

ci

bi

tn1−tn ∞, 3.57

Now, letxtbe an eventually positive solution of3.43such thatxt>0 andxt>0

fort≥ T ≥ t0.Under conditions ofLemma 3.28, letut xt/xt,fort≥ T.Then,3.43

leads to an impulsive Riccati equation:

ut u2t pt 0, t≥T, t /tk,

ut_kdkutk, tk≥T,

3.58

wheredkck/bk, k∈N.

Theorem 3.29see19. Equation3.43is oscillatory if the second-order self-adjoint differential equation

T≤tk<t d_k−1

yt

pt

T≤tk<t d−_k1

yt 0, t > T ≥t0 3.59

is oscillatory, wheredkck/bk, k ∈N.

Proof. Assume, for the sake of contradiction, that3.43has a nonoscillatory solutionxt,

such thatxt>0,fort≥T ≥t0.Now, define

vt

T≤tk<t d−_k1

ut, t > T. 3.60

Then, it can be shown thatvtis continuous and satisfies

vt

T≤tk<t dk

v2t

T≤tk<t d−_k1

pt 0, t > T. 3.61

Next, we define

wt exp

t

T

T≤tk<s dk

vsds

, t > T. 3.62

Thenwt>0 is a solution of3.59. This completes the proof.

By Theorems3.29and2.4, we have the following corollary.

Corollary 3.30. Assume that

_∞

T

T≤tk<t dk

dt∞,

_∞

T

T≤tk<t d_k−1

ptdt∞, 3.63

Example 3.31see19. Consider the equation

where·denotes the greatest integer function, and

_∞

Thus, by Corollary 3.30, 3.64 is oscillatory. We note that the corresponding differential

equation without impulses

y 1

4t2y0 3.67

is nonoscillatory byTheorem 2.3.

In 20, Luo and Shen used the above method to discuss the oscillation and

nonoscillation of the second-order differential equation:

In 21, the oscillatory and nonoscillatory properties of the second-order linear

impulsive differential equation

u −ptu, t≥0 3.69

is investigated, where

pt ∞

n1

an>0 for alln∈N,andδtis theδ-function, that is,

_∞

−∞δtϕtdt ε

−ε

δtϕtdtϕ0 3.71

for allϕtbeing continuous att 0.Before giving the main result, we need the following

lemmas. For eachn∈N,define the sequence{P_kn}∞_k1inductively by

P₁nan1tn1−tn,

P_kn₁ ank1 ank

P_kn

1−Pn

k

anktnk1−tnk

,

3.72

whereP_kn₁∞providedP_kn1 andP_kn₁0 providedP_kn∞.LetSnsup{P_kn:k∈N}.

Lemma 3.32. IfSn0 ≤1for somen0∈N,thenSn01≤1,andSn≤1for alln≥n0.

Proof. By induction and in view of the fact that the functionfx x/1−xis increasing in

x∈0,1,it can be seen that

0< Pn01

k ≤P n0

k1 <1, ∀k∈N. 3.73

Hence,Sn01≤Sn0≤1.

The next lemma can also be proved by induction.

Lemma 3.33. Suppose that0< αk≤βkandλk>0for allk∈N.Define, by induction,

p1 α1λ1, q1 β1λ1,

pk1

αk1

αk

_p

k

1−pk

αkλk1

,

qk1

βk1

βk

_q

k

1−qk

βkλk1

, k∈N.

3.74

If0< qk<1for allk∈N,then

0< pk≤ αk

βk

qk<1, ∀k∈N. 3.75

The following theorem is the main result of21. The proof uses the above two lemmas

and the induction principle.

Theorem 3.34. The following statements are equivalent.

iiThere is n0∈Nsuch thatSn≤1for alln≥n0.

iiiEquation3.69is nonoscillatory.

ivEquation3.69has a nonoscillatory solution.

ApplyingTheorem 3.34, the nonoscillation and oscillation of3.69, in the case oftn

t0λn−1T, λ >1, T >0,andtnt0nT,are investigated in21.

