BRNO UNIVERSITY OF TECHNOLOGY

Faculty of Mechanical Engineering

Institute of Mathematics

Ewa Schmeidel, Ph.D.

Properties of Solutions of Higher OrderDi�erence Equations

Summary of the habilitation dissertation

Vlastnosti řešení diferenčních rovnic vyšších řádů

BRNO 2010

Keywords:

Di�erence equations, higher order, asymptotic behavior, oscillatory solution, nonoscil-latory solution, periodic solution, Volterra di�erence system.

ISSN 1213-418X

© Ewa Schmiedel, 2010 ISBN 978-80-214-4185-9

Contents

Curriculum Vitae 4

1 Introduction 5

2 Motivation and aims of this work 6

3 Fourth order di�erence equations 133.1 Smith-Taylor equation . . . . . . . . . . . . . . . . . . . . . . . . . 133.2 Thandapani-Arockiasamy equation . . . . . . . . . . . . . . . . . . 15

4 Neutral di�erence equations 16

5 Volterra di�erence equations 195.1 Scalar Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195.2 Systems of Volterra equations . . . . . . . . . . . . . . . . . . . . . 21

Bibliography 23

Author publications list 26

Abstract 30

3

Curriculum Vitae

Ewa Schmeidel, nee Kubacka, was born in Pozna«, Poland on26th October 1953. She received the degree in mathematics fromthe Faculty of Mathematics, Physics and Chemistry of AdamMickiewicz University in Pozna« in 1978. During her study hermathematical interest was focused on functional analysis. So,her adviser of master thesis was Professor Andrzej Alexiewicz.On completing her studies she took up a post at the Instituteof Mathematics of the Pozna« University of Technology, whereshe continuously works. At the beginning of the 1980's she �rstcame into contact with the issues involved in the qualitative theory of di�erenceequations, a rapidly developing branch of mathematical analysis. In 1993, shebecame PhD of mathematics upon the thesis on the properties of solutions ofdi�erence equations, under the direction of Professor Jerzy Popenda.

Ewa Schmeidel is the author of approximately �fty scienti�c papers on the qual-itative theory of di�erence equations, published in renowned mathematical jour-nals in Europe, America and Asia, such like Archivum Mathematicum, Advancesin Di�erence Equations, Computers and Mathematics with Applications, Journalof Di�erence Equations and Applications, Mathematical and Computer Modelling,Mathematica Bohemica, Nonlinear Analysis, Rocky Mountain Journal of Math-ematics, and Tatra Mountains Mathematical Publications. In cooperation withengineers from Pozna« University of Technology, researchers on composite mate-rial porosity (metallographic studies), she stated mathematical model of porosityformation in the wall of composite casting while saturating. These results are pub-lished in Archiwum Technologii Maszyn i Automatyzacji journal edited by PolishAcademy of Science. She is a reviewer in many scienti�c journals published both inPoland and in other countries. She is a permanent reviewer of Mathematical Re-views. She has taken part in 35 congresses and scienti�c conferences in Poland andalso in England, China, the Czech Republic, Greece, Germany, Slovakia, Turkey,Hungary and the USA, at many as an invited speaker but also as a member of thescienti�c committee of the conference.

Ewa Schmeidel is widely involved in education, both as a lecturer and as anadvisor for Master's and diploma dissertations. She presently lectures on math-ematical analysis, linear algebra, analytical geometry and di�erential equationsat the Technical Physics Faculty, and on discrete mathematics at the ElectricalFaculty of Pozna« University of Technology.

Ewa Schmeidel is an active member of the Polish Mathematical Society.

4

1 Introduction

Many real world phenomena can be modeled by a recurrence relation. Mathemat-ically, a recurrence relation is an equation which de�nes a sequence recursively,that is, each term of the sequence is de�ned as a function of the preceding terms.Often referred to as di�erence equations, these relations can be used to model awide variety of real world phenomena.

What is especially important in these models is the behavior of the solutionsof di�erence equations for large arguments such as their periodicity, boundedness,stability, oscillation, or convergence to constant which are some of the main topicsof the qualitative theory of di�erence equations. This work is devoted to a rapidlydeveloping branch of this theory. It is an summation the most recent contributionsof the author in this area, which are published in ten papers and will be a stimulusto its further development. All statements in the work are proved.

The main results of this thesis consists of three chapters: 3, 4 and 5.The purpose of Chapter 3 is to establish some necessary and su�cient condi-

tions for the existence of solutions of nonlinear fourth order di�erence equationswithout neutral terms. The Smith-Taylor equations and Thandapani-Arockiasamyequations are considered. Estimations of nonoscillatory solution are established.The necessary and su�cient conditions for existence of minimal and maximal so-lutions are presented. This Chapter collects material connected with �ve papers[a5], [a6], [c10], [c13] and [c17].

Chapter 4 is devoted to higher order di�erence equations with neutral term. Ingeneral, the theory of di�erence equations involves complications with results truefor non-neutral equations being not necessarily true for neutral equations. Theaim of this Chapter is to present some of the asymptotic results that are obtainedfor neutral di�erence equations. The purpose of this Chapter is to establish somenecessary and su�cient or su�cient conditions for the existence of solutions ofthe equation, which can be classi�ed into three distinct categories with each beingnonempty under relatively mild conditions. Particularly, the results publishedfor odd order di�erence equation are presented. The results of this Chapter areestablished in three papers [a4], [c7] and [c8].

Finally, in Chapter 5 we give results for asymptotically periodicity of solutionof Volterra equation. Volterra di�erence equations, whose solution is de�ned bythe whole previous history, are widely used for modeling processes in many �elds.In this Chapter we put together results from two papers [a2] and [c3].

5

2 Motivation and aims of this work

In 1974, Pieter Eykho� de�ned a mathematical model as 'a representation of theessential aspects of an existing system (or a system to be constructed) whichpresents knowledge of that system in usable form'. A mathematical model usuallydescribes a system by a set of variables and a set of equations that establishrelationships between the variables. Mathematical models can take many forms,including but not limited to di�erential equations, or di�erence equations.

Di�erential equations arise in many areas of science and technology, speci�callywhenever a deterministic relation involving some continuously varying quantities(modeled by functions) and their rates of change in space and/or time (expressedas derivatives) is known or postulated. The study of di�erential equations is a wide�eld in pure and applied mathematics, physics, meteorology, and engineering. Allof these disciplines are concerned with the properties of di�erential equations ofvarious types.

There are many examples of sciences modeling by di�erential equations. For ex-ample in model for rolling mills (see [20]). In metalworking, rolling is a metal form-ing process in which metal stock is passed through a pair of rolls. In a rolling mill,the incoming steel strip is rolled to a desired thickness when it is passed througha pair of rollers. The distance between the rollers and, hence, the thickness of therolled strip, can be adjusted by an electric motor. The thickness of the strip is mea-sured by a thickness sensor located downstream of the rollers. Due to the time takesthe strip move from the rollers to the sensor corrective control action is delayed.

Figure 1: Metal rolling system.

