New UMstudentsrepo.um.edu.my/4236/1/All_CHAPTERS_of_THESIS.pdf · 2014. 8. 28. · 1 CHAPTER ONE INTRODUCTION 1.1 Introduction The differential equations (DE) appear naturally in

1

CHAPTER ONE

INTRODUCTION

1.1 Introduction

The differential equations (DE) appear naturally in many areas of science and

humanities. Ordinary differential equations (ODE) have found a wide range of

application in biological, physical, social and engineering systems which are dynamic in

character. They can be used effectively to analyze the evolutionary trend of such systems,

they also aid in the formulation of these systems and the qualitative examination of their

stability and adaptability to external stimuli. Many phenomena in different branches of

sciences are interpreted in terms of second order DE and their solutions. For example, the

so–called Emden-Fowler differential equation arises in the study of gas dynamics and

fluid mechanics. This equation appears also in the study of chemically reacting systems.

Since the classic work of Atkinson (1955), there has much interest in the problem of

determining oscillation criteria for second order non-linear DE. The study of the

oscillation of second–order nonlinear ODE’s with alternating coefficients is of special

interest because of the fact that many physical systems are modeled by second order

nonlinear ODE. Some of the most important and useful tests have involved the average

behavior of the integral of the alternating coefficient. These tests have been motivated by

the averaging criterion of Kameneve (1978) and its generalizations. The use of averaging

functions in the study of oscillation dates back to the work of Wintner (1949) and

Hartman (1952).

Although differential equations of second-order have been studied extensively, the

study of qualitative behavior of third-order differential equations has received

2

considerably less attention in the literature, however certain results for third-order

differential equations have been known for a long time and their applications in

mathematical modeling in biology and physics. In 1961 Hanan (1961) studied the

oscillation and non-oscillation of two different types of third order differential equations

and gave definitions of two types of the solutions. The paper was the starting point for

many investigations to the asymptotic behavior of third-order equations.

The purpose of this thesis is to study the problem of oscillation of second order non-

linear ordinary differential equations of the form

)1.1()(,)()()),(()()()( txtHtxtrtxgtqtxtr

and oscillation of second non-linear equations with damping term of the form

)2.1(,)(),(,)())(()()),(()()()()())(()(

txtxtHtxtxtrtxgtqtxthtxtxtr

where hr , and q are continuous functions on the interval )(,0,, 00 trtt is a positive

function and RRC , , g is continuously differentiable function on the real line R

except possibly at 0 with 0)( xxg and 0)( kxg for all 0x , is a continuous

function on RxR with 0),( vuu for all 0u and ),(),( vuvu for any

),0( and H is a continuous function on ,0t ×R×R with

)())(())(),(,( tptxgtxtxtH

for all x 0 and .0tt

The thesis also deals study of problem of oscillation of third order non-linear

equations of the form

3

)3.1(,)(),(),(,))(()())(()( 1

txtxtxtHtxgtqtxftr

where q and r are defined as above, 1g is continuously differentiable function on the

real line R except possibly at 0 with 0)(1 yyg and 0)(1 kyg for all ,0y f is a

continuous function on R and ,: 01 tH ×R×R×RR is a continuous function such

that )()(),,,( 11 tpygzyxtH for all y 0 and 0tt .

We list some basic definitions and Elementary results which will be needed in the

next chapters.

1.2 The Basic Definitions

Definition 1.2.1

A point 0t is called a zero of the solution )(tx of the differential equation

if 0)( x .

Definition 1.2.2

A solution )(tx of the differential equation is said to be oscillatory if it has arbitrary

large zeros. Otherwise it is said to be non-oscillatory.

Definition 1.2.3

Differential equation is called oscillatory if all its solutions are oscillatory. Otherwise

it is called non oscillatory.

Definition 1.2.4

The differential equations (1.1) and (1.2) are called

4

(1) Sub-linear if the function g satisfies that

0)(

0)(

000

allforug

duand

ug

du

(2) Super-linear if the function g satisfies that

0)(

0)(

0

allforug

duand

ug

du

(3) A mixed type if the function g satisfies that

.)(

0)(

000

ug

duand

ug

du

1.3 Elementary Results

The following theorems play an important role in the theory of oscillation of the

solutions of the linear differential equations:

1.3.1 Sturm’s Comparison Theorem (Bartle (1970))

Let )(),( 21 tqtq and )(tr be continuous functions on ),( ba and 0)( tr on ).,( ba

Assume that )(1 tx and )(2 tx are real solutions of

)4.1(0)()()()( 1

txtqtxtr

and

5

)5.1(0)()()()( 2

txtqtxtr

respectively on ),( ba . Further, let )()( 12 tqtq for ),( bat . Then, between any two

consecutive zeros 21 , tt of )(1 tx in ),,( ba there exists at least one of )(2 tx unless

)()( 21 tqtq on ],[ 21 tt . Moreover, in this case the conclusion is still true if the solution

)(2 tx is linearly independent of ).(1 tx

1.3.2 Sturm’s Separation Theorem (Bartle (1970))

If )(1 tx and )(2 tx are linearly independent solutions of the equation

)6.1(.0)()()()(

txtqtxtr

Then, between any two consecutive zeros of ),(1 tx there is precisely one zero of )(2 tx .

Therefore the solutions of the second order linear differential equations are either all

oscillatory or all non-oscillatory. The story of non-linear equations is not the same. The

nonlinear differential equations may have both oscillatory solutions.

The importance of classification of the second order differential equations into

oscillatory categories is due to the following well-known fact: A non – trivial solution of

the second order ordinary differential equation must change its sign whenever it vanishes,

since )(tx and )(tx

cannot vanish simultaneously (in this case the zeros of )(tx are said to

be isolated).

The following theorem is quite useful element of our study in the following

chapters:

6

1.3.3 The Bonnet's Theorem (Ross (1984))

Suppose that h is a continuous function on [a,b], is a non- negative function and

an increasing function on the interval [a,b]. Then there exists a point c in [a,b] such that

.)()()()(

b

a

b

c

dsshbdsshs

If is a decreasing function on [a,b], then there exists a point c in [a,b] such that

b

a

c

a

dsshadsshs )()()()(

This theorem is a part of the second mean value theorem of integrals (Ross (1984)).

1.4 Riccati Technique

In the study of oscillation theory of differential equations, there are two techniques

which are used to reduce the higher-order equations to the first-order Riccati equation or

inequality. The first one is the Riccati transformation technique. The second one is called

the generalized Riccati technique. This technique can introduce some new oscillation

criteria and can be applied to different equations which cannot be covered by the results

established by the Riccati technique.

Riccati Transformation Technique:

(1) If )(tx is a non-vanishing solution of equation (1.6) on the interval (a,b), then

)()()()( txtxtrt

is a solution of

7

)7.1(),(0)()()()( 21 batforttrtqt

(2) If )(t is a solution of equation (1.7) on (a,b), then

b

a

dssrs

etx)()( 1

)(

is a non-vanishing solution of equation (1.6) on (a,b).

1.5 Applications Of Oscillatory Differential Equations

Ordinary differential equations have a variety of applications in science,

mechanical engineering, aerospace engineering and physical systems, we explore three

of them: Undamped simple pendulum, damped simple pendulum and a half cylinder

rolling on a horizontal plane.

Example 1.1

Consider a pendulum with mass m at the end of a rigid rod of length L attached to

say a fixed frictionless pivot which allows the pendulum to move freely under gravity in

the vertical plane as illustrated in Figure 1.1. The angular equation of motion of the

pendulum is given as a nonlinear differential equation

)8.1(,0sin2

2

Kdt

d

where K = g/L.

8

Figure 1.1: Simple pendulum

The numerical solutions curves of the equation (1.8) when m=100, g =9.81 and L=1 and

for different initial angles π/3 and π/10 with zero initial velocity are

Figure 1.2: Numerical solutions of 0sin/ 22 Kdtd

0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 5-1.5

-1

-0.5

0

0.5

1

1.5

2

Time(t)

Ang

ular

Mot

ion,

(

rad)

(0) = /3

(0) = /10

9

Example 1.2

As in example 1.1, we consider the motion of a simple pendulum that subject to a

frictional force or damping force. We assume that the damping coefficient , so the

nonlinear differential equation represents this motion as follows:

)9.1(,0sin2

2

Kdt

d

dt

d

The numerical solution curve of equation (1.9) when 1.0 is

Figure 1.3: Numerical solution of 0sin// 22 Kdtddtd

0 5 10 15 20 25 30 35 40 45 50

-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

Time(t)

Ang

ular

Mot

ion,

(

rad)

Example 1.3

The general governing differential equation of motion of the half cylinder rolling

on a horizontal plane is

)).(cos()())(cos()())(sin(22

322

txEtxtxmErtxtxmErmr

10

The numerical solution curve of this equation when m =4, 3

4rE and r = 0.1 is

Figure 1.4: Solution curve of half cylinder rolling on a horizontal plane.

0 0.5 1 1.5 2 2.5 3 3.5 40

0.5

1

1.5

2

2.5

3

Time(t)

Beta

11

1.6 Thesis Organization

Besides the introductory chapter (chapter 1) about the oscillation of second and

third order non-linear ordinary differential equations, the thesis is organized as follows:

Chapter 2: This chapter will contain the literature review of the main results of the

oscillation of second and third order ordinary differential equations which are given in

the literature.

Chapter 3: This chapter is devoted to study of the oscillation of the second order

equation (1.1) and contains some oscillation criteria for oscillation equation (1.1). The

oscillation results obtained will be illustrated by some examples and their numerical

solutions that are found by using Runge Kutta method of fourth order.

Chapter 4: This chapter is devoted to study of the oscillation of the second order

equation with damping term (1.2). Some sufficient conditions for oscillation equation

(1.2) will be given in this chapter. The oscillation results obtained will be illustrated by

some examples and their numerical solutions that are found by using Runge Kutta

method of fourth order.

Chapter 5: this chapter is concerned with oscillation of third order ordinary

differential equation (1.3) and includes oscillation results for oscillation of equation

(1.3) and an illustrative example with its numerical solution obtained by using Runge

Kutta method of fourth order for these results presented.

Chapter 6: This chapter contains the conclusion with suggestions for future work

and references.

12

CHAPTER TWO

LITERATURE REVIEW

In this Chapter, we will review the literature within the context of our study of

oscillations of ODEs of the 2nd

Order and the 3rd

Order. We will see that most of the

previous oscillation results depend on the behavior of the integral of the coefficients and

a reduction of order of the ODEs and using the Riccati technique to establish some

sufficient conditions. Our results improve and extend almost of these existing results in

the literature.

2.1 Oscillations Of Second Order Differential Equations

Oscillatory and non-oscillatory behavior of solutions for various classes of second

order has been studied extensively in literature as Atkinson (1955), Bihari (1963), Bhatia

(1966), Grace (1992), Ayanlar & Tiryaki (2000), Elabbasy et al. (2005), Lee & Yeh

(2007), Berkani (2008), Remili (2010). Various researchers have studied particular cases

of the equations (1.1) and (1.2). These particular cases can be classified as follows:

The homogeneous linear equations

)2.2(.0)()()()(

)1.2(.0)()()(

txtqtxtr

txtqtx

13

The homogeneous non-linear equations

)7.2(.0))(()()())(()(

)6.2(.0))(()()()(

)5.2(.0))(,)(()()(

)4.2(.0))(()()(

)3.2(.0,0)(sgn)()()(

txgtqtxtxtr

txgtqtxtr

txtxtqtx

txgtqtx

txtxtqtx

The homogeneous non-linear equations with damping term

)10.2(.0))(()()()()())(()(

)9.2(.0))(()()()()()(

)8.2(.0))(()()()()(

txgtqtxthtxtxtr

txgtqtxthtxtr

txgtqtxthtx

The non-homogeneous non-linear equations

)12.2().())(()()())(()(

)11.2()).(),(,())(,()()(

tHtxgtqtxtxtr

txtxtHtxttxtr

The non-homogeneous non-linear equations with damping term

)13.2(),())(()()()()())(()( tHtxgtqtxthtxtxtr

where, the functions ,,,,, qghr and H are defined as in the equation (1.2).

14

The investigation of the oscillation of (1.1) and (1.2) may be done by following many

criteria. Many of these criteria depend on determining integral tests involving the

function q to obtain oscillation criteria.

For convenience of writing, we adopt the following notations:

is written, it is to be

assumed that

t

tlim and that this limit exists in the extended real numbers.

2.1.1 Oscillation Of Homogenous Linear Equations

2.1.1.1 Oscillation Of Equations Of Type (2.1)

This section is devoted to the oscillation criteria for the second order linear

differential equation of the form (2.1) . The oscillation of equation (2.1) has brought

the attention of many authors as Wintner (1949), Kamenev (1978), Philos (1983) and

Yan (1986), since the early paper by Fite (1918). Among the numerous papers

dealing with this subject we refer in particular to the following :

Theorem 2.1.1 Fite (1918)

If 0)( tq for all 0tt and

0

,)(t

dssq

then, every solution of the equation (2.1) is oscillatory. The following theorem

extended the result of Fite (1918) to an equation in which q is of arbitrary sign.

15

Theorem 2.1.2 Wintner (1949)

Suppose that

.)(1

lim

0 0

dsduuq

t

t

t

s

tt

Then, every solution of the equation (2.1) is oscillatory.

Example 2.1.1

Consider the following differential equation

.0,0)(cos32)(

ttxttx

Theorem 2.1.2 ensures that the given equation is oscillatory, however Theorem 2.1.1

fails.

Hartman (1952) also studied the equation (2.1) and improved Wintner’s result (1949) by

proving the condition given in Theorem 2.1.3.

Theorem 2.1.3 Hartman (1952)

Suppose that

,)(1

suplim)(1

inflim

0 00 0

dsduuqt

dsduuqt

t

t

s

tt

t

t

s

tt

then, equation (2.1) is oscillatory. In the following, Kamenev (1978) has proved a

new integral criterion for the oscillation of the differential equation (2.1) based on the

use of the nth

primitive of the coefficient q(t) which has Wintner’s result (1949) as

a particular case.

16

Theorem 2.1.4 Kamenev (1978)

The equation (2.1) is oscillatory if

,)(1

suplim

0

1

1

dssqstt

t

t

n

nt

for some integer 3n . Philos (1983) improved the above Kamenev’s result (1978).

Theorem 2.1.5 Philos (1983)

Let be a positive continuously differentiable function on the interval

,0t such that

dssstsns

st

ti

t

t

n

nt

23

1)()()()1(

)(

1suplim)(

0

for some integer ,3n

.)()(1

suplim)(

0

1

1

dssqsstt

ii

t

t

n

nt

Then, every solution of the equation (2.1) is oscillatory.

Remark 2.1.1: By setting 1)( t in the above Theorem 2.1.5, Theorem 2.1.5 leads to

Kamenev’s Result (1978) (Theorem 2.1.4).

Yan (1986) presented another new oscillation theorem for equation (2.1).

Theorem 2.1.6 Yan (1986)

Suppose that there exists an integer 3n with

.)(1

suplim

0

1

1

dssqstt

t

t

n

nt

17

Let )(t be a continuous function on ,0t with

.)()(1

inflim

0

0

1

1

t

t

n

nt

tTeveryforTdssqstt

Then equation (2.1) is oscillatory if

0

,)(t

dss

where 0,0,)(max)( tttt .

Also, Philos (1989) extended the Kamenev’s result (1978) as follows


Let H and h be two continuous functions 0:,:, tststDHh R and H

has a continuous and non-positive partial derivative on D with respect to the second

variable such that 0),( ttH for ,0tt 00),( tstforstH and

.),(),(),(),( DstallforstHsthstHs

Then, equation (2.1) is oscillatory if

.),(4

1)(),(

),(

1suplim

0

2

0

dssthsqstHttH

t

tt

18

Also, Philos (1989) extended and improved Yan’s result (1986) in the following theorem:


Let H and h be as in Theorem 2.1.7, moreover, suppose that

),(

),(infliminf0

00 ttH

stH

tts

and

.),(),(

1suplim

0

2

0

dssthttH

t

tt

Assume that )(t as in Theorem 2.1.6 with

.)(

0

2

t

dss

Then, the equation (2.1) is oscillatory if

.)(),(4

1)(),(

),(

1suplim 0

2 tTeveryforTdssthsqstHTtH

t

Tt

2.1.1.2 Oscillation Of Equation Of Type (2.2)

This section is devoted to the study of the oscillation of the equation of type (2.2). It

is interesting to discuss conditions for the alternating coefficient )(tq which are sufficient

for all solutions of equation (2.2) to be oscillated. An interesting case is that of finding

oscillations criteria of equation (2.2) which involve the average behavior of the integral

of q. The problem has received the attention of many authors in recent years as Moore

19

(1955) and Popa (1981). Among numerous papers dealing with such averaging

techniques of the oscillation of equations of type (2.2), we mention the following:

Moore (1955) gave the following oscillation criteria for equations of type (2.2).

Theorem 2.1.9 Moore (1955)

Suppose that the function satisfies 0)(,,0

2 ttC

0)()( 2

tssr

ds

and

.)()()()()(

0

t

dssqsssrs

Then, equation (2.2) is oscillatory.

In fact, Popa (1981) extended Kamenev’s oscillation criterion (1978) to apply on

equation of the form (2.2). He proved the following two theorems:

Theorem 2.1.10 Popa (1981)

If r(t) is bounded above and

,)()(1

suplim

0

1

1

t

t

n

nt

dssqstt

where n is an integer and ,2n then, the equation (2.2) is oscillatory.

20

Theorem 2.1.11 Popa (1981)

If )(

)(

tr

tr

is bounded and

,)(

)()(

1suplim

0

1

1

t

t

n

nt

dssr

sqst

t

where n is an integer and ,2n then the equation (2.2) is oscillatory.

2.1.2 Oscillation Of Homogenous Nonlinear Equations

2.1.2.1 Oscillation Of Equations Of Types (2.3) and (2.4)

This section is devoted to the oscillation criteria for the second order nonlinear

differential equations of the form (2.3) and (2.4). The oscillation of equation (2.4) has

brought the attention of many authors since the earliest work by Atkinson (1955). The

equation (2.3) is also known Emden-Fowler equation (EF). Clearly equation (EF) is

sub-linear if 1 and super linear if 1 .

The oscillation problem for second order nonlinear differential equation is of

particular interest. Many physical systems are modeled by nonlinear ordinary

differential equations. For example, equation (EF) arises in the study of gas dynamics

and fluid mechanics, nuclear physics and chemically reacting systems. The study of

Emden–Fowler equation originates from earlier theorems concerning gaseous dynamics

in astrophysics around the turn of the century. For more details for the equation we refer

to the paper by Wong (1973) for a detailed account of second order nonlinear oscillation

and its physical motivation. There has recently been an increase in studying the

21

oscillation for equations (2.4) and (EF). We list some of more important oscillation

criteria as follows.

The following theorem gives the necessary and sufficient conditions for oscillation

of equation (2.4) with ,...3,2,1,)( 12 nxxg n

Theorem 2.1.12 Atkinson (1955)

Suppose that 0)( tq on ,0t and

,...2,1,)( 12 nxxg n,


.)(

0

t

dsssq

Waltman (1965) extended Wintner's result (1949) for the equation (2.4) without any

restriction on the sign of )(tq .

Theorem 2.1.13 Waltman (1965)

Suppose that

,...2,1,)( 12 nxxg n

and

.)(

0

t

dssq

22

Then every solution of equation (2.4) is oscillatory. Kiguradze (1967) established the

following theorem for the Emden–Fowler equation (2.3).

Theorem 2.1.14 Kiguradze (1967)

The equation (2.3) is oscillatory for 1 if

dttqt )()(

for a continuous, positive and concave function )(t . Wong (1973) extended Wintner's

oscillation criteria (1949) to apply on the equation (2.3).

Theorem 2.1.15 Wong (1973)

Let 1 . Equation (2.3) is oscillatory if

0,)(inflim

0

t

tt

dssq

and

.)()(1

suplim

0

t

tt

dssqstt

Onose (1975) proved the theorem of Wong (1973) (Theorem 2.1.15) for the sublinear

Emden–Fowler differential equation and also study the extension of Wong's result

(1973) to the more general super-linear differential equation of the form equation (2.4)

as in the following three theorems:

23

Theorem 2.1.16 Onose (1975)

Suppose that

(1) ,0,)(inflim

0

t

tt

dssq

(2) ,)(suplim

0

t

tt

dssq

(3)

t

t

s

tt

dudsuqt

0 0

.)(1

suplim

Then, the equation (2.3) is oscillatory for 10 .


Assume that

(1) ,0)(inflim

0

t

tt

dssq

(2) .)(suplim

0

t

tt

dssq

Then, the equation (2.4) is oscillatory.


Suppose that

(1) ,0,)(inflim

0

t

tt

dssq

24

(2) .)(suplim

0 0

t

t

s

tt

dudsuq


Yeh (1982) established new integral criteria for the equation (2.4) which has Wintner's

result (1949) as a particular case.

Theorem 2.1.29 Yeh (1982)

Suppose that

,)(1

suplim

0

1

1

dssqstt

t

t

n

nt

for some integer n > 2. Then, the equation (2.4) is oscillatory.

Philos (1984) gave a new oscillation criteria for the differential equation (2-3) if

10 .


Let be a positive continuous differentiable function on the interval ,0t such

that

.0)()()( 0

2

ttallfortttt

Then, the equation (2.3) is oscillatory if

t

t

n

nt

dssqsstt

0

)()(1

suplim1

1

for some integer 2n .

25

Philos (1985) improved Onose’s result (1975) for equation (2.4).


Suppose that be a positive twice continuously differentiable function on ,0t

such that

,0)(0)( 0tontandt ,

,)()(lim

0

t

tt

dssqs

and

.)()()(1

suplim

0

t

tt

dssqsstt


Wong and Yeh (1992) improved Wong's result (1973) for equation (2.3) to the more

general equation (2.4).

Theorem 2.1.22 Wong and Yeh (1992)

Suppose that

0)(inflim

t

Tt

dssq

for large 0tT and there exists a positive concave function on ,0t such that

26

,)()(1

suplim

0

dssqsstt

t

tt

for some .0 Then the super-linear differential equation (2.4) is oscillatory.

Theorem 2.1.23 Philos and Purnaras (1992)

Suppose that

(1)

t

t

n

nt

dssqstt

0

)(1

inflim1

1for some integer 1n ,

(2) .)(1

suplim

0 0

2

t

t

s

tt

dsduuqt

Then, the super-linear differential equation (2.4) is oscillatory.


This section is devoted to the oscillation criteria for half-linear second order

differential equations of the form (2.5). The oscillation of the equation has brought the

attention of some authors since the early paper by Bihari (1963).

Theorem 2.1.24 Bihari (1963)

If 0tt0 q(t) allfor and

,)(

0

t

dssq

27

then, every solution of the equation (2.5) is oscillatory. The following theorem extended

the result of Bihari (1963) to an equation in which q is of arbitrary sign.

Theorem 2.1.25 Kartsatos (1968)

Suppose that

(i) There exists a constant 0* B such that

m

Bs

dsmG

0

*

),1()( for all m R,

(ii) .)(

0

t

dssq

Then, every solution of equation (2.5) is oscillatory.



differential equation of the form (2.6). Bhatia (1966) presented the following oscillation

criteria for the general equation (2.6) which contains as a special case of Waltman's

result (1965) for the nonlinear case.

Theorem 2.1.26 Bhatia (1966)

Suppose that

(1) ,)(

0

tsr

ds

28

(2) .)(

0

t

dssq


E. M. Elabbasy (1996) improved and extended the results of Philos (1983) to the

equation (2.6).

Theorem 2.1.27 E. M. Elabbasy (1996)

Suppose that

(1) ,)()(inflim

0

t

tt

dssqs

(2) ,)()(1

suplim

0 0

2

t

t

s

tt

dsduuqut

where ,0,: 0t is continuously differentiable function such that

.0)()(0)()(,0)()(,0)(

ttrandttrttrt



Oscillation of the equations of type (2.7) has been considered by many authors who

presented some oscillation criteria for solutions of the equation (2.7). Grace (1992)

29

studied the equation (2.7) and gave some sufficient conditions for oscillation of

equation (2.7) in some theorems for example the following two theorems:

Theorem 2.1.28 Grace (1992)

Suppose that

(1) 0)(

)(

k

x

xg for ,0x

Moreover, there exists a differentiable function ,0,: 0t and the functions

Hh, are defined as in Philos’s result (1989) (in Theorem 2.1.7). Moreover, suppose that

,),(

),(infliminf0)2(

00

ttH

stH

tts

.),()(

)(),()()(

),(

1suplim)3(

00

dsstH

s

ssthssr

ttH

t

tt

If there exists a continuous function )(t on ,0t such that

,)(),()(

)(),(

4

)()()()(),(

),(

1suplim)4(

0

2

0

TdsstHs

ssth

k

ssrsqsstH

ttH

t

tt

for every large ,0tT and

,)()(

)(lim)5(

0

2

t

tds

ssr

s

where ,0,)(max)( tt then, the equation (2.7) is oscillatory.

30


Suppose that the condition (1) from Theorem 2.1.28 holds and functions ,, Hh

are defined as in Theorem 2.1.28 and .0))()((0)( 0ttforttrandt

Moreover, suppose that

,)()(inflim)1(

0

dssqs

t

tt

and

.)()(

lim)2(

0

t

t ssr

ds

Then, the equation (2.7) is oscillatory if the conditions (4) and (5) hold.

2.1.3 Oscillation Of Homogenous Nonlinear Equations With Damping

Term


In last three decades, oscillation of nonlinear differential equations with damping

term has become an important area of research due to the fact that such equations

appear in many real life problems. Oscillation of non-linear equation (2.8) has been

considered by many authors, for example Yeh (1982) considered the equation (2.8) and

presented some oscillation criteria for equation (2.8).

31

Theorem 2.1.30 Yeh (1982)

Suppose that

,)()(suplim

0

1

t

t

n

t

dsssqst

t

t

n

ntdsstn

nshsts

t0

3

2

1)(1)

1)()((

1lim

for some integer ,3n are sufficient conditions for the oscillation of equation (2.8).



differential equation with damping term of the form (2.9). Nagabuchi and Yamamoto

(1988) have extended and improved the result of Yeh (1982) for equation (2.8) to the

equation (2.9).

Theorem 2.1.31 Nagabuchi and Yamamoto (1988)

The equation (2.9) is oscillatory if there exists a continuously differentiable

function )(t on ,0t

and a constant ,1

such that

.)(

)()(

)()()()(

)()()(

4

1)()()(

1suplim

2

2

0

dss

srst

sstssr

shsst

ksqsst

t

t

tt

Grace (1992) established some oscillation conditions for equation (2.9) in some three

theorems for example, the following theorem:

32


Let 00)( xforkxg and the functions ,, Hh are defined as in Theorem

2.1.28 such that the conditions (1) and (2) from Theorem 2.1.29 and

0)()()()()(1

tthttrt and .0)( 01 ttfort

Then, the superlinear equation

(2.9) is oscillatory if there exists a continuous function )(t on ,0t such that the

condition (5) from Theorem 2.1.28 and

,)(),()(),(4

)()()()(),(

),(

1inflim

2

TdsstHssthk

ssrsqsstH

TtH

t

Tt

for every large 0tT and .)()()()()()()(1 ttrtthttrt

Kirane and Rogovchenko (2001) were concerned with the problem of oscillation of

nonlinear second order equation with damping (2.9) and presented some oscillation

theorems for solutions of (2.9). One among the theorems is the next one.

Theorem 2.1.33 Kirane and Rogovchenko (2001)

Assume that

.00)(

xforkx

xg

Suppose further that the functions ,, Hh are defined as in Theorem 2.1.28 and there

exists a function ,0;,0

1 tCf such that

33

,),()(),(),(4

)()()(),(

),(

1suplim

22

0

dsstHsstQsthk

ssrsstH

ttH

t

Tt

where

))(2exp()(,)()()()()()()()()()()( 2

s

duufssshsfsrsfsrsfshskqss

and

.)),(())()((),(),( 211 stHsrshsthstQ


Elabbasy, et al. (2005) have studied the oscillatory behavior of equation (2.9) and

improved a number of existing oscillation criteria.

Theorem 2.1.34 Elabbasy, et al. (2005)

Assume that the condition (1) from Theorem 2.1.29 holds and there exists

,0,: 0t such that

,0)()()()(,0))()((,0))()((,0)(

thtttrttrttrt

.)()(1

suplim

0 0

dsduuqut

t

t

s

tt


34

Lu and Meng (2007) have considered the equation (2.9) and given several oscillation

conditions. They improved and extended result of Philos (1989) and result of Yu

(1993). They needed the following lemma to simplify proofs of their results. First they

recalled a class functions defined on .:, 0tststD A function RDCH , is

said to belong the class W if

(1) 0),( ttH , for 0tt and 0),( stH when ;st

(2) ),( stH has partial derivatives on D such that

.),(),(),(),(,),(),(),( 21 DstallforstHsthstHs

stHsthstHt

for some .,, 1

21 RDLhh loc

Lemma 2.1.1 Lu and Meng (2007)

Let RtCAAA ,,,, 0210 with ,02 A and RtCw ,,0

1 . If there exists

,, 0tba and bac , such that

),,(,)()()( 2

210 baswsAwsAsAw

then

,0),()(4

1)(),(

),(

1),(

)(4

1)(),(

),(

1 2

2

2

0

2

1

2

0

dssb

sAsAsbH

cbHdsas

sAsAasH

acH

b

c

c

a

,WHeveryfor

where

35

),()(),(),(),()(),(),( 122111 sbHsAsbhsbandasHsAashas

Theorem 2.1.35 Lu and Meng (2007)

Suppose that there exists a function ,0,,0

1 tC such that the condition

(1) from Theorem 2.1.29 is satisfied, and

,,0)(,0)( 0tttt

,)(

dss

where .)()(],[,)()(

1)(),()()()()(

1

t

T

dsstTtvttr

ttthttrt

If for every ,0tT

there exists an interval ,,, Tba and that there exists

,,bac WH and for any constant 0D such that

,0),(],[4

1)()(),(

),(

1

),(],[4

1)()(),(

),(

1

2

2

0

2

1

0

dssbtsDv

sqssbHcbH

dsastsDv

sqsasHacH

b

c

c

a

where

.),()()(),(),(),()()(),(),( 2211 sbHsssbhsbandasHssashas


Rogovchenko and Tuncay (2008) have considered the nonlinear equation (2.9) and

established some sufficient conditions for oscillation of solution of equation (2.9) by

giving many theorems for example, the following theorem:

36

Theorem 2.1.36 Rogovchenko and Tuncay (2008)

Suppose that )(xg exists and .00)( xforkxg

Suppose, further, that there exists a function RtC ,,0

1 such that, for some

,1

,),()()(4

)(),(),(

1suplim 2

0

dssthsrsvk

sstHttH

t

Tt

where

t

dsskr

shsksv

)(2

)()(2exp)(

and .)()()()()()()()()( 2 tsrtthttkrtqtvs Then, the equation (2.9) is

oscillatory.