In all the publications mentioned above, the authors have considered differential

equations with fixed moments of impulse actions. That is, it is assumed that the jumps happen at fixed points. However, jumps can be at random points as well. The oscillation of impulsive

differential equations with random impulses was investigated in 11. Below we give the

results obtained in this case.

Letτkbe a random variable defined inDk≡0, dk, k ∈N, 0< dk ≤ ∞,and letτ ∈R

be a constant. Consider the second-order linear differential equation with random impulses:

yatypty0, t∈Rτ, t /ξk,

Δyξk bkτky

ξ−_k, Δyξk bkτky

ξ_k−, k∈N,

3.76

whereRτ τ,∞, a, p:Rτ → Rare Lebesque measurable and locally essentially bounded

functions, ξ0 t0 ∈ Rτ, ξk ξk−1 τk for all k ∈ N, Δyξk yξk−yξ−_k, Δyξk

yξk−yξ_k−,andyξ−_k limt→ξk−0yt.

Definition 3.35. Let X be a real-valued random variable in the probability spaceΩ, F, P,

where Ω is the sample space, F is the σ-field, and P· is the probability measure. If

"

Ω|x|Pdx<∞, then "

ΩxPdxis called theexpectation of Xand is denoted byEX,that is,

EX

ΩxPdx. 3.77

In particular, ifXis a continuous random variable having probability density functionfx,

then

EX _∞

−∞xfxdx. 3.78

Definition 3.36. A stochastic processytis said to be asample path solution to3.76with the initial conditionyt0 y0if for any sample valuet1 < t2 <· · · < tk <· · · of{ξk}k≥1,thenyt

satisfies

yatypty0, t∈Rτ, t /tk,

Δytk bktk−tk−1y

t−_k, ∀k∈N, Δy_t

k bktk−tk−1y

t−_k, ∀k∈N.

Definition 3.37. The exponential distribution is a continuous random variable with the probability density function:

fx ⎧ ⎨ ⎩

λe−λx_{, x >0,}

0, elsewhere,

3.80

whereλ >0 is a parameter.

Definition 3.38. A solutionytof3.76is said to be nonoscillatory in meanifEytis either

eventually positive or eventually negative. Otherwise, it is calledoscillatory.

Consider the following auxiliary differential equation:

xatxptx0, t∈Rτ. 3.81

Lemma 3.39. The functionytis a solution of3.76if and only if

yt ∞

k0

k

i1

1biτiχξk,ξk1t

xt, 3.82

where xt is a solution of 3.81 with the same initial conditions for3.76, andχ is the index function, that is,

χξk,ξk1t ⎧ ⎨ ⎩

1, ifξk≤t < ξk1,

0, otherwise.

3.83

Proof. Ifxtis a solution of system3.81, for anyt0< t1<· · ·< tk<· · ·,we have

yt ∞

k0

k

i1

1biti−ti−1χtk,tk1t

xt. 3.84

It can be seen that

yt atyt ptyt 0, t /tk,

ytk 1bktk−tk−1y

t−_k, ytk 1bktk−tk−1y

t−_k, 3.85

which imply thatytsatisfies3.76, that is,ytis a sample path solution of3.76. Ifytis

a sample path solution of3.76, then it is easy to check thatxtis a solution of3.81. This

completes the proof.

Theorem 3.40see11. Let the following condition hold.

CLetτkbe exponential distribution with parameterλ >0,k∈N,and letτibe independent

If there existsT ∈Rτsuch that

does not change sign for allt≥T,then all solutions of3.76are oscillatory in mean if and only if all solutions of 3.81are oscillatory.

Proof. Letytbe any sample path solution of3.76; thenLemma 3.39implies

yt ∞

Further, it can be seen that

E

are oscillatory in mean if and only if all solutions of3.81are oscillatory. This completes the

Whenbkτk bkis finite,k∈N, then the following result can be proved.