The strip after the rollers is assumed tohave thickness x, width w and velocityv. The distance between the rollers andthe thickness sensor is d. Thickness sen-sor measurements are used to control thedistance u between the rollers. The delayτ(t) between the rollers and the thicknesssensor is given by

d :=

t∫t�τ(t)

v(s)ds

By di�erentiating the above expressionwe obtain

v(t)� v(t� τ(t))(1� τ 0(t)) = 0.

6

Assuming that the width of the strip does not change, mass conservation providesthat

x(t)v(t) = constant.

Hence1

x(t)� 1� τ 0(t)x(t� τ(t))

= 0.

With integral action, the controller output is proportional to the amount of timethe error is present. Suppose that u is generated by an integral controller withgain k so that

u(t) = k

t∫0

(x(t� τ(t))� xd)dt

where xd is the desired thickness of the strip. Integral controller tends to respondslowly at �rst, but over a long period of time it tends to eliminate errors. Takingit into account, we obtain the following system of di�erential equations governingcontrolled process operation

τ 0(t) =�x(t� τ(t)) + x(t)

x(t),

x0(t) = k(x(t� τ(t))� xd).

The theory of di�erential equations is closely related to the theory of di�erenceequations, in which the coordinates assume only discrete values, and the relation-ship involves values of the unknown function or functions and values at nearbycoordinates. Many methods to compute numerical solutions of di�erential equa-tions or study the properties of di�erential equations involve approximation of thesolution of a di�erential equation by the solution of a corresponding di�erenceequation.

The theory of di�erence equations, the methods used, and their wide appli-cations occupy a central position in the broad area of mathematical analysis.Note that new approaches in qualitative research of such equations stimulate newapproaches in applications in modern physics and engineering. Di�erence equa-tions manifest themselves as mathematical models describing real life situation inprobability theory, queuing problems, statistical problems, stochastic time series,combinatorial analysis, number theory, geometry, electrical networks, quanta inradiation.

For a basic introduction to the theory and application of di�erence equations,we refer the reader to the works of Agarwal [1], Agarwal, Bohner, Grace andO'Regan [2], Agarwal and Wong [5], Elaydi [10], Halanay and Samuelson [15],

7

Kelley and Peterson [18], Koci¢ and Ladas [19], Mickens [24], Pierre [27], andSedaghat [28].

In biology, for example, the equation

x(n+ 1) = αx(n) +β

1 + xp(n� k), n = 0, 1, . . . ,

where α 2 (0, 1), p, β 2 (0,1), and k is a nonnegative integer is a discrete analogof an equation that has been used to study haematopoiesis (blood cell production).The equation

x(n+ 1) = αx(n) + βe�γx(n�k), n = 0, 1, . . . ,

where α 2 (0, 1), β, γ 2 (0,1), and k is a nonnegative integer has been used todescribe the survival of red blood cells in an animal.

In the following example which can be found in [10], the national income Y (n)of a country can be viewed as being composed of C(n) consumer expenditures forpurchase of consumer goods, I(n) indicted private investments for buying capitalequipment, and G(n) government expenditures, where n is usually measured inyears. So, the national income Y (n) in a given period n may be written as

Y (n) = C(n) + I(n) +G(n).

Here:a) consumer expenditure C(n) is proportional to the national income Y (n� 1) inthe preceding year n� 1, that is

C(n) = αY (n� 1)

where α > 0 is commonly called the marginal propensity to consume;b) inducted private investment I(n) is proportional to the increase in consumptionC(n)� C(n� 1), that is

I(n) = β[C(n)� C(n� 1)]

where β > 0 is called the relation;c) �nally, the government expenditure G(n) is constant over the years

G(n) = γ.

From above we get the second order di�erence equation

Y (n)� α(1 + β)Y (n� 1) + αβY (n� 2) = γ.

The equilibrium state of the national income Y � =γ

1� αis asymptotically stable

(or just stable in the theory of economics) if and only if the following conditionshold:

α < 1, 1 + α + 2β > 0, and αβ < 1.

8

Y (n)

Y (n)Y �

α <4β

(1 + β)2.

Y (n) = A

(1p2

)cos(nπ

4− ω

)+2.

α = 12, β = 1 γ = 1. Y � = 2,

Y � =Y (n)

Y � = 2,Y (0)

Y (1)Y (n) Y (0) = 1 Y (1) = 2.

A = −p

2 ω = π4

Y (n) = −(

1p2

)n−1cos

(n+ 1

4π

)+ 2.

s1 s2

s1 n1 s2n2 M(n)

n ns1 s2 M(n − n1) �

− � − � � � − � � � �,

M(n− n2) −

− − − � − − � � − − � � � −.

If the message ends with s1, the last signal must start at n� n1 (since s1 takes n1

units of time). Hence there are M(n� n1) possible messages to which the last s1may be appended. So, there are M(n� n1) messages of duration n that end withs1. By a similar argument, one may conclude that there are M(n� n2) messagesof duration n that end with s2. Consequently, the total number of messages M(n)of duration n may be given by

M(n) = M(n� n1) +M(n� n2).

If n1 � n2, then the above equation may be written in the form of an n2th-orderequation

M(n+ n1)�M(n+ n1 � n2)�M(n) = 0.

On the other hand, if n2 � n1, then we obtain the n1th-order equation

M(n+ n2)�M(n+ n2 � n1)�M(n) = 0.

An interesting special case is that in which n1 = 1 and n2 = 2. In this case wehave

M(n+ 2)�M(n+ 1)�M(n) = 0,

orM(n+ 2) = M(n+ 1) +M(n),

which is nothing but a Fibonacci sequence (0, 1, 1, 2, 3, 5, 8, . . . ). The solution ofthe above equation, which ful�lls the sensible assumptionM(0) = 0 andM(1) = 1,is given by the formula

M(n) =1p5

(1 +p

5

2

)n

� 1p5

(1�p

5

2

)n

. (1)

In information theory, the capacity C of a channel is de�ned as

C = limn!1

log2M(n)

n.

From (1) we have

C = limn!1

log21p5

n+ lim

n!1

1

nlog2

[(1 +p

5

2

)n

� 1p5

(1�p

5

2

)n]. (2)

Since(

1�p5

2

)� 0.6 < 1, it follows that

(1�p5

2

)n! 0 as n ! 1. Observe that

the �rst term on the right-hand side of (2) goes to zero as n!1. Thus

C = limn!1

1

nlog2

(1 +p

5

2

)n

,

10

C = log2

(1 +p

5

2

)� 0.7.

In the next example (see [18]) we describe the decrease in the mass of radioac-tive substance by a di�erence equation. It is observed that the decrease in themass of radioactive substance over a �xed time period is proportional to the massthat was present at the beginning of the time period. If the half life of radiumis 1600 years, we can �nd a formula for its mass as a function of time. Let m(t)represent the mass of the radium after t years. Then

m(t+ 1)�m(t) = �km(t),

where k is a positive constant. Hence

m(t+ 1) = (1� k)m(t)

for t = 0, 1, 2, . . . . Using iterations we �nd

m(t) = m0(1� k)t where m0 = m(0).

Since the half life is 1600,

m(1600) = m0(1� k)1600 =1

2m0,

so

1� k =

(1

2

)1/1600

,

and we �nally have that

m(t) = m0

(1

2

)1/1600

.

