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1
A STUDY ON THE ASYMPTOTIC BEHAVIOR OF
SOME DIFFERENCE EQUATIONS
Thesis submitted in Partial fulfilment for the award of
Degree of Doctor of Philosophy in Mathematics
By
V. ANANTHAN
Research Supervisor/Guide
Prof. Dr. S. KANDASAMY M.Sc., M.Phil., Ph.D.,
Professor, Department of Mathematics,
VMKV Engineering College, Salem - 636 308.
VINAYAKA MISSIONS UNIVERSITY
SALEM, TAMILNADU, INDIA
JULY - 2017
2
VINAYAKA MISSIONS UNIVERSITY
DECLARATION
I, V. ANANTHAN declare that the thesis entitled “A STUDY
ON THE ASYMPTOTIC BEHAVIOR OF SOME DIFFERENCE
EQUATIONS” submitted for the Degree of Doctor of Philosophy in
Mathematics is the record of work carried out by me during the period
from January 2011 to July 2017 under the guidance of
Dr. S. KANDASAMY M.Sc., M.Phil., Ph.D., Professor, Department of
Mathematics, VMKV Engineering College, Salem and that has not
formed the basis for the award of any other degree, diploma,
associatesship, fellowship or other titles in this university or any other
university or other similar institutions of higher learning.
Signature of the Candidate
Place: Salem
Date:
3
VINAYAKA MISSIONS UNIVERSITY
CERTIFICATE BY THE GUIDE
I, Dr. S. KANDASAMY M.Sc., M.Phil., Ph.D., certify that the
thesis entitled “A STUDY ON THE ASYMPTOTIC BEHAVIOR OF
SOME DIFFERENCE EQUATIONS” submitted for the Degree of
Doctor of Philosophy in Mathematics by V. ANANTHAN is the record
of research work carried out by him during the period from January
2011 to July 2017 under my guidance and supervision and that this work
has not formed the basis for the award of any other degree, diploma,
associate-ship, fellowship or other titles in this University or any other
University or Institutions of higher learning.
Signature of the Supervisor
Place: Salem
Date:
4
ACKNOWLEDGEMENT
At first, I would like to express my deepest gratitude to my
research supervisor and guide Dr. S. KANDASAMY, without whose
guidance none of this would have been possible. His encouragement,
support, freedom and keen insight have been invaluable and I have indeed
been fortunate to have such an ideal advisor. He played a very important
role in leading me towards scientific maturity. I am indebted to his
support through the years on both scientific and personal matters.
I am grateful to the founders of Vinayaka Missions University,
especially I thank our respected Madam Founder Chancellor, Vinyaka
Missions University Mrs. Annapoorani Shanmugasundaram,
Dato Dr. S. Sharavanan, Vice Chairman, Vinayaka Missions and
respected Chancellor Dr. S. Ganesan, for their valuable support. I thank
the Vice-Chancellor Prof. Dr. V. R. Rajendran and Registrar
Prof. Dr. Y. Abraham, Vinayaka Missions University and I express my
sincere thanks to Dr. S. Prabhavathi, Dean (Research) Vinayaka
Missions University Salem, Tamilnadu, India.
I express my profound thanks to Dr. P. Mohankumar and
Dr. J. Pandurangan, Retired Professors, Aarupadai Veedu Institute of
Technology, Paiyanoor, Chennai. They played a vital role in leading me
5
towards scientific maturity. I am grateful to their support through the
years on scientific matters.
I warmly thank Dr. A. Selvaraj, Professor and Head of the
Department of Mathematics, Nehru Institute of Technology, Coimbatore
for his support throughout my research work. I also thank to my best
friend Dr. A. Ramesh, Principal Incharge, District Institute of Education
and Training, Uthamacholapuram, Salem-636010 for his kind co-
operation.
I thank our Principal Dr. Vemmuri Lakshmi Narayana, Vice
Principal Dr. Vijendrababu, Head of the Department of Science and
Humanities Dr. Jennifer G. Joseph, Head of the Mathematics Division
Dr. L. Tamilselvi and my Department colleagues, Aarupadai Veedu
Institute of Technology Paiyanoor, Kanchipuram District and Vinayaka
Missions Kirupananda Variyar Engineering College, Salem.
Last but not least, I am indebted to my parents, wife, my kid, my
sister, brother, and my friends for their endless support throughout my
studies.
V. ANANTHAN
6
CONTENTS
Chapter No Title Page No
Chapter 1 Introduction …. 1
1.1 Introduction of Difference Equations …. 1
1.2 Asymptotic Behavior of Difference Equation …. 3
1.3 Examples of Difference Equations …. 4
1.4 Examples of Asymptotic Behavior of Difference Equation
…. 6
1.5 Review of Literature …. 7
1.6 Need for the Study …. 34
1.7 Objectives of the Thesis …. 35
1.8 Methodology …. 35
1.9 Plan of the Thesis …. 37
Chapter 2 Asymptotic and Oscillatory Behavior of Third Order Nonlinear Neutral Delay Difference Equation
…. 40
2.1 Introduction …. 40
2.2 Main Results …. 41
2.3 Examples …. 42
Chapter 3 Asymptotic Properties of Third Order Nonlinear Neutral Delay Equation
…. 48
3.1 Introduction …. 48
3.2 Main Results …. 49
3.3 Examples …. 52
7
Chapter 4 Asymptotic Properties Third Order
Nonlinear Neutral Delay Difference
Equation
…. 53
4.1 Introduction …. 53
4.2 Main results …. 53
Chapter 5
Asymptotic of Oscillation Solutions of
Third Order Nonlinear Difference Equations with Delay
…. 57
5.1 Introduction …. 57
5.2 Main Results …. 58
5.3 Boundedness and Oscillation …. 60
Chapter 6 Uncountably Many Positive Solutions of
First Order Nonlinear Neutral Delay Difference Equations
…. 66
6.1 Introduction …. 66
6.2 Krasnoselskii’s Fixed Point Theorem …. 67
6.3 Example with Diagrammatic Representation …. 72
Chapter 7 Asymptotic of Non Oscillatory Behaviour of Third Order Neutral Delay Difference Equations
…. 74
7.1 Introduction …. 74
7.2 Asymptotic and Non-Oscillation Theorem …. 75
Chapter 8 Oscillatory and Non Oscillatory Properties
of Fourth Order Difference Equation
…. 79
8.1 Introduction …. 79
8.2 Main Results …. 82
8.3 Oscillation Theorem by Using Monotonic Property
…. 88
Chapter 9 Conclusion …. 91
Bibliography …. 92
List of Publications …. 104
1
CHAPTER - 1 INTRODUCTION
1.1 INTRODUCTION OF DIFFERENCE EQUATION
Difference equations are of interest both as approximations of
Differential equations and as methods for describing fundamentally
discrete systems in biology, economic, statistics and other problems in
science where the system is described by the discrete variable. For
example in economics the price changes are considered from year to year
or month to month or week to week or from day to day; in every case the
time variable is discrete. In genetics, the genetic Characteristics changes
from generation to generation and the variable representing a generation
is discrete variable.
In population dynamics, we consider the changes in population
from one age group to another and the variable representing the age group
is discrete variable. In finding the probability of ‘n’ success in certain
number of trials, the independent variable is discrete.
A detailed study of difference equations with many examples from
diverse fields can be found as well as in chapter.
In the theory of difference equations oscillatory and non-oscillatory
Behavior of solutions play important role. A nontrivial solution of
2
Difference equation is said to be oscillatory if it is neither eventually
positive or eventually negative. Otherwise it is called non-oscillatory. If
all the solutions of a difference equation are Oscillatory then the
difference equation itself is said to be oscillatory.
Although several results regarding oscillatory theory in the discrete
case are similar to those of already known in the continuous case, The
adaption from the continuous to the discrete is not direct, but it Requires
some special devices. Further it has been show in that there exists some
properties of differential equations which do not carry over directly to
corresponding difference equations. Therefore, it is useful study the
oscillatory and non-oscillatory behavior of solutions of difference
equations. The problem of oscillation and non- oscillation for second and
high order nonlinear difference equations has received more attention in
the Last few years. It is also interesting to studying asymptotic behavior
of difference equation because they are analogues of differential
equations.
In mathematics, delay difference equations are a type of difference
equations in which the variables of the unknown function at a certain
Interval are given in terms of the values of the function at previous
Intervals. A difference equation of the form
( ) 0( , ) 0, 0 .m
n n ky f n y k N
3
1.2 ASYMPTOTIC BEHAVIOR OF DIFFERENCE EQUATION
If
lim 1,t
y t
z t then we say that “ y t is asymptotic to z t as ' 't
tends to infinity” and write ~ ,y t z t .t
Suppose we wish to describe the asymptotic behavior of the
function 3
2 324 ~ 8 , t t t t some other elementary examples are
2 2
1 1~ ,
3 2 3t
t t t
and sin ~ , .
2
teht t the famous nontrivial
example is the prime number theorem.
Let t the number of primes less than t.
~ , .log
tt
tt
We describe the nature and behavior of solutions of difference
systems, without actually constructing or approximating them. Since in
contrast with differential equations, the existence and uniqueness of
solution of discrete initial value problems is already guaranteed we shall
begin with the continuous dependence on the initial conditions and
parameters. This is followed by the as asymptotic behavior of solutions of
linear as well as nonlinear difference systems. In particular, easily
verifiable sufficient conditions are obtained so that the solutions of
perturbed systems remain bounded or eventually tend to zero, provided
the solutions of the unperturbed systems have the same property.
4
For a given difference system one of the pioneer problems is the
study of ultimate behavior of its solutions. In particular, for linear
systems we shall provide sufficient conditions on the known quantities so
that all their solutions remain bounded or tend to zero as k . Thus
from the practical point of view the results we shall discuss are very
important because an explicit form of the solutions is not needed.
1.3 EXAMPLES OF DIFFERENCE EQUATIONS
Example 1.3.1
The difference equation 1 , 0 1
nn
n k
yy n
y
where 1, ,
0, and 0k N is known as delay logistic equation and was
proposed by pie low as model equation of dynamic of single species.
Example 1.3.2
The difference equation 2
1 0, 0n n ny q y n
where α is
quotient of all positive integers 0, nq with nq not eventually equal
to zero is the discrete analogue of the well-known Emden fowler
differential equation, which appear in astrophysics, nuclear physics and
chemical reaction etc.
Example 1.3.3
It is observed that the decrease in the mass of a radioactive
substance over a fixed time period is proportional to the mass that was
5
present at the beginning of the time period. If the half life of radium is
1600 years. Let m t represent the mass of the radium after t years. Then
1m t m t km t , where k is a positive constant. Then
1m t 1 .k m t
Example 1.3.4
Let G(government expenditure), D(government deficit), Y(national
income) be functions of time t related in such a way that
4t t tY G D Make the following inference from this relation:
“Then, an extra dollar of deficit would allow four dollar reduction in tax-
financed government spending without affecting the magnitude of
national income…”
Example 1.3.5
In 1626, Peter Minuit purchased Manhattan Island for goods worth
$24. If the $24 could have been invested at an annual interest rate of 7%
compounded quarterly, what would it have been worth in 1998?.
Solution
Let y t be the value of the investment after t quarters of a year.