Theorem 3.41see11. Let condition (C) hold, and letbkτk≡bkbe finite for allk∈N. Further

assume that there are a finite number of bk such that bk < −1. Then all solutions of 3.76 are

oscillatory in mean if and only if all solutions of 3.81are oscillatory.

3.4. Oscillation of Higher-Order Linear Equations

Unlike the second-order impulsive differential equations, there are only very few papers on

the oscillation of higher-order linear impulsive differential equations. Below we provide some

results for third-order equations given in22. For higher-order liner impulsive differential

equations we refer to the papers23,24.

Let us consider the third-order linear impulsive differential equation of the form

xptx0, t /tk,

xt_kakxtk, x

t_kbkxtk, x

t_kckxtk, k∈N,

xt₀x0, x

t₀x₀, xt₀x₀,

3.91

where ak > 0, bk > 0, ck > 0, tk1 ≥ tk k ∈ N and p ∈ CJ,R is not always zero in

tk, tk1 k∈Nfor sufficiently largek.

The following lemma is a generalization of Lemma 1 in25.

Lemma 3.42see22. Assume thatxtis a solution of 3.91and there existsT ≥t0 such that

for anyt≥T, xt>0<0.Let the following conditions be fulfilled.

C1One has

t1−t0

c1

b1t2−

t1 · · ·

c1· · ·cn

b1· · ·bntn1−

tn · · ·∞. 3.92

C2One has

t1−t0 b1

a1t2−t1 · · ·

b1· · ·bn

a1· · ·antn1−tn · · ·∞. 3.93

Then for sufficiently largeteitherAorBholds, where

Aone has

xt>0<0, xt>0<0; 3.94

Bone has

Theorem 3.43 see 22. Assume that conditions of Lemma 3.42 are fulfilled and for any n ∈

N, an > 0, 1 ≥ bn > 0,and1 ≥ cn >0.Moreover, assume that the sequence of numbers{ni1ai}

has a positive lower bound,∞_n1|an−1|converges, and

"_∞

t2_{ptdt∞holds. Then every bounded}

solution of 3.91either oscillates or tends asymptotically to zero with fixed sign.

Proof. Suppose thatxtis a bounded nonoscillatory solution of3.91andxt>0, t≥T ≥

t0.According toLemma 3.42, eitherAorBis satisfied. We claim thatAdoes not hold.

Otherwise,xtk> 0 for sometk ≥T0.Sincext> 0,it follows thatxtis monotonically

increasing fort∈tki−1, tki, i∈N.For anyt∈tk, tk1,xt> xt_k γ >0.By induction,

it can be seen that

xt> bkn−1· · ·bk1γ >0, 3.96

in particular,

xtkn> bkn−1· · ·bk1γ >0,

xt_knbknxtkn> bkn· · ·bk1γ >0.

3.97

Integratingxt> γfromtktotk1,we obtain

xtk1≥x

t_kγtk1−tk. 3.98

By induction, for any natural numbern≥2,we have

xtkn≥akn−1· · ·ak1

xt_kγtk1−tk γ

bk1

ak1tk2−

tk1

· · ·γbkn−1· · ·bk1 akn−1· · ·ak1tkn−

tkn−1

.

3.99

Considering the conditionC2inLemma 3.42and the sequence of numbers{n_i1ai}has a

positive lower bound, we conclude that the inequality above leads to a contradiction that the

right side tends to ∞ while xtis bounded. Therefore, case B holds. xt < 0 implies

that xt is strictly monotonically decreasing. From the facts that the series ∞_n1|ai −1|

converges andxtis bounded, it follows that∞_i1xt_i−xticonverges and there exists

limit limt→ ∞xt l,where 0≤l <∞.Now, we claim thatl 0.Otherwise,l >0 and there

existsT1 ≥ T0 such thatxt > l/2 for t ≥ T1.From 3.91and the last inequality, we can

deduce

xt −ptxt<−l

2pt,

tsn

ts

t2xtdt≤ −l

2

tsn

ts

t2ptdt.