This problem is traditionally solved in calculus and physics textbooks by settingup and integrating the di�erential equationm0(t) = �km(t). However, the solutionpresented here, using a di�erence equation, is somewhat shorter and employs onlyelementary algebra.

In [17], Iavernaro, Mazzia and Trigiante presented a few examples of problemswhere the struggle between the discrete and continuous conception of the physicalworld is more evident and noted that the continuous world is too smooth for theneeds of applications. The authors observed, that it is obvious that a discreteset is strictly contained in a continuous set and then it is an approximation (inthe sense of a part) of it. Nevertheless, things completely change if one considers

11

discrete functions, i.e., functions de�ned on a discrete set of points. The richnessof possible behaviors drastically grows with respect of continuous functions. Theequations describing the motion of an electron, the Verhulst equation and discretewave equation were investigated from this point of view.

Nevertheless, we also have encountered di�erence equations in one or more ofthe following context: the approximation of solutions of equations by Newton'sMethod, the discretization of di�erential equations, the computation of specialfunctions, in combinatorics, and the discrete modeling of economical or biologicalphenomena.

Some examples which demonstrate applications of di�erence equations to theasymptotic theory of di�erential equations with unbounded lag can be found in [9].The closed connection between both types of equations can be observed especiallyin the oscillation and asymptotic theory.

Di�erence equations are mathematically studied from several di�erent perspec-tives, mostly concerned with their solutions�the set of functions that satisfy theequation. Only the simplest di�erence equations admit solutions given by explicitformulas. However, some properties of solutions of a given di�erence equation maybe determined without �nding their exact form.

For the applications of di�erence equations is important to study the asymp-totic behavior of solutions of the investigate equation. The objectives of this workare following:

1) To �nd the necessary and su�cient conditions for existence of asymptoticallyconstant solution of investigate equations. These conditions are essential forstability of solutions.

2) To present the estimation of solutions eg. minimal and maximal solutions, andsu�cient conditions for existence of such types of solutions.

3) To study conditions under which the equation has asymptotically periodic so-lution.

4) To investigate conditions which guarantee that the equation has bounded orunbounded solution.

5) To give the classi�cation of nonoscillatory solutions of the equation.

12

3 Fourth order di�erence equations

In the last few years there has been an increasing interest in the study of oscillatoryand asymptotic behavior of solutions of di�erence equations. Fourth order lineardi�erence equations were investigated by Smith and Taylor in [29], [30] and [36].The fourth order di�erence equation was studied in many others by Cheng [8],Graef and Thandapani [14], Hooker and Patula [16], Liu and Yan [22], Schmeideland Szmanda [c5], Thandapani and Arockiasamy [34], Taylor and Sun [37], andZhang and Cheng [38].

We study fourth order nonlinear di�erence equation of the form

∆(a(n)∆(b(n)∆(c(n)∆y(n)))) = f(n, y(n)) , n 2 N (3)

where (a(n)), (b(n)) and (c(n)) are sequences of positive real numbers, and functionf : N� R! R.

Let the series1∑i=1

1

a(i),1∑i=1

1

b(i)and

1∑i=1

1

c(i)be divergent. Then this equation

is called Smith-Taylor equation.

Let the series1∑i=1

1

a(i)and

1∑i=1

1

c(i)be divergent, and series

1∑i=1

1

b(i)be conver-

gent. Then this equation (3) is called Thandapani-Arockiasamy equation, sincethese authors studied the equation under the above assumptions.

3.1 Smith-Taylor equation

We present the estimation of nonoscillatory solutions of equation (3) by minimaland maximal solutions. Next, we establish some necessary and su�cient conditionsfor the existence of such kinds of solutions of equation (3).

One or several of the following assumptions will be imposed:

(H1)1∑i=1

1

a(i)=1∑i=1

1

b(i)=1.

(H2) yf(n, y) > 0 for all y 6= 0 and n 2 N.(H3) Function f(n, y) is continuous in second argument,

for each �xed n 2 N.(H4) Function f(n, y) is decreasing in y, for each n 2 N.(H5) Sequence (a(n)) is a nondecreasing sequence .(H6) Sequence (a(n)) ful�ll condition a(1) � 1.(H7) There exist constants K1 and K2,

such that 0 < K1 6 c(n) 6 K2, for each n 2 N.

13

Now we introduce the notation

R(n,N) =n�1∑i=N

1

c(i)

i�1∑j=N

j �Nb(j)

, Q(n,N) =n�1∑k=N

1

c(k)

k�1∑j=N

1

b(j)

j�1∑i=N

1

a(i).

Lemma 3.1. Assume conditions (H1), (H2) and (H7) hold. If (y(n)) is aneventually positive solution of equation (3), then there exist positive constant C1

and C2 and integer N such that

C1 � y(n) � C2Q(n,N)

for enough large n.

We say that a nonoscillatory solution (y(n)) of equation (3) is asymptoticallyconstant if there exists some positive constant α such that yn ! α and asymptot-

ically Q(n,N) if there is some positive constant β such thaty(n)

Q(n,N)! β.

We may regard an asymptotically constant solution as a minimal solution, andan asymptotically Q(n,N) solution as a maximal solution.

Now, we derive a necessary and su�cient condition for the existence of anasymptotically constant solution of equation (3).

Theorem 3.1. Assume that (H1), (H2), (H3) hold and function f is monotonicfunction in second argument. Then a necessary and su�cient condition for equa-tion (3) to have a solution (y(n)) which satis�es

limn!1

y(n) = α 6= 0

is that1∑i=1

P (i, N)jf(i, c)j <1,

for some integer N � 1 and some nonzero constant c.

Next, we present a necessary and su�cient condition for the existence of anasymptotically Q(n,N) solution.

Theorem 3.2. Assume that (H1), (H2), (H3) hold and f is a nondecreasingfunction in second argument. Then a necessary and su�cient condition for equa-tion (3) to have a solution (y(n)) satisfying

limn!1

y(n)

Q(n,N)= β 6= 0

is that1∑n=1

jf(n,CQ(n,N))j <1

for some integer N � 1 and some nonzero constant C.

14

3.2 Thandapani-Arockiasamy equation

In this section the classi�cation of nonoscillatory solutions of equation (3) is given.There are obtained su�cient conditions for the above equation to admit the exis-tence of nonoscillatory solutions with special asymptotic properties.

We consider the Thandapani-Arockiasamy equation. One or several of thefollowing assumptions will be imposed:

(H1) a(n) � c(n), for all large n, and a(n) is bounded away from zero.

(H2)1∑i=1

1

a(i)=1.

(H3)1∑j=1

1

c(j)

1∑i=j

1

b(i)<1.

(H4) Function f(n, y) is continuous on R for each �xed n 2 N.(H5) yf(n, y) > 0 for all y 6= 0 and n 2 N.(H6) For each n 2 N, f(n, x) is monotonic in x.

Set ρ(n) =1∑j=n

1

c(j)

1∑i=j

1

b(i), µ(n,N) =

n�1∑i=N

1

c(i), and ν(n,N) =

1

a(n)

n�1∑i=N

1

c(i).