Then 0 24,y since the interest rate is 1.75% per quarter, y t satisfies
the difference equation 1 1.075 .y t y t
6
1.4 EXAMPLES OF ASYMPTOTIC BEHAVIOR OF
DIFFERENCE EQUATION
Example 1.4.1
The asymptotic behavior of 1
.n
k
k
k
We begin by factoring out the
largest term:
1
nk
k
k
1 21 2 1
1
n n
n
n n n
n nn
n n n
The sum in braces is less than
2 1
1 1 1nn n n
2
0
1 1kn
kn n
11
11
11
n
n
n
n
Since the expression
11
1
11
n
n
n
is bounded, we have
1
nk
k
k
1
1 , nn nn
Then the asymptotic value of 1
nk
k
k
is given by the largest term with
a relative error that approaches zero like 1
n as .n
In a similar way, it can be shown that
1
nk
k
k
1
2
1 1 11 , .
n
n nn n
n n n
7
Consequently, the two largest terms of the series yield an 2
1
n
asymptotic estimate.
Example 1.4.2
The asymptotic behavior of 1
1
2 log .n
k
k
k
We begin by applying
summation by parts.
1
2 logn
k
k
k
1
1
(2 )logn
k
k
k
11
1
2 log 2 .logn
n k
k
kn
By the mean value theorem, 1
log kk
,
11
1
2 logn
k
k
k
1
1
1
12
nk
k k
2
2 1 1 1 1 1 11 . .
1 2 2 3 4 1 2
n
n
n n n
n n n
.
1.5 REVIEW OF LITERATURE
Berry (1967) believed that, “The view of related literature is a must
for scientific approach in all areas of scientific research. One cannot
develop an insight into the problem to be investigating unless and until
one has learnt one has done in a particular area of his own interest. Thus
the related literature forms the foundation upon which all work can be
built.
8
In recent years, there has been considerable interest in the study of
Oscillatory and asymptotic behavior of difference equations. An intensive
survey of the literature related to the present work have been made by the
researcher by referring to a large number of Journals, Books,
Encyclopedias, International Dissertation Abstracts and National Level
publications, etc.
Josef Diblik and Irena Hlavickova [22] discussed the asymptotic
behavior of solutions of systems of difference equations with an
application to delayed discrete equation for k of the form
u k ,F k u k 1.5.1
Where k is the independent variable assuming value from the set
: , 1, . N a a a with a fixed
,a N 1 2, , . , mu u u u 1 ,u k u k u k
and : ,m mF N a R R 1 2, , .. .mF f f f
William F. Trench [63] discussed asymptotic behavior of
solutions of Poincare difference equations of the form
1 1 1 . 0n ny n m a p m y n m a p m y m 1.5.2
Where 0,na the polynomial 1
1 ..q a a n n
nhas
distinct zeros 1 2, , .. ,n and 0,1 .kp m k n
9
M. Maria susai manuel, G.Britto Antony Xavier, D.S.Dillip and
G.Dominic Babu [29] discussed the asymptotic behavior of solutions of
the generalized nonlinear difference equation of the form
0l lp k u k f k F u k g k , ,k a 1.5.3
Where the function , ,p f F and g are defined in their domain of
definition and l is a positive real.
Further,
0uF u for 0,u 0p k for all , k a
for some 0, a and for all 0 ,j i , ,a j kR
Where
,
0
1, ,
1
k l i j
l
i j k
i
R l ap j rl
and 1lk N j l .
Laurens de HAAN, Holger ROOTZEN and Casper G. de. VRJES
[27] discussed extremal behavior of solutions to a stochastic difference
equation with applications to arch processes , 1nY n which satisfies
the stochastic difference equation
1 ,n n n nY A Y B 1,n 0 0Y 1.5.4
Where , , 1n nA B n are i.i.d, 2R valued random pairs. We
study the extremal behavior of nY under rather mild assumptions.
10
William F. Trench [62] discussed asymptotic behavior of solutions
of a linear second order difference equation of the form
2
1ny ,n np y 1, 2, n 1.5.5
Where ∆ is the forward difference operator with unit spacing
nu1n nu u equation of the form (1.5.5) arise in discretizing a second
order linear differential equation to solve it numerically.
E. Thandapani. R. Arul and P.S. Raja [52] discussed the asymptotic
behavior of non-oscillatory solutions of nonlinear neutral type difference
equations of the form
2 , 0,n n k n ly py f n y n N 1.5.6
2 , , 0,n n k n l n ly py f n y y n N 1.5.7
Using some difference inequalities. We establish conditions under
all non-oscillatory solutions are asymptotic to an b as nwith
, .a b R
Vadivel sadhasivam, Pon sundar and Annamalai santhi [60]
discussed on the asymptotic behavior of second order quasilinear
difference equations of the form
1y n y n
1
,p n y n y n
1.5.8
Where 0n N 0 0 0, 1, 2, . ,n n n 0 .n N
11
We classified the solutions into six types by means of their
asymptotic behavior. We establish the necessary and/or sufficient
conditions for such equations to possess a solution of each of these six
types.
Hajnalka Peics and Andrea Roznjik [14] discussed in the
asymptotic behavior of solution of a scalar delay difference equations
with continuous time of the form
x t 1t x t b t x p t 1.5.9
Where , ,a b p are given real functions such that p t t and ‘p’ is
monotone increasing, and the special case of the above equation for
1 .b t a t
E. Thandapani and S. Pandian [48] discussed about the asymptotic
and oscillatory behavior of solutions of some general second order
nonlinear difference equations of the form
1 1 10,n n n n n n n
a h y y p y q f y
n Z 1.5.10
Here ∆ is the forward difference operator defined by
ny1 ,n ny y 0,1 , 2, Z and the real sequences , ,n n np a q ,
n and ,f h are the functions.
12
Eithiraju Thandapani, Zhaoshuang Liu, Ramalingam Arul and
Palanisamy S. Raja [13] discussed the oscillation and asymptotic
behavior certain second order Non-linear neutral difference equations of
the form
1 0n n n k n n la y py q f y
, 0 0n n 1.5.11
Where p is a real number, 0, 0k l are integers, α is a ratio of
odd positive integers, ∆ is the forward difference operator defined by
ny 1n ny y . na is a positive sequence with 0
1, n
n n n
qa
is a non
negative real sequence with a positive subsequence and :f R R is
continuous and non-decreasing with 0uf u for 0.u
Hulima, Hui Feng, Jiaofeng wang and Wandi Ding [17] discussed
the bounded and asymptotic behavior of positive solutions for difference
equations of
1nx 1nx
na bx e 1.5.12
Where a, b are positive constants and the initial values, 1 0,x x are
positive numbers.
J.S.Yu and Z. C. Wang [65] discussed the oscillation of neutral
delay difference equations of the form
0,n n k n n ly py q y 0,1,2,n 1.5.13
13
Whose oscillation and asymptotic behavior have been investigated,
where , 0,1 , 2, np q n real numbers, k and l are positive
constants, the forward difference operator is defined by 1 .n n nx x x
H. Sedaghat and W. Wang [38] discussed the asymptotic behavior
of non linear delay difference equation of
nx 1
1
1 ( ) ,m
p
n i n i
i
x g f x
0, ,p 1, 2, 3,...n 1.5.14
Where g and each if are continuous real functions with g
decreasing and if increasing.
R. P. Agarwal and J. V. Manjlovic [1] discussed the asymptotic
behavior of positive solutions of fourth order nonlinear difference
equations of the form
2 2
3 0,n n n np y q y
n N 1.5.15
Where , are ratios of odd positive integers and ,np nq are
positive real sequences detailed for all 0 .n N n We establish necessary
and sufficient conditions for the existence of non-oscillatory solutions
with specific asymptotic behavior under suitable combinations of
convergence or divergence conditions for the sum
0
2n n
n
n
p
and 0
1
n n n
n
p
1.5.16
14
John R. Graef, Agnes miciano, Paul w.Spikas, P.Sundaram and
E.Thandapani [20] discussed oscillatory and asymptotic behavior of
solutions of non-linear neutral difference equation of
1 1 1 , 0m
n m n m n m k n ly p y F n y 1.5.17
Here k and l are non negative integers, m is a positive integer, p is a
constant and nq is a sequence of real numbers.
G. Papaschinopoulos, M. Radin and C. J. Schinas [32] discussed
the boundedness, the asymptotic behavior, the periodicity and the
stability of the positive solutions of the difference equation of the form
1ny
1
ny
n
e
y
1.5.18
where , , positive constants and the initial values 1 0, x x are
positive numbers.
G.Ladas and C.Qian [26] described the oscillatory behavior of
difference equations with positive and negative coefficients of
1 0,n n n n k n n ly y p y q y 0,1 , 2, n 1.5.19
Where k and l are non-negative integers and the coefficients np
and nq are sequences of nonnegative real numbers and also studied
asymptotic behavior.
15
B. G. Zhang [67] discussed the Oscillation and asymptotic
behavior of second order difference equations of the form
1 0,n n n nC y p y
0,1 , 2, n 1.5.20
where 1n n ny y y γ is a quotient of odd positive integers. Some
necessary and sufficient conditions for oscillation of (1.5.20) with
1 and 1 are obtained. Asymptotic behavior of non-oscillatory
solutions of difference equations with forced term
2
1 ,n n n ny p y f 1, 2, n is considered also.
John R. Grafe and E. Thandapani [21] discussed the
Oscillatory and Asymptotic Behavior of Solutions Of Third
order delay difference equation of the form
1 ,n n n n n m na b y q f y h 0 0,1 , 2, .n N 1.5.21
Where , , ,n n na b q and nh are real sequences, :f R R is
continuous, 0,na 0,nb and 0nq for all 0 0,n n N 0uf u for all
0,u and m is a positive integer. A solution of (1.5.1) is a real sequence
ny defined for all 0 1n n m and satisfying (1.5.1) for all 0n n . In
what follows, we assume that equation (1.5.1) has solutions which are
nontrivial and defined for all large n. A nontrivial solution ny of
equation (1.5.1) is said to be oscillatory if any 0N n there exists
16
n N such that 1 0.n ny y otherwise, the solution is said to be non-
oscillatory. Equation (1.5.1) is said to be oscillatory if every solution of
(1.5.1) is oscillatory, and it is said to be almost oscillatory if every
solution ny is either oscillatory or satisfies lim 0i
nn
y
for 0,1 , 2,...i
J.S. Yu and Z. C. Wang [66] discussed the asymptotic behavior and
oscillation in neutral delay difference equation of the form
1 0,n n k n ny py q y 0,1 , 2, n 1.5.22
Whose oscillation and asymptotic behavior have been investigated,
where , 0,1 , 2, ...np q n are real numbers, k and l are non-negative
integers, and ∆ denotes the forward difference operator
1 .n n nx x x
Jerzy Popenda and Ewa Schmeidel [18] discussed on the
asymptotic behavior of solutions of linear difference equations of the
form
0
,r
i
n n n i
i
x a x
n N 1.5.23
Here by N, R we denote the set of positive integers and reals
respectively. For any function :y N R the difference operator is
defined as follows 1 ,n n ny y y n N and 1 ,i i
n ny y for 1.i
Andrzej Drozdowicz and Jerzy Popenda [3] discussed on the
asymptotic behavior of solutions of difference equations of second order
of the form
17
2 ,n n ny p y n N 1.5.24
We would like to present some elements of qualitative theory of
difference equations. Asymptotic behavior of solutions of second order
difference equations will be investigated.
Let N denote the set of positive integers, R the set of all real
numbers and R the set of non negative reals.
For a function :a N R we introduce the difference operator
by 1 ,n n na a a 2 ,n na a where , .na a n n N
Moreover, let 1
0k
j
j k
a
and 1
1.k
j
j k
a
The sequence 1n n
x
is
called oscillatory if for every n N there exists , m m n such that
1 0.n nx x Otherwise the sequence is called non oscillatory.