Integrating by parts of the above inequality and consideringbn ≤ 1, cn ≤ 1 n ∈ N,and

xt<0, xt>0, we have the following inequality:

−l

2

_tsn

ts

t2ptdt≥2xtsn 2 n−1

i1

1−asixtsi−

*

t2_sxt_s−2tsxts 2xts

+

. 3.101

Since "∞t2_{ptdt ∞ and the series} ∞

i11 − aixti converges, the above inequality

contradicts the fact thatxtis bounded, hencel0, and the proof is complete.

The proof of the following theorem is similar.

Theorem 3.44see22. Assume that conditions ofLemma 3.42hold and for any n ∈ N, an >

0, 1 ≥ bn > 0, 1 ≥ cn > 0, andan/cn ≥ tn1/tn.Moreover, assume that the sequence of numbers

{n

i1ai}is bounded above,

_∞

n1|an−1|converges, and

"_∞

tptdt∞holds. Then every solution of 3.91either oscillates or tends asymptotically to zero with fixed sign.

Some results similar to the above theorems have been obtained for fourth-order linear

impulsive differential equations; see24.

4. Nonlinear Equations

In this section we present several oscillation theorems known for super-liner, half-linear,

super-half-linear, and fully nonlinear impulsive differential equations of second and

higher-orders. We begin with Sturmian and Leighton type comparison theorems for half-linear equations.

4.1. Sturmian Theory for Half-Linear Equations

Consider the second-order half linear impulsive differential equations of the form:

ktϕx_{ptϕx 0,}

t /θi,

Δx|_tθi 0, Δktϕx_tθ

ipiϕx 0,

4.1

mtϕy_qtϕy0,

t /θi,

Δy_tθ

i 0, Δ

mtϕy_tθ iqiϕ

y0,

4.2

whereϕs |s|α−1s, α >0, θi< θi1,{pi}and{qi}are real sequences, andk, m, p, q∈PLCJ

withkt>0 andmt>0.

The lemma below can be found in26.

Lemma 4.1. LetA, B∈Randγ >0be a constant; then

AϕγA γBϕB−

γ1AϕB≥0, 4.3

The results of this section are from16.

Theorem 4.2 Sturm-Picone type comparison. Let xt be a solution of 4.1 having two consecutive zerosaandbinJ. Suppose thatpt≤qtandmt≤ktare satisfied for allt∈a, b, and thatpi≤qifor alli∈Nfor whichθi ∈a, b. If eitherpt/≡qtorkt/≡mtorpi/≡qi, then

any solutionytof4.2must have at least one zero ina, b. Proof. Assume thatytnever vanishes ona, b. Define

ut: x

ϕy

ktϕyϕx−mtϕxϕy, _4.4

where the dependence ontof the solutionsxandyis suppressed. It is not difficult to see that

Integrating4.5fromatoband using4.6, we see that

clear that4.11is not possible under our assumptions, and henceytmust have a zero in

a, b.

Corollary 4.3Separation Theorem. The zeros of two linearly independent solutionsxtandyt of 4.1separate each other.

Corollary 4.4Comparison Theorem. Suppose thatpt≤qtandmt≤ktare satisfied for allt∈t_∗,∞for somet_∗≥t0, and thatpi≤qifor alli∈Nfor whichθi ≥t∗. If eitherpt/≡qtor

kt/≡mtorpi/≡qi, then every solutionytof4.2is oscillatory whenever a solutionxtof4.1

is oscillatory.

Corollary 4.5 Dichotomy Theorem. The solutions of 4.1 are either all oscillatory or all nonoscillatory.

Theorem 4.6Leighton-type Comparison. Letxtbe a solution of4.1having two consecutive zerosaandbinJ. Suppose that

Then any nontrivial solutionytof4.2must have at least one zero ina, b.

Proof. Assume thatythas no zero ina, b. Define the functionutas in4.4.

IfVαx ≥ 0, then we may conclude that eitheryt has a zero ina, bor ytis a

constant multiple ofxt.

As a consequence of Theorems4.2and4.6, we have the following oscillation result.

