A Mathematical Journal
Vol. 7, No 2, (223 - 236). August 2005.

A Survey on the Oscillation of Solutions of
First Order Delay Difference Equations

L. K. Kikina 1

Department of Mathematics, University of Gjirokastra
Gjirokaster, Albania

I.P. Stavroulakis 1

Department of Mathematics, University of Ioannina
451 10 Ioannina, Greece

ipstav@cc.uoi.gr

ABSTRACT
In this paper, a survey of the most interesting results on the oscillation of all

solutions of the first order delay difference equation of the form

xn+1 − xn + pnxn−k = 0, n = 0, 1, 2, ...,

where {pn} is a sequence of nonnegative real numbers and k is a positive integer
is presented, especially in the case when neither of the well-known oscillation
conditions

lim sup
n→∞

n�

i=n−k
pi > 1 and lim inf

n→∞
1

k

n−1�

i=n−k
pi >

kk

(k + 1)k+1

is satisfied.

RESUMEN
En este art́ıculo, hacemos una revisión de los resultados más interesantes

sobre oscilaciones de las soluciones de la ecuación en diferencias de primer orden

1The authors would like to express many thanks to Professor Yuri Domshlak for useful discussions
concerning this paper. Also many thanks to the referee for some helpful comments.



224 L. K. Kikina and I.P. Stavroulakis
7, 2(2005)

con retardo, de la forma

xn+1 − xn + pnxn−k = 0, n = 0, 1, 2, ...,
en donde {pn} es una sucesión de números reales no negativos, k es un entero
positivo, en especial cuando ni siquiera se satisfacen las conocidas condiciones de
oscilación

lim sup
n→∞

n�

i=n−k
pi > 1 y lim inf

n→∞
1

k

n−1�

i=n−k
pi >

kk

(k + 1)k+1

Key words and phrases: Oscillation, nonoscillation, delay difference equation.
Math. Subj. Class.: 39A 11.

1 Introduction

In the last few decades the oscillation theory of delay differential equations has been
extensively developed. The oscillation theory of discrete analogues of delay differential
equations has also attracted growing attention in the recent few years. The reader
is referred to [1-12,14-16, 21, 22, 24-26, 29-46] and the references cited therein. In
particular, the problem of establishing sufficient conditions for the oscillation of all
solutions of the delay difference equation

xn+1 −xn + pnxn−k = 0, n = 0, 1, 2, ..., (1.1)
where {pn} is a sequence of nonnegative real numbers and k is a positive integer,
has been the subject of many recent investigations. See, for example, [2-12, 14, 21,
22, 24-26, 29-39, 42-46] and the references cited therein. Strong interest in (1.1) is
motivated by the fact that it represents a discrete analogue of the delay differential
equation (see [13, 17-20, 23, 27, 28] and the references cited therein)

x′(t) + p(t)x(t− τ) = 0, p(t) ≥ 0, τ > 0. (1.2)
By a solution of (1.1) we mean a sequence {xn} which is defined for n ≥−k and

which satisfies (1.1) for n ≥ 0. A solution {xn} of (1.1) is said to be oscillatory if the
terms xn of the solution are not eventually positive or eventually negative. Otherwise
the solution is called nonoscillatory.

For convenience, we will assume that inequalities about values of sequences are
satisfied eventually for all large n.

In this paper, our main purpose is to present the state of the art on the oscillation
of solutions to (1.1) especially in the case that the oscillation conditions (see below)

lim sup
n→∞

n∑
i=n−k

pi > 1 and lim inf
n→∞

1
k

n−1∑
i=n−k

pi >
kk

(k + 1)k+1

are not satisfied.



7, 2(2005)
A Survey on the Oscillation of Solutions of First Order ... 225

2 Oscillation criteria for Eq. (1.1)

In 1981, Domshlak [7] was the first who studied this problem in the case where k = 1.
Then, in 1989, Erbe and Zhang [14] established the following oscillation criteria for
(1.1).
Theorem 2.1.([14]) Assume that

β := lim inf
n→∞

pn > 0 and lim sup
n→∞

pn > 1 −β (C1)

Then all solutions of (1.1) oscillate.
Theorem 2.2.([14]) Assume that

lim inf
n→∞

pn >
kk

(k + 1)k+1
. (C2)

Then all solutions of (1.1) oscillate.
Theorem 2.3.([14]) Assume that

A := lim sup
n→∞

n∑
i=n−k

pi > 1. (C3)

Then all solutions of (1.1) oscillate.
In the same year 1989 Ladas, Philos and Sficas [22] proved the following theorem.
Theorem 2.4.([22]) Assume that

lim inf
n→∞

1
k

n−1∑
i=n−k

pi >
kk

(k + 1)k+1
. (C4)

Then all solutions of (1.1) oscillate.
Therefore they improved the condition (C2) by replacing the pn of (C2) by the

arithmetic mean of the terms pn−k, ..., pn−1 in (C4).

Concerning the constant k
k

(k+1)k+1
in (C2) and (C4) it should be empasized that,

as it is shown in [14], if

sup pn <
kk

(k + 1)k+1
, (N1)

then (1.1) has a nonoscillatory solution.
In 1990, Ladas [21] conjectured that Eq. (1.1) has a nonoscillatory solution if

1
k

n−1∑
i=n−k

pi ≤
kk

(k + 1)k+1

holds eventually. However this conjecture is false and a counterexample was given in
1994 by Yu, Zhang and Wang [43].



