National College of Business Administration and Economics ...prr.hec.gov.pk/jspui/bitstream/123456789/11224/1...I understand the zero-tolerance policy of the HEC and National College

i

National College of Business

Administration and Economics

Lahore

QUALITATIVE ANALYSIS OF SOME HIGHER ORDER RATIONAL DIFFERENCE EQUATIONS

BY

STEPHEN SADIQ

DOCTOR OF PHILOSOPHY IN

MATHEMATICS

AUGUST, 2018

ii

NATIONAL COLLEGE OF BUSINESS ADMINISTRATION AND ECONOMICS

QUALITATIVE ANALYSIS OF SOME HIGHER ORDER RATIONAL DIFFERENCE EQUATIONS

BY

STEPHEN SADIQ

A dissertation submitted to School of Computer Science

In Partial Fulfillment of the Requirements for the Degree of

DOCTOR OF PHILOSOPHY IN

MATHEMATICS

AUGUST, 2018

iii

“It is because of him that you are in

Christ Jesus, who has become for us

wisdom from God—that is, our

righteousness, holiness and redemption”.

1-Corinthians 1:30

iv

AUTHOR’S DECLARATION

I, Stephen Sadiq hereby state that my PhD thesis titled “Qualitative

Analysis of Some Higher Order Rational Difference Equations” is my own

work and has not been submitted previously by me for taking any degree from

this university, National College of Business Administration and

Economics, (NCBA&E), Lahore or anywhere else in the country/world.

At any time if my statement is found to be incorrect even after my

graduate the university has the right to withdraw my PhD degree.

STEPHEN SADIQ

AUGUST, 2018

v

PLAGIARISM UNDERTAKING

I solemnly declare that research work presented in the thesis titled

“Qualitative Analysis of Some Higher Order Rational Difference

Equations” is solely my research work with no significant contribution from

any other person. Small contribution/help whenever taken has been duly

acknowledged and that complete thesis has been written by me.

I understand the zero-tolerance policy of the HEC and National College

of Business Administration and Economics, Lahore towards plagiarism.

Therefore, I as an Author of the above titled thesis declare that no portion of

my thesis has been plagiarized and any material used as reference is

properly/cited.

I undertake that if I am found guilty of any formal plagiarism in the

above title thesis even after award of PhD degree, the university reserves the

right to withdraw/revoke my PhD degree and that HEC and the University has

the right to publish my name on the HEC/University website on which names

of students are placed who submitted plagiarized thesis.

STEPHEN SADIQ

vi

CERTIFICATE OF APPROVAL

This is to certify that research work presented in the thesis, entitled

“Qualitative Analysis of Some Higher Order Rational Difference Equations”

was conducted by Mr. Stephen Sadiq under the supervision of Prof.

Dr. Muhammad Kalim.

No part of this thesis has been submitted anywhere else for any other degree.

This thesis is submitted to the School of Computer Science in partial fulfillment of

requirements for the degree of requirements for the degree of Doctor of Philosophy

in the field of Mathematics, Faculty of Social Sciences, National College of

Business Administration and Economics, Lahore.

Student Name: Stephen Sadiq Signature:

Examination Committee:

a) External Examiner 1:

Dr. Muhammad Mushtaq Signature:

Professor & HoD of Mathematics

University of Engineering & Technology, Lahore

b) External Examiner 2:

Dr. Nazir Ahmad Chaudhry Signature:

Professor of Mathematics,

Lahore Leads University, Lahore

c) Internal Examiner:

Dr. Adnan Khan Signature:

Assistant Professor, National College of Business

Administration and Economics, Lahore

Supervisor Name: Prof. Dr. Muhammad Kalim Signature:

Name of Dean/HOD: Prof. Dr. Muhammad Kalim Signature:

Name of Rector: Prof. Dr. Munir Ahmad Signature:

vii

DEDICATION

Dedicated

to

My Beloved Parents (late)

&

My Family Angels

Anaiem, Mathew, John Wilber

Shemera & Abner

viii

ACKNOWLEDGEMENT

All the praises are for the Almighty God alone, the omnipresent, who

enabled me with the ability and the potential to complete this research work.

It is a matter of great pleasure and honor for me to convey my deep

sense of gratitude and appreciation to my respected research supervisor

Professor Dr. Muhammad Kalim, Department of Computer Science,

NCBA&E Lahore, under whose kind supervision and sympathetic attitude, the

present research was completed. He continuously encouraged, supported and

guided me throughout this research. His kindness and ability to produce good

ideas always helped me to complete this research thesis quite effectively.

Words are quite insufficient to express my enormous humble gratitude

to all my family members (especially my brothers Javed, Peter & Sohail

Henry) who served as a source of continuous prayers and strong determination

to enabling me to achieve my target.

I am also grateful to Dr. Munir Ahmad, Rector, NCBA&E, Lahore and

Dr. Muhammad Hanif, Director Research, NCBA&E, Lahore for their

selflessness and generous help during my stay and research at NCBA&E.

I offer my heartiest gratitude to all my friends and colleagues for their

support especially Dr. Adnan Khan.

ix

SUMMARY

The study of Qualitative Analysis of rational difference equation is very

important. However, in Physics, Economics and Biological sciences difference

equations are extensively used. Qualitative behavior includes global

attractivity, bounded character and periodicity. The motivation is to study the

Dynamics of some higher order rational difference equations. In this thesis we

have studied the behavior of solutions of some higher order rational difference

equations. To confirm the proved results we have used mathematical program

Matlab to give graphical examples by assigning different numerical values to

initial values.

The first chapter presents brief introduction to the problem of study, its

objectives and methodology used to achieve the goal.

In the second chapter we have presented a comprehensive form of

literature review related to the problem understudy. The current literature is

discussed in detail and results are summarized to draw conclusions and further

directions.

The third chapter is preliminaries, showing notions and basic definitions

which are associated with this study.

In chapter four, we have presented our main results with their

proofs for rational difference equation of order twenty

2

91 9

9 19

, 0,1,2,...nn n

n n

zz z n

z z

With initial conditions 19 18 17 16 15 14 13, , , , , , ,z z z z z z z 12 11, ,z z

10 ,z 9 8 7 6 5 4 3 2 1 0, , , , , , , , ,z z z z z z z z z z R and the coefficients , , ,

are constants. We obtained some special cases of this equation. The numerical

verification and comparison of each graph is also included in this chapter.

In chapter five, we have studied the global stability of the positive

solutions and periodic character of the difference equation

0 1 2 3

1 0 1 24 5 6 7

n t n l n m n pn n n k n s

n t n l n m n p

b z b z b z b zz z z z

b z b z b z b z

x

With non-negative initial conditions 1 1 0, ,..., ,z z z z where

max , , , , ,k s t l m p and the coefficients 0 1 2 0 1, , , , ,b b 2 3 4, , ,b b b

5 6 7, ,b b b R . Numerical examples are also given to confirm the obtained

results.

xi

LIST OF SYMBOLS AND ABBREVIATIONS

Symbol Description

R Set of Real Numbers

R Set of Positive Real Numbers

R Set of non- Zero Real Numbers

Z Set of POSITIVE INTEGERS

Q Set of Positive Rational Numbers

Positive Real Number

0 Positive Real Number

1 Positive Real Number

2 Positive Real Number

Positive Real Number

Positive Real Number

Positive Real Number

, , , , ,Max k s t l m p and Positive Real Number

xii

LIST OF FIGURES

Figure

No. Title Page

1.1 ( 1 0A ) Asymptotically Stable Equilibrium Price 5

1.2 ( 1A ) Stable Equilibrium Price 5

1.3 ( 1A ) Unstable Equilibrium Price 6

1.4 with 0 0.1z and 2.6 7

3.1 Fixed Points of 2( )f x x 31

3.2 Fixed Points of3 2( ) 2 1f x x x x 32

4.1 Behavior of

29

1 99 19

nn n

n n

zz z

z z

45

4.2 Behavior of

29

1 99 19

nn n

n n

zz z

z z

46

4.3 Behavior of

29

1 99 19

nn n

n n

zz z

z z

52

4.4 Behavior of

29

1 99 19

nn n

n n

zz z

z z

55

4.5 Behavior of

29

1 99 19

nn n

n n

zz z

z z

60

4.6 Behavior of

29

1 99 19

nn n

n n

zz z

z z

62

5.1 Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

81

xiii

Figure

No. Title Page

5.2 Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

82

5.3 Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

83

5.4 Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

84

5.5 Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

85

5.6 Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

86

xiv

TABLE OF CONTENTS

AUTHOR’S DECLARATION .......................................................................... iv PLAGIARISM UNDERTAKING ...................................................................... v CERTIFICATE OF APPROVAL ...................................................................... vi DEDICATION .................................................................................................. vii ACKNOWLEDGEMENT .............................................................................. viii SUMMARY ....................................................................................................... ix LIST OF SYMBOLS AND ABBREVIATIONS .............................................. xi LIST OF FIGURES .......................................................................................... xii

CHAPTER 1: INTRODUCTION................................................................... 1 1.1 Background ................................................................................................ 1 1.2 Applications of Difference Equations ....................................................... 3

1.2.1 Application in Economics ............................................................... 3 1.2.2 Application in Biology .................................................................... 6

1.3 Thesis Outline ............................................................................................ 7

CHAPTER 2: LITERATURE REVIEW ...................................................... 9

CHAPTER 3: PRELIMINARIES ................................................................ 28 3.1 Finite Difference Equation ....................................................................... 28 3.2 Linear Difference Equation ..................................................................... 28 3.3 Non-Linear Difference Equation ............................................................. 29 3.4 Rational Difference Equation .................................................................. 29 3.5 Order of a Difference Equation ............................................................... 29 3.6 Invariant Interval ...................................................................................... 30 3.7 Equilibrium POINT ................................................................................. 30 3.8 Periodicity ................................................................................................ 32 3.9 Fibonacci Sequence ................................................................................. 33 3.10 Locally Stable .......................................................................................... 33 3.11 Locally Asymptotically Stable................................................................. 33 3.12 Global Attractor ....................................................................................... 34 3.13 Global Asymptotically Stable .................................................................. 34 3.14 Unstable ................................................................................................... 34 3.15 Semi-Cycle Analysis ................................................................................ 34

3.15.1 Positive Semi Cycle ...................................................................... 34 3.15.2 Negative Semi Cycle .................................................................... 35 3.15.3 Non-Oscillatory Solutions: ........................................................... 35 3.15.4 Strictly Oscillatory Solutions ....................................................... 35

xv

3.15.5 Persistent ....................................................................................... 35 3.15.6 Permanent ..................................................................................... 35

3.16 Useful Results .......................................................................................... 36 3.16.1 Theorem 1 ..................................................................................... 36 3.16.2 Hyperbolic Solution ...................................................................... 36 3.16.3 Non-hyperbolic Solution .............................................................. 36 3.16.4 Saddle Point .................................................................................. 37 3.16.5 Theorem 2 ..................................................................................... 37 3.16.6 Theorem 3 ..................................................................................... 37 3.16.7 Theorem 4 ..................................................................................... 38 3.16.8 Theorem 5 ..................................................................................... 39

3.17 Convergence Theorems ........................................................................... 39 3.17.1 Lemma 1 ....................................................................................... 39 3.17.2 Lemma 2 ....................................................................................... 40 3.17.3 Theorem 6 ..................................................................................... 40

CHAPTER 4: GLOBAL ATTRACTIVITY OF A RATIONAL

DIFFERENCE EQUATION OF ORDER TWENTY ....... 41 4.1 Local Stable Nature of EQ.(4.1) .............................................................. 41

4.1.1 Theorem 1 ...................................................................................... 42 4.2 Global Attractive Behavior of EQ.(4.1) .................................................. 43

4.2.1 Theorem 2 ...................................................................................... 43 4.3 Boundedness Behavior of EQ. (4.1) ........................................................ 44

4.3.1 Theorem 3 ...................................................................................... 44 4.4 GRAPHICAL EXAMPLES .................................................................... 44

4.4.1 Example 1 ...................................................................................... 44 4.4.2 Example 2 ...................................................................................... 45

4.5 Different Cases of EQ. (4.1) .................................................................... 46 4.5.1 FIRST EQUATION ....................................................................... 46 4.5.2 Theorem 4 ...................................................................................... 46 4.5.3 Example 3 ...................................................................................... 52 4.5.4 Second Equation ............................................................................ 53 4.5.5 Theorem 5 ...................................................................................... 53 4.5.6 Example 4 ...................................................................................... 54 4.5.7 Third Equation ............................................................................... 55 4.5.8 Theorem 6 ...................................................................................... 55 4.5.9 Example 5 ...................................................................................... 59 4.5.10 Fourth Equation ............................................................................ 60 4.5.11 Theorem 7 ..................................................................................... 60 4.5.12 Example 6 ..................................................................................... 62

4.6 Conclusion ............................................................................................... 63

xvi

CHAPTER 5: DYNAMICAL BEHAVIOR OF HIGHER ORDER

RATIONAL DIFFERENCE EQUATION ......................... 64 5.1 Local Stability .......................................................................................... 64 5.2 Global Stability ........................................................................................ 67

5.2.1 Theorem 1 ...................................................................................... 67 5.3 Boundedness of Solutions of (5.1) ........................................................... 69

5.3.1 Theorem 2 ...................................................................................... 69 5.3.2 Theorem 3 ...................................................................................... 72

5.4 Periodic Solutions .................................................................................... 72 5.4.1 Theorem 4 ...................................................................................... 73 5.4.2 Theorem 5 ...................................................................................... 80

5.5 Graphical Examples ................................................................................. 80 5.5.1 Example 1 ...................................................................................... 80 5.5.2 Example 2 ...................................................................................... 81 5.5.3 Example 3 ...................................................................................... 82 5.5.4 Example 4 ...................................................................................... 83 5.5.5 Example 5 ...................................................................................... 84 5.5.6 Example 6 ...................................................................................... 85

5.6 Conclusion ............................................................................................... 86

REFERENCES ................................................................................................ 88

1

CHAPTER 1:

INTRODUCTION

1.1 BACKGROUND

The historic review of difference equations proves that it plays very

important role in the field of engineering and applied sciences. The

fundamental concept of linear equations was originated in the eighteenth

century by Langrange, Euler, de Moivre, Laplace, and other mathematicians.