Toshiki Natto, Pham Huu Anh Ngoc and Jong Son Shin [16]
discussed representations and asymptotic behavior of solutions to
periodic linear difference equations of the form
1 ,x n Bx n b n 0 px C 1.5.25
Where 0 : 0 ,n N N B is a complex p p matrix and
pb n C is p-periodic, that is .b n b n p
18
E.Thandapani, K. Mahalingam and John R.Graef [50] discussed
the oscillatory and asymptotic behavior of second order neutral type
difference equations of the form
2
1n n k nx ax bx ,n n m n n rq x p x n N 1.5.26
and 2
1 0, n n k n n rx px Q x n N 1.5.27
Where 1, 2, 3, N , is the forward difference operator defined by
1 .n n nx x x
John W. Hooker and William T. Patula [19] discussed a second
order nonlinear and asymptotic behavior of the form
1 12 0, n n n n ny y y q y
1, 2, 3, n 1.5.28
Where is a quotient of odd positive integers. It is interesting to
study second order nonlinear difference equations because they are
discrete analogues of differential equations, where the forward difference
operator ∆ is defined by the equation
1n n ny y y
and 2
1 1 1 1 12n n n n n n ny y y y y y y
In (1.5.28) , 1, 2, 3, nq q n is a given infinite sequence of real
numbers. By a solution of (1.5.28) we mean a real sequence
, 0,1 , 2, 3, ny y n satisfying (1.5.28). It is clear from (1.5.28) that a
solution of (1.5.1) is uniquely determined if any two successive values
19
1, k ky y are given. Also, it is clear that any solution can be defined for all
0,1 , 2, 3, n equation (1.5.28) is a discrete analogue of the generalized
Emden-Fowler differential equation
'' 0, 0, 0y q t y t
Kenneth S. Berenhaut and Stevo Stevic [23] discussed the behavior
of the positive solutions of the difference equation of the form
2
1
, 0,1,...
p
nn
n
xx A n
x
1.5.29
With , 0, , 1p A p and 2 1, 0, .x x It is shown that:
(a) All solutions converges in the unique equilibrium, 1,x A
whenever min 1, 1 / 2 :p A
(b) All solutions converge to period two solutions whenever
11:
2
Ap
and
(c) There exist unbounded solutions whenever 1.p These results
complement those for the case 1p in A.M.
E. Thandapani, S. Pandian and R. K. Balasubramanian [53]
discussed the oscillatory behavior of second order unstable type neutral
difference equation of the form
( ) 0n c n n k n na y py g f y
1.5.30
20
Where p is a real, α is a ratio of odd positive integer and n is
a sequence of integers.
P. Mohankumar and A. Ramesh [31] discussed the oscillatory
behavior of the solution of the third order nonlinear neutral difference
equation of the form
2
0, 0,n n n n ka x p x f n n n N n 1.5.31
By a solution of equation (1.5.1) we mean real sequence nx
satisfying (1.5.1) 0 0 1 0 2, , ,n n n n a solution nx is said to be
oscillatory if it is neither eventually positive nor eventually negative.
Otherwise it is called non oscillatory. The forward difference operator ∆
is defined by
1 .n n nX X X
Said R. Grace, Ravi P. Agarwal and Sandra Pinelas [37] discussed
on the oscillatory and asymptotic behavior of certain fourth order
difference equations of the form
2 2 0a k x k q k f x g k
1.5.32
With the property that
0x k
k as k are established. Where
∆ is the forward difference operator defined by x k 1x k x k
and is the ratio of positive odd integers. We assume that
21
0, :g a N n R 0, for some 0 0,1,2,...n N and
0 0 0, 1,... ,N n n n 0: :g G g N n N for some 0 :n N g k
,k g k is non-decreasing and limk
g k
. and :f R R is
continuous satisfying 0xf x for 0x and f is non-decreasing.
X. H. Tang [47] discussed the asymptotic behavior of solutions for
neutral difference equations of the form
0, 0,1,2,n n n k n n lx p x q x n 1.5.33
Where is the forward difference operator defined by
1 , ,n n nx x x k l are positive integers, np is a sequence of real
numbers and nq is a sequence of nonnegative real numbers.
Q. Din [11] investigated the qualitative behavior of following
second order fuzzy rational difference equation of the form
11
1
n nn
n n
x xx
p x x
1.5.34
Where ‘p’ is positive fuzzy number and initial conditions 1 0, x x are
positive fuzzy numbers.
A. Aghajani and M. Ejehadi [2] discussed the asymptotic behavior
of a nonautonomous difference equation of the form
1
1
, 0,1 , n nn
n
xx n
x
1.5.35
22
With nonzero initial conditions 1 0, x x where 0n n
is a sequence
of nonzero real numbers. In the case that 0n n
is a constant sequence,
it is easy to see that every solution of equation (1.5.3) is periodic with
period six, and in particular every solution are bounded.
Ravi P. Agarwal, Said R. Grace and Patricia J. Y. Wong [36]
discussed the oscillation of fourth order nonlinear difference equations of
the form
2 21
x ka k
q k f x g k p k h x k 1.5.36
Where is the ratio of two positive odd integers.
By a solution of equation (1.5.1), we mean a real sequence x k
satisfying equation (1.5.1) for all large 0 0.n N A nontrivial solution
x k of (1.5.1) is said to be non-oscillatory if it is either eventually
positive or eventually negative, and it is oscillatory otherwise. The
equation (1.5.1) is said to be oscillatory if all its solutions are oscillatory.
R. Arul and T. J. Raghupathi [6] discussed the some new criteria
are established for the second order neutral difference equation of the
form
0, 0.a n z n a n x n n 1.5.37
23
Where z n x n p n x n and established the extent the
result of the oscillation theorems for second order half-linear neutral
difference equation (1.5.37).
Srinivasam selvarangam, Eithiraju Thandapani and Sandra Peinlas
[43] discussed the oscillation of second order neutral difference equations
of the form
0n n nn n
a x x q x
1.5.38
And established the oscillation of all solutions of this equation
(1.5.38) via comparison theorems.
Vildan kutay and Huseyin Bereketoglu [61] discussed the
oscillation of a class of second order neutral difference equations with
delays of the form
,p n x n q n x n f n x n
, , , 0g n x n x n n 1.5.39
By using Riccati transformation techniques and establish some
oscillation criteria for the second order neutral delay difference equations
(1.5.39).
R. Arul and G. Ayyappan [5] studied the oscillatory behavior of
solution of the second order neutral difference equation of the form
24
00, n n n k n n l n n mr x p q x V x n N
1.5.40
00, n n n k n n l n n mr x p q x V x n N
1.5.41
And established sufficient conditions for the oscillation of all
solutions of equations (1.5.40) and (1.5.41).
S. Lourdu Marian [28] discussed the existence of positive solutions
of nonlinear neutral delay difference equations with positive and negative
co-efficient of the form
1 1[r n x n p n x n q n x n
2 2q n x n e n 1.5.42
Under the following assumption that
0
1, 1, 2, .i
s j s
q j ir s
1.5.43
For various ranges of p n and authors used Banach’s contraction
mapping principle, some sufficient condition are established for the
existence of non-oscillations solutions of the equations (1.5.42).
B. Selvaraj and S. Kaleeswari [42] discussed the sufficient
condition for the oscillation of second order neutral delay difference
equations of the form
2 0,n ny f y 0,1 , 2, n 1.5.44
25
Ethiraju Thandapani, Ramalingam Arul and Palanisamy [12]
discussed the neutral delay difference equation of the form
0,n n n k n n ly h y q y 0,1 , 2,...n 1.5.45
Where 1, a ratio of odd positive integers, is the forward
difference operator defined by 1 .n n ny y y By a solution of equation
(1.5.8) we mean a real sequence ny which satisfies equation (1.5.45) is
said to be oscillatory if it is neither eventually positive nor eventually
negative and non-oscillatory otherwise.
B. Selvaraj, G. Gomathi jawahar [41] studied the sufficient
condition for the oscillation of second order neutral delay difference
equations of the form
2
( ) ( ) 0( ) ( ) 0, ( )n n k n n ly Py q f x n N n 1.5.46
E. Thandapani and S. Selvarangam [59] discussed some new
oscillation results of the difference equations of the form
0n n n nn na x p x q x
1.5.47
And established oscillatory solution of the equation (1.5.47) via
comparison theorem.
B. Selvaraj and J. Daphy Louis Lovenia [40] studied oscillatory
properties of certain first and second order linear difference equations of
the form
26
0n n n na x q x 1.5.48
1 0n n n na x q x 1.5.49
2
1 0n n n n n na x p x q x 1.5.50
And established the some sufficient conditions for the oscillation of
all solutions of difference equations (1.5.12), (1.5.13) and (1.5.14) proved
by using the results on difference inequalities.
E. Thandapani and M. Vijaya [58] classified all solutions of second
order nonlinear neutral delay difference equations with positive and
negative coefficients of the form
1 0n n n n k n n n n ma x c x p f x q g x
1.5.51
And obtained conditions for the existence and non-existence of
solutions of the equations (1.5.51).
L. K. Kikina and I. P. Stavroulakis [24] described the oscillation
criteria for second order delay difference and functional equations of the
form
2 0,x n p n x n for all 0n 1.5.52
E. Thandapani, K. Thangavelu and Chandrasekarn [57] discussed
the oscillatory behavior of second order neutral difference equations with
positive and negative coefficients of the form
0n n n n k n n l n n ma x c x p f x q f x 1.5.53
27
0n n n n k n n l n n ma x c x p f x q f x 1.5.54
Where 0 0 0 1 0, , , n N n n n n is a nonnegative integer, , , k l m
are positive integers , , , n n n na c p q are real sequences and
:f R R is continuous and non-decreasing with 0uf u for 0.u
J. Cheng and Y. Chu [8] discussed the sufficient and necessary
conditions and established for the oscillation theorem for second order
difference equations of the form
1 1 0, 1, 2, n n n nr x p x n
1.5.55
Where is the quotient of odd positive integers.
E. Thandapani and P. Mohankumar [55] described the oscillation
of difference systems of the neutral type of the form
0, , n n n n n nn nx a x p g y y q f x n N n
1.5.56
Where 1, , n na p and nq are real sequences n and
n are real non-negative sequences of integers and , :f g R R are
continuous with 0uf u and 0ug u for 0.u
E. Thandapani and P. Mohankumar [56] discussed the sufficient
conditions for oscillation and non-oscillation of second order nonlinear
neutral delay difference equation of the form
28
2
00, n n n k n n lu p u q f u n n 1.5.57
Where ,n np q are non-negative sequences with 0 1,np and
k and l are positive integers.
Y.G. Sun and S. H. Saker [45] studied the oscillation criteria for
the second order perturbed nonlinear difference equations of the form
1 1 1, , , 1n n n n na x F n x G n x x n
1.5.58
and established the some new oscillation criteria for the second
order perturbed non-linear difference equation (1.5.58) using Riccati
transformation.
A. Murugesan and K. Venkataramanan [53] are discussed
asymptotic behavior of first order delay difference equation with a
forcing term of the form
0,x n p n f x n r n 0n 1.5.59
Where ∆ is the forward operator defined by
1 ,x n x n x n p n is a sequence of positive real numbers,
r n is a sequence of real numbers, is a positive integer and
:f R R is an increasing function.
J. Migda and M. Migda [30] studied the second order difference
equation of the form
29
2 , n n n ku a u 1, 2, n , 0,1 ,2, k 1.5.60
And established asymptotic behavior solution of the second order
difference equation (1.5.60) oscillatory.
J. Deng [9] gave a note on oscillation of second-order nonlinear
difference equation with continuous variable of the form
0,n n n k n n ly h y q y 0,1 , 2, n 1.5.61
And established sufficient conditions for the oscillation of all
solutions of first order neutral delay difference equations. Their approach
is to reduce the oscillation of neutral delay difference equation to the non-
existence of positive solutions of delay difference inequalities (1.5.61).