Lemma 3.2. Assume that (H1), (H2) (H3) and (H5) hold. Let (y(n)) be aneventually positive solution of equation (3), then one of the following four casesholds:

(I) c(n)∆y(n) > 0, b(n)∆(c(n)∆y(n)) > 0, a(n)∆(b(n)∆(c(n)∆y(n))) > 0,(II) c(n)∆y(n) > 0, b(n)∆(c(n)∆y(n)) < 0, a(n)∆(b(n)∆(c(n)∆y(n))) > 0,(III) c(n)∆y(n) > 0, b(n)∆(c(n)∆y(n)) < 0, a(n)∆(b(n)∆(c(n)∆y(n))) < 0,(IV ) c(n)∆y(n) < 0, b(n)∆(c(n)∆y(n)) > 0, a(n)∆(b(n)∆(c(n)∆y(n))) > 0,

for large n.

Lemma 3.3. Assume that (H1), (H2), (H3) and (H5) hold. Let (y(n)) be aneventually positive solution of equation (3). Then there exist positive constants k1and k2 such that

k1ρ(n) � y(n) � k2µ(n,N), for large n.

We say that a nonoscillatory solution (y(n)) of equation (3) is an asymptot-

ically ρ(n) if there exists some nonzero constant α such thaty(n)

ρ(n)! α and an

asymptotically µ(n,N) if there is some nonzero constant β such thaty(n)

µ(n,N)! β.

15

According to Lemma 3.3 we may regard asymptotically ρ(n) solution as aminimal solution, and asymptotically µ(n,N) solution as a maximal solution.

In the following theorem a su�cient condition under which there exists anasymptotically ρ(n) solution of equation (3) are presented.

Theorem 3.3. Assume that (H1)-(H6) hold and

1∑n=1

µ(n+ 1, N)jf(n, cρ(n))j <1

for some nonzero constant c, then there exists a solution (y(n)) of equation (3)such that

limn!1

y(n)

ρ(n)= d 6= 0.

Next, we present a su�cient condition for the existence of an asymptoticallyµ(n,N) solution.

Theorem 3.4. Assume that conditions (H1)-(H6) hold and

1∑n=1

ρ(n+ 1)jf(n, cµ(n,N))j <1

for some nonzero constant c, then there exists a solution (y(n)) of equation (3)such that

limn!1

y(n)

µ(n,N)= d 6= 0.

4 Neutral di�erence equations

We consider the higher order neutral type di�erence equation with delays

∆(a(k�1)(n)∆(a(k�2)(n)...∆(a(1)(n)∆(y(n)+p(n)y(n� τ)))))+f(n, y(n� σ)) = 0,(4)

where k is a positive integer and n 2 fη + 1, η + 2, ...g, and τ , σ are a positiveintegers η = max(τ, σ). Here (a(i)(n)), i = 1, 2, 3, ...k� 1 are sequences of positivereal numbers, and (p(n)) is a sequence of real numbers. Function f : N� R! R.

In recent years, there has been an increasing interest in the study of the os-cillatory and asymptotic behavior of solutions of neutral higher order di�erenceequations, (see for example by Agarwal, Grace and O'Regan [3], Li and Cheng [21],Migda and Migda [23] and Thandapani, Sundaram, Graef, Miciano and Spikes [35])and the references cited therein.

16

Denote the quasidi�erences Li(x(n)) where i = 0, 1, 2, 3, ..., k of a sequence(x(n)) as follows L0(x(n)) = x(n), Li(x(n)) = a(i)(n)∆(Li�1(x(n))), for i =1, 2, 3, ..., k. Assume that a(k)(n) � 1. We rewrite equation (4) in the followingform

Lk(z(n)) + f(n, y(n� σ)) = 0, n 2 N, (5)

where (z(n)) is companion sequence of a sequence (y(n))1n=�τ relative to (p(n)),de�ned by

z(n) = y(n) + p(n)y(n� τ). (6)

We use the following notation

S(n)(f(j, �)) =1

a(1)(n)

n�1∑i2=N

1

a(2)(i2)

i2�1∑i3=N

...1

a(k�1)(ik�1)

ik−1�1∑j=N

f(j, �),

and

S(n) =n�1∑i1=N

1

a(1)(i1)

i1�1∑i2=N

1

a(2)(i2)

i2�1∑i3=N

1

a(3)(i3)...

ik−2�1∑ik−1=N

1

a(k�1)(ik�1).

In this Chapter we assume that1∑j=1

1

a(i)(j)= 1, where i = 1, 2, 3, ..., k � 1, and

yf(n, y) > 0 for all y 6= 0 and n 2 N. The next Lemma gives us the classi�cationof nonoscillatory solutions of equation (5).

Lemma 4.1. Let (y(n)) be a positive sequence and (z(n)) its companion sequencesuch that Lk(z(n)) < 0, for su�ciently large n, where k is a positive integer. Thenexactly one of the following statements holds:

(i) Li(z(n)) > 0, for 0 6 i 6 k � 1,(ii) there exists a positive integer l such that

Li(z(n)) > 0, for 0 6 i < l,andLi(z(n))Li+1(z(n)) < 0, for l 6 i < k � 1,

(iii) Li(z(n))Li+1(z(n)) < 0, for 0 6 i 6 k � 1,(iv) there exists a positive integer l such that

Li(z(n)) < 0, for 0 6 i < l,andLi(z(n))Li+1(z(n)) < 0, for l 6 i 6 k � 1,

(v) Li(z(n)) < 0, for 0 6 i < k.

For k an even positive integer and for k an odd positive integer it is clear thatthe following corollaries hold.

17

Corollary 4.1. In Lemma 4.1 assume that

k is an even positive integer . (7)

Hence the hypothesis of this Lemma holds when the integer l in Class (ii) is aneven positive integer and integer l in Class (iv) is an odd positive integer.

Depending on the sequence (p(n)), the nonoscillatory solutions of equation (5)may belong to two, three or all the above Classes, and the following Lemma holds.

Lemma 4.2. Assume that (7) holds. Let (y(n)) be a positive solution of equa-tion (5) and (z(n)) its companion sequence. If p(n) > 0, for su�ciently large n,then (z(n)) belongs to Class (i) or (ii). If �1 6 p(n) < 0, for su�ciently large n,then (z(n)) belongs to Class (i), (ii) or (iii). If p(n) < �1, for su�ciently largen, then (z(n)) belongs to Class (i) - (v).

Assume thatlimn!1

p(n) = p 2 (�1, 0), (8)

and (7) holds. Under the assumption (8) the Lemma 4.2 states there are ex-actly three types of eventually positive solutions of the equation (5). We say thatnonoscillatory solution (y(n)) of equation (5) is asymptotically zero solution iflimn!1

y(n) = 0, asymptotically constant if there exists some nonzero constant α

such that limn!1

y(n) = α and asymptotically S(n) solution if there exists some

nonzero constant β such that limn!1

y(n)

S(n)= β. From lim

n!1S(n) =1 we have S(n)

solution tends to in�nity.It is of interest to �nd su�cient conditions for the existence of each type of

eventually nonoscillatory solutions. First, we present a su�cient condition for theexistence a solution of equation (5) which tends to zero.

Theorem 4.1. Assume that conditions (7) and (8) hold, and function f(n, x) is

nondecreasing in x 2 (0,1). Let n1∑i=n

S(i)

(f(j,

1

j � σ)

)<1, then equation (5)

has an eventually positive solution (y(n)) which converges to zero.