The concept of using difference equations for approximation of differential

equations developed in 1769 with Euler polygonal method and the

convergence proof was provided by Cauchy in 1840. In 1952 the most useful

application of equation in the field of special functions is for calculating Bessel

functions with the help of Miller algorithm.

During the 1950, different biologist used simple Non-linear equations,

like the logistic equation to examine the increase or decrease in population in

one year by using stability of the iteration.

The idea of computing by recursion is old. About 450 B.C in the

Pythagorean study of pictorial numbers, the triangular numbers and the square

numbers satisfy the equations 1p pz z p , and 2-1p pz z p respectively.

Poincare had laid the foundations for asymptotic properties of solutions of

linear difference equation in 1880’s.

Recently, great interest has been developed in the study of equations.

The reason is that some techniques are needed which can be utilized in

analyzing equations in models which are used in mathematics and real life in

some applied sciences.

Now there is a lot of concern in analyzing the global attractivity,

bounded behavior, the periodic nature and giving the solutions of Non-linear

equations. First result in qualitative theory of difference equations was

obtained by Poincare and Perron in nineteenth century.

The physical and economical problems often lead to difference

equations in the mathematical modeling. The analysis of higher order Non-

linear equations is very important. Several unique approaches have been arisen

2

for determination of global character of such equations. There are many types

of equations which have not been studied uptil now completely.

The investigation of rational equations of order more than one is very

demanding and important. There is not any effective method to cope with the

global behavior of higher order rational equations. Therefore rational

difference equations having order greater than one are of great interest and

studied during last decade. Non-linear such equations are very useful in

handling phenomena in science and engineering modeling.

Difference equations used intrinsically as discrete analogue as well as

numerical solutions of differential equations which have applications in

biology, physics etc. No doubt such equations are very simple in their form but

it’s very hard to study their solutions and behavior.

Many Mathematicians have studied the solutions of such equations, for

example Camouzis (1994) studied global stability, periodic nature and checked

the solutions of different cases of rational equations. Cinar (2004), Elabassy

(2000), Yang (2005), Agarwal et al., (2008), Stevic (2001), Zayed (2005),

Yalcinkaya (2009), El-sayed (2010) had worked on different forms of higher

order rational equations. They found that by making small change in difference

equation, the behavior of equation changes. Many researchers have shown

their obtained results by drawing graphs in using numerical examples with the

help of mathematical program MATLAB. MATLAB is a tool that has reduced

the efforts of engineers and scientists in mathematical calculations and

approximations See Elabbasy and Elsayed (2011).

Difference Equations and Differential Equations are used in population

dynamics in different models (Nedorezov 2012) and (Nedorezov and Sadykov,

2012). In population dynamics exponential equations have many applications

(El- Metwally et al., 2001; Papaschinopoulos et al., 2011). Zhou and Zou

(2003) and Liu (2010) proved that difference equations are better than

differential equations to study the non-overlapping generations. There are

many papers which deal with discrete dynamical systems to study qualitative

behavior of population in literature, see Ahmad (1993); Zhou and Zou (2003);

and Din (2014).

3

1.2 APPLICATIONS OF DIFFERENCE EQUATIONS

Due to numerous applications, difference equations are used in many

fields of sciences. Later many difference equations developed from many

fields in science and engineering.

1.2.1 Application in Economics

Let ( )P n , ( )q n and ( )R n be the supplied no of units, the demanded no

of units and price per unit respectively in time period n . It is noticed that ( )q n depends linearly only on ( )R n and written as

( ) ( )r rR n m R n b with 0, 0r rm b

where rm is the constant of responsiveness of consumers with respect to price.

The equation of supply to price is

( 1) ( )p pP n m R n b with 0, 0p pm b

The constant pm is the responsiveness of supplies to price. A third

supposition, we discuss here that demanded and supplied quantities are equal

at the market price.

i.e ( 1) ( 1)R n P n

( 1) ( )r r p pm R n b m R n b

Probability Theory

Economics

Biology

Difference

Equations

Other Fields

Genetics

Computer Science

Engineering

Psychology

4

( )( 1)

p p r

r r

m R n b bR n

m m

( 1) ( ) ( )R n AR n B f R n (1.1)

where p

r

mA

m

and r p

r

b bB

m

The equilibrium point of Eq. (1.1) is given as

r Ar B

/1r B A

The equilibrium point r in economics is defined as the point of

intersection of supply and demand curves. Cases to be considered here are

(i) 1 0A , prices converge to equilibrium point r . So r is

asymptotically stable. (see Figure 1.1)

(ii) 1A , prices oscillate between two values. Hence r is stable.

(see Figure 1.2)

(iii) 1A , prices oscillate about the equilibrium point infinite many

times. Therefore r is unstable. (see Figure 1.3)

5

Figure 1.1: ( 1 0A ) Asymptotically Stable Equilibrium Price

Figure 1.2: ( 1A ) Stable Equilibrium Price

6

Figure 1.3: ( 1A ) Unstable Equilibrium Price

1.2.2 Application in Biology

Let ( )z n be the population size at any time n . If is the its growth rate

varying from one origination to the next. We can write this model as

( 1) ( )z n z n , 0

If the initial population is 0(0)z z then we find that 0( ) nz n z is the

solution of above equation. If 1 then ( )z n increases unlimitedly and

lim ( )n z n . If 1 then 0( )z n z for all 0n which implies

population remains constant for unlimited future. If 1 , we get

lim ( ) 0n z n and population will exanimate.

None of the above cases is defensible for most biological species as

population increases until to a certain upper limit. Due to limited resources the

creatures will become testy (choleric) under limited resources. This is

proportional to the number of disagreements among them, as given by 2( )z n .

A model is as

2( 1) ( ) ( )z n z n bz n

7

where b is the proportionality constant and greater than 0 .

( ) ( )b

y n z n

Except certain values of , a closed form of solutions of above equation

is not accessible. Inspite of its clarity, this equation show preferably rich and

complex dynamics. The two equilibrium point of above equation are 0y

and ( 1) /y . The stair step diagram (1.4) is given with initial guess of

0.1 and 2.6 . One equilibrium point 0y is unstable and other 0.61y is stable asymptotically.

Figure 1.4: with 0 0.1z and 2.6

1.3 THESIS OUTLINE

The first chapter presents brief introduction to the problem of study, its

objectives, applications and methodology used to achieve the goal.

In the second chapter we have presented a comprehensive form of

literature review related to the problem understudy. The current literature is

8

discussed in detail and results are summarized to draw conclusions and further

directions.

The third chapter is preliminaries, showing notions and basic definitions

which are associated with this study.

The fourth chapter deals with the global attractivity of a rational

difference equation of order twenty

2

91 9

9 19

, 0,1,2,...nn n

n n

zz z n

z z

With initial conditions 19 18 17 16 15 14 13 12 11, , , , , , , , ,z z z z z z z z z

10 ,z 9 ,z 8 7 6 5 4 3 2 1 0, , , , , , , ,z z z z z z z z z R and coefficients , , ,

are constants. We obtained solutions and verified the obtained results by

drawing graphs in solving the different problems. Particularly we had worked

on local stability, global attractivity, bounded character and discussed different

cases of considered equation.

The fifth chapter deals with the global stability of the positive solutions

and the periodic character of the difference equation

0 1 2 31 0 1 2

4 5 6 7

n t n l n m n pn n n k n s

n t n l n m n p

b z b z b z b zz z z z

b z b z b z b z

With non-negative initial conditions 1 1 0, ,..., ,z z z z where

max , , , , ,k s t l m p and 0 1 2 0 7, , , ,...,b b R . Numerical examples are

also given to confirm the obtained results.

9

CHAPTER 2:

LITERATURE REVIEW

This chapter deals with the brief literature survey related to our area of

interest particularly on solving higher order rational difference equations. This

survey shows different methods and formulations to solve rational difference

equations. To get clear understanding the summary of each research done is

presented in this chapter.

CURRENT LITERATURE REVIEW

Difference equations are studied for its vast applications over the years.

Following is the review of research done by different mathematician in the

past years.

Camouzis (1994) studied the global attractive behavior of the equation

2

1 211

nn

n

xx

x

(2.1)

where 1 0,x x R and constant is positive.

Kulenvoic et al. (1998) studied the nature of the equation

11

1

n nn

n n

z zz

z z

(2.2)

where 1 0,x x and all parameters involved are R .

Grove et al. (2000) investigated the behavior and semi-cycles of the

biological equation

1 1nx

n n nx ax bx e

. (2.3)

In (2002) Stevic solved the problem

10

11

nn

n

zz

z

(2.4)

With 1 00, , 0, 0,1,2,...x x n . He also gave results of this equation

by assuming 1 . Moreover he generalize results to the following form

11

( )

nn

n

xx

g x

(2.5)

Yan et al.(2003) studied the attractivity behavior for the rational

equation

11

nn

n

xx

x

(2.6)

where 0, , 0 . They showed that one equilibrium point under specific

conditions restrict on the coefficients is global attractor.

El-Owaidy, et al., (2003) studied the nature of the rational equation

11

nn

n

zz

z

(2.7)

Under specified conditions. They proved that z of these equations under

certain conditions is a global attractor.

El-Owaidy, et al., (2003) studied bounded nature of the equation

11 , 0,1,...

pn

n pn

zz n

z

(2.8)

Under the assumption 0, , 1,p and 1 0,x x R .

Xing-Xue Yan et al. (2003) checked the behavior of equation

11

, 0,1,...nn

n

zz n

z

(2.9)

11

where 0 , , 0 . They proved that the z of above equation under specific

conditions restrict on the involved coefficients is a global attractor.

Kalabusic and Kulenvoic (2003) studied the behavior of solution of the

non- linear rational equation of third order

1 21

1 2

n nn

n n

x xx

Cx Dx

. (2.10)

where all initial conditions 0 1 2, ,x x x and all parameters involved are R .

El. Afifi et al., (2003) investigated invariant intervals, oscillatory and

semi cycles (including positive or negative) of the difference equation. They

found an invariant relation and determined that the solution satisfy the

expression

110

0

1( 1) 1 constant, 0,1,...

( 1)

kk

n jj n

a x k j nx k j

(2.11)

Xiaofan Yang, et al., (2004) considered the equation

2

1 11

2

, 0,1,...n nn

n

a bx cxx n

d x

(2.12)

Under certain conditions they proved that equilibrium point of equation

is an attractor which is determined by the parameters.

El-Owaidy, et al. (2004) checked the behavior of solutions of rational

equation

1n k

nn

xx

x

(2.13)

where 0, and 1 0,x x R .

Xing-Xue Yan et al., (2004) examined the behavior of equation

12

1

0

, 0,1,...nn k

i n ii

a bxx n

A b x

(2.14)

where 1,2,...k , , , 0,kA b b , 0 1, ,..., [0, )ka b b and 0,...,kx x R .