E. Thandapani and K. Mahalingam [51] discussed the second order
difference equation of the form
2
1 1 0, n n k n n ly py q f y 1, 2, 3, n 1.5.62
Where nq is a non-negative real sequences, :f R R is
continuous such that 0uf u for 0,u 0 1,p k and l are positive
integers and author established the necessary and sufficient condition for
the oscillation of all solutions of the equations (1.5.62).
I. Kubiaczyk [25] studied some new oscillation criteria for second
order nonlinear difference equations of the form
30
0, 0,1 , 2, n n n np x q x n
1.5.63
Where 0 1 is a quotient of odd positive integers and
established Kamenek-type oscillation criteria for sub-linear delay
difference equations of (1.5.63) using Riccati transformation techniques.
S.H. Sakar [37] is studied the difference equations of the form
, 0, 0,1 , 2, 3, n n n n r na y p x f n x n
1.5.64
Where 0 is a quotient of odd positive integers and established
some new oscillatory for the second order nonlinear neutral delay
difference equation (1.5.64) using Riccati transformation techniques.
X. H. Tang and J. S. Yu [46] concerned with oscillations of delay
difference equations in a critical state and the equivalence of the
oscillation of the following two difference equations of the form
0n n n kx p x
and
2
1 11
2 10
1
k k
n n nkk
k ky p y
k k
1.5.65
Under the critical state of the form
1inl m fi
1
k
n kn
kp
k
and
1
1
k
n k
kp
k
1.5.66
Where 0np and k is a positive integer, and they obtain some
sharp oscillation and non-oscillation criteria for equation (1.5.66).
31
R. Arul and E. Thandapani [4] considered the difference equations
1, 0, 0,1, 2, n n np x f n x n 1.5.67
When 0
1
n sp
1.5.68
And gave some sufficient conditions for the existence of positive
solutions of equations (1.5.67).
E. Thandapani and K. Ravi [49] discussed the oscillation of second
order half linear difference equations of the form
1 0, 0,1 , 2, n n n np x q x n
1.5.69
Α is a ratio of odd positive integers.
Ravi. P. Agarwal, M. M. S. Maneul and E. Thandapani [35]
discussed about the oscillatory and non-oscillatory behavior of second
order neutral delay difference equations of the form
1 1 0, n n n n k n n lp y h y q f y 0,1 , 2, . n Z 1.5.70
Ravi. P. Agarwal and P. J. Y. Wong [34] discussed about discrete
inequalities and used to offer sufficient conditions for the oscillation
theorems for certain second order non-linear difference equations of the
form
1 1 , 0n n n n na y q f y r n
1.5.71
32
Where 0p
q with p, q are odd positive integers.
Sui-sun Cheng, [44] discussed boundedness and monotoncity
properties of a second order difference equations
1 1 , 1, 2, k k k kp x q x k 1.5.72
And established the necessary and sufficient condition for all
solutions of the equations (1.5.36) bounded via comparison theorem.
J. Popenda [33] discussed about the oscillatory and non-oscillatory
behavior of the solutions of some second order difference equations of
the form
2 , , ,a n n b ny F n y y n N 1.5.73
Where a and b are real constants, considering only nontrivial
solutions,
: 0nsup y n i for every .i N
W. J. Hooker and W.T.Patula [15] studied Atkinson’s oscillation of
the difference equation of the form
2
1 0n n ny q y
1.5.74
And established known results on oscillation growth and
asymptotic behavior of solution of the equation (1.5.74).
33
Blazej Szmanda [7] discussed the oscillatory behavior of solutions
of second order nonlinear difference equation of the form
2
1
, 0, 0,1 , 2, m
n in i n n
i
y a f y y n
1.5.75
Where is the forward difference operator, defined by
1 ,n n ny y y 2
1, ...n n m mny y a a are the real sequence.
J. Diblik and I. Hlavickova [10] discussed the asymptotic behavior
of solutions of delayed difference equations of the form
, , 1 , ..v n f n v n v n v n k 1.5.76
Where n is the independent variable assuming values from the set
aZ with a fixed .a N
Yoshihiro Hamaya and Alexandra Rodkina [64] discussed on
global asymptotic stability of nonlinear stochastic difference equations
with delays of the form
1 1 1 0
1
, , , .. , k
n n l n l n n n k n
l
x aF x b x g n x x x n N
1.5.77
With arbitrary initial conditions 0 1, , . ,kx x x R non-linear
continuous functions F and g, and independent zero mean random
variables .n Equation (1.5.41) describes the dynamics of a neutral
network under stochastic perturbations.
34
Ramalingam Arul and Manvel Angayarkanni [66] are discussed
Oscillatory and asymptotic behavior of solutions of second order Neutral
Delay Difference equations with maxima of the form
,
max 0,n n n n n in
a x p x q x
0n N n 1.5.78
Where ∆ is the forward operator defined by
x n 1x x x n , 0 0 0, 1,...N n n n and 0n is a non-negative
integer.
Charles V. Coffman [54] discussed asymptotic behavior of solution
of ordinary difference equations with “almost constant coefficients”
equations having the form
1y n ,Jy n f n y n 1.5.79
Where y is a d-vector, J is a constant d d matrix and ,f n y is
a vector-valued function which is continuous in y for fixed n and
becomes ‘small’ in some sense as , ,0 .n y
1.6 NEED FOR THE STUDY
Most of the difference equations governing various systems are
non linear in nature and it is well known that these equations cannot be
solved with the aid of computers, but a large number of values of the
solution can be calculated. However, if one is interested in the asymptotic
behavior of the solutions, then more computer time will be required and
this often can be very expensive. Hence in the absence of solutions
known explicitly, it is important to know qualitatively how solutions
35
behave, to obtain approximate solutions. Of particular interest is the
oscillation and asymptotic behavior of solutions of difference equations.
By an oscillatory solution of a difference equation we mean a solution
that is neither eventually positive nor eventually negative and non
oscillatory otherwise. These types of solutions occur in many physical
phenomena such as, vibrating mechanical systems and electrical circuits.
In recent years, there has been considerable interest in the study of
oscillatory and asymptotic behavior of solutions of difference equations.
Hence, there is a need to study the asymptotic behavior of some
difference equations.
1.7 OBJECTIVES OF THE THESIS
The objectives are:
1. To find the asymptotic and non oscillatory behavior of third order
non-linear neutral delay difference equation.
2. To find the asymptotic properties of third order non-linear neutral
delay difference equation.
3. To find the asymptotic and oscillation solution of first order
difference equation with delay.
4. To evaluate the asymptotic and oscillatory behavior of fourth
order nonlinear delay difference equation.
1.8 METHODOLOGY
The following methodology used for this thesis.
36
1.8.1 Double sequences
Double sequences are of the form , | 0H m n m n such that
i , 0H m n for 0 m
ii , 0H m n for 0m n
iii 2 , ) , , :L H m n h m n H m n for 0m n
1.8.2 Schwarz inequality
Let 1 2, , na a a and 1 2, , nb b b be two sequences of real
numbers, then
2
2 2
1 1 1
n n n
i i i i
i i i
a b a b
.
1.8.3 Riccati Transformation
Non-linear difference equations that can be transformed into
equivalent linear difference equations by a change of dependent variables.
1.8.4 Equicontinuous Functions
A set of function 𝔉 on a metric space X is called an Equi-
continuous family if for every 0 there exists a 0 such that
f x f y for all ,x y X such that ,d x y and all .f F
1.8.5 Lipchitz Functions
A function f such that f x f y C x y for all x and y
Where, C is a constant independent of x and y called a Lipchitz function.
37
1.8.6 Krasnoselskii’s Fixed Point Theroem
Let X be a Banach Space. Let Ω be a bounded closed convex subset
of X and let 1 2,s s be maps of Ω into X such that 1s 2 x s y for every
, x y .
1.9 PLAN OF THE THESIS
This thesis presents the results obtained on the oscillatory behavior
of third and fourth order delay difference equations. This thesis contains
seven chapters.
In Chapter – 1– Introduction
Section 1.1 deals with the difference equations’s introduction,
asymptotic behavior, examples, review of literature, objectives, need for
the study, methodology, the plan of the thesis and contribution of authors.
Chapter – 2 – Results and Discussion
Section 2.1 deals with the oscillatory and non oscillatory behavior
of neutral delay difference equations of the form
1 1 0n n n n n k n n la b x c x q f x 0,1,2,...n Z 2.1.1
Section 2.2 deals with main result of (2.1.1). Section 2.3 deals with
the examples. In Section 2.4 sufficient condition for asymptotic behavior
of solution of the equation 2.1.1 were established.
Chapter – 3
Section 3.1 deals the asymptotic properties of third order nonlinear
neutral delay difference equation of the form
38
2 0n n n n k n n la y p y q f y
3.1.1
Section 3.2 finds the asymptotic and oscillatory behavior of all
solutions of a third order nonlinear neutral delay difference equation
(3.1.1). Section 3.3 deals with suitable examples.
Chapter – 4
Section 4.1 concerned with the asymptotic properties of all the
solutions of a third order nonlinear neutral delay difference equation of
the form
2
1 1 0n n n n k n n lp y h y q f y 4.1.1
Section 4.2 deals with the Main results on the asymptotic and
oscillatory properties of all solutions of a third order nonlinear neutral
delay difference equation (1.5.74). Section 4.3 deals with the examples.
Chapter - 5
Section 5.1 deals with the asymptotic of the oscillation solutions of
third order nonlinear difference equations with delay of the form
2 0, 0,1,2,nn n n nr x q f x n 5.1.1
Section 5.2 deals with results and discussions of asymptotic and
oscillation solutions of third order nonlinear difference equation with
delay. Section 5.3 deals the bounded and oscillation of equation of
(5.1.1). Section 5.4 deals with oscillation behavior for third order delay
difference equation using Schwarz’s Inequality.
39
Chapter -6
Section 6.1 deals with the uncountable many positive solutions of
first order neutral delay nonlinear difference equations of the form
0n n n nnx p x q f x 6.1.1
Section 6.2 deals with the Krasnoselskii’s fixed point theorem.
Section 6.3 deals examples with Diagrammatic representation.
Chapter-7
Section 7.1 deals with the asymptotic and non-oscillatory behavior
of third order neutral delay difference equation of the form
2
1 0n n n n k n np y h y q f y 7.1.1
Section 7.2 deals with non-oscillation.
Chapter- 8
Section 8.1 deals with the oscillatory and non-oscillatory properties
of fourth order difference equation of the form
2 2 2
1 1 2 2 0n n n n n np y q y r y 8.1.1
Section 8.2 deals with the main theorem on oscillatory and non-
oscillatory properties of fourth order difference equation. Section 8.3
deals with the oscillation theorem by using monotonic property.
40
CHAPTER - 2
ASYMPTOTIC AND OSCILLATORY BEHAVIOR OF
THIRD ORDER NONLINEAR NEUTRAL
DELAY DIFFERENCE EQUATION
2.1. INTRODUCTION
In this chapter, we concern with the oscillatory behavior of the
solutions of third order Non-linear neutral delay difference equation of
the form
1 1 0,n n n n n k n n la b x c x q f x 0,1,2,...n Z 2.1.1
Subject to the following conditions:
1( ) ,n nH a b and 0nc 0nq for infinitely many values of .n
2( ) :H f R R is continuous and 0xf x for all 0x
3H There exists a real valued function g such that
f u f v , ,g u v u v for all 0u , 0v and
, 0 .g u v L R
4( ) max ,H m k l and 0n be a fixed non-negative integer.