Oscillatory behavior of nonlinear second order differential equation with damping

(2.10) has been studied by many authors. Grace (1992) considered the equation (2.10),

presented some oscillation results for equation (2.10) and extended and improved a

number of previously known oscillation results.


Suppose that the condition (1) from Theorem 2.1.28 holds. Moreover, assume that

the functions h, and H are defined as in Theorem 2.1.28 and

(1) 0)()(,0)(

tthth for .0tt

37

If

,),()(

)(),(

4

)()()()(),(

),(

1suplim

0

2

0

dsstHs

ssth

k

ssrsqsstH

ttH

t

tt

then, superlinear equation (2.10) is oscillatory.


Suppose that the condition (1) from Theorem 1.5.28 holds and

.)(

)(

)(

)(

duug

uanddu

ug

u

Let the functions h, and H are defined as in Theorem 2.1.28 such that the condition

(1) from Theorem 2.1.37 holds and

0)()(0)(

ttrandt for .0tt

If

,),(4

)()()()(),(

),(

1suplim

0

2

0

dssthk

ssrsqsstH

ttH

t

tt

then, superlinear equation (2.10) is oscillatory.

Rogovchenko and Tuncay (2007) were concerned with equation (2.10) and obtained

some oscillation criteria for oscillation of equation (2.10). They gave some oscillation

theorems and proved these theorems by using their following lemma which is a

particular case of Tiryaki and Zafer’s lemma 1.1 (2005).

38

Let .:, tsstD A function ),( stH belongs to the class W if

(1) ;,0, DCH

(2) 0),( ttH , for 0tt and 0),( stH for ; ts

(3) ),( stH has continuous partial derivatives tH and sH satisfying

,),(),(),(,),(),(),( 21 stHsthstHs

stHsthstHt

where .,, 1

21 RDLhh loc

Lemma 2.1.2 Rogovchenko and Tuncay (2007)

Suppose that a function RDCU ,1 satisfies the inequality

),()()()( 2 tUtttU

for all ,,),( 0 tbat where the functions ,,,0 RtC .,0,,0 tC

Then, for any ),( bac and for any ,WH

.0),()(4

1),()(

),(

1),(

)(4

1),()(

),(

1 2

2

2

1

dssbh

ssbHs

cbHdsash

SasHs

acH

b

c

c

a

Theorem 2.1.39 Rogovchenko and Tuncay (2007)

Let g be continuously differentiable and satisfy for all ,Rx

.0)( kxg

39

Assume that, for all ,Rx

.)(0 1CxC

Suppose also that there exists a function RtC ,,0

1 such that, for

some WH and ),,( bac

,0),()()(4

),()(),(

1

),()()(4

),()(),(

1

2

21

2

11

dssbhsrsk

CsbHs

cbH

dsashsrsk

CasHs

acH

b

c

c

a

where

)(4

)(11

)(

))()()(()()()()(

2

11 tkr

th

CCtrC

thtktttqts

and

.)(

)(2)(exp)(

1

ds

srC

sksht

t

Then, every solution of the equation (2.10) has at least one zero in (a,b).

2.1.4 Oscillation Of Nonhomogeneous Nonlinear Equations


This section is devoted to study the oscillation of equation (2.11). Many authors

are concerned with the oscillation criteria of solutions of the homogeneous second order

40

nonlinear differential equations. However, few authors studied the non-homogeneous

equations. Greaf, et al. (1978) considered the non-homogeneous equation (2.11) and

gave some oscillation sufficient conditions for this for the non-homogeneous equation,

for instance, the next three theorems.

Theorem 2.1.40 Greaf, et al. (1978)

Suppose that

(1)

0)(

tsr

ds,

(2)

0

)()(t

dsspsq ,

Then, all solutions of equation (2.11) are oscillatory.

Example 2.1.2

Consider the differential equation:

.0,

1)(

)(sin)(cos2

3

1)(

24

732

tttx

txttxtttx

Theorem 2.1.40 ensures that the given equation is oscillatory.


Suppose that the condition (1) from Theorem 2.1.40 holds and

(1)

0

)()(t

dsspsq ,

41

(2)

t

tt

dsspsq

0

0)()(inflim for all large 0tT ,

(3) .)()()(

1lim

0 0

dudsupuq

sr

t

t

s

tt

Then, the superlinear differential equation (2.11) is oscillatory.


Suppose that

0 0 0

,)()()(

1

)(t t

s

t

dudsupuqsr

dssr

M

for every constant M, then the sub-linear differential equation (2.11) is oscillatory.

Example 2.1.3

Consider the differential equation

0,1)(4

)()(cos

2

1)(

3

4

3

1

2

ttx

txtxttxt ,

Theorem 2.1.42 ensures that the given equation is oscillatory.

Remili (2010) was concerned with non-homogeneous nonlinear equation (2.11) and

presented oscillation criteria for equation (2.11) which contain results of Greaf, et al.

(1978) as particular case. Two theorems are presented here.

42

Theorem 2.1.43 Remili (2010)

Let )(t be a positive continuously differentiable function on ,T

such that

0

on .,T Equation (2-11) is oscillatory if

(1)

,)()(

lim

0

t

tt srs

ds

(2)

,)(

0

t

dssR

where

.)(

)()(

4

1)]()()[()(

2

t

trt

ktptqttR

Theorem 2.1.44 Remili (2010)

If the condition (1) from Theorem 2.1.43 holds and )(t is defined as Theorem

2.1.43 such that

(1) ,)()()(

0

t

dsspsqs

(2) 0)(inflim

t

Tt

dssR for all large 0tT ,

(3) .)()()(

1lim

0

dudsuR

srs

t

t st

Then, the super linear differential equation (2.11) is oscillatory.

43

2.1.4.2 Oscillation Of Equation Of Type (2.12)

Oscillation of solutions of the nonhomogeneous non-linear equation (2.12) has been

studied by many authors. Manojlovic (1991) has considered the nonhomogeneous non-

linear equation (2.12) and established some oscillation theorems for solutions of this

equation. For example the following theorem:

Define the sets 000 :,,:, tststDtststD and introduce the function

)(DCH which satisfies the following conditions:

(1) 0),( ttH for ,0tt 0),( stH for ,),( 0Dst

(2) H has a continuous and non-positive partial derivative on 0D with respect to the

second variable, as well as a continuous function RDh 0: such that

),(),(),( stHsthstHs

for .),( 0Dst

Theorem 2.1.45 Manojlovic (1991)

Suppose that for any 01 there exists 02

such that

2)(

)(

x

xg for .1x

Suppose, furthermore, that

dssthsr

t

t0

),()( 2 for ,0tt

44

dssthsLrsHKsqstHTtH

t

Tt

),()()()(),(),(

1suplim 2

for every 0tT and any positive constants K and L. Then, any solution of equation

(2.12) is either oscillatory or satisfies .0)(inflim

txt

2.1.5 Oscillation Of Non-homogeneous Nonlinear Equations With

Damping Term


This section is devoted to study the oscillation of equation (2.13). Many authors are

concerned with the oscillation criteria of solutions of the non-homogeneous non-linear

second order equations with damping term. Berkani (2008) considered the equation

(2.13) and presented some sufficient condition for all solutions of equation (2.13) to be

oscillatory.

Theorem 2.1.46 Berkani (2008)

Assume that for some constants 1CC, K, and for all ,0x

,)(

Kxx

xg

.)(0 1CxC

45

Suppose further there exists a continuous function )(tu such that

,0)()()( cubuau )(tu is differentiable on the open set ),(),( bcca and satisfies

the inequalities

0)()(2)()(2

)()()(

2

1

22

dssusrCsu

sCr

shsHsKq

c

a

and

.0)()(2)()(2

)()()(

2

1

22

dssusrCsu

sCr

shsHsKq

b

c

Then, every solution of equation (2.13) has a zero in .,ba

2.2 oscillation Of Third Order Differential Equations

In the relevant literature, until now, oscillatory and non-oscillatory behavior of

solutions for various classes of linear and non-linear third order differential equations has

been the subject of intensive investigations for many authors. There are many papers

dealing with particular cases of the equations (1.3). These particular cases can be

classified as follows:

The homogeneous linear equations are given below

)16.2(.0)()()()()()()(

)15.2(.0)()()()()(

)14.2(.0)()()(

txtctxtbtxtatx

txtctxtbtx

txtqtx

46

The non-homogeneous linear equations is given below

)17.2().()()()()()()()( tftxtctxtbtxtatx

The homogeneous non-linear equations are given below

)20.2(.0))(()()(

)19.2(.0))(()()()()(

)18.2(.0)()()()()(

txftqtx

txftctxtbtx

txtctxtbtx

The paper which was presented by Hanan (1961) was the starting point for many

investigations to asymptotic behavior of third order equations.

2.2.1 Oscillation Of Homogenous Linear Equations


This section is devoted to the oscillation criteria for the third order linear differential

equation of the form (2.14). Hanan (1961) considered the equation (2.14) presented the

following oscillation criteria for equation (2.14):

Theorem 2.2.1 Hanan (1961)

The equation (2.14) is non-oscillatory if

0

,)(2

t

dssqs

or

47

,33

2)(suplim 3

tqtt

and the equation (2.14) is oscillatory if

.33

2)(inflim 3

tqtt

Later, in 2001, Adamets and Lomtatidze (2001) also studied oscillatory properties os

solutions of the equation (2.14) where q is eventually of one sign 0, .

Mehri (1976) considered the third equation (2.14) and presented the following

oscillation result.

Theorem 2.2.2 Mehri (1976)


0

.)(t

dssq


This section is devoted to the oscillation criteria for the equation of the form

(2.15). Hanan (1961) also derived some oscillation criteria for equation (2.15) and

proved the following theorem:

Theorem 2.2.3 Hanan (1961)

If ,0)()(2 tbtc and there exists a number 10 k

such that

48

ktbtt

)(inflim 2

and

,)(inflim 3 ktctt

Then, the equation (2.15) is oscillatory. Lazer (1966) studied oscillation of equation

(2.15) and proved the following:

Theorem 2.2.4 Lazer (1966)

Assumed that if 0)( tb and

0

,)(33

2)( 3

2

t

dstbsc


2.2.1.3 Oscillation of Equations of type (2.16)

This section is devoted to the oscillation criteria for the equation of the form

(2.16). Parhi and Das (1993) considered the linear equation (2.16) and presented the

following theorem:

Theorem 2.2.5 Parhi and Das (1993)

Supposed that 0)(),()(,0)(,0)(,0)( tctbtatctbta and proved that if

49

0

,)(3

)(

33

2

3

)()()(

27

)(2 2

3

23

t

dssbsasbsa

scsa

then, the equation (2.16) is oscillatory.

Theorem 2.2.6 Parhi and Das (1993)

Supposed that )()(,0)(,0)(,0)( tbtatctbta and proved that if

0

,)()(3

)(

33

2

3

)()()(

27

)(2 2

3

23

t

dssasbsasbsa

scsa


2.2.2 Oscillation Of Non-homogenous Linear Equations


This section is devoted to the oscillation criteria for the equations of the form

(2.17). Das (1995) studied the equation (2.17) and established some new oscillation

criteria for the equation (2.17).

Theorem 2.2.7 Das (1995)

Supposed that 0)(,0)(),()(,0)(,0)(,0)(,0)( tftftbtatctctbta

and proved that if

50

0

,))()((3

)(

33

2

3

))()()(()(

27

)(2 2

3

23

t

dssasbsasasbsa

scsa

then, every solution of equation (2.17) oscillates.

2.2.3 Oscillation Of Homogenous Non-linear Equations

2.2.3.1 Oscillation Of Equations Of Types (2.18) and (2.19)

This section is devoted to the oscillation criteria for third order non-linear

equations of the form (2.18) and (2.19). Waltman (1966) considered the equations (2.18)

and (2.19) and established two theorems for oscillation.


Supposed that the equation (2.18) is oscillatory if )(tb and )(tc are continuous

functions and ,0)( tb is a ratio of two odd positive integers and

t

t

dssQBtA

0

,0)(

for large t, where ,A B and .)()(0

t

dssctQ


Supposed that the equation (2.19) is oscillatory if )(tb and )(tc are positive

continuous functions such that 0/)(,0)()( kuuftbtkc and

51

.)()(

0

t

dssbskcs

Heidel (1968) also considered the non-linear third order equation (2.18) and

investigated the behavior of non-oscillatory and oscillatory of solutions of equation

(2.18). He proved the following oscillation theorem:

Theorem 2.2.10 Heidel (1968)

Proved that if 0)(,0)( tctb and ,0)(22

tbt

,10,)(

0

2

t

dsscs

,)(2)(

0

4

t

dsscsbs



This section is devoted to the oscillation criteria for third order non-linear

equations of the form (2.20). Ramili (2007) studied non-oscillatory for the third order

non-linear equation (2.20) and presented the following:

Theorem 2.2.11 Ramili (2007)

Supposed that 0)(2)()( 2 ufufuf for every ,0u

52

)(

0uf

du and ,

)(0

uf

du

,)()(inflim

0

Tt

dssqsR

,)()(1

suplim

0 0

t

T

s

Tt

dssqsRt

then, every solution of equation (2.20) is non-oscillatory.

53

CHAPTER THREE

OSCILLATION OF SECOND ORDER NONLINEAR

ORDINARY DIFFERENTIAL EQUATIONS WITH

ALTERNATING COFFICIENTS

3.1 Introduction

In this chapter, we are concerned with the problem of oscillation of second order

non-linear ordinary differential equation of the form

)1.1(,)(,)()()),(()()()( txtHtxtrtxgtqtxtr

where q and r are continuous functions on the interval )(,0,, 00 trtt is a positive

function, g is continuously differentiable function on the real line R except possibly at 0

with 0)( xxg and 0)( kxg for all ,0x is a continuous function on RxR with

0),( vuu for all 0u and ),(),( vuvu for any ),0( and H is a

continuous function on ,0t ×R with )())(())(,( tptxgtxtH for all x 0 and 0tt .

3.2 Second Order Nonlinear ODE Of Type (1.1)

In this chapter, we present the results of our study of finding the sufficient conditions

for oscillation of solutions of ordinary differential equations of second order of type

(1.1). The present oscillation results have among other finding extended and improved

many previous oscillation results, for examples, such as the works of Bihari (1963),

54

Kartsatos (1968), Philos and Purnaras (1992), Philos (1989), El-abbasy (1996), and El-

Abbasy, et al. (2005). We have established some new sufficient conditions which

guarantee that our differential equations are oscillatory. A number of theorems and

illustrative examples for oscillation differential equation of type (1.1) are shown.

Further, a number of numerical examples are given to illustrate the theorems. These

numerical examples are computed by using Runge Kutta of fourth order function in

Matlab version 2009. The present results are compared with existing results to explain

the motivation of proposed research study.

3.3 Oscillation Theorems

Theorem 3.3.1: Suppose that

(1) ),,0(,1

),1(

10

0

CCv

(2) ),0(,),1(

1)( **

0

BBdss

mG

m

for every m R,

(3) ,,)()( 00 tTdsspsqCT

where ,0,: 0tp , then every solution of equation (1.1) is oscillatory.

Proof

Without loss of generality, we assume that there exists a solution 0)( tx of

equation (1.1) such that 0)( tx on ,T for some .00 tT Define

.,))((

)()()( Tt

txg

txtrt

55

This and the equation (1.1) imply

.,)(,1)()()( Ttttqtpt

.),(

)(,1

)(

)(,1

)(Tttq

t

tp

t

t

By the condition (1), we have

.),()(

1

)(,1

)(

0

TttqtpCt

t

Integrate from T to t and from condition (2), we obtain

)).(())(())(()(,1

)()()( 0

*

00000 TGCBCTGCsGCdss

sCdsspsqC

t

T

t

T

Thus, we have

,)()(0

T

dsspsqC

which contradicts to the condition (3). Hence, the proof is completed.

Example 3.3.1


0,)(cos)(

)(cos31)(3

tt

txtxtxttxt

Here uvuxxgttqttr ),(,)(,cos31)(,)( and

).(1)(cos

))((

))(,(33

tptt

tx

txg

txtH

56

All conditions of Theorem 3.3.1 are satisfied and hence every solution of the given

equation is oscillatory. To demonstrate that our result in Theorem 3.3.1 is true, we also

find the numerical solution of the given differential equation in Example 3.3.1 using

the Runge Kutta method of fourth order (RK4) for different step sizes h.

We have

xxxxxtfx 99.3cos),,(

with initial conditions ,1)1( x 5.0)1(

x on the chosen interval [1,50] and finding

the values of the functions r, q and f where we consider )()())(,( xltftxtH at t=1

n =750, n =1500, n=2250 and n=3000 and the step sizes h =0.065, h =0.032, h =0.021

and h = 0.016.

57

Ta

ble

3.1

: C

om

par

ison

of

the

nu

mer

ical

solu

tion

s o

f O

DE

3.1

wit

h d

iffe

ren

t st

eps

size

s

Erro

r %

0

0.0

000

019

6

0.0

000

095

7

0.0

000

029

4

0.0

000

154

3

0.0

000

812

3

0.0

000

223

8

0.0

000

456

3

0.0

000

347

3

0.0

001

071

9

0.0

000

459

0

0.0

000

876

6

0.0

000

010

5

0.0

000

087

5

0.0

000

772

4

0

0.0

000

000

2

0.0

000

000

8

0.0

000

000

3

0.0

000

001

3

0.0

000

002

4

0.0

000

001

9

0.0

000

002

8

0.0

000

002

6

0.0

000

003

9

0.0

000

003

4

0.0

000

000

9

0.0

000

000

5

0.0

000

000

9

0.0

000

004

2

Erro

r %

0

0.0

000

108

0

0.0

000

658

4

0.0

000

225

5

0.0

001

104

4

0.0

005

719

9

0.0

001

555

4

0.0

003

145

7

0.0

002

498

0

0.0

007

531

4

0.0

002

808

1

0.0

000

603

8

0.0

001

102

2

0.0

000

651

6

0.0

007

650

7

0

0.0

000

001

1

0.0

000

005

5

0.0

000

002

3

0.0

000

009

3

0.0

000

016

9

0.0

000

013

2

0.0

000

019

3

0.0

000

018

7

0.0

000

027

4

0.0

000

020

8

0.0

000

006

2

0.0

000

010

8

0.0

000

006

7

0.0

000

041

6

Erro

r %

0

0.0

001

965

4

0.0

011

815

4

0.0

004

364

1

0.0

020

177

0

0.0

100

861

4

0.0

028

704

5

0.0

055

807

9

0.0

044

430

1

0.0

136

088

0

0.0

057

878

4

0.0

012

525

8

0.0

021

278

7

0.0

014

064

5

0.0

146

634

0

0

0.0

000

02

0.0

000

098

7

0.0

000

044

5

0.0

000

169

9

0.0

000

298

0.0

000

243

6

0.0

000

342

4

0.0

000

332

6

0.0

000

495

1

0.0

000

428

7

0.0

000

128

6

0.0

000

208

5

0.0

000

144

6

0.0

000

797

3

h=

0.0

16

x4(t

k)

1

-1.0

17

592

33

0.8

353

477

9

1.0

196

635

4

0.8

420

474

4

-0.2

95

454

92

0.8

486

469

3

-0.6

13

533

10

-0.7

48

590

11

0.3

638

084

8

-0.7

40

690

03

-1.0

26

680

57

0.9

798

485

8

-1.0

28

116

50

-0.5

43

734

55

h

=0.0

21

x3(t

k)

1

-1.0

17

592

31

0.8

353

477

1-

1.0

196

635

1

0.8

420

473

1

-0.2

95

455

16

0.8

486

467

4

-0.6

13

533

38

-0.7

48

590

37

0.3

638

088

7

-0.7

40

690

37

-1.0

26

680

48

0.9

798

486

3

-1.0

28

116

41

-0.5

43

733

97

h=

0.0

32

x2(t

k)

1

-1.0

17

592

22

0.8

353

472

4

-1.0

19

663

31

0.8

420

465

1

-0.2

95

456

61

0.8

486

456

1

-0.6

13

535

03

-0.7

48

591

98

0.3

638

112

2

-0.7

40

692

41

-1.0

26

679

95

0.9

798

496

6

-1.0

28

115

83

-0.5

43

730

39

h=

0.0

65

x1(t

k)

1

-1.0

17

590

33

0.8

353

379

2

-1.0

19

659

09

0.8

420

304

5

-0.2

95

484

72

0.8

486

225

7

-0.6

13

567

34

-0.7

48

623

37

0.3

638

579

9

-0.7

40

732

90

-1.0

26

667

71

0.9

798

694

3

-1.0

28

102

04

-0.5

43

654

82

t k

1

5.9

10.8

12.7

6

17.6

6

20.4

04

24.5

2

27.0

68

30.4

32.3

6

37.2

6

40.2

42.1

6

47.0

6

50

58

Figure 3.1(a): Solution curves of ODE 3.1.

1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 3.1 (b): Solution curves of ODE 3.1.

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

lout

ions

x(t)

x1(t)

x2(t)

x3(t)

x4(t)

Remark 3.3.1: Theorem 3.3.1 is the extension of the results of Bihari (1963) and the

results of Kartsatos (1968) who have studied the equation (2.5) as mentioned in chapter

two. Our result can be applied on their equation, but their oscillation results cannot be

applied on the given equation in Example 3.3.1 because their equation is aparticular

59

case of our equation when ,1)( tr ),())(( txtxg

))(),(())()()),((( txtxtxtrtxg

and 0))(,( txtH .

Theorem 3.3.2

If the conditions (1) and (2) hold and assume that be a positive continuous

differentiable function on the interval [t0,∞) with (t) is a decreasing function on the

interval [t0,∞) and such that

(4) ,)()()(lim 0

t

Tt

dsspsqCs

where p: [t0,∞)(0,∞), then, every solution of equation (1.1) is oscillatory.

Proof


equation (1.1) such that 0)( tx on ,T for some .00 tT Define

.,

)(

)()()()( Tt

txg

txtrtt

Thus and (1.1) imply

.,)()(

)()()(,1)()()()(

)(

)()( Ttt

t

ttttqttpt

t

tt

Thus, we obtain

.,)()(,1)()()()()(

)()( Tttttqttpt

t

tt

After dividing the last inequality by ,0)()(,1 tt integrating from T to t and

using condition (1), we obtain

60

.,

)()(,1

)()()()()()( 00 Ttds

ss

sssCdsspsqCs

t

T

t

T

(3.3.1)

By the Bonnet’s theorem, we see that for each t ≥ T, there exists at[T,t] such that

dsss

ssTds

ss

sss ta

T

t

T

)()(,1

)()()(

)()(,1

)()()(

(3.3.2)

From the inequality (3.3.2) in inequality (3.3.1), we have

,)(

)()()(

)(

)(

)(

)()(

,1)()()()(

0

*

0

0

)()(

)()(

00

T

TGTCBTC

T

TG

a

aGTC

u

duTCdsspsqCs

t

t

aa

TT

t

T

tt

which contradicts to the condition (4). Hence the proof is completed.

Example 3.3.2


.0,

))(sin(()(2

))(()(

)()(

1

cos4)(

510

95

2318

279

5

553

ttt

txtxt

txttx

txtx

t

ttttxt

Here 22

39

5

553 ),(,)(,

1

cos4)(,)(

vu

uuvuxxg

t

ttttqttr

and

).(

2))(sin((2

)((

)(,(510

5

510

5

tptt

t

tt

txt

txg

txtH

61

Taking 5

5 1)(

t

tt

such that

.)()()(lim 0

t

Tt

dsspsqCs

We get all conditions of Theorem 3.3.2 are satisfied and hence every solution of the

given equation is oscillatory. The numerical solutions of the given differential equation

are found out using the Runge Kutta method of fourth order (RK4) for different step

sizes h.

We have

)()(

)()(49.2))(sin()())(),(,()(

218

2799

txtx

txtxtxtxtxtxtftx

with initial conditions 5.0)1( x , 0)1(

x on the chosen interval [1,50] and finding

the values of the functions r, q and f where we consider )()())(,( xltftxtH at t=1,

n =2250, n =2500, n=2750 and n=3000 and the step sizes h =0.0217, h =0.0196,

h =0.0178 and h = 0.0163.

62

Ta

ble

3.2

: C

om

par

ison

of

the

nu

mer

ical

solu

tion

s o

f O

DE

3.2

wit

h d

iffe

ren

t st

ep s

izes

Erro

r %

0

0.0

002

868

0

0.0

007

625

6

0.0

011

490

5

0.0

020

991

1

0.0

007

128

7

0.0

029

285

9

0.0

017

860

6

0.0

013

526

9

0.0

005

718

8

0.0

022

028

7

0.0

251

366

9

0.0

055

914

4

0.0

039

810

7

0.0

045

062

8

0

0.0

000

004

5

0.0

000

014

2

0.0

000

019

5

0.0

000

036

4

0.0

000

048

2

0.0

000

068

9

0.0

000

083

5

0.0

000

102

4

0.0

000

056

8

0.0

000

153

2

0.0

000

18

0.0

000

197

3

0.0

000

238

0.0

000

276

9

Erro

r %

0

0.0

007

520

7

0.0

020

353

0

0.0

030

759

3

0.0

056

514

5

0.0

019

226

8

0.0

079

101

9

0.0

048

319

9

0.0

036

591

5

0.0

015

464

9

0.0

059

658

7

0.0

681

762

9

0.0

151

646

3

0.0

108

007

6

0.0

122

332

1

0

0.0

000

011

8

0.0

000

037

9

0.0

000

052

2

0.0

000

098

0.0

000

13

0.0

000

186

1

0.0

000

225

9

0.0

000

277

0.0

000

153

6

0.0

000

414

9

0.0

000

488

2

0.0

000

535

1

0.0

000

645

7

0.0

000

751

7

Erro

r %

0

0.0

015

615

1

0.0

042

907

7

0.0

064

936

3

0.0

120

122

2

0.0

040

938

3

0.0

168

702

6

0.0

103

121

1

0.0

078

136

7

0.0

033

034

2

0.0

127

642

8

0.1

459

603

8

0.0

324

774

3

0.0

231

471

3

0.0

262

321

7

0

0.0

000

024

5

0.0

000

079

9

0.0

000

110

2

0.0

000

208

3

0.0

000

276

8

0.0

000

396

9

0.0

000

482

1

0.0

000

591

5

0.0

000

328

1

0.0

000

887

7

0.0

001

045

2

0.0

001

146

0.0

001

383

8

0.0

001

611

9

h=

0.0

163

x4(t

k)

-0.5

-0.1

56

899

30

0.1

862

132

6

0.1

697

045

9

-0.1

73

406

63

0.6

761

387

7

0.2

352

660

1

-0.4

67

508

51

0.7

570

063

0

-0.9

93

210

66

-0.6

95

455

98

0.0

716

084

7

-0.3

52

860

38

-0.5

97

827

89

-0.6

14

474

45

h

=0.0

17

8

x3(t

k)

-0.5

-0.1

56

899

75

0.1

862

118

4

0.1

697

065

4

-0.1

73

402

99

0.6

761

339

5

0.2

352

591

2

-0.4

67

500

16

0.7

570

165

4

-0.9

93

216

34

-0.6

95

471

30

0.0

716

264

7

-0.3

52

880

11

-0.5

97

851

69

-0.6

14

446

76

h=

0.0

196

x2(t

k)

-0.5

-0.1

56

900

48

0.1

862

094

7

0.1

697

098

1

-0.1

73

396

83

0.6

761

257

7

0.2

352

474

0

-0.4

67

485

92

0.7

570

340

0

-0.9

93

226

02

-0.6

95

497

47

0.0

716

572

9

-0.3

52

913

89

-0.5

97

892

46

-0.6

14

399

28

h=

0.0

217

x1(t

k)

-0.5

-0.1

56

901

75

0.1

862

052

7

0.1

697

156

1

-0.1

73

385

80

0.6

761

110

9

0.2

352

263

2

-0.4

67

460

30

0.7

570

654

5

-0.9

93

243

47

-0.6

95

544

75

0.0

717

129

9

-0.3

52

974

98

-0.5

97

966

27

-0.6

14

313

26

t k

1

5.9

10.8

12.7

6

17.6

6

20.4

04

24.5

2

27.0

68

30.4

32.3

6

37.2

6

40.2

42.1

6

47.4

72

50

63

Figure 3.2 (a): Solutions curves of ODE 3.2

1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 3.2 (b): Solutions curves of ODE 3.2

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t

)

x1(t)

x2(t)

x3(t)

x4(t)

Remark 3.3.2

Theorem 3.3.2 extends results of Bihari (1963) and Kartsatos (1968), who have

studied the equation (1.1) when ,1)( tr ),())(( txtxg

))(),(())()()),((( txtxtxtrtxg

and .0))(,( txtH Also, Theorem 3.3.2 is the

64

extension of El-Abbasy (2005) who studied the equation (2.6) which is a special case

of the equation (1.1) as mentioned in chapter two. Our result can be applied on their

equation, but their oscillation results cannot be applied on the given equation in

Example 3.3.2 because their equations are particular cases of the equation (1.1).