Corollary 4.7. Suppose for a givenT ≥ t∗there exists an intervala, b ⊂T,∞for which either

conditions ofTheorem 4.2orTheorem 4.6are satisfied, then every solutionytof4.2is oscillatory.

4.2. Oscillation of Second-Order Superlinear and

Super-Half-Linear Equations

Let us consider the forced superlinear second-order differential equation of the following

form:

Assume that the following conditions hold.

A1α >1 is a constant,r:J → 0,∞is a continuous function,p, q∈CJ.

Theorem 4.8 see 27. Assume that conditions (A1)–(A3) hold, and that there exists wt ∈

Proof. Letxtbe a solution of4.14. Suppose thatxtdoes not have any zero inc1, d1∪

c2, d2.Without loss of generality, we may assume thatxt>0 fort∈c1, d1.Define

ut −rtx_xtt, t∈c1, d1. 4.18

Then, by H ¨older’s inequality, fort∈c1, d1andt /τk,we have

ut ptxtα−1−qt xt

1

rtu2t ≥pt1/α_qt1−1/α 1

rtu2t, t∈c1, d1, t /τk.

4.19

Fortτk, k∈N,we obtain

uτ_k bk akuτk.

4.20

If kc1 < kd1, then all impulsive moments are in c1, d1 : τkc11, τkc12, . . . , τkd1.

Multiplying both sides of4.19byw2_{tand integrating onc}

1, d1and using the hypotheses,

we get

kd1

ikc11 ai−bi

ai

w2τiuτi>

d1

c1

pt1/α_qt1−1/α

w2t−rtw2tdt. 4.21

On the other hand, fort∈c1, τkc11,it follows that

rtx_tqt−pt|xt|α−1_{xt≤0, 4.22}

which implies thatrtxtis nonincreasing onc1, τkc11.So, for anyt∈c1, τkc11,one

has

xt−xc1≥

rtxt

rξ t−c1, ξ∈c1, t. 4.23

It follows from the above inequality that

uτkc11

≥ − rξ τkc11−c1

≥ − η1 τkc11−c1

. 4.24

Making a similar analysis onτi−1, τi, ikc1 2, . . . , kd1,we obtain

uτi −rτix

_τ

i

xτi ≥ −

η1

τi−τi−1

From4.21–4.25andA2,we get

_d1

c1 *

pt1/α_qt1−1/α

w2t−rtw2t+dt < Qw, c1, d1, 4.26

which contradicts4.16. Ifkc1 kd1,then Qw, c1, d1 0,and there are no impulse

moments inc1, d1.Similarly to the proof of4.21, we get

0> _d1

c1

pt1/α_qt1−1/α

w2t−rtw2tdt, 4.27

which again contradicts4.16.

In the casext<0,one can repeat the above procedure on the subintervalc2, d2in

place ofc1, d1.This completes the proof.

Corollary 4.9. Assume that conditions (A1) and (A2) hold. If for any T > 0 there exist cj, dj

satisfying (A3) withT ≤ c1 < d1 ≤ c2 < d2,and wt ∈ Ωwcj, djsatisfying4.16,j 1,2,

then4.14is oscillatory.

The proof of following theorem is similar to that ofTheorem 4.8.

Theorem 4.10see27. Assume that conditions (A1)–(A3) hold, and there exists aG∈ΩGcj, dj

such that

dj

cj

pt1/α_qt1−1/α_Gt−rth2_{tdt > RG, c}

j, dj

, 4.28

where RG, cj, dj 0forkcj kdj,and

RG, cj, dj

ηj

⎡ ⎢

⎣Gτkcj1

bkcj1−akcj1

akcj1

τkcj1−cj

kdj

ikcj2

Gτi

bi−ai

aiτi−τi−1

⎤ ⎥

⎦ 4.29

forkcj< kdj, j1,2.Then every solution of 4.14has at least one zero inc1, d1∪c2, d2.