Xing-Xue Yan et al., (2004) explored the global behavior and the

bounded nature of the difference equation

11 , 0,1,...n

nn

zz n

z

(2.15)

by taking and the involved initial conditions are R .

Cinar (2004) studied the behavior of the equations

11

11

nn

n n

zz

az z

(2.16)

11

11

nn

n n

zz

az z

(2.17)

where 0,1,2,...n with 1 0,x x and a are R . For equation (2.16) he checked

the positive solutions and for (2.17) he investigated its solutions.

Li Wantong, et al., (2005) studied the behavior of equation

1 , 0,1,...nn

n k

a bzz n

A z

(2.18)

Li Wantong et al., (2005) studied the periodic behavior, oscillation and

invariant intervals of all solution of equation

1n n k

nn k

px xx

q x

(2.19)

With 0,1,...n and 0, , ,...,kp q x x R .

13

Chatterjee, et al. (2005) examined the behavior of solution of equation

1n n k

nn k

z zz

A z

(2.20)

where the involved parameters , , A and all initial conditions are R while

k Z .

Yang (2005) considered the equation

1 2 31

1 2 3

n n nn

n n n

x x x ax

x x x a

(2.21)

where 0a . They checked the global asymptotic behavior of equilibrium

point.

Alaa (2006) checked the global stability, the persistence and oscillation

of the equation

11 ,n

nn

zz

z

0,1,...n (2.22)

where as well as 1x , 0x are negative initial conditions.

Karatas et al. (2006) studied the behavior of the equation

51

2 51

nn

n n

zz

z z

(2.23)

With 0,1,...n and 5 4 0, ,...,z z z R .

Kalabusic et al., (2006) studied the periodic and bounded nature of

equation

11

2

,nn n

n

xx p

x

0,1,...n (2.24)

With positive initial conditions.

14

Simsek et al. (2006) reviewed the solutions of the difference equation of

order four

31

11

nn

n

xx

x

(2.25)

where 0 1 2 3, , ,x x x x are R initial conditions.

Elabbasy et al. (2006) investigated the behavior of equation. They

obtained the solutions of several forms of following equation

11

nn n

n n

bzz az

cz dz

(2.26)

Elabbasy et al. (2006) studied the attractivity, bounded and periodic

nature. They generated the solutions of some special forms of following

rational equation

1

0

n kn k

n ii

xx

x

. (2.27)

Simsek et al. (2006) studied the solution of six order equation

51

2 51

nn

n n

xx

x x

(2.28)

where 0,1,2,...n and 5 4 0, ,..., (0, )x x x .

Bratislav et al. (2007) examined the global stability of equation

11 ( ,..., )

n kn

n k n n m

zz

z f z z

(2.29)

where ,k m N , 0,1,2,...n and f is a continuous function.

Karatas et al. (2007) investigated the equation

15

(2 2)1 2 2

0

n k

n k

n ii

azz

a z

for n =0,1,2,... (2.30)

where (2 2) (2 1) 0...k kz z z a and initial values are R .

Elabbasy et al. (2007) checked the nature of solutions of different cases

of following equation

1n l n k

nn s

dx xx a

cx b

. (2.31)

Guang et al. (2007) considered the equation

111 ( )

n k n mn

n

a bz czz

g z

, (2.32)

where , ,a b c R , , ,k l m Z and ( )g z is a non-negative real function.

Kenneth et al. (2007) studied the form of equation

1

n k n mn

n k n m

z zz

z z

(2.33)

With 1 1, ,..., (0, )m mz z z , 0,1,2,...n and 1 k m .

Stevic et al. (2007) studied the global attractivity of

1 2 2 1 1( )... ( ) ( )n k k n k n nx f x f x f x (2.34)

where if are continuous function over (0, ) and {1,2,..., }i k .

Stevic et al. (2007) studied the nature of the equation

1 n kn

n m

zz

z

(2.35)

where 0,1,2,...n and 1 1, ,...,s sz z z R .

16

Zayed et al. (2008) studied global attractivity and bounded nature of the

equation

01

0

k

i n ii

n k

i n ii

A z

z

z

(2.36)

where , ,i iA , 1 1 0, ,..., ,k kz z z z R and k Z .

Agarwal et al. (2008) examined the behavior of equation. They also

found the solutions of some cases of the equation

1n l n k

nn s

dx xx a

b cx

. (2.37)

Elsayed (2008) studied the behavior of equation

11

1

n nn n

n n

bz zz az

cz dz

(2.38)

where the involved coefficients and initial conditions are R .

Elsayed (2008) worked on the equation

51

2 5

,1

nn

n n

zz

z z

(2.39)

where initial conditions are R provided 5 2 4 1 3 01, 1, 1z z z z z z . He

gave the solutions and drawn graphs of numerical examples.

Ibrahim Yalcinkaya. (2008) checked the global attractivity of equation

1 / kn n m nz xz z (2.40)

With 0,1,2,...n and initial values are randomly selected R and

, (0, )a k .

17

Elsayed (2009) studied the qualitative behavior of equation

51

2 5

,n 0,1,....1

nn

n n

xx

x x

(2.41)

With the initial values are R.

Elsayed (2009) investigated the rational equation

111

3 7 111

nn

n n n

xx

x x x

. n 0,1,... (2.42)

Ibrahim (2009) got the solutions of the rational equation of order three

21

1 2( )

n nn

n n n

x xx

x a bx x

(2.43)

where 0 1 2, , , ,a b x x x R .

Xiu-Mei Jia, et al. (2010) investigated the equation

1

1, 0,1,...n k

nn

xx n

A x

(2.44)

They showed that the negative equilibrium point of above equation

under certain conditions on the coefficient A is a global attractor.

Elsayed (2010) investigated the convergence, bounded behavior and

periodicity of the equation

1 21

1 2

n nn n

n n

bz czz az

dz ez

(2.45)

With 0,1,...n and 0 1 2, ,z z z R .

Elsayed (2010) investigated the different solutions of forms of the

equation

18

81

2 5 8

, 0,1,...1

nn

n n n

xx n

x x x

(2.46)

With initial conditions are real numbers.

Zayed (2010) studied the solutions of the non-linear equation

1n n k

n n n kn k

px xx Ax Bx

q x

(2.47)

where 0,1,...n and k Z . While all involved coefficients and initial

conditions are R .

Zayed et al. (2010) studied qualitative analysis of rational equation

1n n k

n n kn n k

az bzz z

cz dz

(2.48)

With 0,1,2,...n , 1 0,..., ,kz z z R and , , , ,a b c d R while k Z .

Ibrahim et al. (2010) studied the nature of the equation

21 1 max , /n n n nz z A z z (2.49)

With 0,1,2,...n , 11

rx A and 20

rx A . While A R and 1 2,r r Q .

Zayed et al. (2010) studied the nature of solutions of the equation

0 1 21

0 1 2

, 0,1,2,...n n l n kn

n n l n k

x x xx n

x x x

(2.50)

where the coefficients and ,l k Z with l k . Also 1,..., ,..., ,k lx x x 0x R .

Stevic et al. (2010) studied the behavior of the equation

1 1 1n n nx x x (2.51)

19

where 0,1,2,...n and the involved initial values are real numbers. They

examined the bounded as well as periodic nature of solutions.

Elsayed et al. (2010) studied the max-equation

1 3max{ / , }n n nz A z z (2.52)

where 0n N and 0A . They showed that all solutions of this equation is

periodic with four period.

Vu Van Kn huong, et al., (2011) checked the behavior and non-

oscillatory solution of equation

2 11

1

n nn

n n

z zz

z z

(2.53)

where 2 1 00, , , 0,z z z .

Elsayed (2011) examined the behavior of solutions of the rational

equation

91

4 91

nn

n n

xx

x x

(2.54)

and

31

1 31

nn

n n

xx

x x

(2.55)

where 0,1,2,...n

Vu Van Khuong et al., (2011) studied the equation

1 1 2(1 )n n n nz z z z (2.56)

where 0,1,2,...n .They showed that its solution approaching to 0 with

n ,and they also examined its asymptotic nature.

Yanqin Wang (2011) studied the invariant interval and global

attractivity of equation

20

1n n k

nn n k

a bz zz

A Bz Cz

(2.57)

where 0,1,2,...n and ( 1) 0, , , , , , ,...k ka b A B C z z z R .

Ignacio et al. (2011) studied the second order equation

11

1

, 0nn

n n

xx n

a bx x

(2.58)

With 21 0( , )x x R and parameters ,a b R .

Karatas, et al., (2011) investigated the equation

(2 1)1 2 1

0

n kn k

n ii

Azz

A z

(2.59)

With 0,1,2,...n while k Z and 2 1

0

k

ii

x A

.

Witold (2011) investigated the solutions of equation of minimal

period 5 .

11

1

1

1

nn

n n

zz

z z

(2.60)

With 1,2,3,...n while initial conditions are R and 0 1 1x x .

Ibrahim Yalcinkaya. (2012) investigated the nature of the well-defined

form of the equation

1 1 max 1/ ,n n n nx x A x (2.61)

where the involved initial conditions are R with 0,nA .

21

Mai Nam Phong (2012) investigated the behavior of rational equation of

( 1)thk order

1n k

nn m

a zz

b z

(2.62)

With 0,1,...n and under specific conditions he proved that z is a

global attractor under certain conditions. He also determined invariant interval

of this equation.

Elsayed, et al., (2012) investigated the nature of solutions of system of

equations

1

1n

n p n p

xx y

, 1n p n p

nn q n q

x yy

x y

(2.63)

and

1 1 1

1, ,

n p n p n pn n n

n p n p n p n q n q n q

n q n q n q

n r n r n r

x y zx y z

x y z x y z

x y z

x y z

(2.64)

With initial conditions are non-zero real numbers and p q for system

(2.63) and ,p q q r for system (2.64). They proved that equation system

(2.63) is periodic with period (2 2)q and equation system (2.64) is periodic

with period (2 2)r .

Touafek, et al., (2012) investigated the system of equations

1 11 1

,( 1 ) ( 1 )

n nn n

n n n n

y xx y

x y y x

(2.65)

With non-zero real number initial conditions.

Ibrahim (2012) investigated the periodic nature of equation of six order

2 41

1 3 5

( ), 0,1,....

( )

n n nn

n n n

z z zz n

z z z

(2.66)

22

where 5 4 0, ,...,z z z R while 5 3 1, , 0z z z . He gave graphical behavior by

giving numerical values to initial conditions.

Elsayed (2012) considered the equation of fourth order

2 31

1 2 3( 1 )

n nn

n n n n

z zz

z z z z

(2.67)

Where 0,1,...n and 1 2 3 0, , ,z z z z R . Also he studied qualitative behavior

of these different equations. He obtained the equilibrium points of considered

equation in order to study the local stability of equilibrium points. He gave

numerical graphs by assigning different values to initial values and

coefficients.

Hamdy El-Metwally, et al., (2012) investigated the equation

31

2 3( 1 )

n nn

n n n

z zz

z z z

(2.68)

where 0,1,...n , 1 2 3 0, , ,z z z z R . Qualitative behavior such as bounded

and periodic character of solution has been analyzed. They gave graphical

behavior by giving numerical values to initial conditions to confirm the

obtained results.

Obaid et al (2012) investigated the equation

1 2 31

1 2 3

n n nn n

n n n

bz cz dzz az

z z z

(2.69)

where the involved parameters and initial conditions are in R .

Cinar et al. (2013) checked the nature of the equation

1

1

1

1

, 0,1,...1

n n k

n

n n k

a x xax n

x xa

(2.70)

23

where k is a positive integral number, initial conditions are R and

, (0, )a .

Papaschinopoulos et al. (2013) investigated the equations

1 1ny

n n nx ay bx e

, 1 1nx

n n ny cx dy e

(2.71)

where initial conditions and all constants are R .

Raafat (2013) investigated the global attractive and periodic behavior of

equation

11

2

,nn

n

A Bxx

C Dx

0,1,...n (2.72)

With , , ,A B C D R and for all 0n , denominator is not equal to zero.

Qi Wang et al. (2013) investigated the convergence of systems of two

equations

2 11

1 2 1

n kn

n k n k

xx

Ay x

, 2 11

1 2 1

n kn

n k n k

yy

Bx y

(2.73)

where 0n and the involved parameters and initial conditions are R while

k is positive integer number.

Tarek F. Ibrahim (2013) studied all solutions of system of equations

1 1 1max , , max , ,

max ,

n n n n nn n

nn

x y y z zx y

xz

(2.74)

where is positive real number and initial conditions 0 0 0, , (0, )x y z .