Theorem 3.3.2 is the extension of Theorem 3.3.1 as well.

Theorem 3.3.3

Suppose that condition (1) holds and

(5) .0,0)( tallfortq

Furthermore, suppose that there exists a positive continuous differentiable function on

the interval ,0t with 0)(

t and 0))()((

trt such that condition (4) holds and

(6)

.)()(

0

0

ttforeveryssr

ds

t

Then every solution of superlinear equation (1.1) is oscillatory.

Proof

If )(tx is oscillatory on ,0,, 0 tTT then )(tx

is oscillatory on ,T and if

)(tx

is oscillatory on ,,T then, )(tx

is oscillatory on .,T

Without loss of

generality, we may assume that there exists a solution x(t) of equation (1.1) such that

,0)( Tontx for some .00 tT We have three cases of )(tx

:

(i) .0)( Tteveryfortx

(ii) .0)( Tteveryfortx

(iii) )(tx

is oscillatory.

65

If 0)(

tx for 0, tTTt and we define

.,))((

)()()()( Tt

txg

txtrtt

This and (1.1) imply

.,))((

)()()()()(,1)()()()()( Tt

txg

txtrttttqttptt

From condition (1), we have

.,))((

)()()()()()()( 0 Tt

txg

txtrttptqCtt

Integrate the last inequality from T to t, we obtain

.))((

)()()()()()()()( 0 ds

sxg

sxsrsdsspsqCsTt

t

T

t

T

(3.3.3)

By Bonnet’s theorem since )()( trt

is non-increasing, for a fixed ,Tt there exists

tTt , such that

.)(

)()())((

)()()(

))((

)()()()(

)(

tt x

TxT

t

Tug

duTrTds

sxg

sxTrTds

sxg

sxsrs

Since 0)()(

trt and the equation (1.1) is superlinear, we have

)(

)()(

).()(,)(

)()(,0

)(

tx

Txt

Tx

t

Txxifug

du

Txxif

ug

du

66

We have

,)(

)()()( Ads

sxg

sxsrs

t

T

(3.3.4)

where .)(

)()()(

Tx

ug

duTrTA

Thus, from (3.3.4) in (3.3.3), we obtain

.)()()()()( 0

t

T

dsspsqCsATt

By the condition (4), we get 0)( t , then, 0)(

tx for ., 11 TTTt This is a

contradiction.

If 0)(

tx for every .2 TTt The condition (4) implies that there exists 23 TT such

that

.0)()()( 30

3

TtallfordsspsqCs

t

T

Thus, from equation (1.1) multiplied by ),(t we obtain

.),()()()()(,1)()()()()()( 3Tttptxgttttqtxgttxtrt

By condition (1), we have

.)()())(()()()()( 0 tptqCtxgttxtrt

67

Integrate the last inequality from 3T to t, we obtain

.),()()(

)()()()())((

)()()())(()()()()()()()()()(

3333

0333

33

33

TtTxTrT

dudsupuCqusxsxg

dsspsqCstxgdssxsrsTxTrTtxtrt

s

T

t

T

t

T

t

T

Integrate the last inequality divided by )()( ttr from 3T to t and by condition (6), we

have

,)()(

)()()()()(

3

3333 t

Tssr

dsTxTrTTxtx

as ,t contradicting the fact 0)( tx for all t .T Thus, we have )(tx

is oscillatory

and this leads to (1.1) is oscillatory. Hence the proof is completed.

Example 3.3.3


.0,))(sin()(

)(16)(3

)()(9)(

13

5

2

10

155

tt

txtx

ttxttx

txtxttx

t

t

We have ,)1(

)(t

ttr

,9)( ttq ,)( 5xxg

)(1))(sin(

))((

))(,(

63),(

3322

3

tptt

tx

txg

txtHand

vu

uuvu

for all .00 tandx

68

Taking tt )( such that

.1

9lim)()()(lim

00

300

t

tt

t

tt

dss

sCsdsspsqCs

All conditions of Theorem 3.3.3 are satisfied, then, the given equation is oscillatory.

Also the numerical solutions of the given differential equation are computed using the

Runge Kutta method of fourth order (RK4) for different steps sizes h.

We have

)

)(24)(3

)()((5.4sin)(5.0))(),(,()(

210

1555

txtx

txtxxtxtxtxtftx

with initial conditions 0)1(,5.0)1(

xx on the chosen interval [1,50] and finding the

values of the functions r, q and f where we consider )()(),( xltfxtH at t=1, n =2250,

n =2500, n=2750 and n=3000 and the step sizes h =0.0217, h =0.0196, h =0.0178 and

h = 0.01

69

Ta

ble

3.3

: C

om

par

ison

of

the

nu

mer

ical

solu

tion

s o

f O

DE

3.3

wit

h d

iffe

ren

t st

ep s

izes

Erro

r %

0

0.0

000

075

9

0

0.0

000

376

9

0.0

000

021

4

0

0

0.0

000

036

5

0.0

000

044

3

0

0.0

000

208

1

0.0

000

021

7

0.0

000

050

5

0.0

000

060

8

0

0

0.0

000

000

1

0

0.0

000

000

1

0.0

000

000

1

0

0

0.0

000

000

1

0.0

000

000

1

0

0.0

000

000

1

0.0

000

000

1

0.0

000

000

2

0.0

000

000

2

0

Erro

r %

0

0.0

000

075

9

0.0

000

031

3

0.0

000

376

9

0.0

000

042

8

0.0

000

115

5

0

0.0

000

109

6

0.0

000

088

6

0.0

000

020

9

0.0

000

624

3

0.0

000

043

5

0.0

000

101

0

0.0

000

091

2

0.0

000

071

0

0

0.0

000

000

1

0.0

000

000

1

0.0

000

000

1

0.0

000

000

2

0.0

000

000

1

0

0.0

000

000

3

0.0

000

000

2

0.0

000

000

1

0.0

000

000

3

0.0

000

000

2

0.0

000

000

4

0.0

000

000

3

0.0

000

000

3

Erro

r %

0

0.0

000

075

9

0.0

000

062

7

0.0

000

753

9

0.0

000

064

3

0.0

000

034

6

0

0.0

000

182

7

0.0

000

177

3

0.0

000

041

8

0.0

001

456

7

0.0

000

087

1

0.0

000

202

0

0.0

000

273

6

0.0

000

189

5

0

0.0

000

000

1

0.0

000

000

2

0.0

000

000

2

0.0

000

000

3

0.0

000

000

3

0

0.0

000

000

5

0.0

000

000

4

0.0

000

000

2

0.0

000

000

7

0.0

000

000

4

0.0

000

000

8

0.0

000

000

9

0.0

000

000

8

h=

0.0

163

x4(t

k)

0.5

-0.1

31

729

67

-0.3

18

931

95

-0.0

26

525

81

0.4

663

958

6

0.0

865

644

9

-0.4

79

530

84

-0.2

73

666

94

0.2

255

896

6

0.4

779

933

8

-0.0

48

052

02

-0.4

58

778

43

-0.3

95

864

93

0.3

289

400

2

0.4

219

536

7

h

=0.0

17

8

x3(t

k)

0.5

-0.1

31

729

68

-0.3

18

931

95

-0.0

26

525

82

0.4

663

958

7

0.0

865

644

9

-0.4

79

530

84

-0.2

73

666

95

0.2

255

896

5

0.4

779

933

8

-0.0

48

052

01

-0.4

58

778

42

-0.3

95

864

95

0.3

289

400

0

0.4

219

536

7

h=

0.0

196

x2(t

k)

0.5

-0.1

31

729

68

-0.3

18

931

96

-0.0

26

525

82

0.4

663

958

8

0.0

865

645

0

-0.4

79

530

84

-0.2

73

666

97

0.2

255

896

4

0.4

779

933

7

-0.0

48

051

99

-0.4

58

778

41

-0.3

95

864

97

0.3

289

399

9

0.4

219

537

0

h=

0.0

217

x1(t

k)

0.5

-0.1

31

729

68

-0.3

18

931

97

-0.0

26

525

83

0.4

663

958

9

0.0

865

645

2

-0.4

79

530

84

-0.2

73

666

99

0.2

255

896

2

0.4

779

933

6

-0.0

48

051

95

-0.4

58

778

39

-0.3

95

865

01

0.3

289

399

3

0.4

219

537

5

t k

1

5.9

10.8

12.7

6

17.6

6

20.4

04

24.5

2

27.0

68

30.4

32.3

6

37.2

6

40.2

42.1

6

47.0

6

50

70

Figure 3.3(a): Solutions curves of ODE 3.3.

0 5 10 15 20 25 30-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

Num

erical solu

tions x

(t)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 3.3(b): Solutions curves of ODE 3.3

0 10 20 30 40 50 60-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

t

Num

erical solu

tions x

(t)

x1(t)

x2(t)

x3(t)

x4(t)

Remark 3.3.3

Theorem 3.3.3 extends result of Philos (1983) who has studied the equation (1.1)

as ,1)( tr ),())(( txtxg ))(())()()),((( txgtxtrtxg

and 0))(,( txtH , result of

71

Bhatia (1966) who has considered the equation (1.1)

as ))(())()()),((( txgtxtrtxg

and 0))(,( txtH , and result of Philos and Punaras

(1992) who have studied the equation (1.1) as

,1)( tr ))(())()()),((( txgtxtrtxg

and .0))(,( txtH Our result can be applied on

their equations (2.1), (2.6) and (2.4) respectively, but their oscillation results cannot be

applied on the given equation in Example 3.3.3 because their equations are particular

cases of our equation (1.1).

Theorem 3.3.4

Suppose that the conditions (1), (5) and (6) hold and there exists a continuously

differentiable function which is defined as Theorem 3.3.3 such that

(7) ,)(

0

t

dss

where )(4

)()()()()()(

2

0tk

trttptqCtt

.


Proof: Without loss of generality, we may assume that there exists a solution x(t) of

equation (1.1) such that ,0)( Tontx for some .00 tT Define

.,))((

)()()( Tt

txg

txtrt

72

This, conditions (1), (5) and the equation (1.1) imply

.,))((

)()()()(

))((

)()(2

2

0 Tttxg

txtrktptqC

txg

txtr

(3.3.5)

We multiply the last inequality (3.3.5) by )(t and integrate form T to t, we have

,)()(

)()()()()()(

))((

)()()( 2

01 dsssr

skssdsspsqCsC

txg

txtrtt

T

t

T

where

.))((

)()()(1

Txg

TxTrTC

t

T

t

T

t

T

t

T

dssC

dssk

srsspsqCsC

dssk

srss

sr

skdsspsqCsC

txg

txtrt

,)(

)(4

)()()()()(

)(2

)()()(

)(

)()()()(

))((

)()()(

1

2

01

2

2

01

where .)(2

)()()()(

tk

trttt

By the condition (7), we get

,))((

)()()(lim

txg

txtrt

t

73

and there exists TT 1such that 0)(

tx for .1Tt The condition (7) also implies that

there exists 12 TT such that

.0)()(0)()()( 200

2

2

1

TtfordsspsqCanddsspsqCs

t

T

T

T

Multiplying equation (1.1) by )(t and by conditions (1) and (5), we have

.,)())(()()())(()()()()( 20 TttptxgttqtxgtCtxtrt

Integrate the last inequality from T2 to t, we obtain

.,)()()()())((

)()()())(()()()()()()()()()(

22

22

2

0222

s

T

t

T

t

T

t

T

TtdsduupuCqusxsxg


By the Bonnet’s theorem, for 2Tt there exists tTt ,2 such that

.,)()()(

)()()()())((

)()()())(()()()()()()()()()()(

2222

222222

22

2

TtTxTrT

dsduupuCqusxsxg

dsspsCqstxgTxxTrTTxTrTtxtrt

s

T

t

T

t

T

t

Dividing the last inequality by )()( trt , integrate from T2 to t and the condition (6), we

obtain

,,)()(

)()()()()(

2

2222

tassrs

dsTxTrTTxtx

t

T

74

which is a contradiction to the fact that 0)( tx for .Tt Hence the proof is

completed.

Example 3.3.4


.0,))(sin()(

)(8)(5

)()()(

15

3

4

312

1533

3

tt

txtx

ttxtx

txtxttx

t

We note here

3

1)(

ttr , ,)( 3ttq ,

85),(,)(

44

53

vu

uuvuxxg

)(1))(sin(

))((

))(,(55

tptt

tx

txg

txtH .00 tandxallfor

Taking andtt )(

.)(4

)()()()()()(

0 0

2

0

dssk

srsspsqCsdss

t t

All conditions of Theorem 3.3.4 are satisfied. Thus, the given equation is oscillatory.

We also compute the numerical solutions of the given differential equation using the

Runge Kutta method of fourth order (RK4) for different step sizes h.

75

We have

)(8)(5

)()())(sin()())(),(,()(

4

12

1533

txtx

txtxtxtxtxtxtftx


xx on the chosen interval [1,50] and finding the

values of the functions r, q and f where we consider )()(),( xltfxtH at t=1, n =2250,

n =2500, n=2750 and n=3000 and the step sizes h =0.0217, h =0.0196, h =0.0178 and

h = 0.016.

76

Ta

ble

3.4

: C

om

par

ison

of

the

nu

mer

ical

solu

tion

s o

f O

DE

.3.4

fo

r d

iffe

rent

step

s si

zes

Erro

r %

0

0

0.0

000

058

0

0.0

000

006

2

0.0

000

066

5

0.0

000

007

1

0.0

000

051

7

0.0

000

173

7

0

0.0

000

251

3

0.0

000

012

4

0.0

000

321

1

0.0

000

005

7

0.0

000

062

6

0

0

0.0

000

000

1

0

0.0

000

000

1

0.0

000

000

1

0.0

000

000

1

0.0

000

000

3

0.0

000

000

3

0

0.0

000

000

4

0.0

000

000

2

0.0

000

000

5

0.0

000

000

1

0.0

000

000

5

Erro

r %

0

0

0.0

000

005

8

0.0

000

126

7

0.0

000

012

4

0.0

000

266

3

0.0

000

028

4

0.0

000

120

6

0.0

000

521

2

0.0

000

011

5

0.0

000

691

0

0.0

000

037

2

0.0

000

835

1

0.0

000

017

3

0.0

000

175

4

0

0

0.0

000

000

1

0.0

000

000

2

0.0

000

000

2

0.0

000

000

4

0.0

000

000

4

0.0

000

000

7

0.0

000

000

9

0.0

000

000

2

0.0

000

001

1

0.0

000

000

6

0.0

000

001

3

0.0

000

000

3

0.0

000

001

4

Erro

r %

0

0.0

000

053

4

0.0

000

005

8

0.0

000

253

4

0.0

000

024

8

0.0

000

532

7

0.0

000

064

4

0.0

000

224

1

0.0

000

984

5

0.0

000

023

1

0.0

001

444

9

0.0

000

074

4

0.0

001

734

4

0.0

000

028

9

0.0

000

363

4

0

0.0

000

000

1

0.0

000

000

1

0.0

000

000

4

0.0

000

000

4

0.0

000

000

8

0.0

000

000

9

0.0

000

001

3

0.0

000

001

7

0.0

000

000

4

0.0

000

002

3

0.0

000

001

2

0.0

000

002

7

0.0

000

000

5

0.0

000

002

9

h=

0.0

163

x4(t

k)

-0.5

0.1

871

606

0

1.7

234

751

2

0.5

178

204

1

1.6

078

274

5

-0.1

50

170

01

1.4

040

222

7

0.5

800

895

5

0.1

726

614

2

1.7

244

922

8

-0.1

59

170

22

1.6

122

572

7

-0.1

55

669

91

1.7

251

232

5

-0.7

97

913

52

h

=0.0

17

8

x3(t

k)

-0.5

0.1

871

606

0

1.7

234

751

1

0.5

178

204

1

1.6

078

274

4

-0.1

50

170

00

1.4

040

222

6

0.5

800

895

8

0.1

726

613

9

1.7

244

922

8

-0.1

59

170

26

1.6

122

572

9

-0.1

55

669

86

1.7

251

232

6

-0.7

97

913

47

h=

0.0

196

x2(t

k)

-0.5

0.1

871

606

0

1.7

234

751

1

0.5

178

204

3

1.6

078

274

3

-0.1

50

169

97

1.4

040

222

3

0.5

800

896

2

0.1

726

613

3

1.7

244

922

6

-0.1

59

170

33

1.6

122

573

3

-0.1

55

669

78

1.7

251

232

8

-0.7

97

913

38

h=

0.0

217

x1(t

k)

-0.5

0.1

871

606

1

1.7

234

751

1

0.5

178

204

5

1.6

078

274

1

-0.1

50

169

93

1.4

040

221

8

0.5

800

896

8

0.1

726

612

5

1.7

244

922

4

-0.1

59

170

45

1.6

122

573

9

-0.1

55

669

64

1.7

251

233

0

-0.7

97

913

23

t k

1

5.9

10.8

12.7

6

17.6

6

20.6

24.5

2

27.0

68

30.4

32.3

6

37.2

6

40.2

42.1

6

47.0

6

50

77

Figure 3.4(a): Solutions curves of ODE 3.4

1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

2

t

Num

eric

al s

olut

ions

x(t)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 3.4(b): Solutions curves of ODE 3.4.

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

2

t

Num

eric

al s

olut

ions

x(t)

x1(t)

x2(t)

x3(t)

x4(t)

78

Remark 3.3.4

Theorem 3.3.4 is the extension of the results of Bihari (1963) and the results of

Kartsatos (1968) who have studied the equation (1.1) as

,1)( tr ),())(( txtxg ))(),(())()()),((( txtxtxtrtxg

and 0))(,( txtH and

results of El-abbasy (2000) who has studied the equation (1.1)

as ))(())()()),((( txgtxtrtxg

and .0))(,( txtH Our result can be applied on their

equations (2.5) and (2.6) respectively, but their oscillation results cannot be applied on

the given equation in Example 3.3.4 because their equations are particular cases of our

equation (1.1).

Theorem 3.3.5: Suppose that the conditions (1) and (5) hold and there exists

continuously differentiable function is defined as in Theorem 3.3.3 such that the

condition (6) holds and

(8) ,)()()(

0

0

t

dsspsqCs

(9) 0)(inflim

0

t

tdss for all large t,

(10) .)()()(

1lim

0

t

t st

dudsusrs

Then, every solution of superlinear equation (1.1) is oscillatory.



79

.,))((

)()()( Tt

txg

txtrt


)6.3.3(.,))((

)()()(,1)()(

))((

)()()(

2

2

Tttxg

txtrkttqtp

txg

txtrt

From condition (1), for ,Tt we have

.,))((

)()()()(

))((

)()(2

2

0 Tttxg

txtrktptqC

txg

txtr

We multiply the last inequality by )(t and integrate form T to t, we have

dsssr

skssdsspsqCs

Txg

TxTrT

txg

txtrtt

T

t

T

)()(

)()()()()()(

))((

)()()(

))((

)()()( 2

0

.)(2

)()()()(

))((

)()()(1

tk

trtttand

Txg

TxTrTCLet

Thus, we obtain

)7.3.3(.)(

)(4

)()()()()(

)(2

)()()(

)(

)()()()(

))((

)()()(

1

2

01

2

2

01

t

T

t

T

t

T

t

T

dssC

dssk

srsspsqCsC

dssk

srss

sr

skdsspsqCsC

txg

txtrt

80

From inequality (3.3.7), we have

))((

)()()(

))((

)()()()(

txg

txtrt

Txg

TxTrTdss

t

T

Now, we consider three cases for )(tx

Case 1: If ,0)( 1 TTtfortx

then we get

.))((

)()()(

))((

)()()()(

1

111

1txg

txtrt

Txg

TxTrTdss

t

T

Thus, for all 1Tt , we obtain

.))((

)()()()(

txg

txtrtdss

t

We divide the last inequality )()( trt and integrate from T1 to t, we obtain

.))((

)()(

)()(

1

11

dssxg

sxdsduu

srs

t

T

t

T s

Since the equation (1.1) is superlinear, we get

,)())((

)()(

)()(

1)(

)( 111

tx

Tx

t

T

t

T sug

duds

sxg

sxdsduu

srs

This contradicts condition (10).

81

Case 2: If )(tx

is oscillatory, then there exists a sequence n in ,T such that

.0)(

nx Choose N large enough so that (9) holds. Then from inequality (3.3.7), we

have

.)())((

)()()(

t

n

ndssC

txg

txtrt

So

,0)(inflim)(suplim))((

)()()(suplim

t

t

t

ttn

n

n

ndssCdssC

txg

txtrt

which contradicts the fact that )(tx

oscillates.

Case 3: If 0)(

tx for TTt 2 , the condition (9) implies that there exists

23 TT such that

.0)()()( 30

3

TtfordsspsqCs

t

T

Multiplying the equation (1.1) by )(t and from the condition (1), for ,3Tt we have

30 ,)())(()()())(()()()()( TttptxgttqtxgtCtxtrt


82

.,)()()(

,)()()()())((

)()()())(()()()(

,)()()()())((

)()()())(()()()()()()()()()(

3333

30

0333

30

0333

33

3

33

33

TtTxTrT

TtdsduupuqCusxsxg

dsspsqCstxgTxTrT

TtdsduupuqCusxsxg


s

T

t

T

t

T

s

T

t

T

t

T

t

T

Dividing the last inequality by )()( trt and integrate from T3 to t we obtain

tassrs

dsTxTrTTxtx

t

T

,)()(

)()()()()(

3

3333

which is a contradiction to the fact that 0)( tx for .Tt Hence, the proof is

completed.

Example 3.3.5


.0,1)(

)(

)()(2

)(1)(

4

11

4

28

35

3

ttx

tx

ttxtx

tx

tttx

Here ,1)( ttr ,1)( 3ttq ,)( 7xxg 445 2),( vuuvu and

.00)(01)(

)(

))((

))(,(4

4

tandxallfortptx

tx

txg

txtH

83

Taking 04)( t such that

,2

)()()()1(2

0

00

0

t

CdsspsqCs

t

,02

)(4

)()()()()(inflim)(inflim)2(

2

0

2

0

00

t

Cds

sk

srsspsqCsdss

t

tt

t

tt

.)(4

)()()()()(

)()(

1lim)(

)()(

1lim)3(

2

0

00

duds

uk

uruupuqCu

srsdudsu

srss

t

tt

s

t

tt


equation is oscillatory. To demonstrate that our result in Theorem 3.3.5 is true we also

find the numerical solution of the given differential equation in Example 3.3.5 using the

Runge Kutta method of fourth order.

We have

)(21

))(),(,()(4

28

35

4

11

txx

x

x

xtxtxtftx

with initial conditions 1)1(,1)1(

xx on the chosen interval 50,1 and finding the

values the functions r, q and f where we consider )()(),( xltfxtH at t=1, n=980

and h=0.05.

84

Table 3.5: Numerical solution of ODE 3.5

Figure 3.5: Solution curve of ODE 3.5

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

erical solu

tion x

(t)

k tk x(tk)

1

81

182

261

321

397

461

521

581

702

801

921

981

1

5

10.5

14

17

20.8

24

27

30

36.3

41

47

50

1

-0.22568469

0.13417542

-1.09294017

1.14607460

0.11187711

-0.76853441

1.16599637

-1.06554696

0.13116006

-0.35706690

0.53245699

-0.97421735

85

Remark 3.3.5

Theorem 3.3.5 extends result of Wong and Yeh (1992), result of Philos (1989),

result of Onose (1975) and result of Philos and Purnaras (1992) who have studied the

special case of the equation (1.1) as ,1)( tr ))(())()()),((( txgtxtrtxg

and

0))(,( txtH and result of E. M. Elabbasy (2000) who has studied the special case of

the equation (1.1) as ))(())()()),((( txgtxtrtxg

and .0))(,( txtH Our result can be

applied on their equations (2.4) and (2.9) ,as mentioned in Chapter Two, but their

oscillation results cannot be applied on the given equation in Example 3.3.5 because

their equations are particular cases of our equation (1.1).

Theorem 3.3.6

Suppose that the conditions (1) and (5) hold. Assume that there exists

continuously differentiable function is defined as in Theorem 3.3.3 such that

condition (6) holds and

(11) ),,0(,)(

)(

t

t

tr

(12)

t

Tt

dss)(inflim for all large t.

(13) ,)()(

11suplim

t

T

s

Tt

dudsust

where )(4

)()()()()()(

2

0tk

trttptqCtt

and .,0,: 0 tp Then, every

solution of equation (1.1) is oscillatory.


equation (1.1) such that ,0)( Tontx for some .00 tT We define the function

)(t as

86

.,))((

)()()( Tt

txg

txtrt

This, the equation (1.1) and condition (1), we obtain

Tttxg

txtxgtrtptqC

txg

txtr

,))((

)())(()()()(

))((

)()(2

2

0

We integrate the last inequality multiplied by )(t form T to t, we have

)8.3.3(,))((

)())(()()(

))((

)()()()()()(

))((

)()()(

2

2

01

dssxg

sxtxgsrs

dssxg

sxsrsdsspsqCsC

txg

txtrt

t

T

t

T

t

T

where

.))((

)()()(1

Txg

TxTrTC

Thus, we have

,)(2

)()()(

)(

)()()()(

)()()()(

)()()()(

))((

)()()(

2

2

01

2

01

dssk

srss

sr

skdsspsqCsC

dsssssr

skdsspsqCsC

txg

txtrt

t

T

t

T

t

T

t

T

where .)(2

)()()()(

tk

trttt

87

Thus, we have

)9.3.3(.)()(

)()(

))((

)()()( 2

1 dsssr

skdssC

txg

txtrtt

T

t

T

Also, from the inequality (3.3.9) divided by ),(t we have

)10.3.3().()(

)()(

1 1 tt

Cdss

t

t

T

Now, we have three cases for ).(tx

Case 1: If )(tx

is oscillatory, then, there exists a sequence n in ,T with

nt

lim and such that .0)(

nx Then, from the inequality (3.3.9), we have

.)()()(

)(1

2

nn

TT

dssCdsssr

sk

Hence, by the condition (12), we get

.)()(

)( 2

dsssr

sk

T

This gives, for a positive constant N

Ndsssr

sk

T

)()(

)( 2

for every .Tt (3.3.11)

88

Further, by using the Schwarz’s inequality, for ,Tt we obtain

.

)(

)(

)(

)()(

)(

)(

)(

)()(

)(

)()( 2

22

dss

sr

k

Nds

sk

srdss

sr

skds

sk

srs

sr

skdss

t

T

t

T

t

T

t

T

t

T

By condition (11), the last inequality becomes

.2

)(2

)( 222

2

tk

NTt

k

Ndss

k

Ndss

t

T

t

T

Then,

.2)(2

)()()()( t

k

Nds

sk

srssdss

t

T

t

T

Thus, for ,Tt we have

tk

Ndss

t

T2

)(

(3.3.12)

Integrate the inequality (3.3.10) and from (3.3.12), we obtain

.2)(2

)()(

)()(

)()(

1

11

1

tk

Nt

T

Ct

k

NTt

T

C

dsss

dsCdudsu

s

t

T

t

T

t

T

s

T

Dividing the last inequality by t and taking the limit superior on both sides, we obtain

,2)(

)()(

11suplim 1 k

N

T

Cdudsu

st

t

T

s

Tt

89

which contradicts condition (13).


then, from (3.3.10), we get

.)(

)()(

1 1

1t

Cdss

t

t

T

Integrate the last inequality, dividing by t and taking the limit superior on both sides, we

get

,)(

)()(

11suplim

1

1

1 1

T

Cdudsu

st

t

T

s

Tt

which also contradicts condition (13).

Case 3: if 0)(

tx for .2 TTt Then from inequality (3.3.8), we have

.))((

)())(()()()()()(

))((

)()()(

22

2

2

01 dssxg

sxsxgsrsdsspsqCsC

txg

txtrtt

T

t

T

Now, we have two cases for dssxg

sxsxgsrst

T

2))((

)())(()()(2

2

: If this integral is finite, in

this case, we can get a contradiction by the procedure of case (1). If this integral is

infinite, from condition (12), we obtain

.))((

)())(()()(

))((

)()()(

2

2

2

1 dssxg

sxsxgsrsC

txg

txtrtt

T

Also, from the last inequality, we obtain

90

)13.3.3(,))((

)())(()()(

))((

)())(()()(

))((

)()()(

2

2

2

2

*

2

2

1

dssxg

sxsxgsrsN

dssxg

sxsxgsrsC

txg

txtrt

t

T

t

T

where .1

* CN

We consider a 23 TT such that

0))((

)())(()()(3

2

2

2

*

1

dssxg

sxsxgsrskNN

T

T

Hence, for all 3Tt , we get

.))((

)())(()()(

))((

)()()(

3

2

2

*

dssxg

sxsxgsrsN

txg

txtrtt

T

From the last inequality, we get

.))((

)())((

))((

)())(()()(

))((

)())(()()(

3

2

2

*

2

2

sxg

sxtxgds

sxg

sxsxgsrsN

txg

txtxgtrtt

T

Integrate the last inequality from 3T to t, we have

.))((

))((ln

))((

)())(()()(ln

))((

)())(()()(ln 3

12

2

*

2

2

*

3

3

3

txg

TxgNds

sxg

sxsxgsrsNds

sxg

sxsxgsrsN

t

T

t

T

t

T

91

Thus,

.,))((

))((

))((

)())(()()(3

3

12

2

*

3

Tttxg

TxgNds

sxg

sxsxgsrsN

t

T

From (3.3.13), we have

.,))((

))((

))((

)()()(3

3

1 Tttxg

TxgN

txg

txtrt

Since .0))(( 31 TxgN Thus, from the last inequality, we obtain

.,)()(

))(()()( 3313

3

Ttsrs

dsTxgNTxtx

t

T

which leads to ,)(lim

txt

which is a contradiction to the fact that 0)( tx for t .T

Hence, the proof is completed.