Corollary 4.11. Assume that conditions (A1) and (A2) hold. If for any T > 0 there exist cj, dj

satisfying (A3) with T ≤ c1 < d1 ≤ c2 < d2,and G ∈ ΩGcj, dj satisfying 4.28, j 1,2,

then4.14is oscillatory.

Example 4.12. Consider the following superlinear impulsive differential equation:

1sin2txβcost|x|α−1xsint, t /2kπ±π

4,

xτ_kakxτk, x

τ_kbkxτk, τk2kπ±

π

4, k∈N.

It can be seen that if

then, conditions ofCorollary 4.9are satisfied; hereα > 1, Γ is the gamma function, andak

andbksatisfyA2. So, every solution of4.30is oscillatory.

In28–30, the authors have used an energy function approach to obtain conditions for

the existence of oscillatory or nonoscillatory solutions of the half-linear impulsive differential

equations of the following form:

Define the energy functional

Vx, yyφβ

ΦβandFβare both even and positive definite.

The functionVt Vxt, xtis constant along the solutions of the nonimpulsive

equation

φβx

_φ

βx 0. 4.34

The change in the energy along the solutions of4.32is given by

Vtn1−Vtn Vtn0−Vtn

We see that these impulsive perturbations increase the energy. If the energy increases slowly, then we expect the solutions to oscillate. On the other hand, if the energy increases too fast,

the solutions become nonoscillatory. Letxtbe a solution of4.32,rnVtn−10 Vtn−0,

andFβxtn σnrn.CalculatingFβxtnin terms ofrn1Vtn0,we obtain

Fβxtn σn

bβn11−σn σn

To simplify the notation, we introduce the function

Θu, b: u

bβ1_{1−u u}. 4.37

The function Θ gives the jump in the quantity Fβxt/Vt. Note that Θ0, b 0, 0 <

Θu, b< ufor 0< u <1 andb≥1; Θis monotone increasing with respect touand decreasing

with respect tob.

Theorem 4.13 see29. Assume that there exist a constantN > 0 and a sequence {γn} with

Θγn, bn≤γn1<1such that

tn1−tn≤

γn1

Θγn,βn

dv φβ

F_β−1vΦ_β−11−v 4.38

holds for everyn > N.Then every solution of 4.32is nonoscillatory.

Theorem 4.14see29. Assume that there exist a constantN >0and a sequence{λn}withλn>0

and∞_n1λn∞such that for everyγ∈0,1

tn1−tn≥

_γλn

Θγ,bn

dv φβ

F_β−1vΦ_β−11−v 4.39

holds for everyn > N.Then every solution of 4.32is oscillatory.

Proof. Letxtbe a nontrivial solution of4.32. It suffices to show thatxtxt>0 cannot

hold on any intervalT,∞.Assume that to the contrary,xt >0, xt >0 fort∈ tN,∞.

Let σn be defined by Fβxtn σnrn,where rn Vtn −0. It follows from 4.39 that

σn1≥σnλn.Hence,

σn1≥σN

n

iN

λn. 4.40

Sinceσn Fβxtn/rn ≤ 1,and the right side of the above inequality tends to infinity as

n → ∞,we have a contradiction.

Now, assume thatbn≥b >1 for everyn > N.It can be shown that the integral

γ

Θγ,b

dv φβ

F_β−1vΦ_β−11−v 4.41

takes its maximum in0,1at

γ b

β_b−1

In the special casebnb, tn1−tndn∈N,we have the following necessary and sufficient

condition.

Theorem 4.15see29. Assume thatbnb, tn1−tndn∈N.Then every solution of 4.32

is nonoscillatory if and only if

d≤ γ

Θγ,b

dv φβ

F−1

β v

Φ−1

β 1−v

. _4.43

Remark 4.16. Equation4.32with 0≤bn≤1,was studied in30.

Finally, we consider the second-order impulsive differential equation of the following

form:

rtϕαx

ptϕα

xqtϕβx et, t /θi,

Δrtϕα

xqiϕβx ei, tθi,

4.44

whereϕγs |s|γ−1s,β ≥ α > 0 are real constants,{θi}is a strictly increasing unbounded

sequence of real numbers,{qi}and{ei}are real sequences,r, p, q, e∈PLCJ, andrt>0.