Qamar Din (2014) studied behavior of rational system of equations

24

1n

nn

ayx

b cy

, 1

nn

n

dyy

e fx

(2.75)

where 0,1,2,...n and , , , , ,a d c d e f and initial conditions 0 0, (0, )x y . He

also gave some numerical simulations in order to verify obtained results.

Taixiang et al. (2014) considered the max-type equation

max ,nn n k

n r

Az z

z

(2.76)

With 1,2,...n , gcd( , ) 1k r , , {1,2,...}k r , k r , max ,d r k and

initial Conditions are R .

El-Moneam (2014) investigated the non-linear equation

1n k

n n n k n l nn k n l

bzz Az Bz Cz Dz

dz ez

(2.77)

where

0,1,...n , 1 0,..., ,..., ,..., ,l kz z z z z R and

, , , , , , (0, )A B C D b d e while , ,k l Z such that k l .

Elsayed (2015) studied the equation of order two

11

1

n nn

n n

y xx

x y

, 11 0

1

,n nn

n n

x yy n N

x y

(2.78)

With initial conditions are R.

Elabbasy et al. (2015) investigated the rational equation

2 2

1 2 2, 0,1,...n r n l n k

n

n r n l n k

ax bx xx n

cx dx x

. (2.79)

where max{ , , }t r k l , 0,...,tx x R and , , ,a b c d R .

25

Elsayed et al. (2016) studied the solutions and periodic nature of

equation

1 51

3 1 5

, 0,1,...( 1 )

n nn

n n n

x xx n

x x x

. (2.80)

Elsayed et al. (2016) studied the qualitative and periodic nature of

equation

2

21 2

2 5

nn n

n n

zz z

z z

(2.81)

With 0,1,...n , 5 4 0, ,....,z z z R and coefficients , , , R .

Elsayed (2016) studied the forms of solutions of equation

1 , 0,1,...n l n kn

n l n k

bz czz a n

dz ez

(2.82)

where 1 1 0, ,..., , , , , , ,t tz z z z a b c d e R while max{ , },t l k l k .

Hadi et al. (2016) studied the bounded and oscillatory nature of equation

11

1

n nn n

n n

bz zz az

cz dz

(2.83)

Elsayed et al. (2016) considered the nature of the equation

21

2 3

n nn

n n

az zz

bz cz

(2.84)

where 0,1,...n and 3 0, , , ,...,a b c z z R .

Stevic (2017) considered the two dimensional boundary value problem

, 1, 1, 1m n m n n m nd d f d (2.85)

where 1 n m , While , ,m ma b m N and ,nf n N are complex sequences.

26

Elsayed (2017a) got the nature of equation of systems of fourth order

2 21 1

3 3

, , 0,1,2,...n n n nn n

n n n n

y x x yx y n

y y x x

. (2.86)

Elsayed et al. (2017 b) investigated the dynamical nature of the

difference equation

51

2 5

nn

n n

Cxx

A Bx x

(2.87)

With arbitrary initial conditions while ,A B and C are arbitrary constants.

Changyou et al. (2017) studied nature of equation

1 21

max , ,n nn n

A Az z

z z

(2.88)

where 0n N and 2 1 0, , ,z z z A R .

M.M. El-Dessoky (2017) studied the global stability of equation with

numerical examples

1n k n s

n n n t n ln k n s

dz ezz az bz cz

z z

, 0,1,...n (2.89)

Khaliq (2017) investigated the periodic nature of the equation

1n l n k n s

n nn l n k n s

az bz czz z

z z z

(2.90)

With 0,1,...n while initial values and coefficients are R while

max{ , , }r l k s Z .

Stephen Sadiq and Muhammad Kalim (2018) studied global attractivity

of rational equation

27

1n k

nn

a zz

b z

(2.91)

where 0,1,2,...n , 1 0(0, ), ( ,0), , ,..., ( ,0)k ka b z z z . is non-

negative real number and k Z .

Stephen Sadiq and Muhammad Kalim (2018) studied the global

attractivity of a rational equation of order twenty

2

91 9

9 19

, 0,1,2,...nn n

n n

zz z n

z z

(2.92)

Under certain conditions we obtained solutions and verified the obtained

results by graphical examples.

Stephen Sadiq and Muhammad Kalim (2018) studied the dynamical

nature of some higher order rational equation

201

6 13 201n

nn n n

zz

z z z

(2.93)

With 0,1,2,...n and the involved initial conditions are randomly

selected real numbers. To confirm the obtained results we considered few

examples by assigning different numerical values to initial conditions with

Matlab.

28

CHAPTER 3:

PRELIMINARIES

In this chapter we present preliminaries that help us to recall basic

knowledge about the topic of our research. It includes basic notions,

definitions and fundamental theorems required to understand the work in this

research. We discuss some basic notations and useful results which will be

helpful in this work.

3.1 FINITE DIFFERENCE EQUATION

“A finite - difference equation defines a pattern that gives the next term

in a sequence of numbers in terms of preceding ones. Its generic form is

1 2(z ,z ,..., )n p n p n p nz F z (3.1)

If the first p terms of a sequence are known, then we obtain the next

term of a sequence using (3.1) difference equation. In particular 1 (z )n nz F

is first order difference equation.

Suppose that I is some interval of real numbers and F a continuous

function defined on 1I ( 1k k copies of I ),where k is some natural number.

Throughout this thesis, we consider the following difference equation”

1 1 0,1,...(z ,z ,..., ),nn n n k nz f z (3.2)

For given initial values 0( 1)z , ,...,k kz z I .”

3.2 LINEAR DIFFERENCE EQUATION

“A difference equation is linear if it can be written in the form

1 1 2 2 1 1( ) ( ) ... ( ) ( )n k n k n k k n k n nz b n z b n z b n z b n z Q

where ( ), 1,2,...,ib n i k and nQ are given functions of n .” e.g.

29

1 1( ) ( ) ( ) ( ) 0n n n n nQ x A x B Q x Q x

Is second order linear difference equation.

3.3 NON-LINEAR DIFFERENCE EQUATION

“A difference equation is nonlinear if it is not linear.” e.g.

11

1

  n nn

n n

z z

A Bz Czz

Is a non-linear second order rational difference equation.

3.4 RATIONAL DIFFERENCE EQUATION

“A difference equation in rational form is called rational difference

equation”. e.g.

11

1

n nn

n n

z zz

z z

2

1 211

nn

n

zz

z

1 21

1 2

n nn

n n

z zz

Cz Dz

are all rational difference equations”.

3.5 ORDER OF A DIFFERENCE EQUATION

“The general form of difference equation of order ( 1)thk is given by

1 1 0,1,...(z ,z ,..., ),nn n n k nz f z

30

e.g. 21

1 2(1 )

n nn

n n n

z zz

z az z

(Order 3)

51

2 51

nn

n n

zz

z z

(Order 6)

2 4 61

1 3 5 7

n n n nn

n n n n

z z z zz

z z z z

(Order 8)”

3.6 INVARIANT INTERVAL

“An interval M I is invariant if 0( 1)z , ,...,k kz z M implies that

zn M for all 0n .”

3.7 EQUILIBRIUM POINT

“A point z I is called an equilibrium point of difference equation

(3.2) if

( ,..., )z F z z (3.3)

or a solution of equation (3.2) which remains constant for all n k . That is

nz z for n k is a solution of difference equation (3.2).

The linearized equation associated with the difference equation (3.2)

with equilibrium point z is

10

( ,..., )k

n nF z z

y yz

(3.4)

With its characteristics equation

1

0

( ,..., )kk

kF z z

z

(3.5)

31

e.g. Graphically, an equilibrium point is the point of x -coordinate on the graph

where f intersects the line y x . For example, there are two equilibrium

points for the equation 2( 1) ( )x n x n where 2( )f x x . The equilibrium

points is given by letting ( )f x x , or 2x x . After solving we have two

equilibrium points 0 and 1. Consider another example, where 3 2( ) 2 1f x x x x . Putting 3 22 1x x x x , we find that 1, 1 and

1/ 2 are the equilibrium points.”

Figure 3.1: Fixed Points of 2( )f x x

32

Figure 3.2: Fixed Points of3 2( ) 2 1f x x x x

3.8 PERIODICITY

A solution n n kz

of Equation (3.2) is periodic with period m if there

exists an integer 1m such that zn m nz for all n k . If zn m nz holds

for smallest positive integer m then solution n n kz

of Equation (3.2) is

periodic period of prime m .

e.g. Consider the following eight order difference equation

2 4 61

1 3 5 7

n n n nn

n n n n

z z z zz

z z z z

To find its periodicity we proceed as

1 1 3 52

2 4 6 7

 1n n n n

nn n n n n

z z z z

z z z zz

z

33

2 2 43

1 1 3 5 6

1n n n nn

n n n n n

z z z zz

z z z z z

Inductively we get

16 14 12 1017 1

15 13 11 9 8

1n n n nn n

n n n n n

z z z zz

z z z z zz

17 15 13 1118

16 14 12 10 9

1n n n nn n

n n n n n

z z z zz

z z z z zz

So this difference equation is periodic with period 18 .”

3.9 FIBONACCI SEQUENCE

The 0{ } {1,2,3,5,8,...}n nF that is 1 2 0n n nF F F with initial

conditions 2 10, 1F F is called Fibonacci sequence.”

3.10 LOCALLY STABLE

If for every 0 there exist 0 such that for all

0( 1)z , ,...,k kz z I

with 0

ii k

z z

, we have zn z

for all

n k . Then z of difference equation (3.2) is locally stable.”

3.11 LOCALLY ASYMPTOTICALLY STABLE

If equilibrium point z is locally stable, and there exist 0 such for all

initial values 0( 1)z , ,...,k kz z I with 0

ii k

z z

, we have,

lim zn n z . Then z of difference equation (3.2) is called locally

asymptotically stable.”

34

3.12 GLOBAL ATTRACTOR

“ z of difference equation (3.2) is called global attractor If

0( 1)z , ,...,k kz z I always implies that lim zn n z .”

3.13 GLOBAL ASYMPTOTICALLY STABLE

“If z is locally asymptotically stable as well as a attractor. Then

equilibrium point z of difference equation (3.2) is called global asymptotically

stable.”

3.14 UNSTABLE

“The equilibrium point z of difference equation is unstable if it is not

locally stable.”

3.15 SEMI-CYCLE ANALYSIS

Suppose z is an equilibrium point and n n kz

is a solution of

Eq. (3.2).

3.15.1 Positive Semi Cycle

“A positive semi-cycle of n n kz

comprises on “string” of terms

1,...,{ , },p p qz z z all greater than or equal to z with p k and q such that

Either p k or p k and 1pz z and either q or q and

1qz z .”

35

3.15.2 Negative Semi Cycle

“A negative semi-cycle of n n kz

comprises on “string” of term

1,...,{ , },l l mz z z all less than z with p k and q such that

Either p k or p k and 1pz z and

either q or q and 1qz z .”

3.15.3 Non-Oscillatory Solutions:

“A solution n n kz

of Eq. (3.2) about z is non-oscillatory if there

exists N k such that either nz z or nz z for all n N .Otherwise

n n kz

is oscillatory about z .”

3.15.4 Strictly Oscillatory Solutions

“A solution n n kz

of Eq (3.2).is strictly oscillatory about z if for

every N k there exist ,l m N such that lz z and mz z .”

3.15.5 Persistent

“If there exists a positive constant m then positive solution n n kz

of

Eq. (3.2) persists such that nm z for all n k .”

3.15.6 Permanent

“Eq. (3.2) is permanent if there exists positive real numbers P and Q

such that there exists an integer N k for every solution n n kz

of

Eq. (3.2) depending upon the initial conditions such that nP z Q for all

n N .”

36

3.16 USEFUL RESULTS

Here we state some useful result which will be helpful in our

investigation.

3.16.1 Theorem 1

“Suppose f is a continuously differentiable function defined on open

neighborhood of z . Then the following statements are true;

1. If all the roots of Eq. (3.5) have absolute value less than one, then

the equilibrium point z of Eq. (3.2) is locally asymptotically

stable.

2. If atleast one root of Eq. (3.5) has absolute value greater than one,

then the equilibrium point z of Eq. (3.2) is unstable.

3. If all the roots of Eq. (3.5) have absolute value greater than one,

then the equilibrium point z of Eq. (3.2) is source.”

(See E. A. Grove & G. Ladas (2005, pp. 3))

The above result is Linearized Stability Theorem and is applicable in

analyzing the local stability nature of z of equation (3.2).