Next we present a necessary and su�cient condition for the existence of anasymptotically constant solution of equation (5).

Theorem 4.2. Assume that conditions (7) and (8) hold, and function f is contin-uous and monotonic in the second argument. A necessary and su�cient conditionfor equation (5) to have solution (y(n)) which satis�es lim

n!1y(n) = α 6= 0 is that

1∑i=1

S(i)(jf(j, c)j) <1, for some nonzero constant c.

18

Now, we present a necessary and su�cient condition for the existence a solutionof equation (5) which diverges to positive in�nity.

Theorem 4.3. Assume that conditions (7) and (8) hold, and f is a monotonic

function in the second argument. If1∑i=1

S(i)(jf(j, CS(j � σ))j) < 1, for some

nonzero constant C, then equation (5) has solution (y(n)) such that limn!1

y(n)

S(n)=

β 6= 0.

Theorems 4.1, 4.2 and 4.3 improve and generalize Theorems 1, 2, and 3 pre-sented by Li and Cheng in [21]. To obtain the results presented therein we shouldput a(i)(n) � 1, i = 1, 2, 3, ..., k � 1 in equation (5).

5 Volterra di�erence equations

Uniform asymptotic stability in linear Volterra di�erence equations was studied byElaydi and Murakami in [11]. A su�cient condition for the existence of a boundedsolution of a class of Volterra di�erence equation was presented by Morchaªo andSzma«da in [25]. Periodic and asymptotically periodic solutions of linear di�erenceequations were investigated, e.g., by Agarwal and Popenda in [4], and by Popendaand Schmeidel in [c25], [c27]. Boundedness, attractivity, and convergence of solu-tions were investigated. Convergence rates for solutions of the Volterra equationand su�cient conditions for the asymptotic constancy of solutions are given in [6].

5.1 Scalar Case

In [c3], we consider a Volterra di�erence equation

x(n+ 1) = a(n) + b(n)x(n) +n∑i=0

K(n, i)x(i), (9)

where n 2 N := f0, 1, 2, . . . g, a, b, x : N! R, K : N� N! R.Let ω be a positive integer and b : N! R n f0g be ω-periodic. Then we de�ne

an ω-periodic function β : N! R as

β(n) =

n�1∏j=0

1

b(j)if n � 1,

β(ω) if n = 0.

Further we de�nem := min fjβ(1)j , jβ(2)j , . . . , jβ(ω)jg

19

andM := max fjβ(1)j , jβ(2)j , . . . , jβ(ω)jg .

Theorem 5.1. Let ω be a positive integer and b : N ! R n f0g be ω-periodic.

Assume thatω�1∏i=0

b(i) = 1,1∑i=0

ja(i)j < 1 and1∑j=0

j∑i=0

jK(j, i)j < mM. Then, for any

nonzero constant c, there exists an asymptotically ω-periodic solution x of (9) suchthat

x(n) = u(n) + v(n), n 2 N (10)

with u(n) := cn∗∏k=0

b(k) and limn!1

v(n) = 0 where n� is the remainder of dividing

n� 1 by ω.

Remark 5.1. Tracing the proof of Theorem 5.1 we see that it remains valid evenin the case of c = 0. Then there exists an asymptotically ω-periodic solution xof (9) as well. The formula (10) reduces to x(n) = v(n) = o(1), n 2 N. We canconsider this case as follows. We set (as a singular case) u � 0 with an arbitrary(possibly other than ω) period and with v = o(1) for n!1.

Now, we present su�cient conditions for the nonexistence of asymptoticallyperiodic solution of (9) satisfying some auxiliary conditions. Let x(n) = u(n)+v(n)be an asymptotically periodic solution of (9) such that the sequence u is ω-periodicand lim

n!1v(n) = 0.

Theorem 5.2. If sequences a : N ! R and b : N ! R are bounded and thereexists a positive integer ω such that K(n, i) = K(n + ω, i + ω) for all n, i 2 N,then the equation (9) does not have any asymptotically ω-periodic solution x(n) =u(n) + v(n) such that

ω�1∑i=0

K(ω � 1, i)u(i) 6= 0 (11)

and1∑i=0

jv(i)j <1.

Remark 5.2. We will emphasize the necessity of (11) in Theorem 5.2. If

ω�1∑i=0

K(ω � 1, i)u(i) = 0

then (9) can have an asymptotically ω-periodic solution. Let, e.g., K(j, i) =(1 + (�1)i)/2. Then, taking sequences a and b in (9) in a proper manner, theequation (9) will have an asymptotically 4-periodic solution x(n) = u(n) + v(n)

20

with 4-periodic function u(n) := (0, 1, 0, 2, . . . ). In this caseω�1∑i=0

K(ω � 1, i)u(i) =

ω�1∑i=0

1 + (�1)i

2u(i) = 1 � 0 + 0 � 1 + 1 � 0 + 0 � 2 = 0 and (11) does not hold. Then

lim supn!1

[nω

]�ω�1∑i=0

K(ω � 1, i)u(i) = 0 and we do not get the �nal contradiction in

the proof of Theorem 5.2.

5.2 Systems of Volterra equations

In [a2] we consider a Volterra system of di�erence equations

xs(n+ 1) = as(n) + bs(n)xs(n) +n∑i=0

Ks1(n, i)x1(i) +n∑i=0

Ks2(n, i)x2(i) (12)

where n 2 f0, 1, 2, . . . g, as, bs, xs : N! R, Ksp : N� N! R, s = 1, 2.

ω�1∏i=0

bs(i) = 1. (13)

Then we de�ne an ω-periodic function γs : N! R as

γs(n) =

1 if n = 0,

n�1∏j=0

1

bs(j)if n � 1.

We assume that series1∑i=0

jas(i)j ,1∑j=0

j∑i=0

jKsp(j, i)j , s, p = 1, 2, are convergent and

denote

As =1∑i=0

jas(i)j , Ksp =1∑j=0

j∑i=0

jKsp(j, i)j , s, p = 1, 2.

Let us de�ne determinants

∆ =1

m2

∣∣∣∣m�K11M K12MK21M m�K22M

∣∣∣∣ ,∆1 = ∆1(c1, c2) =

1

m2

∣∣∣∣MA1m+K11Mc1 +K12Mc2 K12MMA2m+K21Mc1 +K22Mc2 m�K22M

∣∣∣∣ ,21

and

∆2 = ∆2(c1, c2) =1

m2

∣∣∣∣m�K11M MA1m+K11Mc1 +K12Mc2K21M MA2m+K21Mc1 +K22Mc2

∣∣∣∣ ,where cs 2 R, s = 1, 2.

Theorem 5.3. Let ω be a positive integer, bs : N! Rnf0g, s = 1, 2, be ω-periodic,and let (13) hold. Assume that

Ksp <m

M,

where s, p = 1, 2, and ∆ 6= 0. Then, for any constant vector c = (c1, c2) withc(i) 6= 0, i = 1, 2 such that ∆1(c1, c2)/∆ > 0, ∆2(c1, c2)/∆ > 0, there exists anasymptotically ω-periodic solution x of (12) such that

x(n) = u(n) + v(n), n 2 N, (14)

with us(n) := csn∗∏k=0

bs(k) and limn!1

v(n) = 0, where s = 1, 2 and n� is the remainder

of dividing n� 1 by ω.