Example 3.3.6


.0,sin)(

)()(2

)()()2()(

8

3

6

43

18

21334

3

tt

ttx

txttx

txtxttxt

We have ,)( 43

ttr ,2)( 3 ttq ,)( 3xxg 66

7

2),(

vu

uuvu

and

92

)(1sin

))((

))(,(88

tptt

t

txg

txtH for all 0x and .0t Taking 5)( t such that

,1,)(

)(

)()(4

3

0

tts

trand

srs

ds

t

,1

)2(inflim

)(4

)()()()()(inflim)(inflim

8

3

0

2

0

dss

sC

dssk

srsspsqCsdss

t

Tt

t

Tt

t

tT

.1

)2(1

suplim

)(4

)()()()(

)(

11suplim)(

)(

11suplim

8

3

0

2

0

t

T

s

Tt

s

T

t

Tt

s

T

t

Tt

dsduu

uCt

dudsuk

uruupuqC

stdudsu

st





We have

)(2

3))(),(,()(6

18

2133

txx

xxxtxtxtftx



values the functions r, q and f where we consider )()(),( xltfxtH at t=1, n=980 and

h=0.05.

93



0 5 10 15 20 25 30 35 40 45 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

Num

erical solu

tion x

(t)

k tk x(tk)

1

81

181

222

313

405

461

251

581

682

821

916

981

1

5

10

12.05

16.6

21.25

24

27

30

35.05

42

48.8

50

0.5

-0.11388331

0.29975744

-0.04307199

0.00691001

-0.04618202

0.45734920

-0.95229708

0.41341535

-0.05082571

0.10467651

-0.00391035

-0.95114725

94

Remark 3.3.6: Theorem 3.3.6 extends result of Popa (1981) for the equation (2.2),

result of Wong (1973) for the equation (2.3), results of Onose (1975), Philos (1985) and

Yeh (1982) for the equation (2.4) and result of E. Elabbasy (2000) for the equation

(2.6). Our result can be applied on their equations (2.2), (2.3), (2.4) and (2.6)

respectively, but their previous oscillation results cannot be applied on the given

equation in Example 3.3.6 because their equations are particular cases of our equation

(1.1).

Theorem 3.3.7: Suppose, in addition to the condition (1) holds that

(14) .0,)(

11

kksr

ds

T

(15) There exists a constant *B such that

m

mandBmBs

dsmG

0

** )0,(,),1(

)( R.

Furthermore, suppose that there exists a positive continuous differentiable function

on the interval ,0t with )(t is a non-decreasing function on the interval ,0t such

that

(16) ,)()()()()(

1suplim 0

t

T

s

Tt

dsduupuqCussr

where, ,0,: 0tp . Then, every solution of superlinear equation (1.1) is

oscillatory.

95

Proof

Without loss of generality, we may assume that there exists a solution x(t) of

equation (1.1) such that .0,0)( 0 tTsomeforTontx Define

.,

)(

)()()()( Tt

txg

txtrtt

Thus and (1.1) imply

.,)()(

)())()(,1()()()()()( Ttt

t

ttttqttptt

Thus, we have

.),)()(,1()()()()()(

)()( Tttttqttpt

t

tt

Dividing the last inequality by 0)()(,1 tt and by condition (1), we obtain

Tttt

tttCtptqCt

,)()(,1

)()()()()()( 00

Integrate from T to t, we have

Ttds

ss

sssCdsspsqCs

t

T

t

T

,)()(,1

)()()()()()( 00

(3.3.14)

By Bonnet’s theorem, since )(t is a non-decreasing function on the

interval ,,0 t there exists tTT ,1 such that

96

.)()(,1

)()()(

)()(,1

)()()(

1

dsss

sstds

ss

ssst

T

t

T

(3.3.15)

Using the inequality (3.3.15) in the inequality (3.3.14), we have

.)(

)()(

)(

)()(

,1,1)(

,1)()()()(

0

1

10

)()(

0

)()(

0

0

)()(

)()(

00

11

11

t

tGtC

T

TGtC

u

du

u

dutC

u

dutCdsspsqCs

ttTT

tt

TT

t

T

Thus, by condition (15), we obtain

).()(

)()()()()( *

0

1

100 tBC

T

TGtCdsspsqCs

t

T

Integrating the last inequality divided by )()( trt from T to t, we obtain

dssxg

sxBC

sr

ds

T

TGCdsduupuqCu

ssr

t

T

t

T

t

T

s

T

))((

)(

)()(

)()()()(

)()(

1 *

0

1

1

00

Taking the limit superior on both sides, by condition (14) and since the equation (1.1) is

superlinear, we have

,)()(

)(suplim)()()(

)()(

1suplim

)(

)(

*

0

1

1

010

tx

Txt

t

T

s

Tt ug

duBC

T

TGCkdsduupuqCu

ssr

tas , which contradicts to the condition (16). Hence the proof is completed.

97

Example 3.3.7


.0,cos)(

)()(

)()(

7

3

2

26

962

tt

xtx

ttxtx

txttxt

We note that ,)( 2ttr ,)( 6ttq 3)( xxg and 22

3

),(vu

uvu

such that

.,1,,)1(1)1(),1(

)(00

2

0

RmallforandRBBthusmmdsdsss

dsmG

mmm

.00),(1cos

)(

)(,77

xandtallfortptt

x

txg

txtH

Let 4)( tt such that

.11

suplim)()()()()(

1suplim

7

6

0

4

60

t

T

s

Tt

t

T

s

Tt

dsduu

uCus

dsduupuqCussr


equation is oscillatory. The numerical solution of the given differential equation using

Runge Kutta method of fourth order (RK4) is as follows:

We have

)()(

)())(cos()())(),(,()(

2

6

93

txtx

txtxtxtxptxtftx



values of the functions r, q and f where we consider )()())(,( xltftxtH at t=1

n=980 and h=0.05.

98



0 5 10 15 20 25 30 35 40 45 50-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t

Num

erical solu

tion x

(t)

k tk x(tk)

1

81

181

221

321

381

521

586

661

721

821

917

981

1

5

10

12

17

20

27

30.3

34

37

42

46.6

50

0.5

-1.23206499

0.40954827

-1.50525503

0.02833177

-0.33103118

0.04981132

-0.04300197

0.43117751

-0.12794647

1.46730560

-0.18820133

0.09644083

99

Remark 3.3.7: Theorem 3.3.7 extends results of Bihari (1963) and Kartsatos (1968)

who have studied the equation (2.5) as mentioned in chapter two. Our result can be

applied on their equation, but their oscillation results cannot be applied on the given

equation in Example 3.3.7 because their equation is a particular case of our equation

(1.1) when ,1)( tr ),())(( txtxg ))(),(())()()),((( txtxtxtrtxg

and

0))(,( txtH .


(17) .0))()((0)( 00 ttallfortqtrandttallfortr

(18) .0),1( vallforvv

(19) ,)()()(1

suplim

0

2

dsduupsqsrAt

t

T

s

tt

where ,0,: 0tp , then, every solution of superlinear equation (1.1) is

oscillatory.


equation (1.1) such that 0,0)( 0 tTsomeforTontx . Define

.,

)(

)()()( Tt

txg

txtrt

From ),(t equation (1.1) and condition (18), we get

.)()()(

)(

)()(ttqtp

txg

txtr

100


)16.3.3(,)(

)()()()(

)(

)()(1

t

T

t

T

dssxg

sxsqsrdsspA

txg

txtr

where

.)(

)()(1

Txg

TxTrA

By condition (17) and the Bonnet’s Theorem, we see that for each ,Tt there exists

tTT ,2 such that

.

)()()(

)(

)()()(

)(

)()()(

)(

)( 22

tx

Tx

t

T

t

Tug

dutqtrds

sxg

sxtqtrds

sxg

sxsqsr

Since 0)()( tqtr and the equation (1.1) is superlinear, we have

)(

)(2

)(

2

2

2

).()(,)(

)()(,0

)(

tx

TxTx

Txtxifug

du

Txtxif

ug

du

Thus, it follows that

),()(

)(

)()()( 2 tqtrAds

sxg

sxsqsr

t

T

where .)(

inf

)(

)(

2

2

tx

Txug

duA

Thus, the inequality (3.3.16) becomes

).()()(

)(

)()(21 tqtrAdsspA

txg

txtrt

T

101

Integrate the last inequality from T to t, we have

.)()()()(

)(

)()(21

t

T

s

T

t

T

dsduupsqsrATtAdssxg

sxsr

Since )(tr is positive and non-increasing for ,Tt the equation (1.1) is superlinear and

by Bonnet’s theorem, there exists Ttt , such that

)()(

)(

)()(3

)(

)(

TrAug

duTrds

sxg

sxsr tx

Tx

t

T

, where

.inf

)(

)(

3 tx

Txug

duA


.)()()()()( 312 TrATtAdsduupsqsrA

t

T

s

T


,)(

suplim)1(suplim)()()(1

suplim 3

12

t

TrA

t

TAdsduupsqsrA

t t

t

Tt

s

Tt


Example 3.3.8


.0,)1)((

))(cos()(

)()(

)()()(

24

7

2

10

1554

ttxt

txtx

ttxtx

txtxtttx

102

Here

,)(,)(,1)( 54 xxgttqttr 22

3

),(vu

uuvu

and

)(1

)1)((

))(cos()(

)(

)(,424

2

tpttxt

txtx

txg

txtH

.00 xandtallfor

.61

suplim)()()(1

suplim4

3

22

t

T

s

Tt

t

T

s

Tt

dsu

dusA

tdsduupsqsrA

t


equation is oscillatory. The numerical solution of the given equation using the Runge

Kutta method of fourth order (RK4) is as follows:

We have

)()()()(1)(

))(cos()())(),(,()(

2

10155

2

7

txtxtxtxtx

txtxtxtxtftx



values of the functions r, q and f where we consider )()())(,( xltftxtH at t=1,

n=980 and h=0.05.

103



0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ion

x(t)

k tk x(tk)

1

80

181

226

321

421

475

521

581

661

761

821

955

981

1

4.95

10

12.25

17

22

24.7

27

30

34

39

42

48.7

50

1

-0.02069572

0.20451305

-0.00138478

0.11785698

-0.04991059

0.36339219

-0.21766245

0.86993097

-0.10477736

0.06293869

-0.71934519

0.05390472

1.15722428

104

Remark 3.3.8:

Theorem 3.3.8 extends results of Bihari (1963), Kartsatos (1968) who have studied

the equation (1.1) as ,1)( tr ),())(( txtxg ))(),(())()()),((( txtxtxtrtxg

and .0))(,( txtH Also, Theorem 3.3.8 extends results of Elabbasy (2000) who has

considered the equation (1.1) as ))(())()()),((( txgtxtrtxg

and .0))(,( txtH Our

result can be applied on their equations (2.5) and (2.6), but their oscillation results

cannot be applied on the given equation in Example 3.3.8 because their equations are

particular cases of our equation (1.1).

Theorem 3.3.9: Suppose that the conditions (1) and (5) hold. Moreover, assume that

there exist a differentiable function ,0,: 0t and the continuous functions

0:),(:, tststDHh R, H has a continuous and non-positive partial derivative

on D with respect to the second variable such that

.0),(,0),( 0tstforstHttforttH o

.),(),(),(),( DststHsthstHs

(20) If ,),()()(),(

1suplim

0

2

0

t

tt

dsstssrttH

where .),()(

)(),(),(

stHs

ssthst

(21) .)()()(),(),(

1suplim

0

0

0

t

tt

dsspsqCsstHttH

105


Proof

Without loss of generality, we assume that there exists a solution x(t) of equation

(1.1) such that 0,0)( 0 tTsomeforTontx . We define the function )(t as

.,

)(

)()()()( Tt

txg

txtrtt

This and the equation (1.1) imply

,,)()()(

)()(

)())(,1()()()()()( 2 Ttt

trt

kt

t

ttvtqttptt

where .)()()( tttv

Then, by condition (1), we have for all Tt

.,)()()(

)()(

)()()()()()( 2

0 Ttttrt

kt

t

ttqtCtptt

Integrate the last inequality multiplied by ),( stH from T to t, we have

,)(),(),()()()(

),()(),(

)()()(

),(

)(),()(

)()(),()(),()()()(),(

2

2

0

t

T

t

T

t

T

t

T

t

T

dssstHstssrs

stkHTTtH

dsssrs

stkH

dssstHs

sdssstH

sTTtHdsspsqCsstH

106

where .),()(

)(),(),( stH

s

ssthst

Hence, we have

)17.3.3(.),(4

)()(

),()()(

2

1)(

)()(

),()(),()()()(),(

2

2

0

dsstk

srs

dsstk

srss

srs

stkHTTtHdsspsqCsstH

t

T

t

T

t

T

Then, for ,Tt we have

.,),()()(4

1)(),()()()(),( 2

0 Ttdsstssrk

TTtHdsspsqCsstH

t

T

t

T

Dividing the last inequality by ),( TtH , taking the limit superior as t and by

condition (20), we obtain

,),()()(),(

1suplim

4

1

)()()()(),(),(

1suplim

2

0

dsstssrTtHk

TdsspsqCsstHTtH

t

Tt

t

Tt


Example 3.3.9: Consider the differential equation

.0,)(cos)(

)()(4

)(3)()(

7

3

2

26

9352

tt

txtx

ttxtx

txtxtttx

107

We have

,

1)(

2ttr ,)( 5ttq ,)( 3xxg

22

3

4

3),(

vu

uuvu

and

.)(1))(cos(

))((

))(,(77

tptt

tx

txg

txtH

Let ,00)(),( 0

2 tstststH then,

.2),()(2),(

sththusandststH

s

Taking2)( tt such that

,4

)(2

2)(

1suplim

),()(

)(),()()(

),(

1suplim),()()(

),(

1suplim)1(

2

2

2

2

00

Tdsst

sTt

dsstHs

ssthssr

TtHdsstssr

ttH

t

Tt

t

t

t

Ttt

.

1)(

)(

1suplim)()()(),(

),(

1suplim)2(

7

5

0

2

20

t

Tt

t

Tt

dss

sCstTt

dsspsqCsstHTtH

All conditions of Theorem 3.3.9 are satisfied, then, the given equation is oscillatory.

Also the numerical solution of the given differential equation is computed using the


We have

)()(4

)(3)())(cos()())(),(,()(

2

6

9

33

txtx

txtxtxtxtxtxtftx

108

with initial conditions 5.0)1(,1)1(


values of the functions r, q and f where we consider )()(),( xltfxtH at t=1, n=980

and h= 0.05.


k tk x(tk)

1

81

226

321

381

461

521

581

661

721

781

841

921

981

1

5

12.25

17

20

24

27

30

34

37

40

43

47

50

-1

0.50348046

-0.00392170

1.05359426

-0.89282510

1.06448349

-1.07378290

0.93949122

-1.04029283

1.08568566

-0.98145745

0.77568867

-1.08892372

1.01773294

109


0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ion

x(t)

Remark 3.3.9: Theorem 3.3.9 extends Kamenev’s result (1978) and Philos’s result

(1989) who have studied a special case of the equation (1.1) as

,1)( tr ),())(( txtxg ))(())()()),((( txgtxtrtxg

and 0))(,( txtH . Our result

can be applied on their equation however; their results cannot be applied to the given

equation in Example 3.3.9.

Theorem 3.3.10

Suppose, in addition to the conditions (1), (5) and (20) hold that there exist

continuous functions h and H are defined as in Theorem 3.3.9 and suppose that

(22) .),(

),(infliminf0

00

ttH

stH

tts

If there exists a continuous function ,0ton such that

110

(23)

t

Tt

Tdsstssrk

spsqCsstHTtH

)(),()()(4

1)()()(),(

),(

1suplim 2

0

for 0tT , where ,),()(

)(),(),( stH

s

ssthst

k is a positive constant and a

differentiable function ),0(,: 0 t and

(24) ,)()(

)(2

T

dssrs

s

where 0),(max)( tt , then every solution of equation (1.1) is oscillatory.

Proof

Without loss of generality, we may assume that there exists a solution x(t) of equation

(1.1) such that ,0)( Tontx for some .00 tT Dividing inequality (3.3.17) by

),( TtH and taking the limit superior as ,t we obtain

.),)()(

2

1)(

)()(

),(

),(

1inflim)(

),()()(

2

1)(

)()(

),(

),(

1suplim

)(),()()(4

1)()()(),(

),(

1suplim

2

2

2

0

t

Tt

t

Tt

t

Tt

dsstk

srss

srs

stkH

TtHT

dsstk

srss

srs

stkH

TtH

Tdsstsrsk

spsqCsstHTtH

By condition (23), we get

.),()()(

2

1)(

)()(

),(

),(

1inflim)()(

2

t

Tt

dsstk

srss

srs

stkH

TtHTT

111

This shows that

,)()( TteveryforTT (3.3.18)

and

.),()()(

2

1)(

)()(

),(

),(

1inflim

2

t

Tt

dsstk

srss

srs

stkH

TtH

Hence,

)19.3.3(.)(),(),(),(

1)(

)()(

),(

),(

1inflim

),()()(

2

1)(

)()(

),(

),(

1inflim

0 0

0

0

2

0

2

0

t

t

t

tt

t

tt

dssstHstttH

dsssrs

stkH

ttH

dsstk

srss

srs

stkH

ttH

Define

0

2

0

,)()()(

),(

),(

1)(

0

ttdsssrs

stkH

ttHtU

t

t

and

.,)(),(),(),(

1)( 0

00

ttdssstHstttH

tV

t

t

Then, (3.3.19) becomes

.)()(inflim

tVtUt

(3.3.20)

112

Now, suppose that

.)()(

)(

0

2

t

dssrs

s

(3.3.21)

Then, by condition (22) we can easily see that

.)(lim

tUt

(3.3.22)

Let us consider a sequence ,...3,2,1nnT in ,0t with

n

nTlim and such that

.)()(inflim)()(lim tVtUTVTUt

nnn

By inequality (3.3.20) there exists a constant N such that

,...3,2,1,)()( nNTVTU nn (3.3.23)


.)(lim

nn

TU (3.3.24)

And hence inequality (3.3.23) gives

.)(lim

nn

TV (3.3.25)

By taking into account inequality (3.3.24), from inequality (3.3.23), we obtain

.2

1

)()(

)(1

nn

n

TU

N

TU

TV

113

Provided that n is sufficiently large. Thus

2

1

)(

)(

n

n

TU

TV,

which by inequality (3.3.25) and inequality (3.3.23) we have

.)(

)(lim

2

n

n

n TU

TV (3.3.26)

On the other hand by Schwarz’s inequality, we have

).(),()()(

),(

1

)()()(

),(

),(

1),(

)()(

),(

1

)(),(),(),(

1)(

2

0

2

0

2

0

2

0

2

2

0

00

0

nn

T

tn

T

t

n

n

n

T

tn

n

T

t

n

n

n

TUdssTk

srs

tTH

dssssr

sTkH

tTHdssT

k

srs

tTH

dsssTHsTtTH

TV

n

nn

n

Thus, we have

dssTk

srs

tTHTU

TVn

T

tnn

nn

),()()(

),(

1

)(

)( 2

0

2

0

for large n.

By inequality (3.3.26), we have

.),()()(),(

1lim

1 2

00

dssTssr

tTHkn

T

tnn

n

Consequently,

t

tt

dsstssrttH

0

,),()()(),(

1suplim 2

0

114

which contradicts to the condition (20). Thus, inequality (3.3.21) fails and hence

.)()(

)(

0

2

t

dsssr

s

Hence from inequality (3.3.18), we have

,)()(

)(

)()(

)(

00

22

tt

dsssr

sds

ssr

s

which, contradicts to the condition (24). Hence the proof is completed.

Example 3.3.10


.0,)1)((

)(sin)(

)(6)(9

)()((

1)(2

29

18

6126

1337

36

ttx

txtx

ttx

tx

txtx

tt

tx

We note that 6

1)(

ttr , and

vu

uuvuxxg

ttq

1818

197

3 69),(,)(,

1)(

.)(0)1)((

)(

)1)((

)(sin)(

))((

))(,(02

2

2

22

ttallfortptx

tx

tx

txtx

txg

txtH

We let ,0)(),( 0

2 tstallforststH thus

),(),()(2),( stHsthststHs

for all 00 tt . Taking 6)( t such that

115

,01

)(

24suplim

),()(

)(),()()(

),(

1suplim),()()(

),(

1suplim

62

2

2

t

Tt

t

Tt

t

Tt

dssTt

dsstHs

ssthssr

TtHdsstssr

TtH

,1)1(inf)(

)(infliminf

),(

),(infliminf

0002

0

2

0

tsttstts tt

st

ttH

stH

,),(

),(infliminf0

00

ttH

stHthus

tts

.4

336)(6

),(

1suplim

),(4

)()()()()(),(

),(

1suplim

2

0

2

0

63

2

0

2

0

T

C

T

Cds

kss

stC

TtH

dsstk

ssrspsqCsstH

TtH

t

Tt

t

Tt

Set 2

0

4

3)(

T

CT , then

2

0

4

3)(

T

CT and

TT

dssC

dsssr

s.

32

3

)()(

)( 2

2

02

All conditions of Theorem 3.3.10 are satisfied. Thus, the given equation is

oscillatory. We also compute the numerical solution of the given differential equation

using the Runge Kutta method of fourth order (RK4). We have

18

126

1337

2

29

)(6)(9

)()(

1)(

)(sin)())(),(,()(

txtx

txtx

tx

xtxtxtxtftx



values of the functions r, q and f where we consider )()(),( xltfxtH at t=1, n=980

and h=0.05.

116


Figure 3.10: Solution curve of ODE 3.10.

0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

erical solu

tion x

(t)

k tk x(tk)

1

81

181

221

321

381

461

561

581

661

721

798

921

981

1

5

10

12

17

20

24

27

30

34

37

40.9

47

50

-1

0.76515384

-0.14404191

1.02261405

-0.56631036

0.93295032

-0.96790375

0.94270152

-0.57991575

0.55142239

-0.15874611

0.06496234

-0.93061112

0.97197772

117

Remark 3.3.10

Theorem 3.3.10 extends and improves the results of Philos (1989) and results of

Yan (1986) who have studied the equation (1.1)

,1)( tr ),())(( txtxg ))(())()()),((( txgtxtrtxg

and 0))(,( txtH , as mentioned

in Chapter Two. Our result can be applied on their equation (2.1), but their oscillation

results cannot be applied on the given equation in Example 3.3.10.

Theorem 3.3.11

Suppose in addition to the condition (1) and (2) hold that assume that there exists

be a positive continuous differentiable function on the interval ,0t with )(t is

increasing on the interval ,0t and such that

(25) ,)()()()(

1suplim 0

t

Tt

dsspsqCst

where ,0,: 0tp , then, every solution of equation (1.1) is oscillatory.


equation (1.1) such that .0,0)( 0 tTsomeforTontx Define

.,

)(

)()()()( Tt

txg

txtrtt

Thus and equation (1.1) imply

118

.),()()(

)()(

)())()(,1()()()()()( 2 Ttt

trt

kt

t

ttttqttptt

Thus, we have

.,)()()(

))()(,1()()()()()(

)()( 2 Ttt

trt

ktttqttpt

t

tt

(3.3.27)

Dividing the last inequality by ,0)()(,1 tt we have

Tttqttt

tpt

tt

ttt

),()()()(,1

)()(

)()(,1

)()()(

By condition (1), for ,Tt we obtain

.,

)()(,1

)()()()()()( 0

0 Tttt

tttCtptqCt


.,

)()(,1

)()()()()()( 00 Ttds

ss

sssCdsspsqCs

t

T

t

T

(3.3.28)

By Bonnet’s theorem, we see that for each ,Tt there exists tTT ,1 such that

.)()(,1

)()()(

)()(,1

)()()(

1

dsss

sstds

ss

ssst

T

t

T

(3.3.29)

119

Using the inequality (3.3.29) in the inequality (3.3.28), we have

.,1

)()()(,1

)()()()()()(

)()(

)()(

000

111

tt

TT

t

T

t

Tu

dutCds

ss

sstCdsspsqCs

By condition (2), dividing the last inequality by )(t and taking the limit superior on both

sides, we obtain

,))(

)(suplim

,1,1suplim

,1suplim)()()(

)(

1suplim

*

1

10

)()(

0

)()(

0

0

)()(

)()(

00

11

11

BT

TGC

u

du

u

duC

u

duCdsspsqCs

t

t

ttTT

t

tt

TTt

t

Tt

,tas which contradicts to the condition (25). Hence the proof is completed.

Example 3.3.11: Consider the differential equation

.0,)(sin)(

)(cos3

)(4

33

2

3

tt

txtxtx

t

tttxt

Here ,)( ttr 2

3 cos3)(

t

tttq

,

3)( xxg , uvu ),( and

)(1)(sin

)(

)(,44

tptt

tx

txg

txtH .00 xandtallfor Taking thatsuchtt 2)(

.1cos31

suplim)()()()(

1suplim

42

0

3

02

20

t

Tt

t

Tt

dsss

sCsCs

tdsspsqCs

t

120

All conditions of Theorem 3.3.11 are satisfied and hence, every solution of the given

equation is oscillatory. To demonstrate that our result in Theorem 3.3.11 is true we also

find the numerical solution of the given differential equation in Example 3.3.11 using

the Runge Kutta method of fourth order (RK4).

We have

xxxtxtxtftx 99.3)sin())(),(,()( 3



values of the functions r, q and f where we consider )()())(,( xltftxtH at t=1,

n=980 and h=0.05.


k tk x(tk)

1

81

181

221

321

400

461

529

581

661

719

785

911

981

1

5

10

12

17

20.95

24

27.4

30

34

36.9

40.2

46.5

50

1

-0.18505453

0.72315333

-0.94485686

1.00637880

-0.15920533

0.23724881

-0.14883373

0.86387004

-0.50741992

0.06342616

-0.12804230

0.07913369

-0.48477630

121


0 5 10 15 20 25 30 35 40 45 50-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ion

x(t)

Remark 3.3.11

Theorem 3.3.11 is the extension of the results of Bihari (1963), Kartsatos (1968)

who have studied the equation (1.1) as ,1)( tr ),())(( txtxg

))(),(())()()),((( txtxtxtrtxg

and 0))(,( txtH and results of Wintiner (1949)

and Kamenev (1978) who have studied the equation (1.1) as ,1)( tr ),())(( txtxg

))(())()()),((( txgtxtrtxg

and ,0))(,( txtH as mentioned in Chapter Two. Our

result can be applied on their equations, but their oscillation results cannot be applied

on the given equation in Example 3.3.11 because their equations are particular cases of

our equation (1.1).


(26) ,0,),1(

111

1

CvC

122

(27) .0),1(

1

vallforv

v

Assume that be a positive continuous differentiable function on the interval ,0t

with )(t is a decreasing function on the interval ,0t and such that

(28) ,)(4

1)()(suplim 2

*

t

Tt

dsspk

sqs

where ,0,: 0tp and *k is a positive constant, then, every solution of equation

(1.1) is oscillatory.

Proof


equation (1.1) such that .0,0)( 0 tTsomeforTontx By conditions (26) and

(27) and from inequality (3.3.27) divided by ,0)()(,1 tt we have

,)()()(

)()()()()()(,1

)()()( 2*

tttr

ktqtttp

tt

ttt

where .1

* Ckk


.)()()()()(

)()()()(,1

)()()( 2*

dsssptssr

kdssqsds

ss

ssst

T

t

T

t

T

123

Thus, we have

.)()()(4

1

)()()(

2

1)(

)()()()(

)()(,1

)()()(

2

*

2

*

*

t

T

t

T

t

T

t

T

dsspssrk

dsspk

ssrt

ssr

kdssqsds

ss

sss

Then, we get

)30.3.3(.)()(,1

)()()()()(

4

1)()( 2

*ds

ss

sssdsspsr

ksqs

t

T

t

T

By Bonnet’s theorem, we see that for each ,Tt there exists tTat , such that

.)()(,1

)()()(

)()(,1

)()()(ds

ss

ssTds

ss

sss ta

T

t

T

(3.3.31)

From inequality (3.3.31) in inequality (3.3.30), the condition (2) and taking the limit

superior on both sides, we obtain

,)(

)(suplim)(

)(

)(

)(

)(suplim)(

,1,1suplim)(

,1suplim)()()(

4

1)()(suplim

*

)()(

0

)()(

0

)()(

)()(

2

*

BT

TGT

a

aG

T

TGT

u

du

u

duT

u

duTdsspsr

ksqs

t

t

t

t

aaTT

t

aa

TTt

t

Tt

tt

tt


124

Example 3.3.12


.0,))(sin()(

)1()(2)(

)()(

1

cos4

1

)(22

9

2

518

279

5

55

5

tt

txtx

ttxtx

txtx

t

ttt

t

tx

Here ,1

2)(

5

ttr

1

cos4)(

5

55

t

ttttq ,

9)( xxg ,22

3

),(vu

uuvu

and

.00)(1))(sin(

)(

)(,22

xandtallfortptt

tx

txg

txtH Let 01

)(5

5

t

tt

such that

.

1

1

2

4

1

1

cos41suplim)()(

4

1)()(suplim

25*5

55

5

52

*

t

Tt

t

Tt

dsssks

sss

s

sdsspsr

ksqs

We get all conditions of Theorem 3.3.12 are satisfied and hence, every solution of the

given equation is oscillatory. The numerical solution of the given differential equation is

found out using the Runge Kutta method of fourth order (RK4). We have

)()(

)()(49.42))(sin()())(),(,()(

218

2799

txtx

txtxtxtxtxtxtftx



values of the functions r, q and f where we consider )()()(, xltftxtH at t=1,

n=980 and h=0.05.