3.16.2 Hyperbolic Solution

“If no roots of Eq. (3.5) has absolute value equal to one then equilibrium

point z of Eq. (3.2) is called hyperbolic.”

3.16.3 Non-hyperbolic Solution

“ z is non-hyperbolic if there exists a root of Eq. (3.5) with absolute

value equal to one.”

37

3.16.4 Saddle Point

"The equilibrium point z of Eq. (3.2) is called saddle point if it is

hyperbolic, and in addition, there exists a root of Eq. (3.5) with absolute value

less and another root of Eq (3.5) with absolute value greater than one. Thus

saddle point is unstable.”

3.16.5 Theorem 2

“Let : [ , ] [ , ] [ , ]g a b a b a b be a continuous function, where a and b are real numbers with a b and consider the difference equation

1 1( , )n n nz g z z , 0,1,...n (3.6)

Suppose that g satisfies the following conditions:

1. ( , )g x y is non-decreasing in [ , ]x a b for each fixed [ , ]y a b and ( , )g x y is non-decreasing in [ , ]y a b for each fixed

[ , ]x a b .

2. If ( , )m M is a solution of the system ( , )m g m m and

( , )M g M M then m M .

Then there exists exactly on equilibrium point z of Eq. (3.6) and every

solution of Eq. (3.6) converges to z .”

(See E. A. Grove & G. Ladas (2005, pp. 16))

3.16.6 Theorem 3

“Let : [ , ] [ , ] [ , ]g a b a b a b be a continuous function, where a and b are real numbers with a b and consider the difference equation

1 1( , )n n nz g z z , 0,1,...n (3.7)

38

Suppose that g satisfies the following conditions:

1. ( , )g x y is non-increasing in [ , ]x a b for each fixed [ , ]y a b and ( , )g x y is non- decreasing in [ , ]y a b for each fixed

[ , ]x a b .

2. If ( , )m M is a solution of the system ( , )m g M m and

( , )M g m M then m M .

Then there exists exactly on equilibrium point z of Eq. (3.7) and every

solution of Eq. (1.7) converges to z .”

(See E. A. Grove & G. Ladas (2005, pp. 16))

3.16.7 Theorem 4

“Let : [ , ] [ , ] [ , ]g a b a b a b be a continuous function, where a and b are real numbers with a b and consider the difference equation

1 1( , )n n nz g z z , 0,1,...n (3.8)

Suppose that g satisfies the following conditions:

1. ( , )g x y is non-decreasing in [ , ]x a b for each fixed [ , ]y a b and ( , )g x y is non- increasing in [ , ]y a b for each fixed

[ , ]x a b .

2. If ( , )m M is a solution of the system ( , )m g m M and

( , )M g M m then m M .

Then there exists exactly on equilibrium point z of Eq. (3.8) and every

solution of Eq. (3.8) converges to z .”

(See E. A. Grove & G. Ladas (2005, pp. 17))

39

3.16.8 Theorem 5

“Let : [ , ] [ , ] [ , ]g a b a b a b be a continuous function, where a and b are real numbers with a b and consider the difference equation

1 1( , )n n nz g z z , 0,1,...n (3.9)

Suppose that g satisfies the following conditions:

1. ( , )g x y is non-increasing in [ , ]x a b for each fixed [ , ]y a b and ( , )g x y is non- increasing in [ , ]y a b for each fixed

[ , ]x a b .

2. If ( , )m M is a solution of the system ( , )m g M M and

( , )M g m m then m M .

Then there exists exactly on equilibrium point z of Eq. (3.9) and every

solution of Eq. (3.9) converges to z .”

(See E. A. Grove & G. Ladas (2005, pp. 17))

The above global attractivity results are very useful in determining

convergence results of difference equations.

The following result extends and modifies theorems 2, 3, 4 and 5.

3.17 CONVERGENCE THEOREMS

The following lemma provides sufficient conditions for establishing the

boundedness of solutions of difference equations.

3.17.1 Lemma 1

“Let J be an interval of real numbers and assume that 1,kf C J J

is non-decreasing in each of its arguments. Suppose also that every point c in

J is an equilibrium point of Eq. (3.1) that is ( , ,..., )f c c c c for everyc J .

40

Let n n kz

be a solution of Eq. (3.1). Set

1 0min{ , ,..., }k km z z z and 1 0max{ , ,..., }k kM z z z

Then nm z M for all n k .”

(See E. A. Grove & G. Ladas (2005, pp. 11))

3.17.2 Lemma 2

“Let J be an interval of real numbers. Assume that the following

statements are true;

1. 1,kf C J J

is non-decreasing in each of its arguments.

2. z is strictly increEq.asing in each of the arguments 1 2, ,...,

li i iz z z

where

1 21 ... 1li i i k and where 1 2, ,..., li i i are relatively prime.

3. Every point c in J is an equilibrium point of Eq. (3.1).

Then every solution of Eq. (3.1) has a finite limit.”

(See E. A. Grove & G. Ladas (2005, pp. 12))

The next result which was given by C. W. Clark provides a sufficient

condition that z of Eq. (3.1).be a sink.

3.17.3 Theorem 6

“Assume that 0 1, ,..., kp p p are real numbers such that

0 1 ... 1kp p p

Then all roots of Eq. (3.5) lie inside the open unit disk 1 .”

(See E. A. Grove & G. Ladas (2005, pp. 7)).

41

CHAPTER 4:

GLOBAL ATTRACTIVITY OF A RATIONAL

DIFFERENCE EQUATION OF ORDER TWENTY

We study qualitative as well as periodic behavior of solutions of form of

difference equation

2

91 9

9 19

, 0,1,2,...nn n

n n

zz z n

z z

(4.1)

with initial conditions

19 18 0, ,...,z z z R (A)

and , , , are constants. We obtained some special cases of consider

equation.

4.1 LOCAL STABLE NATURE OF EQ.(4.1)

The equilibrium point of Eq. (4.1) is given by

2z

z zz z

2

(1 )( )

zz

z

2 2(1 )( )z z

If (1 )( ) then 0z is the unique equilibrium point.

Let : 0, 0, 0,f be continuous and differentiable function

given as

42

2

( , )u

f u v uu v

(4.2)

2

2

(2 )( , ) u v u uf u v

u u v

2 2

22

(2 )( , ) z zf z z

u z

2

2( , )f z z

u

2

( , ) 2f z z

u

2

2

( , )f u v u

v u v

2

22

( , )f z z z

v z

About equilibrium point, the linearized equation of Eq. (4.1) is

1 12 2

20n n ny y y

(4.3)

4.1.1 Theorem 1

Assume that 2( 3 ) ( ) (1 ) , 1 . Then 0z of Eq. (4.1)

is locally asymptotically stable.

43

Proof: Eq. (4.2) is asymptotically stable if

2 2

21

2

31

2

31

2

( 3 ) 1

4.2 GLOBAL ATTRACTIVE BEHAVIOR OF EQ.(4.1)

4.2.1 Theorem 2

The 0z of Eq. (4.1) is global attractor if (1 ) .

Proof: Let , be any real numbers and suppose that

2

: , ,g

be function defined by (4.2). Suppose that ( , )t T is a

solution.

( , )T g T t and ( , )t g t T

From Eq. (4.1), we see that

2T

T TT t

,

2tt t

t T

Subtracting both above equations

2 2 2 2(1 ) ( )T Tt t Tt T t

2 2 2 2(1 )( ) ( )T t T t

If (1 ) , thus T t . It concludes by theorems (2, 3, 4, 5) of

previous chapter that z is a global attractor of Eq. (4.1).

44

4.3 BOUNDEDNESS BEHAVIOR OF EQ. (4.1)

4.3.1 Theorem 3

If 1

, then every solution of Eq. (4.1) is bounded.

Proof: Let 19n n

z

be a solution of Eq. (4.1). Then

2 2

9 91 9 9

9 19 9

n nn n n

n n n

z zz z z

z z z

9nz

Thus 1 9n nz z for all 0n .

Then 10 9 10 1 100 0 0,..., ,n n nn n n

z z z

are decreasing and so are

bounded above by

19 18 17 0max{ , , ,..., }M z z z z

4.4 GRAPHICAL EXAMPLES

To confirm the obtained results we take some numerical examples.

4.4.1 Example 1

By taking different numerical values for the initial conditions stated in

expression (A) in order

2,5,9,8,7,5,0,11,20,9,1,6,11,10,5,8,2,9,5,15, and 0.6, 2, 5, 8.

(see Figure 4.1)

45

Figure 4.1: Behavior of 2

91 9

9 19

nn n

n n

zz z

z z

4.4.2 Example 2

By taking different numerical values for the initial conditions stated in

expression (A) in order

3,9,15,17,11,0,20,13,19,5,7,8,10,14,7,10,3,5,2,and

0.5, 3, 11, 15.

(see Figure 4.2).

46

Figure 4.2: Behavior of

29

1 99 19

nn n

n n

zz z

z z

4.5 DIFFERENT CASES OF EQ. (4.1)

4.5.1 FIRST EQUATION

We discuss some special case of Eq. (4.1)

2

91 9

9 19

nn n

n n

zz z

z z

(4.5.1)

where the same initial conditions as stated in expression (A).

4.5.2 Theorem 4

Let 19n n

z

be a solution of Eq. (4.5.1). Then for 0,1,2,...n

47

2 1 210 9

1 2 2 1

ni i

ni i i

F k F wz w

F k F w

2 1 210 8

1 2 2 1

ni i

ni i i

F j F tz j

F j F t

2 1 210 7

1 2 2 1

ni i

ni i i

F h F sz h

F h F s

2 1 210 6

1 2 2 1

ni i

ni i i

F g F rz g

F g F r

2 1 210 5

1 2 2 1

ni i

ni i i

F f F qz f

F f F q

2 1 210 4

1 2 2 1

ni i

ni i i

F e F pz e

F e F p

2 1 210 3

1 2 2 1

ni i

ni i i

F d F oz d

F d F o

2 1 210 2

1 2 2 1

ni i

ni i i

F c F nz c

F c F n

2 1 210 1

1 2 2 1

ni i

ni i i

F b F mz b

F b F m

2 1 210

1 2 2 1

ni i

ni i i

F a F lz a

F a F l

where , , , , , , , , , , , , , , , , , , ,w t s r q p o n m l k j h g f e d c b a are assigned to initial

conditions stated in expression (A) in order and

0

1,1,2,3,5,8,13,...m mF

48

Proof: We prove by mathematical induction the solutions of Eq. (4.5.1). First

for 0n , the result is true. Suppose the above results hold for 1, 2.n n

1

2 1 210 19

1 2 2 1

ni i

ni i i

F k F wz w

F k F w

1

2 1 210 18

1 2 2 1

ni i

ni i i

F j F tz j

F j F t

1

2 1 210 17

1 2 2 1

ni i

ni i i

F h F sz h

F h F s

1

2 1 210 16

1 2 2 1

ni i

ni i i

F g F rz g

F g F r

1

2 1 210 15

1 2 2 1

ni i

ni i i

F f F qz f

F f F q

1

2 1 210 14

1 2 2 1

ni i

ni i i

F e F pz e

F e F p

1

2 1 210 13

1 2 2 1

ni i

ni i i

F d F oz d

F d F o

1

2 1 210 12

1 2 2 1

ni i

ni i i

F c F nz c

F c F n

1

2 1 210 11

1 2 2 1

ni i

ni i i

F b F mz b

F b F m

49

12 1 2

10 101 2 2 1

ni i

ni i i

F a F lz a

F a F l

2

2 1 210 29

1 2 2 1

ni i

ni i i

F k F wz w

F k F w

2

2 1 210 28

1 2 2 1

ni i

ni i i

F j F tz j

F j F t

2

2 1 210 27

1 2 2 1

ni i

ni i i

F h F sz h

F h F s

2

2 1 210 26

1 2 2 1

ni i

ni i i

F g F rz g

F g F r

2

2 1 210 25

1 2 2 1

ni i

ni i i

F f F qz f

F f F q

2

2 1 210 24

1 2 2 1

ni i

ni i i

F e F pz e

F e F p

2

2 1 210 23

1 2 2 1

ni i

ni i i

F d F oz d

F d F o

2

2 1 210 22

1 2 2 1

ni i

ni i i

F c F nz c

F c F n

2

2 1 210 21

1 2 2 1

ni i

ni i i

F b F mz b

F b F m

2

2 1 210 20

1 2 2 1

ni i

ni i i

F a F lz a

F a F l

50

210 11

10 1 10 1110 11 10 21

nn n

n n

zz z

z z

12 2 1 2 3 2

1 2 2 1 2 1

2 3 2 4 2 1 2 2

12 4 2 5 2 2 2 32 1 2

1 2 2 1 2 1 2

2 2 1

,...,n

i i

i i i

n n n n

nn n n ni i

i i i i i

i i i

F b F m F b F mb

F b F m F b F m

F b F m F b F m

F b F m F b F mF b F mb

F b F m F b F mb

F b F m

1 2

2 1 2

1 1 2 2 1

n ni i

i i i

F b F mb

F b F m

12 1 2 2 1 2 2

11 2 2 1 2 2 2 32 1 2

1 2 2 1 2 1 2 2

2 2 2 3

1

ni i n n

ni i i n ni i

i i i n n

n n

F b F m F b F mb

F b F m F b F mF b F mb

F b F m F b F m

F b F m

12 1 2

2 1 2 211 2 2 12 1 2

1 2 2 1 2 1 2 2 2 2 2 3

( )