5 nH R x Z there exists an integer N Z such that
0,n nx x for all n N
By a solution of equation (2.1.1), we mean a real sequence ny
satisfying (2.1.1) for 0.n n A solution ny is said to be oscillatory if it is
41
neither eventually positive nor eventually negative. Otherwise it is called
non-oscillatory. The forward difference operator ∆ is defined by
1 .n n ny y y
2.2 MAIN RESULTS
Theorem 2.2.1
With respect to the difference equation (2.1.1), assume that the
following hold
1 nC C is non-negative and non-decreasing for all .n Z
0
2 1( ) limsup ;sn
s n
C q
0 .n Z
0 0
3
1 1( ) .
s n s ns s
Ca b
Then .R
Then the equation (2.1.1) is Oscillatory.
Proof
Suppose that the equation (2.1.1) has a solution .nx R since
0,n nx x for all .n N implies that nx is non-oscillatory without loss
of generality we can assume that there exist an integer 1 0n n such that
0,nx 0,nx 0,n mx 0,n mx for all 1.n n In fact, for 0,nx
0,n mx for all large n Z the proof is similar. Set n n n n kz x c x then
42
the view of 1 , 0nC z and 0nz for all 1.n n Dividing equation
(2.1.1) by 1n lf x and summing from 1n to 1n we have
1 1 1
11 1
n n nn n n
n l n l
a b za b z
f x f x
1
11 1 1 2 1 1
2 1
,ns s s s l s l s l
s n s l s l
a b z g x x x
f x f x
1
1
1
n
s
s n
q
.
and hence
1 1 1
11 1
n n nn n n
n l n l
a b za b z
f x f x
1
1
1
n
s
s n
q
Thus, from 2 ,C we find n n na b z as .n
This implies that 1 1, 0n n na b z k k
Summing the last inequality from 2n to 1,n to obtain
2 2
2
1
1
1;
n
n n n n
s n s
b z k b za
2 1n n
Thus from 3 ,C n nb z as ,n which contradicts the
assumption that 0,nz for all large .n
2.3 EXAMPLES
Example 2.3.1
Consider the difference equation
1
1 11 2 3 2 0,n n n
nn y y n f y
n n
1n 2.3.1
43
and f x
3
2.
1 2
x
x x
Here
1 1
1,
n nn
na
1 1
1 1,
1n nnb n
1
n
nc
n
and 1
1 1
2 3 5 .n
n n
q n
Hence all the assumption of theorem 2.2.1 holds. Hence the
equation (2.3.1) has a solution 1
,ny Rn
since 0.ny
Theorem 2.3.1
With respect to the difference equation (2.1.1), assume that in
addition 1 3 ,C C the following hold:
4 1C k and 1 0,nc
5 0nC q for all 0,n n
0
1
6 1lim .n
sn
s n
C q
Then, .R
Proof
As in theorem 2.2.1. Let the equation (2.1.1) has a solution
nx R such that 0,nx 0,nx 0n mx and 0n mx for all
1 0.n n n Again , set ,n n n n kz x c x then in view of 4C and the fact
that nx R , we have 0n n n n kz x c x for all 1.n n Since equation
(2.1.1) is the same as n n na b z 1 1 1,n n lq f x n n from
44
condition 5C it follows that n n na b z is increasing, for all 1.n n
Now suppose that 0,n n na b z for all 1;n n
2 2; 0.n n na b z k k Summing the last inequalities from 3n to
1 ,n we have
n nb z3 3
3
1
2
1;
n
n n
s n s
k b za
3n n
Letting n and because of 3 ,C we see that ,n nb z a
contradiction. Thus 0.n n na b z
Now following as in theorem 2.2.1, and using the condition
5 ,C we obtain
1
lim ,n n n
nn l
a b z
f x
which is the required
contradiction.
Example 2.3.2
Consider the difference equation
2
1 2
11 1( 2 0,
3 1n n n
n ny y f y
n n
1.n 2.3.1
and .f x x
Here 1
,nan
2 1
,3
n
n nb
2nc and
2
1,
1nq
n
1
1 1
1.
2n
n n
qn n
45
For this difference equation, assumption 3C and 5C hold: but
4C and 6C are violated. Hence the equation (2.3.1) has a
solution 1
,n
ny R
n
Since 0.n ny y
Theorem 2.3.2
Let 1n na b and f be non-decreasing.
If 0
2
n
n n
n q
Then equation (2.1.1) has a non-oscillatory
solution that approaches a nonzero real number as .n
Proof
Let 0C be given and choose N so that
2
2 2n
n N
cn q
f c
Let N be the banach space of all real sequences ,nX x n N
with norm || || sup | |nn N
X x
Let : 2 ,N nX c x c n N and define : NT by
1 1
3 11 2 ,
2 2s mN
s n
cTX s n s n q f x n N
Clearly is a bounded closed and convex subset of .N
First we will show that T maps into itself. For any ,X we
have
3
2N
cTX 21
2 ,2 2
s
s n
cS q f c n N
46
Thus .T Next we let nX x and for each 1,2,...i
Let i i
nx x be a sequence in such that lim 0.i
nx x
Then a straight forward argument using the continuity of f shows
that lim 0.i
nnnTX TX
and so T is continuous, Finally, in order to
apply Schauder’s fixed point theorem, we need to show that Ty is
relatively compact. In view of recent result Ty is uniformly Cauchy. To
this end let nX x y and observe that for any k n N we have
2 | | 2n k s
s n
TX TX s q f c
from the hypothesis, it is clear that for a
given 0 there exist an integer 1N such that for all k n N we
have| | .n kTX TX .
Thus Ty is uniformly Cauchy, and so Ty is relatively compact.
Therefore, by Schauder’s a fixed point theorem, there is a fixed point
.X It is clear that nX X is a non-oscillatory solution of (2.1.1)
for n N and has the required properties.
Theorem 2.3.3
Let f u be non-decreasing and 0c be a constant such that na c for
all 0.n n
47
Suppose that 0
1 1 1| |n n n
s n
D A B
0
1 1 1| || |n n n n
s n
D A B q
Then equation (2.1.1) has a bounded non-oscillatory solution that
approaches a nonzero limit.
Proof
Let 0c and let N be so large that
0
1 1 1| |4
n n n
s n
cD A B
0
1 1 1| || |4 2
n n n n
s n
cD A B q
f c
.
Let the Banach space N and the set N be the same as in
above theorem 2.3.2 and define the operator : NT by
Tx 1 1
3,
2s s
s n
ck s n q f y n N
Where ,k s n1 1 1 1s s s n s sD D A B A B
Similar to the proof of above theorem, we show that the mapping T
satisfies the hypothesis of schauder’s fixed point X and it is clear that
nX x is a non-oscillatory solution of equation (2.1.1) for n N and
has the desired properties.
48
CHAPTER - 3
ASYMPTOTIC PROPERTIES OF THIRD ORDER
NONLINEAR NEUTRAL DELAY DIFFERENCE EQUATION
3.1 INTRODUCTION
In the previous chapter, we discussed the oscillatory and non-
oscillatory behavior of neutral delay difference equations. In this chapter,
we concerned with the oscillatory behavior of third order nonlinear
neutral delay difference equation of the form
2 0n n n n k n n la y p y q f y
3.1.1
Where, ∆ is the forward difference operator defined by
1 , ,n n ny y y k l are fixed nonnegative integers and , ,n n na p q are
real sequences with respect to the difference equation (3.1.1) throughout.
It is assumed that the following conditions hold.
1( ) , ,n n nH a p q and 0,nq for infinitely many value of n.
2( ) :H f R R is continuous and 0,uf u for 0.u
3H There exists a real valued function g such that f u f v
, ,g u v u v for all 0,u 0v and , 0 .g u v M R
4 max ,H m k l is a fixed non-negative integer.
5( ) { :nH R x S there exists an integer N Z such that 0,n nx x
,n N Where S is the set of all nontrivial solutions of (3.1.1).
49
By a solution of equation (3.1.1), we mean a real sequence nx
satisfying (3.1.1) for 0.n n A solution nx is said to be oscillatory if it
is neither eventually positive nor eventually negative. Otherwise it is
called non-oscillatory.
3.2 MAIN RESULTS
Theorem 3.2.1
With respect to the difference equation (3.1.1) assumes that the
following hold:
1C np is nonnegative and non decreasing for all .n Z
0
2( ) limsup ,sn
s n
C q
0n Z
0
3
1( ) ,
s n s
Ca
0n Z
Then .R
Proof
Suppose that the equation (3.1.1) has a solution .ny R Since
0,n ny y n N implies that ny is non-oscillatory. Without loss of
generality we can assume that there exists an integer 1 0n n such that
0,ny 0,ny 0,n my 0,n my for all 1.n n Set ' ,n n n n kz y p y
then
in view of 1 ,C 0,nz 0,nz for all 1.n n
50
Dividing equation (3.1.1) by n lf y and summing from 1n to
1,n we obtain
1
1
11
2 22 11 1 2 1 1
2 1
,nn s s s s sn
n n
s nn l s l s ln l
z a z g y y yza a
f y f y f yf y
1
1n
s
s n
q
This implies that
1
1
1
22nn
n n
n l n l
zza a
f y f y
1
1n
s
s n
q
3.2.1
In view of the conditions 2 3,C C and from the inequality (3.2.1),
we obtain 2
n na z as .n Therefore, there exists an integer
2 1n n and 1 0k such that
2
1,n na z k for all 2n n 3.2.2
Summing the inequality (3.2.2) from 2n to 1,n we have
2n nz z 2
1
1
1,
n
s n s
ka
for all 2.n n 3.2.3
In view of the condition 3 ,C and from the inequality (3.2.3), we
obtain nz as ,n which is a contraction to the fact that
0,nz for all large .n z Infact 0,ny 0,n my for all large ,n z the
proof is similar, and hence omitted. Hence this shows that .R
51
Corollary 3.2.1
In addition to the conditions of theorem 3.2.1,
If 4C 1k and 1 0na hold then .R
Where is the forward difference operator defined by
1 .n n ny y y
Where ,k l are fixed non-negative integers ,n np q and nh are
real sequences with respect to the difference equation (3.1.1) throughout
then it is assumed that the following condition hold.
1 :H ,n np q and 0nh for infinitely many value of .n
2 :H :f R R is continuous and 0yf y for all 0y
3 :H There exists a real valued function g such that
f u f v ,g u v u v for all 0u , 0v and
, 1g u v 0 R
4 :H max ,M k l and 0n be a fixed nonnegative integers
5 :H :nR y S there exists a integer n Z such that
0n ny y for all n N and S is the set of all nontrivial solution of
(3.1.1), By a solution of equation (3.1.1), we mean a real sequence ny
satisfying (3.1.1) for 0.n n A solution of ny is said to be oscillatory if
it is neither eventually positive nor negative otherwise it is called
asymptotic behaviour of non-oscillatory.