Remark 5.3. Tracing the proof of Theorem 5.3 we see that it remains valid evenin the case of c1c2 = 0. We consider the case c = 0. Then there exists anasymptotically ω-periodic solution x of (12) as well. The formula (14) reduces toxs(n) = vs(n) = o(1), s = 1, 2, n 2 N. We can consider this case as follows. Weset (as a singular case) u � 0 with an arbitrary (possibly other than ω) period andwith vs = o(1) for s = 1, 2, n ! 1. Remaining cases c1 = 0, c2 6= 0 or c1 6= 0,c2 = 0 can be treated similarly.

Finally, we present su�cient conditions for the nonexistence of asymptoticallyperiodic solution of (12) satisfying some auxiliary conditions.

Let x(n) = u(n) + v(n) be an asymptotically periodic solution of (12) suchthat the sequence u is ω-periodic and lim

n!1v(n) = 0.

Theorem 5.4. If sequences as : N! R and bs : N! R, s = 1, 2 are bounded andthere exists a positive integer ω such that Ksp(n, i) = Ksp(n+ω, i+ω), s, p = 1, 2,for all n, i 2 N, then the equation (12) does not have any asymptotically ω-periodicsolution x(n) = u(n) + v(n) such that

ω�1∑i=0

Ksp(ω � 1, i)up(i) 6= 0, s, p = 1, 2, (15)

and1∑i=0

jvs(i)j <1, s = 1, 2.

For s = 1, the system (12) reduces to one Volterra equation and in this caseresults given in [c3] are a particular case of Theorem 5.3 and Theorem 5.4.

22

References

[1] R.P. Agarwal, Di�erence Equations and Inequalities. Theory, Methods, and

Applications, Second edition, Monographs and Textbooks in Pure and Ap-plied Mathematics, Marcel Dekker, Inc., New York, 2000.

[2] R.P. Agarwal, M. Bohner, S.R. Grace and D. O'Regan, Discrete OscillationTheory, Contemporary Mathematics and Its Applications, Hindawi PublishingCorporation, New York, 2005.

[3] R.P. Agarwal, S.R. Grace and D. O'Regan, Oscillation criteria of comparisontype for higher order order di�erence equations, Nonlinear Anal. Forum, 7,(1) (2002), 73�91.

[4] R.P. Agarwal, J. Popenda, Periodic solutions of �rst order linear di�erenceequations, Math. Comput. Modelling, 22 (1995), 11�19.

[5] R.P. Agarwal and P.J.Y. Wong, Advanced Topics in Di�erence Equations,Kluwer, Dordrecht, 1997.

[6] J. Appleby, I. Györi, D. Reynolds, On exact convergence rates for solutionsof linear systems of Volterra di�erence equations, J.Di�erence Equ. Appl., 12(2006), 1257�1275.

[7] B. Aulbach, S. Elaydi, K. Ziegler, Asymptotic solutions of a discreteSchrödinger equation arising from a Dirac equation with random mass, NewProgress in Di�erence Equations, edited by B. Aulbach, S. Elaydi, G. Ladas,CRC, Boca Raton, FL, (2004), 349�358.

[8] S.S. Cheng, On a class of fourth order linear recurrence equations, Internat.J.Math. and Math. Sci., 7, (1) (1984), 131�149.

[9] J. �ermák, Di�erence equations in the qualitative theory of delay di�erentialequations, New Progress in Di�erence Equations, edited by B. Aulbach, S.Elaydi, G. Ladas, CRC, Boca Raton, FL, (2004), 391�398.

[10] S.N. Elaydi, An Introduction to Di�erence Equations, Third edition. Under-graduate Texts in Mathematics, Springer, New York, 2005.

[11] S.N. Elaydi, S. Murakami, Uniform asymptotic stability in linear Volterradi�erence equations, J.Di�erence. Equ. Appl., 3 (1998), 203�218.

[12] T. Furumochi, Periodic solutions of Volterra di�erence equations and attrac-tivity, Nonlinear Anal., 47 (2001), 4013�4024.

23

[13] T. Furumochi, Asymptotically periodic solutions of Volterra di�erence equa-tions, Vietnam J.Math., 30 (2002), 537�550.

[14] J.R. Graef, E. Thandapani, Oscillatory and asymptotic behavior of fourthorder nonlinear delay di�erence equations, Fasc. Math., 31 (2001), 23�36.

[15] A. Halanay and J. Samuelson, Di�erential Equations, Discrete Systems andControl: Economic Models, Mathematical Modelling: Theory and Applica-tions Vol. 3, Kluwer Academic Publishers, Dordrecht, 1997.

[16] J.W. Hooker, W.T. Patula, Growth and oscillation properties of solutions of afourth order linear di�erence equation, J.Aust. Math. Soc., Ser. B 26 (1985),310�328.

[17] F. Iavernaro, F. Mazzia, D. Trigiante, On the discrete nature of physical laws,New Progress in Di�erence Equations, edited by B. Aulbach, S. Elaydi, G.Ladas, CRC, Boca Raton, FL, (2004), 35�48.

[18] W.G. Kelly, A.C. Peterson, Di�erence Equations, Academic Press, Inc.,Boston-San Diego 1991.

[19] V.L. Koci¢, G. Ladas, Global Behavior of Nonlinear Di�erence Equations ofHigher Order with Applications, Mathematics and its Applications, KluwerAcademic Publishers Group, Dordrecht, 1993.

[20] V. Kolmanovskii, A. Myshkis, Introduction to the Theory and Applications ofFunctional Di�erential Equations, Mathematics and its Applications, KluwerAcademic Publishers Group, Dordrecht, 1999.

[21] W.T. Li, S.S. Cheng, Asymptotic trichotomy for positive solutions of a classof odd order nonlinear neutral di�erence equations, Comput. Math. Appl., 35,(8) (1998), 101�108.

[22] B. Liu, J. Yan, Oscillatory and asymptotic behavior of fourth order nonlineardi�erence equations, Acta Math. Sin., 13 (1) (1997), 105�115.

[23] M. Migda, J. Migda, Asymptotic properties of solutions of second order neu-tral di�erence equations, Nonlinear Anal., 63 (2005), e789�e799 .

[24] R.F. Mickens, Di�erence Equations: Theory and Applications, Van NostrandReinhold Co. New York, 1990.

[25] J. Morchaªo, A. Szma«da, Asymptotic properties of solutions of some Volterradi�erence equations and second-order di�erence equations, Nonlinear Anal.,63 (2005), 801�811.

24

[26] J. Musielak, Wstep do Analizy Funkcjonalnej, PWN, Warszawa 1976 (in Pol-ish).

[27] N.V. Pierre, Dynamical Systems: An Introduction with Applications in Econ-omy and Biology, Springer, Berlin, 1994.

[28] H. Sedaghat, Nonlinear Di�erence Equations: Theory with Applications to

Social Science Models, Kluwer Academic Publishers, Dordrecht, 2003.