All results given in the remainder of this section are from31.

Theorem 4.17. Suppose that for any givent_∗ ≥t0, there exist intervalsI1 s1, t1,I2 s2, t2⊂

t∗,∞, such that

aqt≥0for allt∈ {I1∪I2} \ {θi}andqi≥0for alli∈Nfor whichθi ∈I1∪I2;

bet≤0,t∈I1\ {θi},et≥0,t∈I2\ {θi};ei≤0, θi ∈I1,ei ≥0,θi∈I2 for alli∈N.

If there existsH∈ Dsk, tksuch that

_tk

sk

6

qt|Ht|α1_−α1−α1_rtα1Ht−pt

rtHt α1

dt

sk≤θi<tk 6

qi|Hθi|α1>0, k1,2,

4.45

where

6

qt βα−α/ββ−αα/β−1qtα/β|et|1−α/β, 4.46

6

qiβα−α/β

β−αα/β−1qi

α/β_|e

i|1−α/β, 4.47

Proof. Suppose that there exists a nonoscillatory solutionxtof4.44so thatxt/0 for all

t≥t_∗for somet_∗. Let

νt: rtϕαx

_t

ϕαxt

, t≥t∗. 4.48

It follows that fort≥t∗,

ν−p rν−α

|ν|11/α

r1/α −

q|x|β−α₋ e

ϕαx

, t /θi;

Δν−

qi|x|β−α−

ei

ϕαx

, tθi,

4.49

wheretdependence is suppressed for clarity.

Define a functionEz:0,∞ → 0,∞by

Ez:A1zβ−αA2z−α, A1≥0, A2≥0. 4.50

It is not difficult to see that ifβ > α, then

min

z∈0,∞Ez βα

−α/β_β−αα−β/β

Aα/β₁ A₂β−α/β. 4.51

Clearly, ifβα, then we haveEz≥A1. Thus, with our convention that 00 1,4.51holds

forβ≥α.

Suppose thatxt>0 for allt≥t_∗. Chooses1≥t∗and consider the intervalI1 s1, t1.

Fromb, we see thatet≤0 onI1\ {θi}andei ≤0 for alli∈Nfor whichθi ∈I1. Applying

4.51to the terms in the parenthesis in4.49we obtain

6

q≤ −ν−p rν−α

|ν|11/α

r1/α , t∈I1\ {θi};

Δνθi q6i≤0, θi∈I1,

4.52

whereq6andq6iare defined by4.46and4.47, respectively.

LetH∈ Ds1, t1. Multiplying4.52by|H|α1and integrating overI1give

_t1

s1 6

q|H|α1_{dt≤ −}

_t1

s1 p r |H|

α1_νdt−α

_t1

s1 |ν|11/α

r1/α |H| α1_dt−

_t1

s1

In view of4.52and the assumptionHs1 Ht1 0, employing the integration by parts

formula in the last integral we have

t1

We useLemma 4.1with

γ 1

which obviously contradicts4.45.

Ifxtis eventually negative then we can considerI2and reach a similar contradiction.

This completes the proof.

Example 4.18. Consider

whereγis a positive real number.

LetHt sin23t,I1 2n5/3π,2n2π, andI2 2n2π,2n7/3π. For

a-bare satisfied. It is also easy to see that, forj2n5/3 andj 2n2,

Note that if there is no impulse then the above integrals are negative, and therefore no conclusion can be drawn.

Whenβ α, then4.44reduces to forced half-linear impulsive equation with

dam-ping

then4.60is oscillatory.

Takingα1 in4.44, we have the forced superlinear impulsive equation with

Corollary 4.20. Letβ0ββ−11/β−1, β >1. Suppose that for any givent∗≥t0, there exist intervals

then4.62is oscillatory.

Letβ1 in4.62. Then we have the forced linear equation:

then4.64is oscillatory.

Example 4.22. Consider

whereσis a positive real number.