52
3.3 EXAMPLES
Example 3.3.1
Consider the difference equation
52 3
1 2
1 12 3 2 0,
1 2
nn n
n n
n yy y n
n n y y
3n 3.3.1
Here 1
,nan
1
,n
np
n
2 3 2nq n
and
5
2,
1 2
nn
n n
yf y
y y
3, 1, 0k l
All the conditions of the theorem 3.2.1 are satisfied, and hence all
solutions of equation (3.1.1) are not in R. One such solution of equation
(3.1.1) is 1
.ny Rn
53
CHAPTER 4
ASYMPTOTIC PROPERTIES THIRD ORDER NONLINEAR
NEUTRAL DELAY DIFFERENCE EQUATION
4.1 INTRODUCTION
To concerned with the oscillatory properties of all the solutions of
a third order nonlinear neutral delay difference equation of the form
2
( ) ( 1) ( 1 ) 0n n n n k n n lp y h y q f y
4.1.1
4.2 MAIN RESULTS
Theorem 4.2.1
With respect to the difference equation (4.1.1) assumes that the
following hold:
1 : nC h is non-negative and non-decreasing for all n Z
0
2 1lim p: su sn
s n
qC
0: n Z
0 0
3
1 1:
s n s ns s
Cq h
Then R
Proof
Suppose that the equation (3.1.1) has a solution Rny since
0n ny y for all n N implies that ny is a non-oscillatory without loss
of generality we can assume that exists an integer 1 0n n such that
0,ny 0,ny 0n my and 0n my for all 1.n n
54
In fact, for all 0, 0n n my y for all large ,n z the proof is similar
set n n n n kz y h y then in view of 1 0,nC z 0nz and
2 0nz for
all 1.n n Dividing the equation (4.1.1) by 1 1nf x and summing from
1n to 1,n we have
1 1
1
2 22 11
1 1 1
,nn n s s s l s l s ln n
s nn l n l s l s l
p z p z g y y yp z
f x f x f y f y
1
1
1
n
s
s n
q
and hence
1 1
22
1 1
n nn n
n l n l
p zp z
f x f x
1
1
1
n
s
s n
q
Thus from 2C we find 2
n np z as .n This implies that
2
1,n np z k 1 0k . summing this last inequality from 2n to 1n , we
obtain 2
1,n np z k 1 0k summing the last inequality from 2n to 1n .
We obtain
2
nz2 2
2
12
1
1,
n
n n
s n s
k p zp
2n n
Thus from 3C 2
nz as n
Which is contradict to the assumption that 2 0nz for all large n.
Example 4.2.1
Consider the difference equation
2
1
11 2 3 2 0n n n
nn y y n f y
n
4.2.1
55
1 1
1 ,n
n n
p n
1
1 1
2 3 2n
n n
q n
Hence all the assumption of theorem holds. Hence the equation
(4.2.1) has a solution 1 Rnny such 0.n ny y
Theorem 4.2.2
With respect to the difference equation (4.1.1) assume that in
addition to the condition 3C the following holds
4C 1k and 1 0nh
5C 0nq for all 0n n
0
1
6 1limn
sn
s n
C q
Then R
Proof
As in theorem as have a solution ny R such that 0,ny
0,ny 0n my and 0n my for all 1 0.n n n Again set
n n n n kz y h y then in view of 4C and the fact ny R we have
n n k n n kz y h y then in view of 4C and the fact that ny R we have
0n n k n n kz y h y for all 1.n n
Since equation (4.1.1) is the same as 2
n np z 1 1 1 ,n nq f x
1.n n From condition 3C it follows that 2
n np z is non-increasing for
56
all 1.n n now suppose that 2 0n np z for all 1,n n 2
2 ,n np z k
2 0,k summing the last inequality from 3n to 1.n We have
2
nz2
3
12
2 3
1,
n
n
s n s
k z n np
Let n and because of 3C we see that 2
3,nz k 3 0k
summing the last inequality from 4n to 1.n We obtain
nz4
4 4
1 1
3
1n m
n
s n k n ks
k zp
Let nand because of 3C we see that nz contradicts.
Thus 2 0n np z now following as in the theorem and using the
condition 5 ,C we obtain
2
1
lim n n
nn l
p z
f x
.
Which is the required contradiction.
57
CHAPTER - 5
ASYMPOTIC OF OSCILLATION SOLUTIONS OF THIRD
ORDER NONLINEAR DIFFERENCE
EQUATIONS WITH DELAY
5.1 INTRODUCTION
Sufficient conditions for the asymptotic oscillation of some third
order nonlinear difference equations of the form
2 0,nn n n nr x q f x 0,1,2,n 5.1.1
Where, ∆ denotes the forward difference operator 1 ,n n ny y y
nq is a sequence of real numbers, n is a sequence of integers such
that
1 : lim ,nn
C n
where nr is a sequence of positive numbers
1
2
0
1:
n
n
k
C Rrk
asn
3 : :C f R R is a continuous with 0 0uf u u
By a solution of equation (5.1.1) we mean a sequence ,nx which
is defined for 0
min ii
N i r
and satisfies equation (5.1.1).
A nontrivial solution nx of (5.1.1) is said to be oscillatory if for
every 0 0n there, exists 0n n such that
1 0.n nx x Otherwise it is called
non-oscillatory.
58
5.2 MAIN RESULTS
Theorem 5.2.1
Assume that
4 :C 0nq and 1
n
n
q
5 :C inli fm 0u
f u
Then every solution of equation (5.1.1) is oscillatory.
Proof
Assume that equation (5.1.1) has non-oscillatory solution nx and
we assume that nx is eventually positive. Then there is a positive integer
0n such that
0nnx for 0n n 5.2.1
From the equation (5.1.1) we have,
2
n nr x 0,nn nq f x 0n n
and so 2
n nr x is an eventually non-increasing sequence.
We first show that 2
00, .n nr x n n In fact, if there is an 1 0n n
such that
1
2 0n nr x c
and 2
n nr x c for 1n n
59
that is 2
n
n
cx
r
and hence nx1
1
1 1n
n
k n k
x cr
nx2
1 1 1
1 1 1 1m m n
s n
s m s m k n ks
x c xr
as ,n m
which contradicts the fact that 0nx for 1.n n
hence 00, .n nr x n n therefore we obtain,
0,nnx 2 0,nx 2 0n nr x for 0.n n
Let lim .nn
L x
Then 0L is finite or infinite.
Case (i)
If 0L is finite
From the continuity of function f u we have
limnn
nf x
0.f L
Thus we may choose a positive integer 3 0n n such that
nnf x
1,
2f L 3n n 5.2.2
By substituting (5.2.2) into equation (5.1.1) we obtain
2 10,
2n n nr x f L q 3n n 5.2.3
60
Summing up both sides of (5.2.3) from 3n to 3 .n n We obtain
3 3
3
1 1
10
2
n
n n n n i
i n
r x r x f L q
and so 3
1
2
n
i
i n
f L q
3 3,n nr x 3n n contradicts.
Case (ii)
If L
For this case, from the condition 1C
We have liminf 0nn
nx
and so we may choose a positive
constant ‘c’ and a positive integer 4n sufficiently large such that
nnf x c for 4n n 5.2.4
Substituting (5.2.4) into equation (5.1.1), we have
2 0,n n nr x cq 4.n n
Using the similar argument as that of case 1 we may obtain a
contradiction to the condition 1 .C This completes the proof.
5.3 BOUNDEDNESS AND OSCILLATION
Theorem 5.3.1
Assume that 6 : 0nC q and 0
n n
n
R q
, then every bounded
solution of (5.1.1) is oscillatory.
61
Proof
Proceeding as in the proof of theorem 5.2.1 with assumption that
nx is a bounded non-oscillatory solution of (5.1.1) we get the inequality
(5.2.3) and so we obtain
2 10,
2n n n n nR r x f L R q 3n n 5.3.1
It is easy to see that
2
n n nR r x 2 2
n n n n n nR r x r x R 5.3.2
From inequalities (5.3.1) and (5.3.2) we deduce
3 3
2 2 10,
2
n n n
k k k k k k
k n k nk n
R r x x f L R q
3.n n
This implies
3
1
2
n
k k
k n
f L R q
3 3 3 3
2
1 3,n n n n nx R r x x n n
Hence there exists a constant ‘c’ such that 3
,n
k k
k n
R q c
3.n n
Contrary to the assumption of the theorem.
Theorem 5.3.2
Assume that
7( ) : nC n is non-decreasing, where 0,1,2,n there is a
subsequence of ,n say kn such that 1,
knr 0,1,2,k
62
8
0
( ) : ,n
n
C q
9( ) :C f is non-decreasing and there is a nonnegative constant M
such that
0
limsupu
uM
f u 5.3.3
Then the difference 2
nx of every solution nx of equation
(5.1.1) oscillates.
Proof
If not, then equation (5.1.1) has a solution nx such that its
difference 2
nx is non-oscillatory. Assume that the sequence 2
nx is
eventually negative.
Then there is positive integer 0n such that 2 0,nx
0n n and so
nx decreasing for 0n n which implies that nx is also non-oscillatory.
Set nw
2
,
n
n n
n
r x
f x
1 0n n n 5.3.4
Then nw
1
2 2
1 1
1 n n
n n n n
n n
r x r x
f x f x
1
1
212
1 1
1
n n
n n n
n nn nn n
n n n
f x f xr xr x
f x f x f x
5.3.5
63
2
,
n
n nn
n
r xq
f x
1.n n
Summing up both sides of (5.3.5) from 1n to ,n we have
11n nw w
1
n
i
i n
q
and by (5.3.1), we get
lim nn
w
5.3.6
This implies that eventually
0nnf x 5.3.7
and therefore 0.nnx
By (5.3.6), we can choose 2 1 ,n n such that
1 ,nw M 2.n n
2 1 0,nn nr M f x
2n n 5.3.8
Set lim nn
x L
then 0.L
Now we prove that 0.L If 0L then we have
limnn
nf x
0f L
By the continuity of ,f u choosing an 3n sufficiently large, such that
nnf x
1,
2f L 3n n 5.3.9
64
And substituting (5.3.9) into (5.3.8) we have
2 11 0,
2nx M f L
rn 3n n 5.3.10
Summing up both sides of (5.3.10) from 3n to n , we get
3
3
1
1 11 0
2
n
n n
i n i
x x M f Lr
This implies that lim .nn
x
This contradicts (5.3.7).
Hence lim 0.nn
x
By the assumptions, we have
limsup .n
n
n
nn
xM
f x
From this we can choose 4n such that
n
n
n
n
x
f x
1,M
that is 41 ,n nn nx M f x n n
and so from (5.3.8) we get
2 0,
nn n nr x x 4n n 5.3.11
In particular, from (5.3.11) for a subsequence knr satisfying the
condition 5C of theorem 5.2.1, We have
65
1k k k kn n n nx x x r 1 0,
k k k k kn n n n nr x x x r
For k sufficiently large, implies that
10 0k k k kn n n nx x r x for all large .k
This is a contradiction.
The case that 2
nx is eventually positive can be treated in a
similar fashion and so the proof of theorem 5.3.2 is completed.
66
CHAPTER 6
UNCOUNTABLY MANY POSITIVE SOLUTIONS OF FIRST
ORDER NONLINEAR NEUTRAL DELAY
DIFFERENCE EQUATIONS
6.1 INTRODUCTION
We consider the difference equation of the form
0n n n nnx p x q f x 6.1.1
where, 0,n n 0 are integers.
also 0 , , 0, ,a C t , 0,p R
and ,f C R R
where f is non decreasing function for 0,f x 0.x
we are concern with the first order nonlinear neutral delay
difference equation (6.1.1).
1 0( ) , , 0, ,nH r C n 1
s n rs
2 0( ) , , 0, 0nH p C n p
3( ) , , 0,H C 0 / 0)x
4( ) , , , ,H f x C 0 / 0)f x x
5( )H G x
0 0 :f x
x
G x is non decreasing in ,0
6 0( ) , 0, ,H G n C n g n n
67
A non trivial solution nx is said to be oscillatory if it has
arbitrarily large zeros otherwise nx is said to be non-oscillatory. The
proof is an adaptation of that given (6.1.1) where the special case
g n n was considered.