[29] B. Smith, W.E. Taylor, Oscillatory and asymptotic behavior of certain fourthorder di�erence equations, Rocky Mountain J.Math., 16, (2) (1986), 403�406.

[30] B. Smith, W.E. Taylor, Oscillation properties of fourth order linear di�erenceequations, Tamkang J.Math., 18, (4) (1987), 89�95.

[31] B. Szmanda, Oscillatory and asymptotic behaviour of higher order di�erenceequations, Matematiche (Catania), 70, (1) (1997), 171�178.

[32] E. Thandapani, M. Arockiasamy, Some oscillation and non-oscillation theo-rems for fourth order di�erence equations, Z. Anal. Andendungen, 19 (2000),863�872.

[33] E. Thandapani, M. Arockiasamy, Fourth-order nonlinear oscillations of dif-ference equations, Comput. Math. Appl., 42 (2001), 357�368.

[34] E. Thandapani, M. Arockiasamy, Oscillatory and asymptotic properties ofsolution of nonlinear fourth order di�erence equations, Glas. Mat., 37, (37)(2002), 119�131.

[35] E. Thandapani, P. Sundaram, J.R. Graef, A .Miciano, P.W. Spikes, Classi�-cation of nonoscillatory solutions of higher neutral type di�erence equations,Arch. Math., 31, (4) (1995), 263�277.

[36] W.E. Taylor, Oscillation properties of fourth order di�erence equations, Port.Math., 45, (1) (1988), 105�144.

[37] W.E. Taylor, M. Sun, Oscillation properties of nonlinear di�erence equations,Port. Math., 52, (1) (1995), 15�24.

[38] B.G. Zhang, S.S. Cheng, On a class of nonlinear di�erence equations,J.Di�erence Equ. Appl., 1 (1995), 391�411.

25

Author publications list

a) Original scienti�c paper published in a scienti�c journal with impactfactor (IF) bigger than 0,500.

[a1] J. Diblík, M. R·ºi£ková, E. Schmeidel, Asymptotically periodic solutions ofVolterra systems of di�erence equations, Comput. Math. Appl., 59 (2010),2854�2867.

[a2] J. Diblík, M. R·ºi£ková, E. Schmeidel, Existence of asymptotically periodicsolutions of Volterra system di�erence equations, J.Di�erence Equ. Appl.,15 (2009), 1165�1177.

[a3] J. Diblík, M. Migda, E. Schmeidel, A version of retract method for discreteequations, Nonlinear Anal., 65, (4) (2006), 845�853.

[a4] E. Schmeidel, Asymptotic trichotomy for positive solutions of a class of fourthorder nonlinear neutral di�erence equations with quasidi�erences, NonlinearAnal., 63 (2005), e899�e907.

[a5] M. Migda, A. Musielak, E. Schmeidel, On a class of fourth order nonlineardi�erence equations, Adv. Di�erence Equ., 1 (2004), 23�36.

[a6] M. Migda, E. Schmeidel, Asymptotic properties of fourth order nonlineardi�erence equations, Math. Comput. Modelling, 39 (2004), 1203�1211.

[a7] E. Schmeidel, Nonoscillation and oscillation theorems for a fourth order non-linear di�erence equations, Fields Inst. Commun., 42 (2004), 313�321.

[a8] E. Schmeidel, B. Szmanda, Oscillatory and asymptotic behavior of certaindi�erence equation, Nonlinear Anal., 47 (2001), 4731�4742.

[a9] J. Popenda, E. Schmeidel, On the asymptotic behaviour of solutions of integervalued di�erence equations, Nonlinear Anal., 30 (2) (1997), 1119�1124.

b) Original scienti�c paper in a scienti�c journal with IF 0,100-0,500

[b1] E. Schmeidel, Oscillation and nonoscillation theorems for fourth order di�er-ence equations, Rocky Mountain J.Math., 33 (3) (2003), 1083�1094.

[b2] J. Popenda, E. Schmeidel, On the asymptotic behavior of solutions of nonho-mogeneous linear di�erence equations, Indian J. Pure Appl. Math., 28 (3)(1997), 319�327.

26

c) Original scienti�c paper in a scienti�c journal with IF lower than0,100 or in a scienti�c journal without IF

[c1] E. Schmeidel, Oscillation of nonlinear third-dimensional di�erence systemswith delays, Math. Bohem., 135 (2010), 163�170.

[c2] E. Schmeidel, Boundedness of solutions of nonlinear three-dimensional di�er-ence systems with delays, Fasc. Math., 44 (2010), 107�113.

[c3] J. Diblík, M. R·ºi£ková, E. Schmeidel, Asymptotically periodic solutions ofVolterra di�erence equations, Tatra Mt. Math. Publ., 43 (2009), 43�61.

[c4] J. Elyseeva, E. Schmeidel, Generalized Kiguradze's lemma on time scaleswith application for oscillation of higher order nonlinear dynamic equations,Fundamental'nye Fiziko-Matematiceskie Problemy i Modelirovanie Tehniko-Tehnologiceskih Sistem, 11 (2008), 18�21.

[c5] K. Janglajew, E. Schmeidel, Periodic and asymptotically periodic solutionsof linear di�erence equations, Ru�ng, A. (ed.) et al., Ru�ng, A. (ed.) etal.,Communications of the Laufen colloquium on science, Laufen, Austria,April 1�5, 2007. Aachen: Shaker. Berichte aus der Mathematik, 14 (2007),1�11.

[c6] J. Diblík, M. R·ºi£ková, E. Schmeidel, Asymptotically periodic solutions ofVolterra di�erence equations, Ru�ng, A. (ed.) et al.,Communications of theLaufen colloquium on science, Laufen, Austria, April 1�5, 2007. Aachen:Shaker. Berichte aus der Mathematik, 5 (2007), 1�12.

[c7] E. Schmeidel, Asymptotic trichotomy of solutions of a class of even ordernonlinear neutral di�erence equations with quasidi�erences, Proceedings ofthe International Conference on Di�erence Equations, Special Functions and

Orthogonal Polynomials, World Scienti�c Publishing Co, (2007), 600�609.

[c8] E. Schmeidel, J. Schmeidel, Asymptotic behavior of solutions of a class offourth order nonlinear neutral di�erence equations with quasidi�erences,Tatra Mt. Math. Publ., 39 (2007), 243�254.

[c9] E. Schmeidel, Oscillatory and asymptotically zero solutions of third orderdi�erence equations with quasidi�erences, Opuscula Math., 26, (2) (2006),361�369.

[c10] M. Migda, A. Musielak, E. Schmeidel, Oscillatory of fourth order nonlineardi�erence equations with quasidi�erences, Opuscula Math., 26, (2) (2006),371�380.

27

[c11] J. Mikoªajski, E. Schmeidel, Comparison of properties of solutions of dif-ferential equations and recurrence equations with the same characteristicequation, Opuscula Math., 26, (2) (2006), 343-349.

[c12] M. Migda, E. Schmeidel, M. Zb�aszyniak, On the existence of solutions ofsome second order nonlinear di�erence equations, Arch. Math., 42 (2005),379�388.