( )

ni i

n nni i ii i

i i i n n n n

F b F mb F b F m

F b F mF b F mb

F b F m F b F m F b F m

Using Fibonacci sequence

2 2 1 2 2

2 1 2 2 2 3

n n n

n n n

F b F b F b

F m F m F m

12 1 2

2 1 2 211 2 2 12 1 2

1 2 2 1 2 2 1

( )

( )

ni i

n nni i ii i

i i i n n

F b F mb F b F m

F b F mF b F mb

F b F m F b F m

1

2 1 2 2 1 2 2

1 2 2 1 2 2 1

( )1

( )

ni i n n

i i i n n

F b F m F b F mb

F b F m F b F m

1

2 1 2 2 2 1 2 1 2 2

1 2 2 1 2 2 1( )

ni i n n n n

i i i n n

F b F m F b F m F b F mb

F b F m F b F m

51

1

2 1 2 2 1 2

1 2 2 1 2 2 1( )

ni i n n

i i i n n

F b F m F b F mb

F b F m F b F m

2 1 2

1 2 2 1

ni i

i i i

F b F mb

F b F m

Similarly

210 10

10 10 1010 10 10 20

nn n

n n

zz z

z z

12 2 1 2 3 2

1 2 2 1 2 1

2 3 2 4 2 1 2 2

12 4 2 5 2 2 2 32 1 2

1 2 2 1 2 1 2

2 2 1

,...,n

i i

i i i

n n n n

nn n n ni i

i i i i i

i i i

F a F l F a F la

F a F l F a F l

F a F l F a F l

F a F l F a F lF a F la

F a F l F a F la

F a F l

1 2

2 1 2

1 1 2 2 1

n ni i

i i i

F a F la

F a F l

12 1 2 2 1 2 2

11 2 2 1 2 2 2 32 1 2

1 2 2 1 2 1 2 2

2 2 2 3

1

ni i n n

ni i i n ni i

i i i n n

n n

F a F l F a F la

F a F l F a F lF a F la

F a F l F a F l

F a F l

12 1 2

2 1 2 211 2 2 12 1 2

1 2 2 1 2 1 2 2 2 2 2 3

( )

( )

ni i

n nni i ii i

i i i n n n n

F a F la F a F l

F a F lF a F la

F a F l F a F l F a F l

Using Fibonacci sequence

2 2 1 2 2

2 1 2 2 2 3

n n n

n n n

F a F a F a

F l F l F l

12 1 2 2 1 2 2

1 2 2 1 2 2 1

( )1

( )

ni i n n

i i i n n

F a F l F a F la

F a F l F a F l

52

12 1 2 2 2 1 2 1 2 2

1 2 2 1 2 2 1( )

ni i n n n n

i i i n n

F a F l F a F l F a F la

F a F l F a F l

12 1 2 2 1 2

1 2 2 1 2 2 1( )

ni i n n

i i i n n

F a F l F a F la

F a F l F a F l

2 1 2

1 2 2 1

ni i

i i i

F a F la

F a F l

4.5.3 Example 3

We now confirm result by taking numerical example for initial

conditions stated in expression (A) in order

10,5,8,9,2,7,1,5,11,2,3,10,8,15,2,10,7,5,1,8

(See Figure 4.3).

Figure 4.3: Behavior of

29

1 99 19

nn n

n n

zz z

z z

53

4.5.4 Second Equation

We take form of Eq. (4.1)

2

91 9

9 19

nn n

n n

zz z

z z

(4.5.3)

where the initial conditions are same as stated in expression (A).

4.5.5 Theorem 5

Let 19n n

z

be solution of Eq. (4.5.3). Then for 0,1,2,...n

210 9

1 2

ni i

ni i i

F k F wz w

F k F w

210 8

1 2

ni i

ni i i

F j F tz j

F j F t

210 7

1 2

ni i

ni i i

F h F sz h

F h F s

210 6

1 2

ni i

ni i i

F g F rz g

F g F r

210 5

1 2

ni i

ni i i

F f F qz f

F f F q

210 4

1 2

ni i

ni i i

F e F pz e

F e F p

210 3

1 2

,n

i in

i i i

F d Foz d

F d F o

54

210 2

1 2

ni i

ni i i

F c F nz c

F c F n

210 1

1 2

ni i

ni i i

F b F mz b

F b F m

210

1 2

ni i

ni i i

F a F lz a

F a F l

where

, , , , , , , , , , , , , , , , , , ,w t s r q p o n m l k j h g f e d c b a are assigned to initial

conditions stated in expression (A) in order and

0

1,1,2,3,5,8,13,...m mF

Proof: Proof is same and omitted.

4.5.6 Example 4

We will take some numerical examples by assigning different numerical

values to initial conditions stated in expression (A) in order to confirm result.

5,15,18,20,19,17,18,20,13,4,11,8,10,12,9,7,5,1,3,7

(See Figure 4.4)

55

Figure 4.4: Behavior of

29

1 99 19

nn n

n n

zz z

z z

4.5.7 Third Equation

We discuss the form of solutions of Eq. (4.1) in this section

2

91 9

9 19

nn n

n n

zz z

z z

(4.5.7)

where the initial conditions are as stated in expression (A).

4.5.8 Theorem 6

The solution of Eq. (4.5.7) will take the following formulas for 0,1,2,...n

56

10 91

nn n

kwz

F k F w

10 81

nn n

jtz

F j F t

10 71

nn n

hsz

F h F s

10 61

nn n

grz

F g F r

10 51

nn n

fqz

F f F q

10 41

nn n

epz

F e F p

10 31

nn n

doz

F d F o

10 21

nn n

cnz

F c F n

10 11

nn n

bmz

F b F m

101

nn n

alz

F a F l

Proof:

For 0n the result is obvious. Suppose that 0n and the assumption is

true for 1, 2n n .

10 191

nn n

kwz

F k F w

10 181

nn n

jtz

F j F t

57

10 171

nn n

hsz

F h F s

10 161

nn n

grz

F g F r

10 151

nn n

fqz

F f F q

10 141

nn n

epz

F e F p

10 131

nn n

doz

F d F o

10 121

nn n

cnz

F c F n

10 111

nn n

bmz

F b F m

10 101

nn n

alz

F a F l

10 292 1

nn n

kwz

F k F w

10 282 1

nn n

jtz

F j F t

10 272 1

nn n

hsz

F h F s

10 262 1

nn n

grz

F g F r

58

10 252 1

nn n

fqz

F f F q

10 242 1

nn n

epz

F e F p

10 232 1

nn n

doz

F d F o

10 222 1

nn n

cnz

F c F n

10 212 1

nn n

bmz

F b F m

10 202 1

nn n

alz

F a F l

Now from (4.5.7)

210 11

10 1 10 1110 11 10 21

nn n

n n

zz z

z z

1 1

1

1 2 1

n n n n

n n

n n n n

bm bm

F b F m F b F mbm

F b F m bm bm

F b F m F b F m

2 1

1 1 2 1 1

n n

n n n n n n n n

F b F mbm bm

F b F m F b F m F b F m F b F m

2 1

1 1

1 n n

n n n n

F b F mbm

F b F m F b F m

59

1n n

bm

F b F m

Similarly

210 10

10 10 1010 10 10 20

nn n

n n

zz z

z z

1 1

1

1 2 1

n n n n

n n

n n n n

al al

F a F l F a F lal

F a F l al al

F a F l F a F l

2 1

1 1 2 1 1

n n

n n n n n n n n

F a F lal al

F a F l F a F l F a F l F a F l

2 1

1 1

1 n n

n n n n

F a F lal

F a F l F a F l

1n n

al

F a F l

4.5.9 Example 5

We will take numerical example to confirm result by assigning these

numerical values to initial conditions in order stated in expression (A).

20,1,2,25,19,17,9,14,11,0,5,9,7,8,12,13,0,2,5,3

(See Figure 4.5)

60

Figure 4.5: Behavior of 2

91 9

9 19

nn n

n n

zz z

z z

4.5.10 Fourth Equation

In this part we discuss the solutions of form of Eq. (4.1)

2

91 9

9 19

nn n

n n

zz z

z z

(4.5.9)

where the initial conditions are stated in expression (A).

4.5.11 Theorem 7

Assume that 19n n

z

be a solution of Eq. (4.5.9). Then every solution

of Eq. (4.5.9) is periodic of period 60. Moreover 19n n

z

gets the form

61

, , , , , , , , , , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , , , , , ,

, , , , , , , , , , , , ,

kw ep jt do hsw t s r q p o n m l k j h g f e d c b a

k w e p j t d o h s

cn gr bm fq alw t s r q p o n m l k j

c n g r b m f q a l

kw ep jt do hsh g f e d c b a

k w e p j t d o h s

,

, , , , , , , , , , , , , , , , , , , , , , ,....

cn gr

c n g r

bm fq alw t s r q p o n m l k j h g f e d c b a

b m f q a l

or

60 19nz w , 60 18nz t , 60 17nz s

60 16nz r , 60 15nz q , 60 14nz p

60 13nz o , 60 12nz n , 60 11nz m

60 10nz l , 60 9nz k , 60 8nz j

60 7nz h , 60 6nz g , 60 5nz f

60 4nz e , 60 3nz d , 60 2nz c

60 1nz b , 60nz a , 60 1n

kwz

k w

60 2n

epz

e p

, 60 3n

jtz

j t

, 60 4n

doz

d o

60 5n

hsz

h s

, 60 6n

cnz

c n

, 60 7n

grz

g r

60 8n

bmz

b m

, 60 9n

fqz

f q

, 60 10n

alz

a l

60 11nz w , 60 12nz t , 60 13nz s

60 14nz r , 60 15nz q , 60 16nz p

60 17nz o , 60 18nz n , 60 19nz m

60 20nz l , 60 21nz k , 60 22nz j

60 23nz h , 60 24nz g , 60 25nz f

60 26nz e , 60 27nz d , 60 28nz c

60 29nz b , 60 30nz a , 60 31n

kwz

k w

60 32n

epz

e p

, 60 33n

jtz

j t

, 60 34n

doz

d o

60 35n

hsz

h s

, 60 36n

cnz

c n

, 60 37n

grz

g r

62

60 38n

bmz

b m

, 60 39n

fqz

f q

, 60 40n

alz

a l

where , , , , , , , , , , , , , , , , , , ,w t s r q p o n m l k j h g f e d c b a are assigned to initial

conditions stated in expression (A) in order.

Proof: Same proof as theorem (4.5.7) and will be omitted therefore.

4.5.12 Example 6

We will take numerical example to confirm result by assigning these

numerical values to initial conditions in order stated in expression (A).

5,9,2,3,2,1,4,6,0,10,12,15,17,18,0,10,7,8,1,2

(See Figure 4.6)

Figure 4.6: Behavior of 2

91 9

9 19

nn n

n n

zz z

z z

63

4.6 CONCLUSION

We studied the global stability, bounded behavior and forms of solutions

of few cases of difference equation (4.1) 1 9n nz z 2

9

9 19

,n

n n

z

z z

0,1,2,...n and concluded that if

(1 )( ) , then the unique

equilibrium point of above equation is 0z . The equilibrium point 0z of

Eq. (4.1) is locally asymptotically stable when 2( 3 ) ( ) (1 ) . The

equilibrium point 0z of Eq. (4.1) is global attractor if (1 ) . Eq. (4.1)

has every solution bounded if 1

. In the end we obtained solution of

four different types of Eq. (4.1) and provided graphical examples in different

cases by assigning different initial values by using Matlab.