6.2 KRASNOSELSKII’S FIXED POINT THEOREM
Let X be a Banach space. Let Ώ be a bounded close convex subset
of X and let 1 2,s s be maps of Ώ into X such that 1 2s x s y for every
, .x y
If 1s is contractive and 2s is completely continuous. Then the
equation 1 2 .s x s x x
Theorem 6.2.1
Suppose that there exist bounded from below and from above by
the function 0, , , 0,n nu v c n constant 2 10, 0c k k and
1 0n n m such that
0,n nu v n n 6.2.1
1 1 0,n n n nv v u u 0 1n n n 6.2.2
1
1( n s s n
s n
u k p f v au n
2
1( ) 1,n s sv k p f u c
v n
1n n 6.2.3
68
Then equation (6.1.1) has uncountable many positive solution
which are bounded by the function , .u v
Proof
Let 0 , , )c k R be the set of all continuous bounded functions with
the norm 0
|| || sup .n n nx x Then is Banach space.
We define a close bounded and convex subset of 0 , ,c n R as
0 0Ώ , , ) : ,n n n nx x c n R u x v n n
For 1 2,k k k we define two maps 1 2 0& : Ώ ,s s c n as follows
1 ns x 1
1
1 0 1
n n
n
k a x n n
s x n t n
6.2.4
2 ns x
1 1
1
2 0 1
s s
s n
n n n
p f x n n
s x v v n t n
We will show that for any , Ώx y we have 1 2 Ώs x s y for
every , Ώx y and 1t t with regard to (6.2.3) we obtain
1 2n ns x s y n n s s
s n
k a x p f y
n n s s
s n
k a v p f y
2n nk v k v
69
For 0 1,n n n we have
1 2n ns x s y 1 1 11 2n n n ns x s y v v
1 1n n nv v v nv
Further more for 1n n we get
1 2n ns x s y n n s s
s n
k a u p f v
nk u t k u
Let 0 1,n n n with regards to (6.2.2) we get
1 1,n n n nv v u u 0 1n t n
Then 0 1,n n n and any , Ώx y we obtain
1 2n ns x s y 1 1 11 2n n t ts x s y u u
1 1n n n nu u u u
Then we have prove that 1 2 Ώs x s y for any , Ώx y .
We will show that 1s is a contraction mapping on Ώ for , Ώx y
and 1.n n We have
1 2n ns x s y | || | || ||n na x c x y
This implies 1 2s x s y || ||c x y
Also for 0 1,n n n the above inequalities are valid.
We conclude that 1s is a contraction mapping on Ώ.
70
We now show that 2s is completely continuous.
First we show that 2s is continuous. Let Ώii
nx x be such that
in nx x as .n Because x is close Ώnx x for 1n n we have
2 2
i
n ns x s x i
s s s
s n
p fx f x
1
i
s s
s n
fx f x
Since 0i
s sf x f x as i by applying the lebeague
dominant convergence theorem we obtain 2 2lim 0.i
is x s x
This means
2s is continuous.
We now show that 2s is relatively compact in Ώ, it is sufficient to
show by Arzela ascolic theorem that the family of functions 2 : Ώs x x
is uniformly continuous. Then
2
s s
s n
p f x
Then Ώ,x 2 1N N n
where 2 2 2 1s x N s x N 2 1
s s s s
s N s N
p f x p f x
2 2
2 2 2 1s x N s x N 1
s s
s N
p f x
2 1max ,s sp f x N N 1 2N N n
71
then there exist 1s
M
where max s sM p f x such that
1 2N N n
2 2 2 1s x N s x n if 2 1 10 N N s .
Next we show that equation (6.1.1) has uncountable many positive
solutions Ώ. Let 1 2,k k k
be such that .k k
we assume that , Ώx y
1 2 1 2,s x s x x s y s y y
,n n s sn
s n
x k a x p f x
1n n
,n n s sn
s n
y k a y p f y
1n n .
It follow that there exist a 2 1n n satisfying
2
s s s
s n
p f x f y
|| k k
In order to prove that the set of bounded positive solution of
equation (6.1.1) is constant. It is sufficient to verify that x y for 2n n ,
we get
(1 ) || ||x x y | | [ ]s s s
s n
k k p f x f y
Corollary 6.2.1
Suppose that there exist bounded from below and from above by
function 0, , 0,u v C n that 0,C 2 1 0,k k 1 0n n m such
that (6.2.1), (6.2.3) holds.
72
Proof
0,n nv u 1 2n n n
H t 1 1n n n nv v u u
H t 0n nv u
nH t 0 .
6.3 EXAMPLE WITH DIAGRAMMATIC REPRESENTATION
1 1
10n n nx x x
n
Input
1
1 2 1 1 0x n x n x n x nn
Graph of the difference equation is:
74
CHAPTER - 7
ASYMPTOTIC OF NON OSCILLATORY BEHAVIOUR
OF THIRD ORDER NEUTRAL DELAY
DIFFERENCE EQUATIONS
7.1 INTRODUCTION
We are concerned with the oscillatory properties of all the
solutions of a third order nonlinear neutral delay difference equation of
the form
2
1 0n n n n k n np y h y q f y 7.1.1
Where ∆ is the forward difference operator defined by
1 ,n n ny y y where ,k l are fixed nonnegative integers and np , nq
and nh are real sequences with respect to the difference equation
(7.1.1).
Throughout we shall assume that the following conditions hold:
1 : n nH p q and 0nh and 0nq for infinitely many value of n
2 :H f R R is continuous and 0yf y for all 0y
3H There exists a real valued function g such that f u f v
,g u v u v for all 0u and 0v and ,g u v 0L R
4H M max ,k l and 0n be a fixed non-negative integer
5H R :ny S there exists an integer N S such that 0n ny y
for all n N and ‘S’ is the set of all nontrivial solution of (7.1.1).
75
By a solution of equation (7.1.1), real sequence ny satisfying
(7.1.1) for 0.n n
A solution nh is said to be oscillatory if it is neither eventually
positive nor negative, otherwise it is called non-oscillatory.
7.2 ASYMPTOTIC AND NON-OSCILLATION THEOREMS
Theorem 7.2.1
With respect to the difference equation (7.1.1) assumes that the
following hold:
1 nC h is non-negative and non-decreasing for all n Z
0
1
2 1limsup ;n
sn
s n
C q
0n Z
0 0
3
1 1( )
s n s ns s
Cq h
then R
and equation (7.1.1) is asymptotic behaviour of non-oscillatory.
Proof
Suppose that the equation (7.1.1) has a solution ny R . since
0n ny y for all n N implies that ny is a non-oscillatory. Without
loss of generality we can assume that there exists an integer 1 0n n such
that
0,ny 0.ny
0n my and 0n my for all 1n n
76
In fact for all 0,ny 0n my for all large ,n z the proof is
similar.
Set n n n n kz y h y then in view of 1 , 0nC z , 0nz and
2 0nz for all 1.n n
Dividing the equation (7.1.1) by n lf x and summing from 1n to
1n , we have
1 1
1
22
1
n nn n
n n
p zp z
f x l f x l
1
211 1,
1
ns s s s l s l
s n s l
p z g y y y
f y f s l
1
1
1
n
s
s n
q
and hence
1 1
1
22n nn n
n l n l
p zp z
f x f x
1
1
1
n
s
s n
q
Thus from 2C we find 2
n np z as n . This implies that
2
1 1, 0n np z k k .
Summing this last inequality from 2n to 1n , we obtain
2
1 1, 0n np z k k , summing the last inequality from 2N to 1n , we
obtain
2
nz2 2
2
12
1 1
1n
n n
s n s
k p z np
Thus from 2
3 nC z as .n Which is a contradict to the
assumption that 2 0nz for all large n.
77
Theorem 7.2.2
With respect to the difference equation (7.1.1) assumes that in
addition to the condition 3C the following holds:
4( ) 1C k and 1 0nh
5( ) 0nC q for all 0n n
0
1
6 1( ) limn
sn
s n
C q
Then R
Then equation (7.1.1) is asymptotic behaviour of non-oscillatory.
Proof
As in theorem (7.2.1)we have a solution ny R such that
0,ny 0ny and 0n my and 0n my for all 1 0n n n .
Again set n n n n kz y h y then in view of 4C and the fact
that ny R .
We have n n k n n kz y h y then in view of 4C and the fact
that ny R .
We have 0n n k n n kz y h y for all 1.n n Since equation (7.1.1)
is the same as
2
1,n n n n lp z q f x n n
78
From condition 5C it follows that 2
n np z is non increasing for
all 1n n .
Now suppose that 2 0n np z for all 1n n ,
2
2 2, 0n np z k k
Summing the last inequality from 3n to 1n , We have
2
nz2
3
12
2 3 1
1,
n
n
s n s
k z n np
Let n and because of 3C we see that
2
nz that 2
3 3, 0nz k k summing the last inequality
from 4n to 1n , we obtain
nZ4
4 4
1 1
3 4
1,
n m
n
s n k n s
k z n npk
We see that nz , which a contradiction. Let n and
because of 3C we have 2 0n np z . Now following as in theorem 7.2.1
and using the condition 5C we obtain
2
lim n n
nn l
p z
f x
Which is the required contradiction.
79
CHAPTER - 8
OSCILLATORY AND NON OSCILLATORY PROPERTIES
OF FOURTH ORDER DIFFERENCE EQUATION
8.1 INTRODUCTION
Consider the fourth order difference equation of the form
2 2 2
1 1 2 2 0n n n n n np y q y r y 8.1.1
where ,n np q and nr are real sequences satisfying 0,np 0nq
and 0nr for each 0n and the forward difference operator ∆ is defined
by 1n n ny y y also .ny y n
Definition 8.1.1
Let ny be a function defined on ,N we say k N is a generalized
zero for ny if one of following holds:
i 0ny
ii 1k N and 1 0,n ny y 1 ,k N and there exists an integer
,m such that 1 .m k
iii 1 0m
k m ny y and 0jy for all 1, 1 .j N k m k
A generalized zero for ny is said to be of order 0, 1, or 1m
according to whether condition (i), (ii) or (iii), respectively holds. In
particular, a generalized zero of order 0 will simply be called a zero, and
a generalized zero of order one will again be called a node.
80
Obviously, if y a 1y a 2y a 3y a 0 , for some
a N then 0ny is the only solution of (8.1.1). Thus, a nontrivial
solution of (8.1.1) can have zeros at not more than three consecutive
values of k. In definition 8.1.1, we shall show that a nontrivial solution of
(8.1.1) cannot have a generalized zero of order 3.m However, a
solution of (8.1.1) can have arbitrarily many consecutive nodes, as it is
clear from 1n
ny which is a solution of (8.1.1).
The following properties of the solutions of (8.1.1) are fundamental
and will be used subsequently.
1S If ny is a nontrivial solution of (8.1.1) and if
a 0ny b 0ny c 2
1 0ny d 3
2 0ny
for some 2k a N then 1 1 1, ,a b c and 1d holds for all
,k N a with strict inequality in a for all 2 ,k N a strict
inequality in 1b for all 1k N a and strict inequality in 1c and
1d for all 3k N a . Furthermore,
2 2 2
1 1 2 2 0n n n n n np y q y r y for all k N a 8.1.2
With strict inequality for all 2 ,k N a and ,ny ny , 2
ny all
tends to as .k
In 2S , If ny is a nontrivial solution of (8.1.1) and if
81
1a 0ny 1b 0ny 1c2 0ny 1d
3 0ny
for some k a N , then 1 1 1, ,a b c and 1d holds for all
k N a with strict inequality in 1 1 1, ,a b d for all 3k N a and
in 1c for all 4 .k N a
Furthermore, 4 0ny for all k N a 8.1.3
With strict inequality for all 2k aa N and ,ny ny and
2
ny all tends to as .k
3S If ny is a nontrivial solution of (8.1.1) and if
2 0na y 2 1 0nb y 2
2 1 0nc y 3
2 1 0nd y
for some 3k a N then (8.1.2) holds for all 2,k N a and
2 2 2
1 1 2 2 0n n n n n np y q y r y for all 2,k N a 8.1.4
Furthermore, 0 1 0y y and 0 0y . Strict inequality
holds in 2a and (8.1.3) for all 2, 2k N a , if 4a N in 2b for
all 2, 1k N a and in 2C for all 2, 3k N a , if 5 .a N
4S Let 2a N
If ny is a solution of (8.1.1) with 0,y a 1 0,y a
1 0,y a 1y a and 1y a not both zero, then at least one of the
following conditions must be true.