125



0 5 10 15 20 25 30 35 40 45 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

Num

erical solu

tion x

(t)

k tk x(tk)

1

83

191

226

333

381

479

521

581

661

721

821

921

981

1

5.1

10.5

12.25

17.6

20

24.9

27

30

34

37

42

47

50

-0.5

0.02786237

-0.06638742

0.03073821

-0.01683783

0.62716312

-0.16418655

0.47451786

-0.10418323

0.31823882

-0.26359395

0.63188040

-0.75244510

-0.19568525

126

Remark 3.3.12

Theorem 3.3.12 extends and improves the results of Bihari (1963) and the results of

Kartsatos (1966) who have studied the equation (2.5) as mentioned in chapter two. Our

result can be applied on their equation, but their oscillation results cannot be applied on

the given equation in Example 3.3.12 because their equation is a particular case of our

equation (1.1) when ,1)( tr ),())(( txtxg ))(),(())()()),((( txtxtxtrtxg

and

0))(,( txtH .

127

3.4 Conclusion

In this section, oscillation of second order nonlinear differential equation with

alternating coefficients of type (1.1) has been investigated. Some oscillation results have

been presented. These results contain the sufficient conditions for oscillation of

solutions of the equation of type (1.1) which have been derived by using the generalized

Riccati technique. Our results extend and improve many previous results that have been

obtained before, for example, such as the works of Fite (1918), Wintner (1949),

Atkinson (1955), Bihari (1963), Kartsatos (1968), Philos (1989), Philos and Purnaras

(1992), El-abbasy (1996), and El-abbasy et al. (2005). All these previous results have

been studied for particular cases of the equation (1.1) whereas our sufficient conditions

have been derived for the generalized equation (1.1). A number of theorems and

illustrative examples for oscillation differential equation of type (1.1) are given. Further,

a number of numerical examples are given to illustrate the theorems which are

computed by using Runge Kutta of fourth order function in Matlab version 2009. The

present results are compared with existing results to explain the motivation of proposed

research study.

128

CHAPTER FOUR

OSCILLATION THEOREMS FOR SECOND ORDER

NONLINEAR DIFFERENTIAL EQUATIONS

WITH DAMPING

4.1 Introduction

In this chapter, we consider the second order nonlinear ordinary differential

equation of the form

)2.1()),(),(,())())(()()),((()()()()())(()( txtxtHtxtxtrtxgtqtxthtxtxtr

where hr , and q are continuous functions on the interval ),(,,0

RRCt and

)(tr is a positive function. g is a continuous function for ,x , continuously

differentiable and satisfies 0)( xxg and 0)( kxg for all 0x . The function is

a continuous function on RxR with 0),( vuu for all u 0 and

),(),( vuvu for any ),0( and H is a continuous function on ,0t ×R×R

with )())(())(),(,( tptxgtxtxtH

for all x 0 and 0tt .

129

4.2 Second Order Nonlinear ODE With Damping Term of Type (1.2)

We consider a problem of finding the sufficient conditions for oscillation of

solutions of ordinary differential equations of second order. The obtained oscillation

results are motivated extended and improved many previous oscillation results, for

examples, Bihari (1963), Kartsatos (1968), Greaf, et al. (1978), Grace (1992), Elabbasy

et al. (2005), Lu & Meng (2007), Berkani (2008) and Remili (2010). Some new

sufficient conditions are established which guarantee that our differential equations are

oscillatory. A number of theorems and illustrative examples for oscillation differential

equation (1.2) are given. Also, a number of numerical examples are given to illustrate

the theorems. These numerical examples are computed by using Runge Kutta of fourth

order in Matlab. The obtained results are compared with existing results to explain the

motivation of proposed research study.


We state and prove here our oscillation theorems.

Theorem 4.3.1: Suppose that

(1) 0,,)( 2121 aaaxa and for ,Rx

(2)

m

BBs

dsmG

0

** 0,),1(

)( for every m R.

Assume that there exists a positive continuous differentiable function on the

interval ,0t , )(t is an increasing function on the interval ,0t and such that

130

(3) ,)(4

)()()()(

)(

1suplim

*

2

0

t

Tt

dssra

shspsqCs

t

where ,0,: 0tp , then, every solution of equation (1.2) is oscillatory.


equation (1.2) such that ,0)( Tontx for every 00 tT . Define

.,

)(

)())(()()()( Tt

txg

txtxtrtt

By equation (1.2) and condition (1), we obtain

)1.3.4(.

)(

)())(()()(

)()(

)())()(,1()()(

))((

)()()()()()(

2

2

txg

txtxtrtk

tt

ttttqt

txg

txthttptt

Thus, we have

.,

)(

)())(()()(

))((

)()()())()(,1()()()()(

)(

)()(

2

2

Tttxg

txtxtrtk

txg

txthttttqttpt

t

tt

Dividing the last inequality by ,0)()(,1 tt we have

)2.3.4(.

)()()(,1

)()()(

))(()()(,1

)()()()()(

)()(,1

)()(

)()(,1

)()()(

2

2

1

txgtt

txtrtka

txgtt

txthttqt

tt

tpt

tt

ttt

131

By condition (1) and since 0))()(,1( tt then, there exists a positive constant C0

such that 0))()(,1( Ctt thus, 0

1

)()(,1

10

Ctt

.

Then, for ,Tt we

obtain

.

)()()(,1

)()()(

))(()()(,1

)()()(

)()(,1

)()()()()()(

2

2

10000

txgtt

txtrtkaC

txgtt

txthtC

tt

tttCtptqCt


)3.3.4(.

)(

)(

)()(,1

)()(

))((

)(

)()(,1

)()(

)()(,1

)()()()()()(

2

2

10

00

dssxg

sx

ss

srska

sxg

sx

ss

shsC

dsss

sssCdsspsqCs

t

T

t

T

t

T

Since )(t in the first integral in R. H. S. of the inequality (4.3.3) is an increasing

function and by applying the Bonnet’s theorem, we see that for each ,Tt there exists

tTT ,1 such that

.)()(,1

)()()(

)()(,1

)()()(

1

dsss

sstds

ss

ssst

T

t

T

(4.3.4)

132

From the second integral in R. H. S. of (4.3.3), we have

)5.3.4(,)(

)()(

4

1

)()(,1)(

)()(

4

)()()(,1)(

)(

2

1

))((

)(

)()(,1

)()(

)(

)(

)()(,1

)()(

))((

)(

)()(,1

)()(

2

*

2

1

0

2

1

10

2

2

10

dssr

shs

a

dssssr

shs

ka

C

dsshssskra

s

sxg

sx

ss

srskaC

dssxg

sx

ss

srska

sxg

sx

ss

shsC

t

T

t

T

t

T

t

T

where kaa 1

* .

From inequalities (4.3.4) and (4.3.5) in inequality (4.3.3), we have

.

,1)(

)()(,1

)()()(

)(4

)()()()(

)()(

)()(

0

0*

2

0

11

1

tt

TT

t

T

t

T

u

dutC

dsss

sstCds

sra

shspsqCs

By condition (2), the last inequality divided by )(t and taking the limit superior on both

sides, we obtain

,))(

)(suplim

)(

)(

)(

)(suplim

,1,1suplim

,1suplim

)(4

)()()()(

)(

1suplim

*

1

1

0

1

1

0

)()(

0

)()(

0

0

)()(

)()(

0*

2

0

11

11

BT

TGC

t

tG

T

TGC

u

du

u

duC

u

duCds

sra

shspsqCs

t

t

t

ttTT

t

tt

TTt

t

Tt


133

Example 4.3.1


.0,)(

))(cos()(2)(

cos3)()(

1)(

2)(46

525

2

3

32

2

ttt

txtxttx

t

tt

t

txtx

tx

tx

We have 2

3

3

cos3)(,

1)(,1)(

t

tttq

tthtr

,

5)( xxg , uvu ),( and

.21)(

11)(10

1)(

2)()()1(

22

2

Rxallfortx

xandtx

txx

)(

2

)(

))(cos(2

)(

))(),(,()2(

446

2

tpttt

txt

txg

txtxtH

for all 0t and .0x

Taking2)( tt such that

.4

12cos31suplim

)(4

)()()()(

)(

1suplim)3(

6*42

0

3

02

2

*

2

0

t

Tt

t

Tt

dssass

sCsCs

t

dssra

shspsqCs

t



find the numerical solutions of the given differential equation in Example 4.3.1 using

the Runge Kutta method (RK4) for different steps sizes.

We have

55 9.3)cos())(),(,()( xxxtxtxtftx

134


xx on the chosen interval 50,1 , the functions

0)(,1)( thx and finding the values of the functions r, q and f , where we

consider ),()(),,(

xxltfxxtH at t=1, n =2250, n =2500, n=2750 and n=3000 and

the steps sizes h =0.021, h =0.019, h =0.017 and h = 0.016.

135

Ta

ble

4.1

: C

om

par

ison

of

the

nu

mer

ical

solu

tion

s o

f O

DE

4.1

wit

h d

iffe

ren

t st

eps

size

s

Erro

r %

0

0.0

000

018

6

0.0

001

395

8

0.0

000

229

9

0.0

000

054

3

0.0

000

073

9

0.0

000

008

6

0.0

000

098

7

0.0

000

018

6

0.0

000

010

4

0.0

000

018

2

0

0.0

000

154

1

0.0

000

049

7

0.0

000

051

9

0

0.0

000

000

2

0.0

000

000

7

0.0

000

000

7

0.0

000

000

5

0.0

000

000

6

0.0

000

000

1

0.0

000

000

6

0.0

000

000

2

0.0

000

000

1

0.0

000

000

2

0

0.0

000

000

2

0.0

000

000

5

0.0

000

000

5

Erro

r %

0

0.0

000

027

9

0.0

003

190

6

0.0

000

558

5

0.0

000

141

3

0.0

000

184

8

0.0

000

025

8

0.0

000

197

5

0.0

000

018

6

0.0

000

020

8

0.0

000

027

3

0.0

000

547

7

0.0

000

847

5

0.0

000

169

2

0.0

000

155

8

0

0.0

000

000

3

0.0

000

001

6

0.0

000

001

7

0.0

000

001

3

0.0

000

001

5

0.0

000

000

3

0.0

000

001

2

0.0

000

000

2

0.0

000

000

2

0.0

000

000

3

0.0

000

000

7

0.0

000

001

1

0.0

000

001

7

0.0

000

001

5

Erro

r %

0

0.0

000

055

8

0.0

006

181

8

0.0

001

084

1

0.0

000

261

0

0.0

000

345

0

0.0

000

051

7

0.0

000

329

2

0.0

000

009

3

0.0

000

010

4

0.0

000

027

3

0.0

002

191

0

0.0

002

928

0

0.0

000

418

2

0.0

000

426

0

0

0.0

000

000

6

0.0

000

003

1

0.0

000

003

3

0.0

000

002

4

0.0

000

002

8

0.0

000

000

6

0.0

000

002

0

0.0

000

000

1

0.0

000

000

1

0.0

000

000

3

0.0

000

002

8

0.0

000

003

8

0.0

000

004

2

0.0

000

004

1

h=

0.0

16

x4(t

k)

1

-1.0

73

949

84

0.0

501

469

4

-0.3

04

383

37

0.9

194

908

9

-0.8

11

591

30

-1.1

59

675

73

0.6

073

582

0

1.0

702

391

0

-0.9

61

418

22

1.0

980

865

4

-0.1

27

790

29

-0.1

29

778

63

-1.0

04

159

21

-0.9

62

439

37

h

=0.0

17

x3(t

k)

1

-1.0

73

949

82

0.0

501

468

7

-0.3

04

383

30

0.9

194

908

4

-0.8

11

591

24

-1.1

59

675

72

0.6

073

582

6

1.0

702

391

2

-0.9

61

418

23

1.0

980

865

2

-0.1

27

790

29

-0.1

29

778

65

-1.0

04

159

16

-0.9

62

439

42

h=

0.0

19

x2(t

k)

1

-1.0

73

949

81

0.0

501

467

8

-0.3

04

383

20

0.9

194

907

6

-0.8

11

591

15

-1.1

59

675

70

0.6

073

583

2

1.0

702

391

2

-0.9

61

418

24

1.0

980

865

1

-0.1

27

790

22

-0.1

29

778

74

-1.0

04

159

04

-0.9

62

439

52

h=

0.0

21

x1(t

k)

1

-1.0

73

949

78

0.0

501

466

3

-0.3

04

383

04

0.9

194

906

5

-0.8

11

591

02

-1.1

59

675

67

0.6

073

584

0

1.0

702

391

1

-0.9

61

418

21

1.0

980

865

1

-0.1

27

790

01

-0.1

29

779

01

-1.0

04

158

79

-0.9

62

439

78

t k

1

5.9

10.8

12.7

6

18.6

4

20.4

04

24.5

2

27.0

68

30.4

32.3

6

37.2

6

40.2

42.1

6

47.0

6

50

136

Figure 4.1(a): Solution curves of ODE 4.1

1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t

)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 4.1(b): Solution curves of ODE 4.1

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t

)

x1(t)

x2(t)

x3(t)

x4(t)

Remark 4.3.1: Theorem 4.3.1 is the extension of the results of Bihari (1963), Kartsatos

(1968), who have studied the equation (1.2) when ,1)( tr ,0)(,1))(( thtx

137

),())(( txtxg ))(),(())())(()()),((( txtxtxtxtrtxg

and 0))(),(,(

txtxtH and

results of Kamenev (1978) and Wintiner (1949) who have studied the equation (1.2) as

,1)( tr ,0)(,1))(( thtx ),())(( txtxg ))(())())(()()),((( txgtxtxtrtxg

and

0))(),(,(

txtxtH . Our result can be applied on their equations, but their oscillation

results cannot be applied on the given equation in Example 4.3.1 because their equations

are particular cases of our equation (1.2).

Theorem 4.3.2: Suppose, in addition to the conditions (1) and (2) hold that

(4) .,0)( 0ttallforth

(5) 0,),1( 11 CCv for ,Rv

(6) vv

),1(

1 for all .Rv

Assume that be a positive continuous differentiable function on the interval ,0t

with )(t is a decreasing function on the interval ,0t and such that

(7) ,)(

)()()()()(suplim

2

2

2

1

t

Tt

dssr

shkspsrksqs

where 0

2

122121 4,4 CkaakkCak and

,0,: 0tp , then, every solution of

the equation (1.2) is oscillatory.


equation (1.2) such that ,0)( Tontx for some .00 tT From inequality (4.3.2)

and by conditions (1) and (6), we have

138

.)(

)()(,1)()(

)()()()()(,1))(()(

)()(

)()(,1

)()()(

2

2

ttttrta

k

tqtttttxtr

thtp

tt

ttt


.)()(,1))(()(

)()()()(,1)()(

4

)()(,1)()(2

)()(,1))(()()()()(

)()(,1)()(

)()()()(,1

)()()(

2

2

2

22

t

T

t

T

t

T

t

T

dssssxsr

shspsssrs

k

a

dssssrsak

sssxsrshsps

sssrsa

k

dssqsdsss

sss

Since 0)()(,1 Css and by conditions (4) and (5), we get

t

T

t

T

t

T

dssrCa

shspsrCs

k

adssqsds

ss

sss.

)(

)()()()(

4)()(

)()(,1

)()()(

0

2

1

22

12

Thus,

)6.3.4(.)()(,1

)()()(

)(

)()()()()(

2

2

2

1 dsss

sssds

sr

shkspsrksqs

t

T

t

T

Since )(t is a decreasing function and by the Bonnet’s theorem, we see that for each

,Tt there exists tTat , such that

.)()(,1

)()()(

)()(,1

)()()(ds

ss

ssTds

ss

sss ta

T

t

T

(4.3.7)

139

From inequality (4.3.7) in inequality (4.3.6), condition (2) and taking the limit superior

on both sides, we obtain

,)(

)(suplim)(

,1,1suplim)(

,1suplim)(

)(

)()()()()(suplim

*

)()(

0

)()(

0

)()(

)()(

2

2

2

1

BT

TGT

u

du

u

duT

u

duTds

sr

shkspsrksqs

t

aaTT

t

aa

TTt

t

Tt

tt

tt


Example 4.3.2


.))(sin()(

)()3)((

)4)(()(

)()(

1

cos4)()(

)3)((

)4)((9

9

2

4

4918

279

5

552

4

49

t

txtx

txtx

txttx

txtx

t

ttttxttx

tx

txt

Here ,)(,)( 29 tthttr 1

cos4)(

5

55

t

ttttq ,

9)( xxg , 22

3

),(vu

uuvu

,

Rxallfortx

txx

3

4

3)(

4)()(1

4

4

and

)(1))(sin(

)(

))(),(,(88

tptt

tx

txg

txtxtH

for all

.00 xandt Let 05

)(,1

)(65

5

tallfort

tt

tt and such that

.

1

cos41suplim

)(

)()()()()(suplim

5

2

7

1

5

55

5

52

2

2

1

t

Tt

t

Tt

dss

k

s

k

s

sss

s

sds

sr

shkspsrksqs

140


equation is oscillatory. The numerical solutions of the given differential equation are

found out using the Runge Kutta method of fourth order (RK4) for different steps sizes.

We have

.

)()(

)()(49.2))(sin()())(),(,()(

2

18

2799

txtx

txtxtxtxtxtxtftx



0)(1)( thandx and finding the values of the functions r, q and f where we

consider ),()(),,(

xxltfxxtH at t=1, n =2250, n =2500, n=2750 and n=3000 and the

steps sizes h =0.021, h =0.019, h =0.017 and h = 0.016.

141

Ta

ble

4.2

: C

om

par

ison

of

the

nu

mer

ical

solu

tion

s o

f O

DE

4.2

wit

h d

iffe

ren

t st

eps

size

s

Erro

r %

0

0.0

000

028

5

0.0

007

417

3

0.0

006

687

8

0.0

000

886

6

0.0

009

502

3

0.0

017

655

4

0.0

008

010

6

0.0

127

839

6

0.0

549

120

3

0.0

040

436

6

0.0

039

315

2

0.0

034

853

9

0.0

178

146

2

0.0

073

575

7

0

0.0

000

000

3

0.0

000

029

3

0.0

000

033

3

0.0

000

009

3

0.0

000

086

1

0.0

000

132

5

0.0

000

081

7

0.0

000

168

5

0.0

000

225

8

0.0

000

295

0.0

000

326

6

0.0

000

300

8

0.0

000

389

5

0.0

000

521

3

Erro

r %

0

0.0

000

095

2

0.0

019

872

5

0.0

017

954

8

0.0

002

393

1

0.0

025

692

7

0.0

047

809

7

0.0

021

698

2

0.0

346

342

9

0.1

488

558

8

0.0

109

699

9

0.0

106

678

2

0.0

094

608

5

0.0

483

853

5

0.0

199

895

0

0

0.0

000

000

1

0.0

000

078

5

0.0

000

089

4

0.0

000

025

1

0.0

000

232

8

0.0

000

358

8

0.0

000

221

3

0.0

000

456

5

0.0

000

612

1

0.0

000

800

3

0.0

000

886

2

0.0

000

816

2

0.0

001

057

9

0.0

001

416

3

Erro

r %

0

0.0

000

209

5

0.0

042

023

3

0.0

038

078

8

0.0

005

119

9

0.0

054

839

9

0.0

102

162

0

0.0

046

406

6

0.0

741

090

3

0.3

186

503

2

0.0

235

094

7

0.0

228

644

5

0.0

202

843

5

0.1

038

096

5

0.0

429

076

5

0

0.0

000

002

2

0.0

000

166

0.0

000

189

6

0.0

000

053

7

0.0

000

496

9

0.0

000

766

7

0.0

000

473

3

0.0

000

976

8

0.0

001

310

3

0.0

001

715

1

0.0

001

899

4

0.0

001

750

6

0.0

002

269

7

0.0

003

040

1

h=

0.0

16

x4(t

k)

0.5

1.0

497

434

4

0.3

9501

821

-0.4

97

914

41

1.0

488

427

7

-0.9

06

091

77

-0.7

50

474

60

1.0

198

969

3

0.1

318

057

9

0.0

411

203

1

-0.7

29

535

75

0.8

307

219

1

-0.8

63

029

64

-0.2

18

640

55

-0.7

08

521

62

h

=0.0

17

x3(t

k)

0.5

1.0

497

434

1

0.3

950

211

4

-0.4

97

917

74

1.0

488

418

4

-0.9

06

083

16

-0.7

50

461

35

1.0

198

887

6

0.1

317

889

4

0.0

411

428

9

-0.7

29

506

25

0.8

307

545

7

-0.8

63

059

72

-0.2

18

679

50

-0.7

08

469

49

h=

0.0

19

x2(t

k)

0.5

1.0

497

433

4

0.3

950

260

6

-0.4

97

923

35

1.0

488

402

6

-0.9

06

068

49

-0.7

50

438

72

1.0

198

748

0

0.1

317

601

4

0.0

411

815

2

-0.7

29

455

72

0.8

308

105

3

-0.8

63

111

29

-0.2

18

746

34

-0.7

08

379

99

h=

0.0

21

x1(t

k)

0.5

1.0

497

432

2

0.3

9503

481

-0.4

97

933

37

1.0

488

374

0

-0.9

06

042

08

-0.7

50

397

93

1.0

198

496

0

0.1

317

081

1

0.0

412

513

4

-0.7

29

364

24

0.8

309

118

5

-0.8

63

204

70

-0.2

18

867

52

-0.7

08

217

61

t k

1

5.9

10.8

12.7

6

18.6

4

20.4

04

24.5

2

27.0

68

30.4

32.3

6

37.2

6

40.2

42.1

6

47.0

6

50

142

Figure 4.2(a): Solution curves of ODE 4.2

1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t

)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 4.2(b): Solution curves of ODE 4.2

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

erical solu

tions x

(t)

x1(t)

x2(t)

x3(t)

x4(t)

143

Remark 4.3.2:

Theorem 4.3.2 is the extension of the results of Bihari (1963) and Kartsatos

(1968) who have studied the equation (1.2) when

,1)( tr ,0)(,1))(( thtx ),())(( txtxg

))(),(())())(()()),((( txtxtxtxtrtxg

and .0))(),(,(

txtxtH Our result can be

applied on their equation, but their oscillation results cannot be applied on the given

equation in Example 4.3.2 because their equation are particular cases of our equation

(1.2).

Theorem 4.3.3

Suppose, in addition to the conditions (1) and (4) hold that

(8) 0)( tq for .0tt

(9) .0)(

)(

allforug

duu

Assume that there exist a differentiable function ,0,: 0t such that

0

r and

,)(

)()(

)(

)()(1suplim)10(

0 0

2

1

2

t

t

s

tt

dudsura

uhu

u

uru

t

.)()()(1

suplim)11(

0 0

0

t

t

s

tt

dudsupuqCut


144

Proof


(1.2) such that ,0)( Tontx for some 00 tT . From the inequality (4.3.1) and

condition (1), we have

).()()(

)()(

)())()(,1()()()(

))(()(

)()()()( 2

2

ttrta

kt

t

ttttqtt

txtr

thtptt

Since 0))()(,1( Ctt and integrating the last inequality from T to t we have

.)())(()(

)(

)(

)()(

)()()()()()()( 2

2

0 dsssxsr

sh

s

ss

srsa

kdsspsqCsTt

t

T

t

T

Then, for t T and by the condition (4), we have

.)(

)()(

)(

)()(

4)()()()(

))(()(

)(

)(

)()()(

4)()()()(

)(

)())(()()(

2

1

22

20

2

20

dssra

shs

s

srs

k

adsspsqCsT

dssxsr

sh

s

ssrs

k

adsspsqCsT

txg

txtxtrt

t

T

t

T

t

T

t

T


)8.3.4(.)(

)()(

)(

)()(

4

)()()())(()(

)())(()()(

2

1

22

2

0

dsduura

uhu

u

uru

k

a

dudsupuqCuTtTdssxg

sxsxsrs

t

T

s

T

t

T

s

T

t

T

Since )()( trt is a non-increasing function and by the Bonnet’s theorem, we see that

for each Tt , there exists tTt , such that

145

.

)(

)()()(

)(

)())(()()(

)(

)())(()()()(

)(

tt x

TxT

t

Tug

duuTrTds

sxg

sxsxTrTds

sxg

sxsxsrs

Since 0)()( trt and the condition (9), we have

,)(

)())(()()( 3Ads

sxg

sxsxsrs

t

T

where .)(

)()()(inf

)(

)(

3

2

tx

Txug

duuTrTA


.)(

)()(

)(

)()(

4))(()()()(

2

1

22

230 dsdu

ura

uhu

u

uru

k

aATtTdudsupuqCu

t

T

s

T

t

T

s

T

Dividing the last inequality by t , taking the limit superior as t , we obtain

,)(

)()(

)(

)()(1suplim

4

)(1)(suplim)()()(

1suplim

2

1

2

2

2

3

0

dsduura

uhu

u

uru

tk

a

t

ATTdudsupuqCu

t

t

T

s

Tt

t

t

T

s

Tt

which contradicts to the condition (11). Hence the proof is completed.

146

Example 4.3.3


.))(sin()(

)2)(()()2)((

)(

)()(

)()(

1)(

2)(10

5

2

26

210

1553

926

2

t

txtx

txttxtx

tx

txtxt

t

txtx

txt

tx

Here ,1

)(,01

)(96 t

tht

tr 3)( ttq ,

5)( xxg ,22

3

),(vu

uuvu

,

Rxallfortxtx

txxand

tx

txx

2

1)(

11

1)(

2)()(10

1)(

2)()(

22

2

2

2

and

.02

12

)(

)(0

45

allforx

dx

xg

dxx

.00)(

1))(sin(

)(

))(),(,(1010

xandtallfortptt

tx

txg

txtxtH

Taking .002

)()(04)(,00)(3

34

tallfort

trtandtttfortt

.7

1

3

16

1161suplim

)(

)()(

)(

)()(1suplim)1(

7

0

3

0

842

1

2

0 00 0

tt

dudsuut

dudsura

uhu

u

uru

t

t

t

s

tt

t

t

s

tt

.11

suplim)()()(1

suplim)2(

0 00 0

10

3

0

4

0

t

t

s

tt

t

t

s

tt

dudsu

uCut

dsduupuqCut

147

It follows from Theorem 4.3.3 that the given equation is oscillatory. The numerical

solutions of the given differential equation are found out using the Runge Kutta method

of fourth order (RK4) for different steps sizes.

We have

)()(

)()())(sin()())(),(,()(

2

10

1555

txtx

txtxtxtxtxtxtftx



0)(1)( thandx and finding the values of the functions r, q and f where we

consider ),()(),,(



148

Ta

ble

4.3

: C

om

par

ison

of

the

num

eric

al s

olu

tion

s o

f O

DE

4.3

wit

h d

iffe

rent

step

s si

zes

Err

or

%

0

0.0

000

13

43

0.0

000

18

86

0.0

000

24

91

0.0

003

74

64

0.0

000

82

43

0.0

000

75

65

0.0

001

30

52

0.0

002

54

93

0.0

008

19

95

0.0

017

89

23

0.0

002

00

03

0.0

005

94

72

0.0

004

15

10

0.0

003

86

63

0

0.0

000

00

11

0.0

000

00

21

0.0

000

00

29

0.0

000

00

68

0.0

000

00

88

0.0

000

00

83

0.0

000

01

19

0.0

000

00

51

0.0

000

01

66

0.0

000

00

21

0.0

000

02

26

0.0

000

02

6

0.0

000

03

14

0.0

000

03

41

Err

or

%

0

0.0

000

36

63

0.0

000

51

20

0.0

000

66

16

0.0

009

91

72

0.0

002

18

25

0.0

002

02

36

0.0

003

47

70

0.0

031

29

19

0.0

021

88

19

0.0

048

05

37

0.0

005

35

48

0.0

016

01

18

0.0

011

17

08

0.0

010

40

85

0

0.0

000

00

03

0.0

000

00

57

0.0

000

00

77

0.0

000

01

8

0.0

000

02

33

0.0

000

02

22

0.0

000

03

17

0.0

000

06

26

0.0

000

04

43

0.0

000

05

64

0.0

000

06

05

0.0

000

07

0.0

000

08

45

0.0

000

09

18

Err

or

%

0

0.0

000

74

50

0.0

001

05

99

0.0

001

37

48

0.0

020

66

08

0.0

004

57

11

0.0

004

26

60

0.0

007

31

60

0.0

062

73

38

0.0

046

08

53

0.0

101

56

03

0.0

011

34

68

0.0

033

89

93

0.0

023

71

64

0.0

022

09

82

0

0.0

000

00

61

0.0

000

01

18

0.0

000

01

6

0.0

000

03

75

0.0

000

04

88

0.0

000

04

68

0.0

000

06

67

0.0

000

12

55

0.0

000

09

33

0.0

000

11

92

0.0

000

12

82

0.0

000

14

82

0.0

000

17

94

0.0

000

19

49

h=

0.0

16

x4(t

k)

0.5

0.8

187

86

20

1.1

132

73

56

-1.1

637

48

01

-0.1

815

02

82

1.0

675

55

65

1.0

970

37

94

-0.9

116

90

27

-0.2

000

51

47

-0.2

024

50

32

0.1

173

68

59

-1.1

298

24

17

0.4

371

75

97

0.7

564

36

13

-0.8

819

71

24

h

=0

.01

7

x3(t

k)

0.5

0.8

187

86

09

1.1

132

73

35

-1.1

637

47

72

-0.1

815

03

50

1.0

675

56

53

1.0

970

37

11

-0.9

116

91

46

-0.2

000

49

13

-0.2

024

51

98

0.1

173

66

49

-1.1

298

21

91

0.4

371

73

37

0.7

564

32

99

-0.8

819

74

65

h=

0.0

19

x2(t

k)

0.5

0.8

187

85

90

1.1

132

72

99

-1.1

637

47

24

-0.1

815

04

62

1.0

675

57

98

1.0

970

35

72

-0.9

116

93

44

-0.2

000

45

21

-0.2

024

54

75

0.1

173

62

95

-1.1

298

18

12

0.4

371

68

97

0.7

564

27

68

-0.8

819

80

42

h=

0.0

21

x1(t

k)

0.5

0.8

187

85

59

1.1

132

72

38

-1.1

637

46

41

-0.1

815

06

57

1.0

675

60

53

1.0

970

33

26

-0.9

116

96

94

-0.2

000

38

29

-0.2

024

59

65

0.1

173

56

67

-1.1

298

11

35

0.4

371

61

15

0.7

564

18

19

-0.8

819

90

73

t k

1

5.9

10

.8

12

.7 6

18

.6 4

20

.4

04

24

.5 2

27

.0

68

30

.4

32

.3 6

37

.2 6

40

.2

42

.1 6

47

.0 6

50

149

Figure 4.3(a): Solution curve of ODE 4.3

1 2 3 4 5 6 7 8 9 10-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t

)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 4.3(b): Solution curve of ODE 4.3

0 10 20 30 40 50 60-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ions

x(t

)

x1(t)

x2(t)

x3(t)

x4(t)

Theorem 4.3.4

Suppose that conditions (1) and (8) hold and

(12) .0)( 0ttforth

150

Furthermore, suppose that there exists a positive continuous differentiable function on

the interval ,0t with ,0)(

t ,0))()((

trt 0))()()()(( 2

thttrta and

0))()()()(( 2

thttrta such that

(13)

.,)()(

0

0

ttforeveryssr

ds

t

.)()()(lim)14( 0

t

Tt

dsspsqCs

Then, every solution of super-linear equation (1.2) is oscillatory.