64

CHAPTER 5:

DYNAMICAL BEHAVIOR OF HIGHER ORDER

RATIONAL DIFFERENCE EQUATION

In this chapter we discuss the global stability of the nature of positive

solutions and the periodicity of the difference equation

1 0 1 2

0 1 2 3

4 5 6 7

n n n k n s

n t n l n m n p

n t n l n m n p

z z z z

b z b z b z b z

b z b z b z b z

(5.1)

with non-negative initial conditions 1 1 0, ,..., ,z z z z where

max , , , , ,k s t l m p and coefficients

0 1 2 0 1 2 3 4 5 6 7, , , , , , , , , ,b b b b b b b b R (B)

Numerical examples are also given to confirm the obtained results.

5.1 LOCAL STABILITY

Here we discuss the local stability of Eq. (5.1).The equilibrium point of

Eq. (5.1) is given by

0 1 2 30 1 2

4 5 6 7

b z b z b z b zz z z z

b z b z b z b z

0 1 2 30 1 2

4 5 6 7

(1 )b b b b

zb b b b

If 0 1 2 1 , then the unique equilibrium point is

0 1 2 3

4 5 6 7 0 1 2

( )

( )(1 )

b b b bz

b b b b

65

Let 7: (0, ) (0, )f be a continuous function given by

0 1 2 3 4 5 6

0 3 1 4 2 5 3 60 0 1 1 2 2

4 3 5 4 6 5 7 6

( , , , , , , )f q q q q q q q

b q b q b q b qq q q

b q b q b q b q

0 1 2 3 4 5 60

0

( , , , , , , )f q q q q q q q

q

0 1 2 3 4 5 61

1

( , , , , , , )f q q q q q q q

q

0 1 2 3 4 5 62

2

( , , , , , , )f q q q q q q q

q

0 1 2 3 4 5 6

3

5 0 1 4 4 6 0 2 4 5 7 0 3 4 62

4 3 5 4 6 5 7 6

( , , , , , , )

( ) ( ) ( )

( )

f q q q q q q q

q

b b b b q b b b b q b b b b q

b q b q b q b q

0 1 2 3 4 5 6

4

1 4 0 5 3 6 1 2 5 5 7 1 3 5 62

4 3 5 4 6 5 7 6

( , , , , , , )

( ) ( ) ( )

( )

f q q q q q q q

q

b b b b q b b b b q b b b b q

b q b q b q b q

0 1 2 3 4 5 6

5

2 4 0 6 3 5 2 1 6 4 7 2 3 6 62

4 3 5 4 6 5 7 6

( , , , , , , )

( ) ( ) ( )

( )

f q q q q q q q

q

b b b b q b b b b q b b b b q

b q b q b q b q

0 1 2 3 4 5 6

6

3 4 0 7 3 5 3 1 7 4 6 3 2 7 52

4 3 5 4 6 5 7 6

( , , , , , , )

( ) ( ) ( )

( )

f q q q q q q q

q

b b b b q b b b b q b b b b q

b q b q b q b q

66

Thus 0 10

( , , , , , , )f z z z z z z zu

q

1 21

( , , , , , , )f z z z z z z zu

q

2 32

( , , , , , , )f z z z z z z zu

q

3

0 5 6 7 4 1 2 3 0 1 24

4 5 6 7 0 1 2 3

( , , , , , , )

( ( ) ( ))(1 )

( )( )

f z z z z z z z

q

b b b b b b b bu

b b b b b b b b

4

1 4 6 7 5 0 2 3 0 1 25

4 5 6 7 0 1 2 3

( , , , , , , )

( ( ) ( ))(1 )

( )( )

f z z z z z z z

q

b b b b b b b bu

b b b b b b b b

5

2 4 5 7 6 1 2 3 0 1 26

4 5 6 7 0 1 2 3

( , , , , , , )

( ( ) ( ))(1 )

( )( )

f z z z z z z z

q

b b b b b b b bu

b b b b b b b b

6

3 4 5 6 7 0 1 2 0 1 27

4 5 6 7 0 1 2 3

( , , , , , , )

( ( ) ( ))(1 )

( )( )

f z z z z z z z

q

b b b b b b b bu

b b b b b b b b

The linearized equation of equation (5.1) about z is

1 1 2 3 4

5 6 7

n n n k n s n t

n l n m n p

z u z u z u z u z

u z u z u z

(5.2)

67

5.2 GLOBAL STABILITY

5.2.1 Theorem 1

The z is a global attractor of Eq. (5.1) if one of following conditions

holds

i) 0 1 2 1 , 0 1 2 3b b b b and 4 5 6 0b b b .

ii) 0 1 2 1 , 0 1 3 2b b b b and 4 5 7 0b b b .

Proof: We consider two cases.

Case 1: If function 0 1 2 3 4 5 6( , , , , , , )f v v v v v v v is increasing in

0 1 2 3 4 5, , , , ,v v v v v v and decreasing in 6v . Suppose ( , )m M is a solution of the

system.

( , , , , , , )M g M M M M M M m and ( , , , , , ,, )m g m m m m m m M

From Eq. (5.1)

0 1 2 30 1 2

4 5 6 7

b M b M b M b mM M M M

b M b M b M b m

and

0 1 2 30 1 2

4 5 6 7

b m b m b m b Mm m m m

b m b m b m b M

Then

2

0 1 2 4 5 6 7 0 1 2

0 1 2 3

(1 )( ) (1 )

( )

M b b b b Mm

M b b b b m

and

2

0 1 2 4 5 6 7 0 1 2

0 1 2 3

(1 )( ) (1 )

( )

m b b b b Mm

m b b b b M

68

Subtracting above two equations

0 1 2

4 5 6 0 1 2 3

10

M mM m

b b b b b b b

By conditions 0 1 2 1 , 4 5 6 0b b b and 0 1 2 3b b b b .

We see that M m . Hence z is a global attractor of Eq. (5.1).

Case 2: If function 0 1 2 3 4 5 6( , , , , , , )f v v v v v v v is decreasing in

0 1 2 3 4 6, , , , ,v v v v v v and increasing 5v . Suppose ( , )m M is a solution of the

system.

From Eq. (5.1)

( , , , , , , )M g M M M M M m M and ( , , , , , , )m g m m m m m M m

Then

0 1 2 30 1 2

4 5 6 7

b M b M b m b MM M M M

b M b M b m b M

and

0 1 2 30 1 2

4 5 6 7

b m b m b M b mm m m m

b m b m b M b m

Then

20 1 2 4 5 7

6 0 1 2 0 1 3 2

1

1

M b b b

b Mm M b b b b m

and

20 1 2 4 5 7

6 0 1 2 0 1 3 2

1

1

m b b b

b Mm m b b b b M

Subtracting above two equations

69

0 1 2

4 5 7 0 1 3 2

10

M mM m

b b b b b b b

By using conditions 0 1 2 1 , 0 1 3 2b b b b and 4 5 7 0b b b . We

see that M m . Hence z is a global attractor of Eq. (5.1).

5.3 BOUNDEDNESS OF SOLUTIONS OF (5.1)

Here we study the bounded nature of the solutions of Equation (5.1).

5.3.1 Theorem 2

Every solution of equation (5.1) is bounded if 0 1 2 1 .

Proof: Let n nz

be a solution of Eq. (5.1).It follows from Eq. (5.1)

0 1 2 31 0 1 2

4 5 6 7

n t n l n m n pn n n k n s

n t n l n m n p

b z b z b z b zz z z z

b z b z b z b z

00 1 2

4 5 6 7

1

4 5 6 7

2

4 5 6 7

3

4 5 6 7

1

n tn n k n s

n t n l n m n p

n l

n t n l n m n p

n m

n t n l n m n p

n p

n t n l n m n p

b zz z z

b z b z b z b z

b z

b z b z b z b z

b z

b z b z b z b z

b z

b z b z b z b z

30 1 2

0 1 24 5 6 7

n pn t n l n mn n k n s

n t n l n m n p

b zb z b z b zz z z

b z b z b z b z

0 31 20 1 2

4 5 6 7n n k n s

b bb bz z z

b b b b 1n

70

We have 1 1n nz y , where 1 0 1 2n n n k n sy z z z

0 31 2

4 5 6 7

b bb b

b b b b is non-homogenous linear equation. It is easy to check that

solution of this equation is locally asymptotically stable and converges to

0 1 2 3

4 5 6 7 0 1 2

( )

( )(1 )

b b b bz

b b b b

if 0 1 2 1 .

By comparison, we see

0 5 6 7 1 4 6 7 2 4 5 7 3 4 5 6

4 5 6 7 0 1 2

lim sup(1 )

n n

b b b b b b b b b b b b b b b bz M

b b b b

.

Hence solution is bounded. Now we will prove that there also exist

0m such that nz m 1n .

For this use the transformation 1

nn

zx

. So Eq. (5.1) becomes

0 1 2 30 1 2

1 4 5 6 7

/ / / /1

/ / / /

n t n l n m n p

n n n k n s n t n l n m n p

b x b x b x b x

x x x x b x b x b x b x

0 1 2

1

0 1 2 3

4 5 6 7

1

n n n k n s

n l n m n p n t n m n p n t n l n p n t n l n m

n l n m n p n t n m n p n t n l n p n t n l n m

x x x x

b x x x b x x x b x x x b x x x

b x x x b x x x b x x x b x x x

0 1 2

4 5

6 7

n k n s n n s n n k

n l n m n p n t n m n p

n t n l n p n t n l n m

x x x x x x

b x x x b x x x

b x x x b x x x

0 1

2 3

4 51

6 7

1

n l n m n p n t n m n p

n n k n sn t n l n p n t n l n m

n l n m n p n t n m n pn

n n k n sn t n l n p n t n l n m

b x x x b x x xx x x

b x x x b x x x

b x x x b x x xxx x x

b x x x b x x x

71

4 5

6 71

0 1 2

4 5

6 7

0

(

)

( )

( )(

n n k n s n l n m n p n t n m n p

n t n l n p n t n l n mn

n k n s n n s n n k

n l n m n p n t n m n p

n t n l n p n t n l n m

n n k n s n l n m n

x x x b x x x b x x x

b x x x b x x xx

x x x x x x

b x x x b x x x

b x x x b x x x

x x x b x x x

1 2 3 )

p

n t n m n p n t n l n p n t n l n mb x x x b x x x b x x x

It follows

4 5

6 71

0 1

2 3

(

)

( )(

)

n n k n s n l n m n p n t n m n p

n t n l n p n t n l n mn

n n k n s n l n m n p n t n m n p

n t n l n p n t n l n m

x x x b x x x b x x x

b x x x b x x xx

x x x b x x x b x x x

b x x x b x x x

4

0 1

2 3

(

)

n l n m n p

n l n m n p n t n m n p

n t n l n p n t n l n m

b x x x

b x x x b x x x

b x x x b x x x

5

0 1

2 3

(

)

n t n m n p

n l n m n p n t n m n p

n t n l n p n t n l n m

b x x x

b x x x b x x x

b x x x b x x x

6

0 1

2 3

(

)

n t n l n p

n l n m n p n t n m n p

n t n l n p n t n l n m

b x x x

b x x x b x x x

b x x x b x x x

7

0 1

2 3

(

)

n t n l n m

n l n m n p n t n m n p

n t n l n p n t n l n m

b x x x

b x x x b x x x

b x x x b x x x

72

4 5 6

0 1 2

7 5 6 74

3 0 1 2 3

n l n m n p n t n m n p n t n l n p

n l n m n p n t n m n p n t n l n p

n t n l n m

n t n l n m

b x x x b x x x b x x x

b x x x b x x x b x x x

b x x x b b bb

b x x x b b b b

1 2 3 4 0 5 2 3 0 1 6 3 0 1 2 7

0 1 2 3

b b b b b b b b b b b b b b b bT

b b b b

1n

Thus we get 1 1

nn

zx T

0 1 2 3

1 2 3 4 0 5 2 3 0 1 6 3 0 1 2 7

b b b bm

b b b b b b b b b b b b b b b b

1n

Hence every solution of (5.1) is bounded and persistent.

5.3.2 Theorem 3

If 0 1 or 1 1 or 2 1 , every solution of Eq. (5.1) is unbounded.

Proof: Let n nz

be a solution of Eq. (5.1). It follows from Eq. (5.1)

1 0 1 2n n n k n sz z z z

0 1 2 30

4 5 6 7

n t n l n m n pn

n t n l n m n p

b z b z b z b zz

b z b z b z b z

, 1n

The right side can be written as

1 0 0 0n

n n nx x x y

It is unstable as 0 1 and limn nz . Hence n nz

is

unbounded above by ratio test.The remaining cases can be prove by same

technique.