[c13] M. Migda, A. Musielak, E. Schmeidel, Oscillation theorems for a class offourth order nonlinear di�erence equations, Proceedings of the Eight Inter-national Conference on Di�erence Equations and Applications, edited by S.Elaydi, G. Ladas, B. Aulbach, O. Doslý, CRC, Boca Raton, FL, (2005),201�208.

[c14] E. Schmeidel, Asymptotic behavior of certain second order nonlinear di�er-ence equations, Proceedings of the Eight International Conference on Di�er-ence Equations and Applications, edited by S. Elaydi, G. Ladas, B. Aulbach,O. Doslý, CRC, Boca Raton, FL, (2005), 245�252.

[c15] E. Schmeidel, Nonscillation and oscillation properties for fourth order non-linear di�erence equations, New Progress in Di�erence Equations, edited byB. Aulbach, S. Elaydi, G. Ladas, CRC, Boca Raton, FL, (2004), 531�538.

[c16] J. Jackowski, E. Schmeidel, G. Schmeidel, Komputerowy program progno-zowania porowato±ci w metalowych odlewach kompozytowych z nasycanymzbrojeniem, Archiwum Odlewnictwa, 4 (11) (2004), 200�208 (in Polish).

[c17] E. Schmeidel, Nonoscillation properties of certain fourth order nonlineardi�erence equations, Funct. Di�er. Equ., 11 (1-2) (2004), 141�146.

[c18] M. Migda, E. Schmeidel, M. Zb�aszyniak, Some properties of solutions ofsecond order nonlinear di�erence equations, Funct. Di�er. Equ., 11 (1-2)(2004), 147�152.

[c19] J. Jackowski, E. Schmeidel, Model kszta³towania sie porowato±ci w nasy-canych odlewach kompozytowych, Archiwum Technologii Maszyn i Automatyza-cji, 23 (1) (2003), 71�83 (in Polish).

[c20] E. Schmeidel, On the asymptotic behavior of solutions of second order non-linear di�erence equations, Folia Fac. Sci. Natur. Univ. Masaryk. Brun.Math., 13 (2003), 287�293.

[c21] A. Drozdowicz, M. Migda, E. Schmeidel, Nonoscillation results for somethird order nonlinear di�erence equation, Folia Fac. Sci. Natur. Univ.Masaryk. Brun. Math., 13 (2003), 185�192.

28

[c22] E. Magnucka-Blandzi, M. Migda, E. Schmeidel, Mathematical works ofJerzy Popenda, Appl. Math. E-Notes, 2 (2002), 155�170.

[c23] M. Migda, E. Schmeidel, On the asymptotic behaviour of solutions of non-linear di�erence equations, Fasc. Math., 31 (2001), 63�69.

[c24] M. Migda, E. Schmeidel, Some properties of solutions of a class of nonlineardi�erence equations, Proceedings of the International Conference on Di�er-ential Equations-Berlin 1999, World Scienti�c, Singapore (2000), 1476�1478.

[c25] J. Popenda, E. Schmeidel, Asymptotically periodic solution of some lineardi�erence equations, Facta Univ. Ser. Math. Inform., 14 (1999), 31�40.

[c26] M. Migda, E. Schmeidel, Some properties of solutions of a class of nonlineardi�erence equations, Acta Univ. Palack. Olomuc. Fac. Rerum Natur.Math., 38 (1999), 131�137.

[c27] J. Popenda, E. Schmeidel, On the asymptotically periodic solution of somelinear di�erence equations, Arch. Math., 35, (1) (1999), 13�19.

[c28] M. Migda, E. Schmeidel, Asymptotic behavior of solutions of higher orderdi�erence equations, Demonstratio Math., 31 (1) (1998), 71�80.

[c29] E. Schmeidel, On the asymptotic behaviour of solutions of di�erence equa-tions, Demonstratio Math., 30 (1) (1997), 193�197.

[c30] E. Schmeidel, Asymptotic properties of solution of nonlinear di�erence equa-tion, Journal of Communications of Applied Nonlinear Analysis, 2 (4) (1997),87�92.

[c31] J. Popenda, E. Schmeidel, On the asymptotic behavior of solutions of lineardi�erence equations, Publications Matematiques, 38 (1994), 3�9.

[c32] E. Schmeidel, Asymptotic behaviour of solutions of the second order di�er-ence equation, Demonstratio Math., 26 (3-4) (1993), 811�819.

[c33] J. Popenda, E. Schmeidel, Nonoscillatory solutions of third order di�erenceequations, Port. Math., 49 (2) (1992), 233�239.

[c34] J. Popenda, E. Schmeidel, Some properties of solutions of di�erence equa-tions, with Fasc. Math., 13 (1981), 89�98.

[c35] J. Popenda, E. Schmeidel, On the asymptotic behaviour of nonoscillatorysolutions of di�erence equations, Fasc. Math., 12 (1980), 43�53.

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Abstract

Presented work deals with the qualitative theory of di�erence equations. The mainresults are presented in chapters 3, 4 and 5.

In Chapter 3 the nonlinear fourth order di�erence equations of the form

∆(a(n)∆(b(n)∆(c(n)∆y(n)))) = f(n, y(δ(n))),

where (a(n)), (b(n)) and (c(n)) are sequences of positive real numbers and functionf : N�R! R are studied. This Chapter is devoted to Smith-Taylor equations andto Thandapani-Arockiasamy equations for which some asymptotic and oscillatoryresults are presented. The main purpose of this Chapter is to establish some nec-essary and su�cient conditions for the existence of solutions of the above equationwith special asymptotic properties, which are minimal and maximal solutions.

Chapter 4 is devoted to even order di�erence equations with neutral term

∆(a(k�1)(n)∆(a(k�2)(n)...∆(a(1)(n)∆(y(n)+p(n)y(n� τ)))))+f(n, y(n� σ)) = 0,

where k is a positive integer and n 2 fη + 1, η + 2, ...g, and τ , σ are positiveintegers η = max(τ, σ). Here (a(i)(n)), i = 1, 2, 3, ...k� 1 are sequences of positivereal numbers, (p(n)) is a sequence of real numbers and function f : N � R ! R.The asymptotic behavior of nonoscillatory solution of the equations is investigated.Firstly, the results obtained for nonlinear fourth order di�erence equations withneutral term, next the results obtained for higher order di�erence equation arepresented. The necessary and su�cient or su�cient conditions for the existenceof solutions of the equation, which can be classi�ed into three distinct categorieswith each being nonempty under relatively mild conditions are given.

Finally, in Chapter 5 the author gives new results for periodicity of solution ofVolterra equations of the form

xs(n+ 1) = as(n) + bs(n)xs(n) +n∑i=0

Ks1(n, i)x1(i) +n∑i=0

Ks2(n, i)x2(i),

where as, bs, xs : N ! R, Ksp : N � N ! R in scalar case and two-dimensionalsystem of Volterra equations as well. Su�cient conditions for the existence andnonexistence of an asymptotically ω-periodic solution of the above equation areestablished.

In general, the results which are applied to higher order di�erence equationsincluding Volterra di�erence equations are studied. The obtained criteria containsresults known for special cases of investigated equations. Many presented resultsdirectly imply stability of solutions of the considered equation.

AMS Subject classi�cation 39A11, 39A10

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