Proof

If )(tx is oscillatory on ,0,, 0 tTT then, )(tx

is oscillatory on ,T and if

)(tx

is oscillatory on ,,T then, )(tx

is oscillatory on .,T

Without loss of

generality, we may assume that there exists a solution x(t) of equation (1.2) such that

,0)( Tontx .for some 00 tT We have three cases of )(tx

:

(i) .0)( Tteveryfortx

(ii) .0)( Tteveryfortx

(iii) )(tx

is oscillatory.

If 0)(

tx for 0, tTTt and we define

.,

)(

)())(()()()( Tt

txg

txtxtrtt

151

Then, by equation (1.2) and condition (1), we get

.,))((

)()()())()(,1()()(

))((

)()()()()()( 2 Tt

txg

txtrtatttqt

txg

txthttptt

Since ,0))()(,1( tt then, there exists a positive constant C0 such

that 0))()(,1( 0 Ctt for ,Tt we have

.,))((

)()()()()()()()()( 20 Tt

txg

txthttrtatptqCtt


.))((

)()()()()()()()()()( 20 ds

sxg

sxshssrsadsspsqCsTt

t

T

t

T

(4.3.9)

Since ))()()()(( 2 thttrta

is a non-increasing and by Bonnet’s Theorem for a

fixed ,Tt there exists tTt , such that

)(

)(

2

22

)()()()()(

))((

)()()()()(

))((

)()()()()(

t

t

x

Tx

T

t

T

ug

duThTTrTa

dssxg

sxThTTrTads

sxg

sxshssrsa

Since 0))()()()(( 2

thttrta and the equation (1.2) is superlinear, we have

)(

)()(

).()(,)(

)()(,0

)(

tx

Txt

Tx

t

Txxifug

du

Txxif

ug

du

152

We have

,)(

)()()()()( 12 Ads

sxg

sxshssrsa

t

T

(4.3.10)

where .)(

)()()()()(

21

Txug

duThTTrTaA

Thus, from (4.3.10) in (4.3.9), we obtain

t

T

dsspsqCsATt .)()()()()( 01

By the condition (14), we get 0)( t , then, 0)(

tx for ., 11 TTTt This is a

contradiction.

If 0)(

tx for every ,2 TTt the condition (14) implies that there exists 23 TT such

that

.0)()()( 30

3

TtallfordsspsqCt

t

T

Thus, from equation (1.2) multiplied by ),(t we obtain

.),()()()()(,1)()()()()()()())(()()( 3Tttptxgttttqtxgttxthttxtxtrt


.)()())(()()())(()()( 0 tptqCtxgttxtxtrt

153

Integrate the last inequality from 3T to t and by condition (1), we obtain

.,)())(()()(

,)()()()())((

)()()())(()()()()())(()()(

)()()())(()()()()())(()()()()()(

33333

0

013333

0133331

33

33

33

TtTxTxTrT

dudsupuqCusxsxg

dsspsqCstxgdssxsrsaTxTxTrT

dsspsqCssxgdssxsrsaTxTxTrTtxtrta

s

T

t

T

t

T

t

T

t

T

t

T

Integrate the last inequality divided by )()( trt from 3T to t and by condition (13), we

have

,

)()()())(()()()()(

3

3333322 t

Tssr

dsTxTxTrTTxatxa

as ,t contradicting the fact 0)( tx for all t .T Thus, we have )(tx

is oscillatory

and this leads to (1.2) is oscillatory. Hence the proof is completed.

Example 4.3.4


.0,

))(sin()(

)4)((

)()5)((6)(3

)()(9

)()(

)4)((

)5)((3

5

2

2

210

155

22

2

tt

txtx

txt

txtxtx

txtxt

t

txtx

txt

tx

154

We have ,9)(,1

)(,1

)(2

ttqt

tht

tr 5)( xxg ,

22

3

63),(

vu

uuvu

and

.0

0)(1))(sin(

))((

))(),(,())(sin()())(),(,()1(

333

5

tand

xallfortptt

tx

txg

txtxtHand

t

txtxtxtxtH

.4

5

4)(

11

4)(

5)()(10

4)(

5)()()2(

22

2

2

2

Rxallfortxtx

txxand

tx

txx

Taking ,02

))()((,02

))()((,02)(,02)(2

ttrt

ttrtttt

02

9))()()()((,0

2

9))()()()((

222

tthttrta

tthttrta for all 0t such

that

(3) ,2

1lim

2lim

2lim

)()(lim

)()(0

0000

ttdsds

srs

ds

srs

ds

t

t

tt

t

tt

t

tt

t

.2

6lim1

92lim)()()(lim)4(

000

3

0300

t

tt

t

tt

t

tt s

sCdss

sCsdsspsqCs

All conditions of Theorem 4.3.4 are satisfied. Then the given equation is

oscillatory. Also the numerical solutions of the given differential equation are computed

using the Runge Kutta method of fourth order (RK4) for different steps sizes.

We have

)(6)(3

)()(9))(sin()())(),(,()(

2

10

1555

txtx

txtxtxtxtxtxtftx

155



1)( x and 0)( th and the finding the values of the functions r, q and f where we

consider ),()(),,(



156

Ta

ble

4.4

: C

om

par

ison

of

the

nu

mer

ical

solu

tion

s o

f O

DE

4.4

wit

h d

iffe

ren

t st

eps

size

s

Erro

r %

0

0.0

000

993

8

0.0

000

682

9

0.0

000

848

5

0.0

000

321

7

0.0

001

704

0

0.0

002

053

4

0.0

022

977

7

0.0

006

021

9

0.0

007

334

7

0.0

011

411

5

0.0

018

205

0

0.0

332

111

6

0.0

008

955

0

0.0

093

133

9

0

0.0

000

003

6

0.0

000

005

3

0.0

000

006

6

0.0

000

002

7

0.0

000

013

4

0.0

000

016

4

0.0

000

034

7

0.0

000

038

3

0.0

000

045

8

0.0

000

062

7

0.0

000

073

4

0.0

001

075

0.0

000

067

3

0.0

000

102

4

Erro

r %

0

0.0

002

595

0

0.0

001

816

7

0.0

002

237

0

0.0

000

846

0

0.0

004

527

1

0.0

005

446

7

0.0

061

251

9

0.0

016

068

9

0.0

019

618

0

0.0

030

576

4

0.0

048

736

9

0.0

062

158

9

0.0

024

030

8

0.0

250

206

5

0

0.0

000

009

4

0.0

000

014

1

0.0

000

017

4

0.0

000

007

1

0.0

000

035

6

0.0

000

043

5

0.0

000

092

5

0.0

000

102

2

0.0

000

122

5

0.0

000

168

0.0

000

196

5

0.0

000

201

2

0.0

000

180

6

0.0

000

275

1

Erro

r %

0

0.0

005

300

4

0.0

003

813

9

0.0

004

705

4

0.0

001

775

4

0.0

009

537

6

0.0

011

456

9

0.0

129

390

5

0.0

033

961

7

0.0

041

542

1

0.0

064

865

7

0.0

103

401

6

0.0

131

886

9

0.0

051

068

8

0.0

531

882

4

0

0.0

000

019

2

0.0

000

029

6

0.0

000

036

6

0.0

000

014

9

0.0

000

075

0.0

000

091

5

0.0

000

195

4

0.0

000

216

0.0

000

259

4

0.0

000

356

4

0.0

000

416

9

0.0

000

426

9

0.0

000

383

8

0.0

000

584

8

h=

0.0

16

x4(t

k)

0.5

0.3

622

312

4

-0.7

76

096

88

0.7

778

136

3

-0.8

39

200

82

0.7

863

584

0

0.7

986

403

4

-0.1

51

015

68

-0.6

36

008

86

0.6

244

262

4

-0.5

49

442

65

0.4

031

852

0

-0.3

23

686

29

0.7

515

346

6

-0.1

09

949

16

h

=0.0

17

x3(t

k)

0.5

0.3

622

316

0

-0.7

76

096

35

0.7

778

129

7

-0.8

39

201

09

0.7

863

570

6

0.7

986

419

8

-0.1

51

019

15

-0.6

36

012

69

0.6

244

308

2

-0.5

49

436

38

0.4

031

925

4

-0.3

23

793

79

0.7

515

279

3

-0.1

09

959

40

h=

0.0

19

x2(t

k)

0.5

0.3

622

321

8

-0.7

76

095

47

0.7

778

118

9

-0.8

39

201

53

0.7

863

548

4

0.7

986

446

9

-0.1

51

024

93

-0.6

36

019

08

0.6

244

384

9

-0.5

49

425

85

0.4

032

048

5

-0.3

23

706

41

0.7

515

166

0

-0.1

09

976

67

h=

0.0

21

x1(t

k)

0.5

0.3

622

331

6

-0.7

76

093

92

0.7

778

099

7

-0.8

39

202

31

0.7

863

509

0

0.7

986

494

9

-0.1

51

035

22

-0.6

36

030

46

0.6

244

521

8

-0.5

49

407

01

0.4

032

268

9

-0.3

23

728

98

0.7

514

962

8

-0.1

10

007

64

t k

1

5.9

10.8

12.7

6

18.6

4

20.4

04

24.5

2

27.0

68

30.4

32.3

6

37.2

6

40.2

42.1

6

47.0

6

50

157

Figure 4.4(a): Solution curve of ODE 4.4

1 2 3 4 5 6 7 8 9 10-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

Num

erical solu

tions x

(t)

x1(t)

x2(t)

x3(t)

x4(t)

Figure 4.4(b): Solution curve of ODE 4.4

0 10 20 30 40 50 60-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

Num

eric

al s

olut

ions

x(t

)

x1(t)

x2(t)

x3(t)

x4(t)

158

Remark 4.3.3

Theorem 4.3.4 extends results of Grace (1992) and El-abbasy et al. (2005), who

have studied the equation (1.2) as

,1))(( tx ))(())())(()()),((( txgtxtxtrtxg

and 0))(),(,(

txtxtH . Our result

can be applied on their equation, but their oscillation results cannot be applied on the

given equation in Example 4.3.4 because their equation is a particular case of our

equation (1.2).

Theorem 4.3.5: Suppose that the conditions (1), (8), (12) and (13) hold and there exists

a continuously differentiable function ,0,: 0t such that

,0))()(( tht ,0)()(

trt

,0)()( 0tontrt and

(15) ,)(

0

t

dss

where )(4

)()()()()()(

2

0tk

trttptqCtt

. Then, every solution of superlinear

equation (1.2) is oscillatory.



.,))((

)())(()()( Tt

txg

txtxtrt

This, condition (8) and (1.2) imply

159

.,)(

)())(()(

))((

)()()()(

))((

)())(()(2

2

0 Tttxg

txtxtkr

txg

txthtptqC

txg

txtxtr

(4.3.11)

We multiply the last inequality (4.3.11) by )(t and integrate form T to t, we have

,)()(

)()()(

))((

)()()()()()(

))((

)())(()()(

2

2

01

dsssr

s

a

kss

dssxg

sxshsdsspsqCsC

txg

txtxtrt

t

T

t

T

t

T

where .))((

)())(()()(1

Txg

TxTxTrTC

Thus

,)(2

)()()(

)(

)(

))((

)()()()()()(

))((

)())(()()(

2

22

2

01

dssk

srsas

sr

s

a

k

dssxg

sxshsdsspsqCsC

txg

txtxtrt

t

T

t

T

t

T

where .)(2

)()()()( 2

tk

trtatt


)12.3.4(.))((

)()()(

)(4

)()()()()(

))((

)())(()()(

2

2

01 dssxg

sxshsds

sk

srsaspsqCsC

txg

txtxtrtt

T

t

T

The second integral in R. H. S. of the inequality (4.3.12) is bounded from above. This

can be by using the Bonnet’s Theorem, for all ,Tt there exists tTat , such that

160

.)(

)()())((

)()()(

))((

)()()()(

)(

tt ax

Tx

a

T

t

Tug

duThTds

sxg

sxThTds

sxg

sxshs

Since 0)()( tht and the equation (1.2) is superlinear, we have

)13.3.4(,

)(

)()()( 1Bds

sxg

sxshs

t

T

where

)(

1 .)(

)()(Tx

ug

duThTB

By inequality (4.3.13) and the condition (15), the inequality (4.3.12) becomes

.)())((

)())(()()(11 dssBC

txg

txsxtrtt

T


.))((

)())(()()(lim

txg

txtxtrt

t

Thus, there exists TT 1 such that 0)(

tx for .1Tt The condition (15) also implies

that there exists 12 TT such that

.0)()()(0)()()( 200

2

2

1

TtfordsspsqCsanddsspsqCs

t

T

T

T

Multiplying equation (1.2) by ,)(t from the conditions (12) and (8), we have

,,)())(()()())(()()())(()()( 20 TttptxgttqtxgtCtxtxtrt

161

where .)(,1min0)(

0 tCRt


s

T

t

T

t

T

t

T

dsduupuqCusxsxgdsspsqCstxg

dssxsxsrsTxTxTrTtxtxtrt

222

2

.)()()()())(()()()())((

)())(()()()())(()()()())(()()(

00

2222

By the condition (1) and the Bonnet’s Theorem, for 2Tt there exists tTt ,2 such

that

.,)()()()())((

)()()())(()()()()()())(()()()()()(

2

222122222

22

2

TtdsduupuCqusxsxg

dsspsCqstxgTxxTrTaTxTxTrTtxtrta

s

T

t

T

t

T

t

Thus

.,)())(()()()()()( 222222 TtTxTxTrTtxtrta

Dividing the last inequality by )()( trt , integrate from T2 to t and the condition (7), we

obtain

,,)()(

)())(()()()()(

2

2222222

tassrs

dsTxTxTrTTxatxa

t

T

which is a contradiction to the fact that x(t) >0 for tT . Hence the proof is completed.

162

Example 4.2.5


.))()(sin()(

)()1)((

2)(8)(5

)()()()(

)1)((

2)(5

3

4

63

612

15332

63

6

t

txtxtx

txtxt

txtx

txtxttxttx

txt

tx

We note that

3

1)(

ttr ,

32 )(,)( ttqtth , ,85

),(,)(44

53

vu

uuvuxxg

,))()(sin()(

))(),(,(5

3

t

txtxtxtxtxtH

)(1))()(sin(

))((

))(),(,(55

tptt

txtx

txg

txtxtH

for

all 0x and .0t1

2)(

6

6

x

xx

and 2)(1 x

for all x R

.)()(

0

031

)()(03)()(,01

)()(,)(

00

2

43

2

3

dsssrs

dsandtallfor

tttrtandttht

ttrtttTaking

tt

.6

1

3

1

5

2

11

)(4

)()()()()()()1(

0

0

0 0

33

5

0

45

3

0

2

20

t

t

t t

kss

sC

dskss

sCs

dssk

srsaspsqCsdss

163


equation is oscillatory. We also compute the numerical solution of the given differential

equation using the Runge Kutta method of fourth order (RK4). We have

)(8)(5

)()())()(sin()())(),(,()(

4

12

1533

txt

txtxtxtxtxtxtxtftx



1)( x and 0)( th and finding the values of the functions r, q and f where we

consider ),()(),,(

xxltfxxtH at t=1, n=980 and h=0.05.


k tk x(tk)

1

62

181

221

321

381

461

561

603

685

726

781

821

933

981

1

4.05

10

12

17

20

24

29

31.1

35.2

37.25

40

42

47.6

50

-0.5

0.22021420

-1.39354243

1.53595848

-1.01339800

2.04072897

2.19425748

-0.01291195

0.14141788

0.10102608

-0.08118400

1.86554418

-1.78419073

0.28301496

-1.28617425

164


0 5 10 15 20 25 30 35 40 45 50-2.5

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

2.5

t

Num

eric

al s

olut

ion

x(t)

Remark 4.3.4

Theorem 4.3.5 is the extension of the results of Greaf, et al. (1978) and Remili

(2010) who have studied the equation (1.2)

when ,1))(( tx 0)( th and .))(,())())(()()),((( txttxtxtrtxg

Our result can

be applied on their equation, but their oscillation results cannot be applied on the given

equation in Example 4.3.5 because their equation is a particular case of our equation

(1.2).

Theorem 4.3.6

Suppose, in addition to the conditions (1), (8) and (9) hold that

(16) 0,)(

11

kksr

ds

T

.

165

(17) There exists a constant *B such that

m

meveryforBmBs

dsmG

0

** ,0,),1(

)( R.

Furthermore, suppose that there exists a positive continuous differentiable function

on the interval ,0t with )(t is a non-decreasing function on the interval ,0t such

that

(18) ,)(4

)()()()(

)()(

1suplim

*

2

0

t

T

s

Tt

dsduura

uhupuqCu

ssr

where ,0,: 0tp . Then, every solution of super-linear equation (1.2) is

oscillatory.

Proof


equation (1.2) such that 0)( tx on ,T for some .00 tT

Define

.,

)(

)())(()()()( Tt

txg

txtxtrtt

This and by condition (1) and equation (1.2), we have

.,

)(

)()()()(

)(

)())()(,1()()(

))((

)()()()()()(

2

2

1 Tttxg

txtrtkat

t

ttttqt

txg

txthttptt

166

Thus for ,Tt we have

.,

)(

)()()(

))((

)()()())()(,1()()()()(

)(

)()(

2

2

1 Tttxg

txtrtka

txg

txthttttqttpt

t

tt

Dividing the last inequality by 0)()(,1 tt , then, there exists a positive constant

C0 such that 0))()(,1( Ctt then, 0

1

)()(,1

10

Ctt

. Thus, for Tt , we

obtain

.,

)()()(,1

)()()(

))(()()(,1

)()()(

)()(,1

)()()()()()(

2

2

1000

0 Tttxgtt

txtrtkaC

txgtt

txthtC

tt

tttCtptqCt

Integrate from T to t, we obtain

)14.3.4(.

)(

)(

)()(,1

)()(

))((

)(

)()(,1

)()(

)()(,1

)()()()()()(

2

2

10

00

dssxg

sx

ss

srska

sxg

sx

ss

shsC

dsss

sssCdsspsqCs

t

T

t

T

t

T

From the second integral in R. H. S. of (4.3.14), we have

)15.3.4(,)(

)()(

4

1

)()()(,1

)()(

4)(

)()(,1)(

)(

2

1

))((

)(

)()(,1

)()(

)(

)(

)()(,1

)()(

))((

)(

)()(,1

)()(

2

*

2

1

0

2

1

10

2

2

10

t

T

t

T

t

T

t

T

dssr

shs

a

dssrss

shs

ka

Cdssh

ssskra

s

sxg

sx

ss

srskaC

dssxg

sx

ss

srska

sxg

sx

ss

shsC

where kaa 1

* .

167

By the Bonnet’s Theorem, since )(t is a non-decreasing function on the interval

,,0 t there exists tTT ,1 such that

.)()(,1

)()()(

)()(,1

)()()(

1

dsss

sstds

ss

ssst

T

t

T

(4.3.16)

From inequalities (4.3.16) and (4.3.15) in inequality (4.3.14), we have

.)(

)()(

)(

)()(

,1,1)(

,1)(

)()(,1

)()()(

)(4

)()()()(

0

1

10

)()(

0

)()(

0

0

)()(

)()(

00*

2

0

11

111

t

tGtC

T

TGtC

u

du

u

dutC

u

dutCds

ss

sstCds

sra

shspsqCs

ttTT

tt

TT

t

T

t

T

By the condition (17), we obtain

).()(

)()(

)(4

)()()()( *

0

1

10*

2

0 tBCT

TGtCds

sra

shspsqCs

t

T

Integrating the last inequality divided by )()( trt from T to t, taking the limit superior

on both sides and by conditions (16) and (17), we have

,)(

)(suplim

)()(

)(suplim

)(4

)()()()(

)()(

1suplim

)(

)(

*

0

1

1

0*

2

0

tx

Txt

t

Tt

t

T

s

Tt

ug

duuBC

sr

ds

T

TGCdsdu

sra

shupuqCu

ssr


168

Example 4.3.6


.0,))(cos()(

)1)(()()2)(()(

)()()(

)1)((

)2)((7

3

2

4246

96

34

42

tt

txtx

txttxtxtx

txt

t

txtx

tx

txt

We note that ,)(,)(,1

)(,)( 36

3

2 xxgttqt

thttr 01)(

2)()(

4

4

tx

txx and

2)(1 x for all Rx and 22

3

),(vu

uvu

such that

.

,1,)1()1(),1(

)()1(00

2

0

Rmallfor

andRBBmmdsdsss

dsmG

mmm

.00)(

1))(cos(

)(

))(),(,()2(

77

xandtallfortptt

tx

txg

txtxtH

Let 4)( tt such that

.5)4

1

2

1

11(

96

1

14

1

66suplim

4

111suplim

)(4

)()()()(

)()(

1suplim)3(

5

*2

11

08*7

6

0

6*7

6

0

4

6

*

2

0

t

Tt

t

T

s

Tt

t

T

s

Tt

sTaT

TC

sas

sC

dsduuau

uCus

dsduura

uhupuqCu

ssr

169


equation is oscillatory. The numerical solution of the given differential equation is

found out using the Runge Kutta method of fourth order (RK4). We have

)()(

)())(cos()())(),(,()(

26

93

txtx

txtxtxtxtxtftx


xx on the chosen interval 50,1 ,

0)(,1)( thx and finding the values of the functions r, q and f where we consider

),()(),,(

xxltfxxtH at t=1 n=980 and h=0.05.


k tk x(tk)

1

81

181

221

321

381

461

521

587

653

721

781

821

921

981

1

5

10

12

17

20

24

27

30.3

33.6

37

40

42

47

50

0.5

-1.23206499

0.40954827

-1.50525503

0.02833177

-0.33103118

-0.35258529

0.04981132

-0.43001970

0.03619157

-0.12794647

-0.17456968

1.46730560

0.20608737

0.09644083

170


0 5 10 15 20 25 30 35 40 45 50-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

t

Num

eric

al s

olut

ion

x(t)

Remark 4.3.5

Theorem 4.3.6 is the extension of the results of Bihari (1963) and Kartsatos (1968),

who have studied the equation (1.2) when ,1)( tr ,0)(,1))(( thtx


and .0))(),(,(

txtxtH Our

result can be applied on their equation, but their oscillation results cannot be applied on

the given equation in Example 4.3.6 because their equation is a particular case of our

equation (1.2).

Theorem 4.3.7

Suppose in addition to the conditions (1), (8) and (12) hold that there exists

continuously differentiable function ,0,: 0t such that ,0)(

t condition

(13) holds and

171

(19) 0)(inflim

0

1

t

t

dss for all large t,

(20) ,)()()(

1lim

0

1

t

t st

dudsusrs

where .)(

)()()(

)(

)(

4)()()()(

2

1

201

tra

thtt

t

tr

k

atptqCtt




.,))((

)())(()()( Tt

txg

txtxtrt


)17.3.4(.,))((

)())(())(()()(,1)(

))((

)()()(

))((

)())(()(2

2

Tttxg

txtxgtxtrttq

txg

txthtp

txg

txtxtr

Since ,0))()(,1( tt then there exists a positive constant C0 such that

,))()(,1( 0Ctt thus, we have

.,))((

)())(())(()(

))((

)()()()(

))((

)())(()(2

2

0 Tttxg

txtxgtxtr

txg

txthtptqC

txg

txtxtr

We multiply the last inequality by )(t and integrate from T to t, we have

172

)18.3.4(,))((

)())(())(()()(

))((

)())(()()(

))((

)()()()()()(

))((

)())(()()(

2

2

01

t

T

t

T

t

T

t

T

dssxg

sxsxgsxsrsds

sxg

sxsxsrs

dssxg

sxshsdsspsqCsC

txg

txtxtrt

where ))((

)())(()()(1

Txg

TxTxTrTC

.

Thus, for we have

)19.3.4(,)(

)()(

)()(

))(()(

)()()(

)(4

))(()()(

))(()(

)(

)()()(

)())(()(

)()()()(

))(()(

)()()()(

))((

)())(()()(

11

2

2

11

2

22

222

01

2

01

t

T

t

T

t

T

t

T

t

T

t

T

t

T

dssC

dsssra

skdssC

dssxsr

shss

sk

sxsrs

sxsr

sk

dsspsqCsC

dsssxsr

shsss

sxsr

skdsspsqCsC

txg

txtxtrt

where .))(()(

)()()(

)(2

))(()()()(

txtr

thtt

tk

txtrtt


.))((

)())(()()(

))((

)())(()()()(1

txg

txtxtrt

Txg

TxTxTrTdss

t

T

173

Now, we consider three cases for )(tx

.


then, we get

.))((

)())(()()(

))((

)())(()()()(

1

11111

1txg

txtxtrt

Txg

TxTxTrTdss

t

T

Thus, for all 1Tt , we obtain

.))((

)())(()()()(1

txg

txtxtrtdss

t

Integrate the last inequality divided by )()( trt from T1 to t and by condition (1), we

obtain

.))((

)()(

)()(

1

11

21 dssxg

sxadsduu

srs

t

T

t

T s

Since the equation (1.2) is superlinear, we get

.)(

)()()(

1)(

)(

21

11

tx

Tx

t

T sug

duadsduu

srs

This contradicts condition (20).

Case 2: If )(tx

is oscillatory, then there exists a sequence n in ,T such

that .0)(

nx Choose M large enough so that (19) holds. Then from inequality (4.3.19),

we have

174

.)())((

)())(()()(1

t

n

ndssC

txg

txtxtrt

So

,0)(inflim)(suplim))((

)())(()()(suplim 11

t

t

t

ttn

n

n

ndssCdssC

txg

txtxtrt

which contradicts the fact that )(tx

oscillates.

Case 3: If 0)(

tx for ,2 TTt the condition (19) implies that there exists 23 TT

such that

.0)()( 30

3

TtfordsspsqC

t

T

Multiplying the equation (1.2) by )(t and by condition (12), we have

,,)())(()()())(()()())(()()( 30 TttptxgttqtxgtCtxtxtrt

where .)(,1min0)(

0 tCRt


175

.,)())(()()(

,)()()()())((

)()()())(()()()()())(()()()())(()()(

33333

3

013333

33

33

TtTxTxTrT

TtdsduupuCqusxsxg

dsspsqCstxgdssxsrsaTxTxTrTtxtxtrt

s

T

t

T

t

T

t

T

By condition (1), the last inequality divided by )()( trt and integrate from T3 to t, we

obtain

,,

)()()())(()()()()(

3

3333322

tassrs

dsTxTxTrTTxatxa

t

T

which is a contradiction to the fact that 0)( tx for .Tt Hence the proof is

completed.

Example 4.3.7


.0,1)(

))((cos)()

)()5)((

6)()(2

)(1)(

)5)((

6)(4

211

4

2

228

35

22

2

ttx

txtx

txtxt

txtx

tx

ttx

txt

tx

Here we have

,1

)(t

tr ,)(,1

)(,0)( 7

2xxg

ttqth 0

5)(

6)()(,2),(

2

2445

tx

txxvuuvu

and

.00)(01)(

))((cos)(

)(

))(),(,(4

24

xandtallfortptx

txtx

txg

txtxtH

176

Taking 4)( t and

.0,044

inflim

)(

)()()(

)(

)(

4)()()(inflim)(inflim)1(

0

0

2

0

2

1

2

01

TT

Cds

s

C

dssra

shss

s

sr

k

aspsqCsdss

t

Tt

t

Tt

t

Tt

.lim4

1

4lim

)(

)()()(

)(

)(

4)()()(

)()(

1lim

)()()(

1lim)2(

0002

0

2

1

20

1

tCtCdsduu

Cs

dudsura

uhuu

u

ur

k

aupuqCu

srs

dudsusrs

ts

t

Tt

s

t

Tt

s

t

Tt





We have

)(21

))((cos))(),(,()(

4

28

35

4

211

txx

x

x

txxtxtxtftx


xx on the chosen interval 50,1 , the function

1)( x and finding the values of the functions r, q and f where we consider

),()(),,(


177



0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ion

x(t)

k tk x(tk)

1

81

195

250

321

381

461

521

581

624

721

781

821

921

981

1

5

10.75

13.45

17

20

24

27

30

31.15

37

40

42

47

50

1

-0.38710915

0.02100274

-0.04535638

0.94181375

-1.21862863

0.11404729

0.12824333

-0.55051467

0.01009389

-0.50610435

0.17389288

-0.86814628

-1.19426955

0.89817500

178

Remark 4.3.6: Theorem 4.3.7 extends Result of Wong and Yeh (1992), Result of

Philos (1985), Result of Onose (1975), Result of Philos and Purnaras who have studied

the equation (1.2) as ,1)( tr ,0)(,1))(( thtx ))(())())(()()),((( txgtxtxtrtxg

and 0))(),(,(

txtxtH and Result of Elabbasy, et al. (2005) who have studied the

equation (1.1) as ,1))(( tx ))(())())(()()),((( txgtxtxtrtxg

and

0))(),(,(

txtxtH . Also, Theorem 4.3.7 extends and improves the results of Greaf, et

al. (1978) and Remili (2010) who have considered the equation (1.2) as ,1))(( tx

0)( th and .))(,())())(()()),((( txttxtxtrtxg

Our result can be applied on their

equations, but their oscillation results cannot be applied on the given equation in

Example 4.3.7 because their equations are particular cases of our equation (1.2).