82
(i) Either 0ny for all 2k N a or
(ii) 0ny for all 0, 1 .k N a In particular, ny cannot have
generalized zeros of any order at both and where
0, 1N a and 2 .N a An analogous statement
holds for the hypothesis 1 0y a and 1 0.y a
8.2 Main Theorem on Oscillatory and Non-Oscillatory Properties of
Fourth Order Difference Equation
Theorem 8.2.1
If ny is a nontrivial solution of (8.1.1) with zeros at three
consecutive values of k, say ,a 1a and 2a then ny has no other
generalized zeros. If 3 0 0y a then 0 0ny for all k, and the
inequality is strict if 2k N a or 0, 1k N a In particular, if
0, 1N a and 3 ,N a then 0.y y
Proof
Clearly 2 0.y a y a Since the solution ny is nontrivial, we
may assume that 3 0.y a Thus, 3 0y a and by 2S ny is
positive and strictly increasing on 3 .N a Next, Let n nv y , then
1 0,v a 0,v a 2 0v a and 3 0.v a If 2a N , then
3S implies that nv is positive and strictly decreasing on ,0 .N a
83
Thus ny is negative and strictly increasing on ,0 .N a If 1,a
then we again assume that 3 4 0.y a y Then by (8.1.1)
4 0 2 ,y r 2 0.y
But 4 0 4 0 ,y y y
So 0 4 0y y and 0 1 0 0y y y as claimed.
If 0,a then the part of the conclusion concerning 1k a is
empty. This completes the proof.
Theorem 8.2.2
Let 1 ,a N suppose that ny is a solution of (8.1.1) with
0 0,y 1 0,y a 2 0,y a but 2a is a generalized zero for
.ny Then ny has no other generalized zeros.
If 2 0 0 ,y a then 0 0ny for all ,k N with strict
inequality for all 2k N a or 0, 1 .k N a In particular, if
0, 1N a and 2 ,N a then 0.y y
Proof
Since 2 0,y a we can assume that 2 0.y a
Since y a 1 0,y a 2a cannot be a generalized zero of
order 1 or 2, and theorem (8.2.1) implies that the order cannot be greater
than 3.
84
Thus, 2a is a generalized zero of order 3, which implies that
1 0.y a now since from (8.1.1), we have 2 0,n np y it follows
that 3 0,y a clearly 2 0,y a 0y a and 0,y a thus by
2 ,Sny is positive and strictly increasing on 3 .N a
For 0, ,k N a Let .n nv y
Then 0,v a 1 0v a , 2 1 0v a and
3 1 0v a . If 3 ,a N
then as in equation (8.1.1), 3S yields the results.
If 2,a then 2 3 0,y y 1 0y , 4 0y and 1 0y
By (8.1.1) we have 4 0 0.y
But, 4 0y 4 4 3 6 2 4 1 0y y y y y
4 4 1 0 ,y y y
and so 4 1 0y y 4 0.y
Hence, 0 4,y 1 0y and 0 1 3 1 0.y y y
Therefore, 0 0y and 0 0y as claimed. If 1,a then
1 2 0,y y 3 0y and 2 3a is a generalized zero. It follows
from the definition of a generalized zero that this must be a generalized
zero of order 3, so that if 3 0y then 0 0.y
Hence 0 0.y Which completes the proof.
85
Corollary 8.2.1
If ny is a nontrivial solution of (8.1.1) with generalized zeros at
and and zero at ,a where 1 1,a then 1 1 0.y a y a
In particular, ny does not have a generalized zero at 1.a
Proof
Since 1 1a from theorem (8.2.1) it follows that 1y a
and 1y a both cannot be zero. If 1 1 0,y a y a then 4S
implies that ny cannot have generalized zeros at both and which is a
contradiction. Thus, 1 1 0.y a y a
Corollary 8.2.2
If ny is a nontrivial solution of (8.1.1) with
0,y y a y where 1a then 1 0.y a
Corollary 8.2.3
If a nontrivial solution ny of theorem (8.2.1) has a zero at and a
generalized zero at where , then cannot have consecutive
zeros at , 1a a where 1.a
Theorem 8.2.3
If two nontrivial solutions ny and nv of (8.1.1) have three zeros in
common, then ny and nv are linearly dependent, i.e., specifying any three
zeros uniquely determines a nontrivial solution up to a multiplicative
constant.
86
Proof
If y y a 1y a v v a 1 0v a for some
and ,a where 0 ,a then by theorem 8.2.1, 2 0u a and
2 0.v a
Define w n 2v a y n 2 .y a v n Since w n is a linear
combination of y n and v n , it is a solution of (8.1.1). However,
w w a 1w a 2 0,w a and so w n must be the trivial
solution of (8.1.1) by theorem (8.2.1). Since 2u a and 2v a are
nonzero, u n and v n must be constant multiples of each other.
Next, if y y a y v v a 0,v where
1a then by corollary 8.2.3, 1 0y a and 1 0.v a
Define w n 1 1 .v a y n y a v n
Clearly, w w a 1w a 0,w which contradicts
corollary 8.2.2 unless 0w n . But this means y n and v n are
constant multiples of each other. This completes the proof.
Definition 8.2.1
A solution y n of (8.1.1) is called recessive if there exists an
a N such that for all .k N a
0,y n 0,y n 2 0y n and 3 0y n 8.2.1
87
Let my n be the solution of (8.1.1) satisfying my m
1my m 2 0my m and 0 1my .where 1 .m N
For each , mm y n exists and is unique. The existence is clear from
theorem 8.2.1 and normalization. While the uniqueness follows from
theorem 8.2.3.
Note that by construction
0 1my n for all 0, 2k N m 8.2.2
Also, Theorem (8.2.1) implies that
1m my n y n for all .k N 8.2.3
We now consider m sequence 1 .my
By (8.2.1), 0 1 1my for all 1m N
Thus lim sup 1m
my
exists, we call it 1 .y Then, there exists a
subsequence 1 1lm N such that
2 22 1m m
m k m k m k m k m k m ky k p y y k q y r y 8.2.4
Consider (8.2.4) with 2k and m replaced by 3 .lm we can
conclude that 3lim 5 5 .lm
ly y
Proceeding inductively, we conclude
that 3lim lm
ly k y k
exists for any .k N
88
Replacing m by 3lm in (8.2.4) and letting ,l we conclude that
y k is a solution of (8.1.1).
Also, 1 0y k y k 8.2.5
This follows from (8.2.3) by replacing m by 3 ,lm fixing ,k and
letting .l From (8.2.5) we conclude that
limk
y k
exists, and we shall call it L 8.2.6
We will now show that this y k is a recessive solution of (8.1.1).
8.3 OSCILLATION THEOREM BY USING MONOTONIC
PROPERTY
Theorem 8.3.1
The solution y k constructed above is a recessive solution of
(8.1.1). In addition 2,y k y k and 3 y k all monotonically approach
zero as .k
Proof
We will first show that (8.2.1) is satisfied. By (8.2.3) and theorem
8.2.1, 3
3 3 0.lm
ly m
Choosing 3 3lm and using 3S with 3 1,la m we can conclude
that for any k such that 32 1,lk m 3 1 0,lm
y k 32 1 0lmy k
and 33 1 0.lmy k
89
Letting l implies that y k satisfies (8.2.1) for 1a and is
recessive. We note that y k also satisfies (8.2.1) for 0.a Concerning
the monotonicity, we choose any 2k N and any 3 .lm k
Then, 32 1 0lmy k which means 3lm
y k 3 1 ,lmy k and
hence 3 30 1 .l lm my k y k Taking the limit as l implies
that y k is monotonically decreasing in absolute value. By (8.1.1),
Since y k monotonically approaches a finite limit, 0y k as
.k The argument that 2 y k and 3 y k monotonically approach
zero is similar.
By theorem 8.3.1 this recessive solution y k of (8.1.1) can be
written as
2 2 2
1 1n n n np y q y 1
1 2 36 l k
l l k l k l k r l y l
8.3.1
Corollary 8.3.1
If 3
1
,l r l
then the recessive solution of (8.1.1)
constructed above approaches zero as .k
Corollary 8.3.2
Suppose that y k and v k are two recessive solutions of (8.1.1)
such that .y a v a If y k v k for all k N a then .y k v k
90
Proof
Let limk
l y k
and limk
h v k
.
By hypothesis, l h .
Thus, if w k y k v k , then
From (8.3.1) with 2 k a , we have
2
10 1 1 0
6 l a
l m l a l a l a r l w l
From this we conclude that .y k v k
Conclusion
The oscillatory property of fourth order difference equation
becomes Oscillate.
91
CHAPTER - 9 CONCLUSIONS
1. The asymptotic Behaviour of the solution of the third order nonlinear
Neutral Delay difference equation using Riccati transformations was
found to be asymptotic.
2. Asymptotic Behaviour of third order nonlinear Neutral Delay
difference equation using Riccati transformations were found to obtain
several oscillation criteria.
3. The Asymptotic property of third order Nonlinear Delay Difference
Equations were found to become Oscillate using Schwarz’s Inequality.
4. The Asymptotic property of Third order Nonlinear Neutral Delay
difference Equation was found to become Oscillate using Schwarz’s
Inequality.
5. Uncountably many positive solutions of first order Nonlinear Neutral
Delay Difference Equations were found using Krasnoselskii’s fixed
point theorem.
6. Asymptotic of Non- oscillatory Behaviour of fourth order Neutral
Delay Difference Equations were found using Double sequence
Techniques.
92
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LIST OF PUBLICATIONS
1) Dr. B. Selvaraj, Dr. P. Mohankumar and V. Ananthan, Asymptotic and
Oscillatory Behavior of third order nonlinear Neutral Delay Difference
Equations, International Journal of Nonlinear Science,
Volume.13(2012) No.4.pp.472-474.
2) Dr. B. Selvaraj, Dr. P. Mohankumar and V. Ananthan, Asymptotic
properties of third order nonlinear Neutral Delay difference equation,
Journal.Comp.& Math.Sci.Vol.4(5), 356-359(2013).
3) Dr. P. Mohankumar, V. Ananthan and Dr. A. Ramesh, Asymptotic
properties of third order nonlinear Neutral Delay difference Equation,
International Journal of Mathematical Archive-5(8), 2014, 188-190.
4) Dr. P. Mohan Kumar, V. Ananthan and Dr. A. Ramesh, Asymptotic
of Oscillation solutions of third order nonlinear difference equations
with delay, International Journal of Mathematics And Computer
Research, Volume 2 Issue 8 August 2014, Page No 581-586,
ISSN2320-7167.
5) V. Ananthan and Dr. S. Kandasamy, Uncountably Many Positive
solutions of First Order Non-Linear Neutral Delay difference
105
equations, International Journal of Mathematical Archive-7(8), 2016,
144-147.
6) V. Ananthan and Dr. S. Kandasamy, Asymptotic of Non-oscillatory
Behavior of Third Neutral Delay Difference Equations, Journal of
Chemical and Pharmaceutical Sciences, ISSN: 0974-2115(2016).
7) V. Ananthan and Dr. S. Kandasamy, Oscillatory and Non-oscillatory
properties of fourth order difference equation, Int.J.Chem.Sci:14(4),
2016, 3005-3012. ISSN.0972-768X.