5.4 PERIODIC SOLUTIONS

Here we discuss that periodic solutions of Eq. (5.1) exists.

73

5.4.1 Theorem 4

If , , , ,k s t l m are even and p is odd then Eq. (5.1) has a prime period

two solutions iff

3 0 1 2 3 0 1 2 4 5 6 7b b b b b b b b b b b b

0 1 2 3 4 5 6 0 1 2 7 0 1 21 4 0b b b b b b b b

(5.3)

Proof: Suppose first there exists a prime period two solutions

.., , , , ,...

If , , ,k s t l and m are even then

n n k n s n t n l n mz z z z z z and 1n n pz z

From Eq. (5.1)

0 1 2 30 1 2

4 5 6 7

b b b b

b b b b

0 1 2 30 1 2

4 5 6 7

b b b b

b b b b

On simplifying

24 5 6 7 0 1 2 4 5 6 7

0 1 2 3

b b b b b b b b

b b b b

24 5 6 7 0 1 2 4 5 6 7

0 1 2 3

b b b b b b b b

b b b b

Then

74

2 24 5 6 7 0 1 2 4 5 6

0 1 2 7 0 1 2 3

b b b b b b b

b b b b b

2 24 5 6 7 0 1 2 4 5 6

0 1 2 7 0 1 2 3

b b b b b b b

b b b b b

On subtracting

2 2 2 27 0 1 2 4 5 6

0 1 2 3

b b b b

b b b b

3 0 1 2

7 0 1 2 4 5 6

( )

( )( )

b b b b

b b b b

(5.4)

Adding

4 5 6 0 1 2 7

2 20 1 2 4 5 6 7

0 1 2 3

2 b b b b

b b b b

b b b b

4 5 6 0 1 2 7

2

0 1 2 4 5 6 7

0 1 2 3

2

2

b b b b

b b b b

b b b b

4 5 6 0 1 2 7 0 1 2 4 5 6 7

0 1 2 4 5 6 7

2 b b b b b b b b

b b b b

2

3 0 1 2

7 0 1 2 4 5 6

3 0 1 20 1 2 3

7 0 1 2 4 5 6

b b b b

b b b b

b b b bb b b b

b b b b

75

4 5 6 7 0 1 2

2

0 1 2 4 5 6 7 3 0 1 2

7 0 1 2 4 5 6 0 1 2 3

3 0 1 2

2

7 0 1 2 4 5 6

2 1b b b b

b b a b b b b b

b b b b b b b b

b b b b

b b b b

3 0 1 2

0 1 2 4 5 6 7

3 0 1 2 7 0 1 2

4 5 6 0 1 2 3

2

7 0 1 2 4 5 6

(

b b b b

b b b b

b b b b b

b b b b b b b

b b b b

3 0 1 2

0 1 2 4 5 6 3 0 1 2

7 3 0 1 2 7 3 0 1 2

0 1 2 4 5 6 3 0 1 2

2

7 0 1 2 4 5 6

b b b b

b b b b b b b

b b b b b b b b b b

b b b b b b b

b b b b

3 0 1 2

3 0 1 2 4 5 6 7 0 1 2

2

7 0 1 2 4 5 6

2 b b b b

b b b b b b b b

b b b b

3 0 1 2

3 0 1 2 4 5 6 7 0 1 2

4 5 6 7 0 1 2

2

7 0 1 2 4 5 6

1

b b b b

b b b b b b b b

b b b b

b b b b

(5.5)

Let , be the roots of quadratic equation which are positive, real and

distinct.

76

2 ( ) 0t t

2 3 0 1 2

7 0 1 2 4 5 6

3 0 1 2

3 0 1 2 4 5 6 7 0 1 2

4 5 6 7 0 1 2

2

7 0 1 2 4 5 6

01

b b b bt t

b b b b

b b b b

b b b b b b b b

b b b b

b b b b

(5.6)

Thus discriminant is

2

3 0 1 2

7 0 1 2 4 5 6

3 0 1 2

3 0 1 2 4 5 6 7 0 1 2

4 5 6 7 0 1 2

2

7 0 1 2 4 5 6

4 01

b b b b

b b b b

b b b b

b b b b b b b b

b b b b

b b b b

or

3 0 1 2 3 0 1 2 4 5 6 7b b b b b b b b b b b b

0 1 2

3 4 5 6 0 1 2 7 0 1 2

1

4 0b b b b b b b b

Hence inequality (5.3) holds. Now suppose inequality (5.3) is true. Now

we will show that Eq. (5.1) has prime period two solutions.

Let

3

72( )

b A

b GH

and 3

72( )

b A

b GH

where

77

2 3 3 7

37

4 ( )

( )(1 )

b A b GH b Ab A

H b G

, 0 1 2A b b b ,

0 1 2G

and

4 5 6H b b b .

Now set

3 2 1 0

, , , , ,

,..., , , , .

k s t l m pz z z z z z

z z z z

We will prove 1 1z z and 2 0z z .

From Eq. (5.1)

0 1 2 31 0 1 2

4 5 6 7

b b b bz

b b b b

0 1 2 3

4 5 6 7

( )

( )

b b b bG

b b b b

3 33

7 7

3 37

7 7

2( ) 2( )

2( ) 2( )

b A b AA b

b GH b GHG

b A b AH b

b GH b GH

Dividing numerator and denominator by 72( )b GH

3 3 3

3 7 3

( ) ( )

( ) ( )

A b A b b AG

H b A b b A

3 3 3

7 3 7

( )( )

( )( )

A b b A A bG

H b b A H b

3 3

7 3 7

( )( )

( )( ) ( )

b A A bG

H b b A b H

78

Multiplying and dividing by 7 3 7( )( ) ( )H b b A b H

3 3 7 3 7

7 3 7 7 3 7

( )( ) ( )( ) ( )

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

b A A b H b b A b HG

H b b A b H H b b A b H

2

3 7 3 3 3 72 2 2 2

7 3 7

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

( ) ( ) ( )

b A H b A b b A A b b HG

H b b A b H

3 3 7 3 3 7

22 2 23 3 77 3 3 7

7

( ){( )( )( ) ( ) ( )}

4 ( )( ) ( ) ( )

( )(1 )

b A b A H b A b A b b HG

b A b GH b AH b b A b A b H

H b G

2 23 7 3 7

32

3 7 7

2 3 7 3 73 7

4 ( )4

(1 )

b A H b b A H bb A

A b b H b HG

b A H b b GH b Ab A Hb

G

2 23 7 3 7

3 2 3 3 73 7

7

7 3 7 3 73

2 2

4

(1 )

4 1 4

(1 )

b A H b Hb Ab

b A b A b GH b Ab A b H

H b GG

Hb b A G H b b GH b Ab A

G

2 23 7 3 7

3 2 3 3 73 7

7

7 3 73

2 2

4

( )(1 )

4( )

(1 )

b A H b Hb Ab

b A b A b GH b Ab A b H

H b GG

b HG Hb b Ab A

G

79

2 23 7 3 7

3 2 3 3 73 7

7 3 73

2 2

4

(1 )

4(( )

(1 )

b A H b Hb Ab

b A b A b GH b Ab A b H

GG

b HG Hb b Ab A

G

3 3 7 3 7

3 3 3 7

7 3 73

2 2

4

(1 )

4

(1 )

b A b H Ab Hb Ab

b A b A b GH b A

GG

b HG Hb b Ab A

G

3 3 7 3 7

3 3 7

7 3 7

2 1 2

1 4

4( )( )

b A b H Ab G Hb Ab

G b A b GH b AG

b HG Hb b A

3 3 3 7 7 3 7

3 7

7 3 7

2 2

( )(1 )

2( )( )

b A b H b HG Ab Ab G b GH b A

Hb Ab GG

b HG Hb b A

3

7

3 3 7 3 7

7 3 7

2

1 1

2

b AG

b GH

b A G b H Ab Hb Ab G

b HG Hb b A

3 3

7 7

( )(1 ) (1 )

2( ) 2( )

b A b A G GG

b GH b HG

3 3

3 3

7 72 2

G b A G G b A

b A G b A

b HG b GH

80

In the same way, its quite easy to prove that 2z . By mathematical

induction for all n , we get 2nz and 2 1nz . Thus Eq. (5.1) has

prime period two solution which are distinct roots of Eq. (5.6).

.., , , , ,...

5.4.2 Theorem 5

Eq. (5.1) has no prime period two solutions if , , , , ,k s t l m p are even and

0 1 2( ) 1 .

Proof: Proof is same as previous and therefore omitted.

5.5 GRAPHICAL EXAMPLES

Here we give some numerical examples to confirm the obtained results.

These examples give some different types of solutions of Eq. (5.1)

5.5.1 Example 1

5, 4, 2, 3, 0, 1,k s t l m p and values to coefficients stated in

expression (B) in order 0.1,0.2,0,2,0.5,0.6,0.1,0.3,1,0.2,0.5 and the initial

conditions taken in order 0.2,0.7,0.5,2.1,1.15,0.4,0.1,0.3. (See Figure 5.1)

81

Figure 5.1: Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

5.5.2 Example 2

5, 4, 4, 0, 0, 1,k s t l m p and values to coefficients stated in

expression (B) 0.9,0.2,0.23,2,0.5,0.06,0.15,0.3,1.5,0.2,0.25, and the initial

conditions taken in order 1.22,0.7,0,2.1,1.15,1.4,0.1,0.(See Figure 5.2)

82

Figure 5.2: Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

5.5.3 Example 3

5, 4, 2, 3, 0, 1,k s t l m p and values to coefficients stated in

expression (B) 0.1,0.13,0.15,1.5,0.06,0.2,1.1,0.3,1,0.2,0.5, and the initial

conditions taken in order 0.2,0.7,0.5,2.1,1.15,0.4,0.1,0.3. (See Figure 5.3)

83

Figure 5.3: Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

5.5.4 Example 4

5, 4, 2, 3, 0, 1,k s t l m p and values to coefficients stated in

expression (B) 0.1,0.03,0.05,1.02,0.18,0.02,0.1,0.3,1,0.2,0.05, and the initial

conditions taken in order0.2,0.7,0.5,2.1,1.15,0.4,0.1,0.3. (See Figure 5.4)

84

Figure 5.4: Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

5.5.5 Example 5

4, 4, 4, 4, 4, 5,k s t l m p and values to coefficients stated in

expression (B) 0.03,0.04,0.02,0.01,0.55,0.01,6,0.01,0.1,0.02,0.5, and the

initial conditions taken in order 0.5,1.5,0.5,1.5,0.5,1.5,0.5,1.5 . (See Figure

5.5)

85

Figure 5.5: Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

5.5.6 Example 6

4, 4, 4, 4, 4, 5,k s t l m p and values to coefficients stated in

expression (B) 0.03,0.04,0.02,0.01,0.5,0.2,2,0.2,0.1,0.12,0.5, and the initial

conditions taken in order 0.5,1.5,0.1,1.2,0.5,2.5,0.2,0.5. (See Figure 5. 6)

86

Figure 5.6: Behavior of 1 0 1 2n n n k n sz z z z

0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

5.6 CONCLUSION

We studied the global stability, bounded behavior and forms of solutions

of 1 0 1 2n n n k n sz z z z 0 1 2 3

4 5 6 7

n t n l n m n p

n t n l n m n p

b z b z b z b z

b z b z b z b z

with

non-negative initial conditions 1 1 0, ,..., ,z z z z where

max , , , , ,k s t l m p and the coefficients 0 1 2 0 1, , , , ,b b 2 3 4 5, , , ,b b b b

6 7,b b R . It is concluded that if 0 1 2 1 , then the unique

equilibrium point is 0 1 2 3

4 5 6 7 0 1 2

( )

( )(1 )

b b b bz

b b b b

.Using conditions

0 1 2 1 and 0 1 3 2b b b b .We see that M m . Hence z is a global

attractor of Eq. (5.1). Every solution of equation (5.1) is bounded

if 0 1 2 1 . Every solution of (5.1) is unbounded if 0 1 or 1 1 or

2 1 . If , , , ,k s t l m are even and p is odd then Eq. (5.1) has a prime period

two solutions iff

87

3 0 1 2 3 0 1 2 4 5 6 7

0 1 2

3 4 5 6 0 1 2 7 0 1 2

[ ( )] ( ) ( )

1

4{ ( )( ) ( )} 0

b b b b b b b b b b b b

b b b b b b b b

In the end we obtained solution of four different types of Eq. (5.1) and

gave numerical examples of each case by assigning different initial values by

using Matlab.

88

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