Theorem 4.3.8

Suppose, in addition to the conditions (1), (8) and (12) hold that there exists the

function such that ,0

h condition (13) holds and

(21) .0,)(

)(

t

t

tr

(22)

t

Tt

dss)(inflim 1

for all large t.

(23) .)()(

11suplim 1

t

T

s

Tt

dudsust


179

Proof


equation (1.2) such that ,0)( Tontx for some .00 tT

From the inequality (4.3.19) divided by )(t , we have

)20.3.4().()(

)()(

1 11 t

t

Cdss

t

t

T

Now, we have three cases for )(tx

.

Case 1: If )(tx

is oscillatory, then, there exists a sequence n in ,T with

nt

lim and such that .0)(

nx Then, from the inequality (4.3.20), we have

.)()()(

)(11

2

2

nn

TT

dssCdsssra

sk

Hence, by the condition (22), we get

.)()(

)( 2

2

dsssra

sk

T

This gives, for a positive constant N

Ndsssra

sk

T

)()(

)( 2

2

for every .Tt (4.3.21)

180

Further, by using the Schwarz’s inequality, for ,Tt we obtain

.)(

)(

)(

)()(

)(

)(

)(

)()(

)(

)()( 222

2

2

2

2

2

dss

sr

k

Nads

sk

sradss

sra

skds

sk

sras

sra

skdss

t

T

t

T

t

T

t

T

t

T

By the condition (22), the last inequality becomes

.2

)(2

)( 222222

2

tk

NaTt

k

Nadss

k

Nadss

t

T

t

T

Then,

.2))(()(

)()()(

)(2

))(()()()( 2 t

k

Nads

sxsr

shss

sk

sxsrsdss

t

T

t

T

Thus, for Tt we have

.2

)( 2 tk

Nadss

t

T

(4.3.22)

Integrate the inequality (4.3.20) and from the inequality (4.3.22), we obtain

.2)(2

)()(

)()(

)()(

1

2121

11

tk

Na

T

Ct

k

NaTt

T

C

dsss

dsCdudsu

s

t

T

t

T

t

T

t

T


,2)(

)()(

11suplim 21

1 k

Na

T

Cdudsu

st

t

T

s

Tt

181

which contradicts the condition (23).


then, from (4.3.20), we get

.)(

)()(

1 11

1t

Cdss

t

t

T

Integrate the inequality, dividing by t and taking the limit superior on both sides, we get

,)(

)()(

11suplim

1

11

1 1

T

Cdudsu

st

t

T

t

Tt

which also contradicts the condition (23).

Case 3: If 0)(

tx for ,2 TTt from inequality (4.3.18), we have

.))((

)())(())(()()(

))((

)()()()()()(

))((

)())(()()(

2

22

2

2

01

dssxg

sxsxgsxsrs

dssxg

sxshsdsspsqCsC

txg

txtxtrt

t

T

t

T

t

T

Since )()( tht is non-increasing and by Bonnet’s Theorem for a fixed ,2Tt there

exists tTt ,2 such that

.)(

)()(

))((

)()()(

))((

)()()(

)(

)(

22

22

2

22

t

t

x

Tx

T

t

T

ug

duThT

dssxg

sxThTds

sxg

sxshs

Since the equation (1.2) is superlinear, we have

182

)(

)(2

)(

2

2

2

),()(,)(

)()(,0

)(

tx

Txt

Tx

t

Txxifug

du

Txxif

ug

du

and 0)()( tht , it follows that

,))((

)()()( 3

2

Bdssxg

sxshs

t

T

where .)(

)()()(

223

2

Tx

ug

duThTB

Hence, we have

.))((

)())(())(()()()()()(

))((

)())(()()(

22

2

2

031 dssxg

sxsxgsxsrsdsspsqCsBC

txg

txtxtrtt

T

t

T

Now, we have two cases for dssxg

sxsxgsxsrst

T

2))((

)())(())(()()(2

2

: If this integral is

finite, in this case, we can get a contradiction by the procedure of case (1). If this

integral is infinite, from condition (22), we obtain

.))((

)())(())(()()(

))((

)())(()()(

2

2

2

31 dssxg

sxsxgsxsrsBC

txg

txtxtrtt

T

Also, from the last inequality, we obtain

)23.3.4(,))((

)())(())(()()(

))((

)())(()()(

2

2

2

* dssxg

sxsxgsxsrsN

txg

txtxtrtt

T

where .31

* BCN

183

We consider a 23 TT such that

.0))((

)())(())(()()(3

2

2

2

*

1

dssxg

sxsxgsxsrskNN

T

T

Hence, for all 3Tt , we get

.))((

)())(())(()()(

))((

)())(()()(

2

2

2

*

dssxg

sxsxgsxsrsN

txg

txtxtrtt

T

From the last inequality, we get

.))((

)())((

))((

)())(())(()()(

))((

)())(())(()()(

2

2

2

*

2

2

txg

txtxgds

sxg

sxsxgsxsrsN

txg

txtxgtxtrtt

T

Integrate the last inequality from 3T to t , we have

.,))((

))((ln

))((

)())(())(()()(ln 3

3

12

2

*

2

Tttxg

TxgNds

sxg

sxsxgsxsrsN

t

T

Thus,

.,))((

))((

))((

)())(())(()()(3

3

12

2

*

2

Tttxg

TxgNds

sxg

sxsxgsxsrsN

t

T

From (4.3.23), we have

.,))((

))((

))((

)())(()()(3

3

1 Tttxg

TxgN

txg

txtxtrt

184

By condition (1) and the last inequality, we obtain

.,)()(

1))(()( 3

2

31 Tttrta

TxgNtx

Since ,0))(( 31 TxgN thus, from the last inequality, we obtain

,,)()(

))(()()( 3

2

313

3

Ttsrs

ds

a

TxgNTxtx

t

T

which leads to ,)(lim

txt

which is a contradiction to the fact that 0)( tx for .Tt

Hence the proof is completed.

Example 4.3.8


.0,))(sin()(

)3)((

)()4)(()(2

)()2()()2()(

)3)((

)4)((8

3

6

2

243

18

21333

2

243

tt

txtx

tx

txtxttx

txttxttx

tx

txt

We have

,)( 43

ttr 2)(,0)( 3 ttqth , 3)( xxg , 3

4)(

2

2

x

xx and

3

4)(1 x for all

x R.

,2

),(66

7

vu

uuvu

.00)(1))(sin(

))((

))(),(,(88

tandxallfortptt

tx

txg

txtxtH

185

Taking 5)( t such that

.1

)2(inflim5

)(

)()()(

)(

)(

4)()()(inflim)(inflim)1(

8

3

0

2

1

2

01

dss

sC

dssra

shss

s

sr

k

aspsqCsdss

t

Tt

t

Tt

t

tT

.1

)2(1

suplim

)(

)()()(

)(

)(

4)()(

)(

11suplim)(

)(

11suplim)2(

8

3

0

2

1

201

t

T

s

Tt

s

T

t

Tt

s

T

t

Tt

dsduu

uCt

dudsura

uhuu

u

ur

k

aupuqC

stdudsu

st



the Runge Kutta method of fourth order (RK4) is as follows:

We have

)()(2

)()(3))(sin()())(),(,()(

6

18

2133

txtx

txtxtxtxtxtxtftx




consider ))(),(()())(),(,( txtxltftxtxtH

at t=1, n=980 and h=0.05.

186



0 5 10 15 20 25 30 35 40 45 50-1

-0.8

-0.6

-0.4

-0.2

0

0.2

0.4

0.6

0.8

1

t

Num

eric

al s

olut

ion

x(t)

k tk x(tk)

1

71

181

221

322

381

461

521

581

641

734

781

821

881

981

1

4.5

10

12

17.05

20

24

27

30

33

37.65

40

42

45

50

1

-0.13945104

0.89757769

-0.84742563

0.05022124

-0.93139166

-0.89616464

0.38338342

0.64126671

-0.5028129

0.03915567

-0.40817452

0.27436989

-0.83142722

0.02214472

187

Remark 4.3.7

Theorem 4.3.8 is the extension of the results of Greaf, et al. (1978) and Remili

(2010) who have studied the equation (1.2) when ,1))(( tx 0)( th

and .))(,())())(()()),((( txttxtxtrtxg

Also, Theorem 4.3.8 extends and

improves results of Grace (1992) and Elabbasy, et al. (2005) who have

studied the equation (1.2) as ,1))(( tx ))(())())(()()),((( txgtxtxtrtxg

and

0))(),(,(

txtxtH . Our result can be applied on their equations, but their oscillation

results cannot be applied on the given equation in Example 4.3.8 because their

equations are particular cases of our equation (1.2).

Theorem 4.3.9

Suppose, in addition to the conditions (8), (9) and (12) hold that

(24) .0))()((0)( 00 ttallfortqtrandttallfortr

(25) 0)(

th for all .0t

(26) .0),1( vallforvv

(27) ,)()()(1

suplim 2

dsduupsqsrAt

t

T

s

Tt

where, ,0,: 0tp , then every solution of superlinear equation (1.2) is

oscillatory.

188



By (4.3.17), conditions (8) and (25), we obtain

.)()(

))((

)()()(

)(

)())(()(ttq

txg

txthtp

txg

txtxtr


)24.3.4(,

)(

)())(()()(

)(

)()()(

)(

)())(()(1

t

T

t

T

t

T

dssxg

sxsxsqsrds

sxg

sxshdsspA

txg

txtxtr

where )(

)())(()(1

Txg

TxTxTrA

.

By the Bonnet’s theorem, we see that for each ,Tt there exists tTt , such that

.)(

)(

)()(

)(

)()()(

)(

tt x

TxT

t

Tug

duThds

sxg

sxThds

sxg

sxsh

Since the equation (1.2) is superlinear, so we have

)(

)()(

).()(,)(

)()(,0

)(

tx

Txt

Tx

t

Txxifug

du

Txxif

ug

du

)25.3.4(,)(

)(

)()(2AThds

sxg

sxsht

T

where

)(

2

Txug

duA .

189

By condition (24) and the Bonnet’s Theorem, we see that for each Tt , there exists

tTT ,2 such that

.

)(

)()()(

)(

)())(()()(

)(

)())(()()(

)(

)( 22

tx

Tx

t

T

t

Tug

duutqtrds

sxg

sxsxtqtrds

sxg

sxsxsqsr

Since 0)()( tqtr and the condition (9), we have

,)()()(

)())(()()( 3 tqtrAds

sxg

sxsxsqsr

t

T

(4.3.26)

where .)(

)(inf

)(

)(

3

2

tx

Txug

duuA

From inequalities (4.3.25) and (4.3.26), the inequality (4.3.24) becomes

).()()()(

)(

)())(()(321 tqtrAdsspThAA

txg

txtxtrt

T


.)()()()))((()(

)())(()(321

t

T

s

T

t

T

dsduupsqsrATtThAAdssxg

sxsxsr

Since )(tr is positive and non-increasing for ,Tt the condition (24) and by Bonnet’s

Theorem, there exists tTt , such that

),(

)()(

)(

))(()(

)(

)())(()(4

)(

)(

TrAug

duuTrds

sxg

sxTrds

sxg

sxsxsr tt x

TxT

t

T

190

where

.)(

inf

)(

)(

4

tx

Txug

duuA

Thus, for Tt we have

.)()))((()()()( 4213 TrATtThAAdsduupsqsrA

t

T

s

T


,)(

suplim)1))(((suplim)()()(1

suplim 4

213

t

TrA

t

TThAAdsduupsqsrA

t t

t

Tt

s

Tt


Example 4.3.9


.0,)1)((

))(cos()(

)()(

)()()()(

124

7

2

10

1554

ttxt

txtx

ttxtx

txtxttxttx

t

We note that ,1

)(t

tr ,1)( Rxallforx ,)( tth 4)( ttq and

5)( xxg .

)(),( 223 vuuuvu ,

)(1

)1)((

))(cos()(

)(

))(),(,(424

2

tpttxt

txtx

txg

txtxtH

for all 0t and

,0x and

.61

suplim)()()(1

suplim4

3

33

t

T

s

Tt

s

T

s

Tt

dsu

dusA

tdsduupsqsrA

t

191



the Runge Kutta method of fourth order (RK4) is as follows:

We have

)()(

)()(

1)(

))(cos()())(),(,()(

2

10

155

2

7

txtx

txtx

tx

txtxtxtxtftx



0)( th and finding the values of the functions r, q and f where we consider

))(),(()())(),(,( txtxltftxtxtH

at t=1, n=980 and h=0.05.


k tk x(tk)

1

81

184

232

321

405

461

521

579

627

730

781

829

921

981

1

5

10.15

12.55

17

21.2

24

27

29.9

32.3

37.45

40

42.4

47

50

1

-0.10579952

0.10131997

-0.0022419

0.60071833

1.20586650

-1.08911367

0.53416290

-0.05451221

0.15360619

-0.05346404

0.12804823

-0.02894594

0.40081430

0.19478376

192


0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

erical solu

tion x

(t)

Remark 4.3.8:

Theorem 4.3.9 is the extension of the results of Bihari (1963), Kartsatos (1968),

who have studied the equation (1.2) when ,1)( tr ,0)(,1))(( thtx


and 0))(),(,(

txtxtH and

results of Kamenev (1978) and Wintiner (1949) who have studied the equation (1.2) as

,1)( tr ,1))(( tx ,0)( th ),())(( txtxg ))(())())(()()),((( txgtxtxtrtxg

and 0))(),(,(

txtxtH . Our result can be applied on their equations, but their

oscillation results cannot be applied on the given equation in Example 4.3.9 because

their equations are particular cases of our equation (1.2).

193

Theorem 4.3.10

Suppose the conditions (1), (4) and (8) hold. Moreover, assume that there exists a

differentiable function 0)()(,,0,: 0 tht for 0tt and the continuous

functions 0:),(:, tststDHh R, H has a continuous and non-positive partial

derivative on D with respect to the second variable such that

.0),(,0),( 0tstforstHttforttH o

.),(),(),(),( DstallforstHsthstHs

If

(28) ,),()()(),(

1suplim

0

2

0

t

tt

dsstssrttH

where .),()(

)(),(),(

stHs

ssthst

(29) ,)()()(),(),(

1suplim

0

0

0

t

tt

dsspsqCsstHttH

where ,0,: 0tp

and 0C is a positive constant, then, every solution of

superlinear equation (1.2) is oscillatory.

Proof


(1.2) such that 0,0)( 0 tTsomeforTontx . We define the function as

194

.,

)(

)())(()()()( Tt

txg

txtxtrtt

From )(t , Eq. (1.2), condition (1) and since ,0))()(,1( tt then, there exists a

positive constant 0C such that 0))()(,1( Ctt , we have

.,)()()(

)()(

)()()(

))((

)()()()()()( 2

2

0 Ttttrta

kt

t

ttqtC

txg

txthttptt

Integrating the last inequality multiplied by ),( stH from T to t, we have

)27.3.4(.)()()(

),()(),(

)(

)(

))((

)()()(),()(),()()()(),(

2

2

0

dsssrsa

stkHdssstH

s

s

dssxg

sxshsstHdssstHdsspsqCsstH

t

T

t

T

t

T

t

T

t

T

From the first integral in the R. H. S. for ,Tt we have

t

T

t

T

t

T

TtdssstHsthTTtH

dssstHt

TTtHdssstH

)28.3.4(.,)(),(),()(),(

)(),()(),()(),(

Since H has a continuous and non-positive partial derivative on D with respect to the

second variable and h is non-increasing. The second integral in the R. H. S. is by

using the Bonnet’s theorem twice as follows: for ,Tt there exists tTat , such that

ta

T

t

T

dssxg

sxshsTtHds

sxg

sxshsstH

))((

)()()(),(

))((

)()()(),(

195

and tt aTb , such that

)(

)(

.)(

)()(),(

))((

)()()(),(

))((

)()()(),(

t

tt

bx

Tx

b

T

a

T

ug

duThTTtH

dssxg

sxThTTtHds

sxg

sxshsTtH

Since H and )(t are positive functions, by condition (4) and the equation (1.2) is

superlinear, we have

)(

)()(

).()(,)(

);()(,0

)(

tbx

Txt

Tx

t

Txbxifug

du

Txbxif

ug

du

Thus, it follows that

)29.3.4(),()(),())((

)()()(),(

))((

)()()(),(1 ThTTtHAds

sxg

sxshsTtHds

sxg

sxshsstH ta

T

t

T

where

)(

)(

1 .)(

inftbx

Txug

duA

Thus, from (4.3.28) and (4.3.29), the inequality (4.3.27) becomes

.)(),(),()(

)(),()(

)()(

),(

)()(),()(),()()()(),(

2

2

10

dssstHstHs

ssths

srsa

stkH

ThTTtHATTtHdsspsqCsstH

t

T

t

T

Since 0)()(),(1 ThTTtHA and for ,Tt we have

196

t

T

t

T

dssstHstssrsa

stkHTTtHdsspsqCsstH ,)(),(),()(

)()(

),()(),()()()(),( 2

2

0

where .),()(

)(),(),( stH

s

ssthst

Hence, we have

)30.3.4(.),()()(

2

1)(

)()(

),(

),(4

)()()(),()()()(),(

2

2

2

220

dsstk

srsas

srsa

stkH

dsstk

srsaTTtHdsspsqCsstH

t

T

t

T

t

T

Then, for ,Tt we have

.,),()()(4

)(),()()()(),( 220 Ttdsstssr

k

aTTtHdsspsqCsstH

t

T

t

T

Dividing the last inequality by ),( TtH , taking the limit superior and by condition (28),

we obtain

,),()()(),(

1suplim

4)()()()(),(

),(

1suplim 22

0

dsstssrTtHk

aTdsspsqCsstH

TtH

t

Tt

t

Tt


Theorem 4.3.11

Suppose, in addition to the conditions (1), (4), (8) and (28) hold that there exist

continuous functions h and H are defined as in Theorem 4.3.10 and

197

(30) .),(

),(infliminf0

00

ttH

stH

tts

If there exists a continuous function ,0ton such that

(31)

t

Tt

Tdsstssrk

aspsqCsstH

TtH)(),()()(

4)()()(),(

),(

1suplim 22

0

for 0tT , where ,),()(

)(),(),( stH

s

ssthst

k is a positive constant and a

differentiable function ),0(,: 0 t and

(32)

T

dssrs

s

)()(

)(2

where 0),(max)( tt , then, every solution of superlinear equation (1.2) is

oscillatory.

Proof

Without loss of generality, we may assume that there exists a solution )(tx of


Dividing inequality (4.3.30) by ),( TtH and taking the limit superior as ,t we

obtain

198

t

Tt

t

Tt

t

Tt

dsstk

srsas

srsa

stkH

TtHT

dsstk

srsas

srsa

stKH

TtH

Tdsstsrsk

aspsqCsstH

TtH

2

2

2

2

2

2

220

),)()(

2

1)(

)()(

),(

),(

1inflim)(

),()()(

2

1)(

)()(

),(

),(

1suplim

)(),()()(4

)()()(),(),(

1suplim

By condition (31), we get

.),()()(

2

1)(

)()(

),(

),(

1inflim)()(

2

2

2

t

Tt

dsstk

srsas

srsa

stkH

TtHTT

This shows that

,)()( TteveryforTT (4.3.31)

and

.),()()(

2

1)(

)()(

),(

),(

1inflim

2

2

2

t

Tt

dsstk

srsas

srsa

stkH

TtH

Hence,

)32.3.4(.)(),(),(),(

1)(

)()(

),(

),(

1inflim

),()()(

2

1)(

)()(

),(

),(

1inflim

0 0

0

0

2

20

2

2

20

t

t

t

tt

t

tt

dssstHstttH

dsssrsa

stkH

ttH

dsstk

srsas

srsa

stkH

ttH

Define

199

0

2

0

,)()()(

),(

),(

1)(

0

ttdsssrs

stkH

ttHtU

t

t

and

.,)(),(),(),(

1)( 0

00

ttdssstHstttH

tV

t

t

Then, (4.3.32) becomes

.)()(inflim

tVtUt

(4.3.33)

Now, suppose that

.)()(

)(

0

2

t

dssrs

s

(4.3.34)

Then, by condition (30) we can easily see that

.)(lim

tUt

(4.3.35)

Let us consider a sequence ,...3,2,1nnT in

n

nTwitht lim,0 and such that

.)()(inflim)()(lim tVtUTVTUt

nnn

By inequality (4.3.33), there exists a constant N such that

,...3,2,1,)()( nNTVTU nn (4.3.36)


200

,)(lim

nn

TU (4.3.37)

and hence inequality (4.3.36) gives

.)(lim

nn

TV (4.3.38)

By taking into account inequality (4.3.37), from inequality (4.3.36), we obtain

,2

1

)()(

)(1

nn

n

TU

N

TU

TV

provided that n is sufficiently large. Thus

2

1

)(

)(

n

n

TU

TV,

which by inequality (4.3.38) and inequality (4.3.37) we have

.)(

)(lim

2

n

n

n TU

TV (4.3.39)

On the other hand by Schwarz’s inequality, we have

).(),()()(

),(

1

)()()(

),(

),(

1),(

)()(

),(

1

)(),(),(),(

1)(

22

0

2

20

22

0

2

0

2

2

0

00

0

nn

T

tn

T

t

n

n

n

T

tn

n

T

t

n

n

n

TUdssTk

srsa

tTH

dssssra

sTkH

tTHdssT

k

srsa

tTH

dsssTHsTtTH

TV

n

nn

n

Thus, we have

201

dssTk

srsa

tTHTU

TVn

T

tnn

nn

),()()(

),(

1

)(

)( 22

0

2

0

for large n.

By inequality (4.3.39), we have

.),()()(),(

1lim 2

0

2

0

dssTssr

tTHk

an

T

tnn

n

Consequently,

t

tt

dsstssrttH

0

,),()()(),(

1suplim 2

0

which contradicts to the condition (28), Thus inequality (4.3.34) fails and hence

.)()(

)(

0

2

t

dsssr

s

Hence from inequality (4.3.31), we have

,)()(

)(

)()(

)(

00

22

tt

dsssr

sds

ssr

s

which, contradicts to the condition (32), hence the proof is completed.

Example 4.3.10


.0,)1)((

))((sin)(

)()1)((

)2)((6)(9

)()(

1)()(

)1)((

)2)((2

29

18

26

2126

1337

3226

2

ttx

txtx

txtxt

txtx

txtx

tt

txtx

txt

tx

202

Here we note

1818

197

322

2

6 69),(,)(,

1)(,

1)(,

)1)((

)2)(()(,

1)(

vu

uuvuxxg

ttq

tthRxallfor

tx

txx

ttr

and .)(0)1)((

)(

)1)((

))((sin)(

))((

))(),(,(02

2

2

22

ttallfortptx

tx

tx

txtx

txg

txtxtH

Let ,0)(),( 0

2 tstallforststH thus

),(),()(2),( stHsthststHs

for all 00 tt . Taking 6)( t such that

,01

)(

24suplim

),()(

)(),()()(

),(

1suplim),()()(

),(

1suplim

62

2

2

t

Tt

t

Tt

t

Tt

dssTt

dsstHs

ssthssr

TtHdsstssr

TtH

,1)1(inf)(

)(infliminf

),(

),(infliminf

0002

0

2

0

tsttstts tt

st

ttH

stH

,),(

),(infliminf0

00

ttH

stHthus

tts

.4

3312)(6

)(

1suplim

),(4

)()()()()(),(

),(

1suplim

2

0

2

0

63

2

02

220

T

C

T

Cds

kss

stC

Tt

dsstk

ssraspsqCsstH

TtH

t

Tt

t

Tt

Set 2

0

4

3)(

T

CT , then

2

0

4

3)(

T

CT

and

TT

dssC

dsssr

s.

32

3

)()(

)( 2

2

02

203

All conditions of Theorem 4.3.11 are satisfied, thus, the given equation is


using the Runge Kutta method of fourth order (RK4). We have

18

126

1337

2

29

)(6)(9

)()(

1)(

))((sin)())(),(,()(

txtx

txtx

tx

txtxtxtxtftx


xx on the chosen interval [1,50] , the functions


consider ),()(),,(



k tk x(tk)

1

81

161

221

301

381

461

521

581

642

721

781

821

925

981

1

5

9

12

16

20

24

27

30

33.05

37

40

42

74.2

50

-1

0.59598105

-0.15335506

0.45164375

-0.00824951

-0.43550908

0.87072286

-0.58206656

0.28505118

-0.02507562

0.43312716

-0.13643410

-0.89118231

-0.43426037

-0.01036663

204


0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

t

Num

eric

al s

olut

ion

x(t)

Remark 4.3.9

Theorem 4.3.10 and Theorem 4.3.11 extend and improve Results of Kamenev

(1978), Results of Philos (1989) and Results of Yan (1986) who studied the equation

(1.2) as )())())(()()),(((,0)(,1))((,1)( txtxtxtrtxgthtxtr

and

.0))(),(,(

txtxtH Our result can be applied on their equation, but their oscillation

results cannot be applied on the given equation in Example 4.3.10 because their

equation is particular case of our equation (1.2).

We need the following lemma which will significantly simplify the proof of our next

Theorem.

Let },:),{( 0tststD we say that a function ),( RDCH belongs to the class W if

(1) 0),( ttH for 0tt and 0),( stH when ;st

205

(2) ),( stH has partial derivatives on D such that

,),(),(),( 1 stHsthstHt

,),(),(),( 2 stHsthstHs

for all ,),( Dst and some ).,(, 1

21 RDLhh loc

Lemma 4.3.1: Let ),,(,, 0210 RtCAAA with 02 A and ).,,( 0

1 RtCz If

there exist ,),( 0tba and ),( bac such that

)40.3.4(),,(,)()()( 2

210 baszsAzsAsAz

then,

)41.3.4(,0),()(4

)()(),(),(

1

),()(4

)()(),(),(

1

2

2

2

0

2

1

2

0

b

c

c

a

dssbsA

ksAscbH

cbH

dsassA

ksAsasH

acH

for all WH where

),()(),(),( 111 asHsAashas

and

.),()(),(),( 122 sbHsAsbhsb

206

The proof of this lemma is similar to that of Lu and Meng (2007) and hence will be

omitted.

Theorem 4.3.12

Suppose in addition to the condition (8) holds that 1)( x for x R and assume

that there exist ,, Tbac and WH such that

b

c

c

a

dssbsrs

kspsqCscbH

cbH

dsassrs

kspsqCsasH

acH

,0),()()(4

)()()(),(),(

1

),()()(4

)()()(),(),(

1)33(

2

20

2

10

where

),()(

)(

)(

)(),(),(

,),()(

)(

)(

)(),(),(

22

11

tbHtr

th

t

ttbhtb

atHtr

th

t

tathat

and the function is defined as in Theorem 4.3.10. Then, every solution of equation

(1.2) is oscillatory.

Proof

Without loss of generality, we assume that there exists a solution )(tx of equation

(1.2) such that .0,0)( 0 tTsomeforTontx We define the function as

.,

)(

)()()()( Tt

txg

txtrtt

207


,),()()(

)()(

)())(,1()()()(

)(

)()()()( 2

1 Ttttrt

kt

t

ttvtqtt

tr

thtptt

where .)()()(1 tttv

Since 0))(,1( 1 tv then there exists 0C such that ,))(,1( 01 Ctv we have

.,)()()(

)()(

)(

)(

)()()()()( 2 Ttt

trt

kt

tr

th

t

ttptqtt

From the last inequality and by lemma 4.3.1, we conclude that for any bac , and

WH

b

c

c

a

dssbsrs

kspsqCscbH

cbH

dsassrs

kspsqCsasH

acH

,0),()()(4

)()()(),(),(

1

),()()(4

)()()(),(),(

1

2

20

2

10

which contradicts (33). Thus, the equation (1.2) is oscillatory.

Remark 4.3.10

Theorem 4.3.12 is the extension of results of Lu and Meng (2007) who have

studied the equation (1.2) when ,1))(( tx ))(())())(()()),((( txgtxtxtrtxg

and

.0))(),(,(

txtxtH Our result can be applied on their equation, but their oscillation

results cannot be applied on our equation (1.2) because their equation is a particular

case of our equation (1.2).

208

4.4 Conclusion

In this section, the problem of finding the sufficient conditions for oscillation of

solutions of ordinary differential equations of second order with damping term of type

(1.2) is considered. We present some oscillation results that contain the sufficient

conditions for oscillation of solutions of the equation of type (1.2). These sufficient

conditions have been derived by using the generalized Riccati technique. Our results

extend and improve many previous results that have been obtained before, for example,

such as the works of Fite (1918), Wintner (1949), Atkinson (1955), Bihari (1963),

Kartsatos (1968), Greaf, et al. (1978), Grace (1992), Elabbasy et al. (2005), Lu & Meng

(2007), Berkani (2008), and Remili (2010). All these previous results have been studied

for particular cases of the equation (1.2) whereas our sufficient conditions have been

derived for the generalized equation (1.2). A number of theorems and illustrative

examples for oscillation differential equation of type (1.2) are given. Further, a number

of numerical examples are given to illustrate the theorems which are computed by using

Runge Kutta of fourth order function in Matlab version 2009. The present results are

compared with existing results to explain the motivation of proposed research study.

209

CHAPTER FIVE

OSCILLATION OF THIRD ORDER NONLINEAR

ORDINARY DIFFERENTIAL EQUATIONS

5.1 Introduction

In this chapter, we are concerned with the problem of oscillation of third order non-

linear ordinary differential equation of the form

)3.1()),(),(),(,())(()())(()( 1 txtxtxtHtxgtqtxftr

where q and r are continuous functions on the interval )(,0,, 00 trtt is a positive

function, 1g is continuously differentiable function on the real line R except possibly at 0

with 0)(1 yyg and 0)(1 kyg for all ,0y f is a continuous function on R and

,: 0tH ×R×R×RR is a continuous function such that )()(),,,( 1 tpygzyxtH for

all y 0 and 0tt .

5.2 Third Order Nonlinear ODE Of Type (1.3)

In this chapter, we present the oscillation results of our study of finding the

sufficient conditions for oscillation of solutions of ordinary differential equations of

third order of type (1.3). The present oscillation results have among other finding

extended and improved many previous oscillation results, for examples, such as the

210

works of Hanan (1961), Lazer (1966), Jones (1973), Mehri (1976), Parhi & Das (1990),

Parhi & Das (1993), Adamets & Lomtatidze (2001) and Remili (2007). We have

established some new sufficient conditions which guarantee that our differential

equations are oscillatory. A number of theorems and an illustrative example for

oscillation differential equation of type (1.3) are shown. Further, a numerical example is

given to illustrate the theorems. This numerical example is computed by using Runge

Kutta of fourth order function in Matlab version 2009. The present results are compared

with existing results to explain the motivation of proposed research study.


Theorem 5.2.1

Suppose that

(1) ,0)(

01

allforug

du

(2) 21

)(0 k

z

zfk for all ,0z

(3) ,,)(

01 tTallforMsr

ds

T

(4) ,)()()(

1suplim

t

T

s

Tt

dsduupuqsr

where ,0,: 0tp . Then, every solution of equation (1.3) is oscillatory.

Proof


equation (1.3) such that 0)( tx and 0)(

tx on ,T for some .00 tT Define

211

.,

))((

))(()()(

1

Tt

txg

txftrt

This and (1.3) and condition (2), we have

.),()(

))((

))(()()(

1

Tttqtp

txg

txftrt

Integrate the last inequality from T to t and also by condition (2), we have

)1.3.5(.,)()()(

))((

)()(

1

1 TtdsspsqT

txg

txtrkt

T

Integrate (5.3.1) divided by )(tr from T to t, we obtain

)2.3.5(.,)()()(

1

)()(

))((

)(

1

1 Ttdsduupuqsrsr

dsT

sxg

dssxk

s

T

t

T

t

T

t

T

By condition (1), we obtain

)(

)()(

)()(,)(

)()(,0

)(

tx

TxTx

Txtxifug

du

Txtxif

ug

du

This follows that

,

))((

)(1

1

Ads

sxg

sxt

T

where

)(1

1)(

inf

Txug

duA .

212


.,)(

)()()()(

111 TtAk

sr

dsTdsduupuq

sr

s

T

t

T

t

T

By condition (3) and taking the limit superior on both sides, we have

,})({suplim)()()(

1suplim 111

AkMTdsduupuqsr

s

T t

t

Tt

as ,t which contradicts to the condition (4). Hence, the proof is completed.

Theorem 5.3.2

Suppose, in addition to the conditions (1) and (2) hold that there exists the

differentiable function ,0,: 0t and 0

such that

(5) ,)()(

02 tTallforMssr

ds

T

(6) .)(

)()()()()(

)()(

12

3

dsduu

urukupuqu

srs

s

TT


Proof




213

.,

))((

))(()()()(

1

Tt

txg

txftrtt

This, (1.3) and the condition (2), we have

.,

))((

)()()(

))((

)()()()()()()()(

2

1

2

1

1

2 Tt

txg

txtrtkk

txg

txtrtktqttptt

Integrate the last inequality, we get

)3.3.5(.)(

))((

)()()(

))((

)()()()()()(

))((

))(()()(

1

22

2

1

1

1

dssx

sxg

srsksx

sxg

srskkdsspsqsT

txg

txftrtt

T

t

T

From the second integral in the R. H. S., we have

)4.3.5()(

)()(

)(

)()(

4

)(2

)()()(

))((

)()()(

))((

)()()(

))((

)()(

2

3

2

1

2

2

2

1

2

1

1

1

2

2

1

1

ds

srsk

dss

srs

kk

k

dsskk

srsksx

sxg

srskkdssx

sxg

srssx

sxg

srskk

t

T

t

T

t

T

t

T

where .4 1

2

23

kk

kk

From (5.3.4) in (5.3.3), we obtain

.)(

)()()()()()(

))((

))(()()(2

3

1

t

T

dss

srskspsqsT

txg

txftrt

214

By condition (2) and integrating the last inequality divided by )()( trt , we obtain

s

T

t

T

t

T

t

T

dsduu

urukupuqu

srssrs

dsTds

sxg

sxk

)(

)()()()()(

)()(

1

)()()(

))((

)(2

3

1

1

By using the condition (1) and as in Theorem 5.3.1 the first integral in the L. H. S. is

bounded and the condition (5), then, we get

,)()(

)()()()()(

)()(

1112

2

3

AkMTdsduu

urukupuqu

srs

s

T

t

T

as ,t which contradicts to the condition (6). Hence, the proof is completed.

Theorem 5.3.3

Suppose, in addition to the conditions (1) and (2) hold that there exists the

differentiable function ,0,: 0t , 0)(,0

r and 0)(

r such that

(7)

T

srs

ds,

)()(

(8) ,)()()(inflim

t

Tt

dsspsqs

(9) ,)()()(1

suplim

t

T

s

Tt

dsduupuqut

then, every solution of equation (1.3) is oscillatory.

215

Proof: Without loss of generality, we assume that there exists a solution 0)( tx of



.,

))((

))(()()()(

1

Tt

txg

txftrtt

This and the equation (1.3), we have

.,

))((

))(()())(()()(

))((

))(()()()()()()()(

2

1

1

1

Tt

txg

txgtxtxftrt

txg

txftrttqttptt


)5.3.5(.),()()(

))((

))((

)()()()()()()( 2

2

1

1

2 Ttttrtk

txg

txg

txtrtktptqtt

Integrating (5.3.5) from T to t, we get

)6.3.5(.)()()(

1

))((

)()()()()()()()( 2

21

2 dsssrsk

kds

sxg

sxsrsktTdsspsqs

t

T

t

T

t

T

Since )()( trt

is a decreasing function, then by the Bonnet’s Theorem there exists a

tTat , such that the first integral in the R. H. S. becomes

)(

)(1

2

1

2 .)(

)()(

))((

)()()( tax

Tx

t

Tug

duTrTkds

sxg

sxsrsk

216


)(

)()(

2

2).()(,

)()()(

),()(,0

)()()(

tax

Txt

Tx

t

Txaxifug

duTrTk

Txaxif

ug

duTrTk

Hence

)7.3.5(,)(

)()(0

)(

)(

22

tax

Tx

Aug

duTrTk

where

)(

22)(

)()(

Txug

duTrTkA .

From (5.3.7) in (5.3.6), we get

)8.3.5(.)()()(

1)()()()()( 2

2

2 dsssrsk

kAtTdsspsqs

t

T

t

T

We have two cases for the integral

.)()()(

1 2 dsssrs

t

T

Case 1:

dsssrs

T

)()()(

1 2

is finite.

Thus, there exists a positive constant B such that

.)()()(

1 2 TtforBdsssrs

t

T

217

Thus, (5.3.8) becomes

,,)()()()( 3 TttAdsspsqs

t

T

where .)( 223 kkBATA

Integrating the last inequality from T to t, we get

)9.3.5(.)()()()()( 3 dssTtAdsduupuqu

t

T

t

T

s

T

By using the Schwarz’s inequality, we obtain

)10.3.5(.)()(

)()()()(

)()()(

)()(

)()()(

222

t

T

t

T

t

T

t

T

t

T

t

T

dssrsB

dssrsdssrs

sdssrs

srs

sdssdss

From (5.3.10) in (5.3.9), we obtain

).)(()()(

)()()()()()(

3

3

TtTrTBTtA

dssrsBTtAdsduupuqu

t

T

t

T

s

T

Dividing the last inequality by t and taking the limit superior on both sides, we have

,1)()(suplim)()()(1

suplim 3

t

TTrTBAdsduupuqu

t t

t

T

s

Tt

as ,t which contradicts to the condition (9).

Case 2: If the integral

dsssrs

T

)()()(

1 2

is infinite,

218

by condition ,))((1 ktxg

we get

.)()()(

))(( 21

dsssrs

sxg

T

(5.3.11)

Integrating (5.3.5), from (5.3.7) and condition (8), it follows that here exists a constant

A4 such that

)12.3.5(,)()()(

))((1)( 21

2

5 dsssrs

sxg

kAt

t

T

where .)( 425 AATA

From (5.3.11), we get )(t is negative on ,T . Furthermore, we choose a TT 1 such

that

.)()()(

))((1 1

21

2

56 dsssrs

sxg

kAA

T

T

Thus, for ,1Tt we have

.

))((

)())((

))((

))(())(()(

)()(

))((1

)()(

)())((

1

1

12

1

1

21

2

5

2

2

1

txg

txtxg

txgk

txftxgdss

srs

sxg

kA

trtk

ttxgt

T

Thus

.

))((

))((log)(

)()(

))((1log

1

116

21

2

5

txg

TxgAdss

srs

sxg

kA

t

T

Hence

219

.

))((

))(()(

)()(

))((1

1

116

21

2

5

txg

TxgAdss

srs

sxg

kA

t

T

By (5.3.12), we get

.

))((

))(()(

1

116

txg

TxgAt

From the last inequality and condition (2), we obtain

.)()(

1))(()( 1161

trtTxgAtxk

Integrate from 1T to t and condition (7), we have

,)()(

))(()()(

1

116111 t

Tsrs

dsTxgATxktxk

as ,t

this contradicts to the assumption that 0)(

tx . This completes the proof of Theorem

5.3.3.

Example 5.3.3


.0,))((sin)(

)()(1

6

25

5

4

4

tt

txtxtxttx

t

We have 4

4)(,

1)( ttq

ttr and 0)(,)(

65

xxgxxxg for all 0

x and

220

.4

1

)(0

4

ug

du

(1) )())(( txtxf

and 21

)(

))((10

tx

txf for all .0

x

(2) 6

2

5

))((sin)())(),(),(,(

t

txtxtxtxtxtH

and

)(1))((sin

))((

))(),(),(,(66

2

1

tptt

tx

txg

txtxtxtH

for all 0

x and .0t

Taking 01

)()(,2)(,)( 2

ttrtttt and 0

1)()(

2

ttr for 0t such that

(3)

T T

dsssrs

ds,

)()(

2

(4) ,1

inflim)()()(inflim6

42

t

T

t

Ttt

dss

ssdsspsqs

,7

6

6

1

3

1

66

1

42

1suplim

11suplim)()()(

1suplim)5(

7

23

6

2

7

6

42

T

Tt

T

T

t

t

t

dsduu

uut

dsduupuqut

t

t

T

s

Tt

t

T

s

Tt

All conditions of Theorem 5.3.3 are satisfied, thus, the given equation is


using Runge Kutta method of fourth order. We have

),())((sin))((

))(),(),(,()(5

2

2

5

txt

txttxtxtxtxtftx

221

with initial conditions 0)1(,1)1(,5.0)1(

xxx on the chosen interval [1,50] and

finding the values of the functions r, q and f where we consider

).,,()(),,,(

xxxltgxxxtH


k tk x(tk)

1

80

181

221

321

379

461

521

615

721

800

868

929

981

1

4.954

10.009

12.011

17.016

20.92

24.023

27.026

31.731

37.036

40.99

44.444

47.447

50

0.5

-0.01467758

0.76997118

-0.77145176

1.56655689

-0.02744315

-0.93017943

1.21756203

-0.02268732

0.41913726

1.92358434

-0.00644259

-1.29549185

-0.45092350

222


0 5 10 15 20 25 30 35 40 45 50-1.5

-1

-0.5

0

0.5

1

1.5

2

t

Num

erical solu

tion x

(t)

Remark 5.3.1

Our theorems extend and improve the obtained results by Hanan (1961) and

Adamets and Lomtatidze (2001) for the equation (2.14), results of Lazer [1966] and

Jones (1973) for the equation (2.15) and result of Kiguradze (1992) for the equation

(2.18), as mentioned in Chapter Two. Our results can be applied on their equations but,

their oscillation results cannot be applied on the equation (1.3) because their equations

are special cases of the equation (1.3).

223

5.4 Conclusion

In this section, we present the oscillation results of our study of finding the

sufficient conditions for oscillation of solutions of ordinary differential equations of

third order of type (1.3). Some oscillation have been introduced which contain the

sufficient conditions for oscillation of solutions of the equation of type (1.3). These

sufficient conditions have been established by using the generalized Riccati technique.

Our results extend and improve many previous results that have been obtained before,

for example, such as the works of Hanan (1961), Lazer (1966), Jones (1973), Mehri

(1976), Parhi & Das (1990), Parhi & Das (1993), Adamets & Lomtatidze (2001) and

Remili (2007). All these previous results have been studied for particular cases of the

equation (1.3) whereas our sufficient conditions have been derived for the generalized

equation (1.3). A number of theorems and an illustrative example for oscillation

differential equation of type (1.3) are given. Further, the numerical example is given to

illustrate the theorem which is computed by using Runge Kutta of fourth order function

in Matlab version 2009. The present results are compared with existing results to

explain the motivation of proposed research study.

224

CHAPTER SIX

CONCLUSION AND FUTURE WORK

6.1 Conclusion

Over the past three decades, there have been many studies which have dealt with

the oscillatory properties of nonlinear ordinary differential equations. The problem of

finding oscillation criteria for second order nonlinear differential equations has received

a great deal of attention in the 20 years from the publication of the classic paper by

Atkinson (1955). The study of the oscillation of second order nonlinear ordinary

differential equations with alternating coefficients is of special interest because of the

fact that many physical systems are modeled by second order nonlinear ordinary

differential equations.

In this thesis, we are concerned with oscillation behavior of solutions of non-

linear ordinary differential equations of second order and third order with variable

coefficients. The main results are presented in chapter three, chapter four and five.

Oscillation of second order nonlinear differential equation with alternating coefficients

of type (1.1) has been investigated in chapter three. The present oscillation results

contain the sufficient conditions for oscillation of solutions of the equation of type (1.1)

which have been derived by using the generalized Riccati technique. Our results extend

and improve many previous results that have been obtained before, for example, such as

the works of Fite (1918), Wintner (1949), Philos (1989) for the equation (2.1), Atkinson

(1955) for the equation (2.4), Bihari (1963), Kartsatos (1968) for equation (2.5),

225

Elabbasy (1996) for the equation (2.6), and Elabbasy et al. (2005) for the equation (2.9).

All these previous results have been studied for particular cases of the equation (1.1)

whereas our sufficient conditions have been derived for the generalized equation (1.1).

A number of oscillation theorems differential equation of type (1.1) are given. Further, a

number of numerical examples are given to illustrate the theorems which are computed

by using Runge Kutta of fourth order function in Matlab version 2009. The present

results are compared with existing results to explain the motivation of our oscillation

results for equation (1.1).

In chapter four, the problem of finding the sufficient conditions for oscillation of

solutions of ordinary differential equations of second order with damping term of type

(1.2) is considered. The present oscillation results contain the sufficient conditions for

oscillation of solutions of the equation of type (1.2). These sufficient conditions have

been derived by using the generalized Riccati technique. Our results extend and

improve many previous results that have been obtained before, for example, such as the

works of Fite (1918), Wintner (1949) for the equation (2.1), Atkinson (1955) for the

equation (2.4), Bihari (1963), Kartsatos (1968) for equation (2.5), Greaf, et al. (1978),

Grace (1992), Elabbasy et al. (2005), Lu & Meng (2007), Berkani (2008), and Remili

(2010). All these previous results have been studied for particular cases of the equation

(1.2) (as mentioned in chapter two) whereas our sufficient conditions have been derived

for the generalized equation (1.2). A number of theorems and illustrative examples for

oscillation differential equation of type (1.2) are given. Further, a number of numerical

examples are given to illustrate the theorems which are computed by using Runge Kutta

of fourth order function in Matlab version 2009. The present results are compared with

existing results to explain the motivation of our oscillation results for the equation (1.2).

226

Therefore, we present the oscillation results of our study of finding the sufficient

conditions for oscillation of solutions of ordinary differential equations of third order of

type (1.3) in chapter five. Some oscillation results have been introduced which contain

the sufficient conditions for oscillation of solutions of the equation of type (1.3). These

sufficient conditions have been established by using the generalized Riccati technique.

The present results extend and improve many previous results that have been obtained

before, for example, such as the works of Hanan (1961), Lazer (1966), Jones (1973),

Mehri (1976), Parhi & Das (1990), Parhi & Das (1993), Adamets & Lomtatidze (2001)

and Remili (2007). All these previous results have been studied for particular cases of

the equation (1.3) (as mentioned in chapter two) whereas our sufficient conditions have

been derived for the generalized equation (1.3). A number of theorems and an

illustrative example for oscillation differential equation of type (1.3) are given. Further,

the numerical example is given to illustrate the theorem which is computed by using

Runge Kutta of fourth order function in Matlab version 2009. The present results are

compared with existing results to explain the motivation of proposed research study.

We compare our results with other previous oscillation results in the literature to show

that our oscillation results are more general where our sufficient conditions are derived

to more general equations. These results can be applied to many particular cases of our

general equations but many previous oscillation results cannot applied to our equations

since all terms of our equations are not included in their studies. This is the main

advantage of our research work.

227

6.2 Future Work

Many phenomena in different branches of sciences are interpreted in terms of

second order differential equations and their solutions. In future, this research work will

be continued to the study of oscillation behavior of the higher nonlinear ordinary

differential equations and also partial differential equations.

228

REFERENCES

Adamets, L. and Lomtatidze, A. (2001). Oscillation conditions for a third order linear

equation. Differ. Uran., 37(6), 723-729, 861. [Russian]. Translation in Differ. Equ.,

37(6).

Atkinson, F. V. (1955). On second order nonlinear oscillations. Pacific J. Math., 5, 643-

647.

Ayanlar, B. and Tiryaki, A. (2000). Oscillation theorems for nonlinear second order

differential equations with damping. Acta Math. Hungar., 89, 1-13.

Bartle, N. G. (1970). The elements of real analysis. Seven Edition. John Wiley and

Sons, p. 233.

Berkani, A. (2008). Sufficient conditions for the oscillation of solutions to nonlinear

second order differential equations. Elect. J. of Differ. Eq., Vol. 2008, No. 03, 1–6.

Bhatia, N. P. (1966). Some oscillation theorems for second order differential equations.

J. Math. Anal. Appl., 15, 442-446.

Bihari, I. (1963). An oscillation theorem concerning the half linear differential equation

of the second order. Magyar Tud. Akad.Mat. Kutato Int.Kozl. 8, 275-280.

Coles, W. J. (1968). An oscillation criterion for the second order differential equations.

Proc. Amer. Math. Soc., 19, 755-759.

229

Das, P. (1995). On oscillation of third order forced equations. J. Math. Anal. Appl., 196,

502-513.

Elabbasy E. M. (1996). On the oscillation of nonlinear second order differential

equations. Pan Amer. Math. J., 4, 69-84.

Elabbasy, E. M. (2000). On the oscillation of nonlinear second order differential

equations. Appl. Math. Comp., 8, 76-83.

Elabbasy, E. M. and Elhaddad, W. W. (2007). Oscillation of second order non-linear

differential equations with a damping term. Elect. J. of Qual. Theory of Differ. Eq., 25,

1-19.

Elabbasy, E. M. and Elzeiny, Sh. R. (2010). Oscillation theorems concerning nonlinear

differential equations of the second order. Opuscula Mathematica, vol. 31, No. 3, 373-

390.

Elabbasy, E. M., Hassan, T. S. and Saker, S. H. (2005). Oscillation of second order

nonlinear differential equations with a dampind term. Elec. J. of differ. Eq., 76, 1-13.

Erbe, L. (1976). Existence of oscillatory solutions and asymptotic behavior for a class of

a third order linear differential equations. Pacific J. Math., 64, 369-385.

Erbe, L. (1976). Oscillation, non-oscillation and asymptotic behavior for third order

nonlinear differential equations. Ann. Math. Pura Appl., 110, 373-393.

Fite, W. B. (1918). Concerning the zeros of the solutions of certain differential

equations. Trans. Amer. Math. Soc., 19, 341-352.

230

Grace, S. R. (1992). Oscillation theorems for nonlinear differential equations of second

order. J. Math. Anal. Appl., 171, 220-241.

Grace, S. R. and Lalli, B. S. (1980). Oscillation theorems for certain second perturbed

differential equations. J. Math. Anal. Appl., 77, 205-214.

Greaf, J. R., Rankin, S. M. and Spikes, P. W. (1978). Oscillation theorems for perturbed

nonlinear differential equations. J. Math. Anal. Appl., 65, 375-390.

Hanan, M. (1961). Oscillation criteria for a third order linear differential equations.

Pacific J. Math., 11, 919-944.

Hartman (1952). Non-oscillatory linear differential equations of second order. Amer. J.

Math., 74, 389-400.

Heidel, J. W. (1968). Qualitative behavior of solutions of a third order nonlinear

differential equation. Pacific J. Math., 27, 507-526.

Jones, G. D. (1973). An asymptotic property pf solution of .0)()( yxqyxpy Pacific J. Math., 48, 135-138.

Kamenev, I.V. (1978). Integral criterion for oscillation of linear differential equations of

second order. Math. Zametki, 23, 249-251.

Kartsatos, A. G. (1968). On oscillations of nonlinear equations of second order. J. Math.

Anal. Appl., 24, 665-668.

231

Kiguradze, I. T. (1967). A note on oscillations of nonlinear equations of second

order .0sgn)(

uutau n Casopis Pest. Math., 92, 343-350.

Kiguradze, I. T. (1992). An oscillation criterion for a class of ordinary differential

equations. Differentsial’nye Uravneniya, Vol. 28, No. 2, 207–219.

Kim, W.J. (1976). Oscillatory properties of linear third order differential equations.


Kirane, M. and Rogovchenko, Y. V. (2001). On oscillation of nonlinear second order

differential equation with damping term. Appl. Math. and Compu., 117, 177-192.

Lazer, A. C. (1966). The behavior of solutions of the differential equation

.0)()( yxqyxpy Pacific J. Math., 17, 435-466.

Lee, Ch. F., and Yeh, Ch. Ch. (2007). An oscillation theorem, Appl. Math. Letters. 20,

238-240.

Lu, F. and Meng, F. (2007). Oscillation theorems for superlinear second-order damped

differential equations. Appl. Math. and Compu., 189, 796-804.

Manojlovic, J. V. (1991). Oscillation theorems for nonlinear second order differential

equations. Computers and Math. Applic., 1-14.

232

Manojlovic, J. V. (2000). Oscillation criteria of second order sublinear differential

equations. Computers and Math. Applic., 39, 161-172.

Manojlovic, J. V. (2001). Integral averages and oscillation of second order nonlinear

differential equations. Computers and Math. Applic., 41, 1521-1534.

Mehri, B. (1976). On the conditions for the oscillation of solutions of nonlinear third

order differential equations. Cas. Pest Math., 101, 124-129.

Moore, R. A. (1955). The behavior of solutions of a linear differential equation of

second order. Pacific J. Math., 51, 125-145.

Nagabuchi and Yamamoto, M. (1988). Some oscillation criteria for second order

nonlinear ordinary differential equations with damping. Proc. Japan Acad. Ser. A

Math. Sci., 64, 282-285.

Onose, H. (1975). Oscillations criteria for second order nonlinear differential equations.


Parhi, N. and Das, P. (1990). Asymptotic property of solutions of a class of third-order

differential equations. Proc. Amer. Math. Soc., 110, 387-393.

Parhi, N. and Das, P. (1993). On asymptotic property of solutions of linear

homogeneous third order differential equations. Bollettino U. M. I. 7-B, 775-786.

233

Philos, Ch. G. (1983). Oscillation of second order linear ordinary differential equations

with alternating Coefficients. Bull Astral. Math. Soc., 27, 307-313.

Philos, Ch. G. (1984). An oscillation criterion for second order sub-linear ordinary

differential equations. Bull. Pol. Aca. Sci. Math., 32, 567-572.

Philos, Ch. G. (1985). Integral averages and second order super-linear oscillation, Math.

Nachr., 120, 127-138.

Philos, Ch. G. (1989). Oscillation criteria for second order super-linear differential

equations, Canada. J. Math., 41, 321-340.

Philos, Ch. G. (1989). Oscillation theorems for linear differential equations of second

order. Sond. Arch. Math., 53, 482-492.

Philos, Ch. G. and Purnaras, I. K. (1992). On the oscillation of second order nonlinear

differential equations. Arch. Math., 159, 260-271.

Popa, E. (1981). Sur quelques criterias d oscillations pour les equation differentials

linears du second order. Rev. Roumaine. Math. Pures apple., 26, 135-138.

Remili, M. (2010). Oscillation criteria for second order nonlinear perturbed differential

equations. Elect. J. of Quali. Theory of Differ. Eq., No. 24, 1-11.

234

Remili, M. (2007). Nonoscillatory properties of nonlinear differential equations of third

order. Int. Journal of Math. Analysis, Vol. 1, No. 26, 1291-1302.

Rogovchenko, Y. V. (2000). Oscillation theorems for second order differential

equations with damping. Nonlinear anal., 41, 1005-1028.

Rogovchenko, Y. V. (2000). Oscillation theorems for second order differential

equations with damping, Nonlinear anal., 41, 1005-1028.

Rogovchenko, Y. V. and Tuncay, F. (2007). Interval oscillation of a second order

nonlinear differential equation with a damping term. Discrete and Contin. Dynam.

Systems Supplement, 883-891.

Rogovchenko, Y. V., and Tuncay, F. (2008). Oscillation criteria for second order

nonlinear differential equations with damping. Discrete and Contin. Nonlinear Anal.,

69, 208-221.

Ross, S. L. (1984). Differential equations 3rd

ed. John Wiley and Sons, p. 580.

Sevelo, V. N. (1965). Problems, methods and fundamental results in the theory of

oscillation of solutions of nonlinear non-autonomous ordinary differential equations.

Proc. 2nd

All-Union Conf. Theo. Appl. Mech., Moscow, 142-157.

Travis, C. C. (1972). Oscillation theorems for second order differential equation with

functional arguments. Proc. Math. Soc., 31,199-202.

235

Tiryaki, A. and Cakmak, D. (2004). Integral averages and oscillation criteria of second

order nonlinear differential equations. Computers and Math. Applic., 47, 1495-1506.

Tiryaki, A. and Yaman, S. (2001). Asymptotic behavior of a class of nonlinear

functional differential equations of third order. Appl. Math. Letters, 14, 327-332.

Tiryaki, A. and Zafer, A. (2005). Interval oscillation of a general class of second-order

nonlinear differential equations with nonlinear damping. Nonlin, Anal., 60, 49-63.

Waltman, P. (1965). An oscillation criterion for a nonlinear second order equation. J.

Math. Anal. Appl., 10, 439-441.

Waltman, P. (1966). Oscillation criteria for third order nonlinear differential equations.

Pacific J. Math., 18, 385-389.

Wintner, A. (1949). A criterion of oscillatory stability. Quart. Appl. Math., 7, 115-117.

Wong, F. H. and Yeh, C. C. (1992). An oscillation criteria for second order super-linear

differential equations. Math. Japonica., 37, 573-584.

Wong, J. S. (1973). A second order nonlinear oscillation theorems. Proc. Amer. Math.

Soc., 40, 487-491.

Wong, J. S. W. and Yeh, C. C. (1992). An oscillation criterion for second order sub-

linear differential equations. J. Math. And. Appl., 171, 346-351.

236

Yan, J. (1986). Oscillation theorems for second order linear differential equations with

damping. Proc. Amer. Math. Soc., 98, 276-282.

Yeh, C. C. (1982). Oscillation theorems for nonlinear second order differential

equations with damped term. Proc. Amer. Math. Soc., 84, 397-402.

Yu, Y. H. (1993). Oscillation criteria for second order nonlinear differential equations

with damping. Acta. Math. Appl., 16, 433-441 (in Chinese).

237

LIST OF PUBLICATIONS

1. List of Papers Submitted and Accepted for Publication

1.1 List of Papers Submitted and Accepted for Publication in Journals

1- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Oscillation of Second Order

Nonlinear Ordinary Differential Equation with Alternating Coefficients, Submitted to

International Conference on Mathematical Modelling & Scientific Computation 2012,

India. Published in Springer Journal, Communications in Computer and Information

Science, 283 (2012), 367-373. (ISI publication).

2- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Oscillation Theorems for Nonlinear

Second Order Equations with Damping. Accepted to publish in the Bulletin of

Malaysian Mathematical Sciences Society. (ISI publication).

3- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Oscillation Criterion for Second

Order Nonlinear Equations With Alternating Coefficients, Accepted to be published in

American Institute of Physics. (ISI publication).

238

1.2 List of Papers Submitted and Accepted for Publication at

International Conferences

1- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Oscillation of Second Order

Nonlinear Ordinary Differential Equation with Alternating Coefficients, Presented at the

International Conference on Mathematical Modelling & Scientific Computation 2012,

India.

2- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Oscillation Criterion for Second

Order Nonlinear Equations With Alternating Coefficients, Accepted to present at The

2013 International Conference on Mathematics and Its Applications (ICMA-2013),

Malaysia.

2. Publications Under Review

1- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Oscillation Theorems For Second

Order Nonlinear Differential Equations. Submitted to Filomat Journal. (Under review).

2- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Some Oscillation Theorems for

Nonlinear Second Order Ordinary Differential Equations with Alternating Coefficients.

Submitted to Sains Malaysiana Journal. (Under Review).

239

3- M. J. Saad, N. Kumaresan and Kuru Ratnavelu, Oscillation of Third Order

Nonlinear Ordinary Differential Equations. Submitted to Applied and Computational

Mathematics Journal. (Under Review).

Documents

New UMstudentsrepo.um.edu.my/4236/1/All_CHAPTERS_of_THESIS.pdf · 2014. 8. 28. · 1 CHAPTER ONE INTRODUCTION 1.1 Introduction The differential equations (DE) appear naturally in