226 L. K. Kikina and I.P. Stavroulakis
7, 2(2005)

It is interesting to establish sufficient conditions for the oscillation of all solutions
of (1.1) when (C3) and (C4) are not satisfied. (For the equation (1.2) this question
has been investigated by many authors, see, for example, [13, 17-20, 23, 27, 28] and
the references cited therein.)

In 1993, Yu, Zhang and Qian [42] and Lalli and Zhang [24], trying to improve
(C3), established the following (false) sufficient oscillation conditions for (1.1)

0 < α := lim inf
n→∞

n−1∑
i=n−k

pi ≤
(

k

k + 1

)k+1
and A > 1 − α

2

4
(F1)

and

n∑
i=n−k

pi ≥ d > 0 for large n and A > 1 −
d4

8

(
1 − d

3

4
+

√
1 − d

3

2

)−1
(F2)

respectively.
Unfortunately, the above conditions (F1) and (F2) are not correct. This is due to

the fact that they are based on the following (false) discrete version of Koplatadze-
Chanturia Lemma. (See [6] and [10]).

Lemma A (False). Assume that {xn} is an eventually positive solution of (1.1) and
that

n∑
i=n−k

pi ≥ M > 0 for large n. (1.3)

Then

xn >
M2

4
xn−k for large n.

As one can see, the erroneous proof of Lemma A is based on the following (false)
statement. (See [6] and [10]).

Statement A (False). If (1.3) holds, then for any large N, there exists a positive
integer n such that n−k ≤ N ≤ n and

N∑
i=n−k

pi ≥
M

2
,

n∑
i=N

pi ≥
M

2
.

It is obvious that all the oscillation results which have made use of the above
Lemma A or Statement A are incorrect. For details on this problem see the paper by
Cheng and Zhang [6].

Here it should be pointed out that the following statement (see [22], [31]) is correct
and it should not be confused with the Statement A.
Statement 2.1.([22], [31]) If

n−1∑
i=n−k

pi ≥ M > 0 for large n, (1.4)



7, 2(2005)
A Survey on the Oscillation of Solutions of First Order ... 227

then for any large n, there exists a positive integer n∗ with n−k ≤ n∗ ≤ n such that
n∗∑

i=n−k
pi ≥

M

2
,

n∑
i=n∗

pi ≥
M

2
.

In 1995, Stavroulakis [31], based on this correct Statement 2.1, proved the following
theorem.
Theorem 2.5.([31]) Assume that

0 < α ≤
(

k

k + 1

)k+1

and

lim sup
n→∞

pn > 1 −
α2

4
. (C5)

Then all solutions of (1.1) oscillate.
In 1999, Domshlak [10] and in 2000, Cheng and Zhang [6] established the follow-

ing lemmas, respectively, which may be looked upon as (exact) discrete versions of
Koplatadze-Chanturia Lemma.
Lemma 2.1.([10]) Assume that {xn} is an eventually positive solution of (1.1) and
that the condition (1.4) holds. Then

xn >
M2

4
xn−k for large n. (1.5)

Lemma 2.2.([6]) Assume that {xn} is an eventually positive solution of (1.1) and
that the condition (1.4) holds. Then

xn > M
kxn−k for large n. (1.6)

Based on these lemmas the following theorem was established in [32].
Theorem 2.6.([32]) Assume that

0 < α ≤
(

k

k + 1

)k+1
.

Then either one of the conditions

lim sup
n→∞

n−1∑
i=n−k

pi > 1 −
α2

4
(C6)

or

lim sup
n→∞

n−1∑
i=n−k

pi > 1 −αk (C7)



228 L. K. Kikina and I.P. Stavroulakis
7, 2(2005)

implies that all solutions of (1.1) oscillate.
Remark 2.1.([32]) From the above theorem it is now clear that

0 < α := lim inf
n→∞

n−1∑
i=n−k

pi ≤
(

k

k + 1

)k+1
and lim sup

n→∞

n−1∑
i=n−k

pi > 1 −
α2

4

is the correct oscillation condition by which the (false) condition (F1) should be
replaced.
Remark 2.2.([32]) Observe the following:

(i) When k = 1, 2,

αk >
α2

4
,

(since, from the above mentioned conditions, it makes sense to investigate the case

when α <
(

k
k+1

)k+1
) and therefore condition (C6) implies (C7).

(ii) When k = 3,

α3 >
α2

4
when α >

1
4

while

α3 <
α2

4
when α <

1
4
.

So in this case the conditions (C6) and (C7) are independent.
(iii) When k ≥ 4,

αk <
α2

4
,

and therefore condition (C7) implies (C6).
(iv) When k < 12 condition (C6) or (C7) implies (C3).
(v) When k ≥ 12 condition (C6) may hold but condition (C3) may not hold.
We illustrate these by the following examples.

Example 2.1.([32]) Consider the equation

xn+1 −xn + pnxn−3 = 0, n = 0, 1, 2, ...,
where

p2n =
1
10
, p2n+1 =

1
10

+
64
95

sin2
nπ

2
, n = 0, 1, 2, ....

Here k = 3 and it is easy to see that

α = lim inf
n→∞

n−1∑
i=n−3

pi =
3
10

<

(
3
4

)4

and

lim sup
n→∞

n−1∑
i=n−3

pi =
3
10

+
64
95

> 1 −α3.



7, 2(2005)
A Survey on the Oscillation of Solutions of First Order ... 229

Thus condition (C7) is satisfied and therefore all solutions oscillate. Observe, however,
that condition (C6) is not satisfied.

If, on the other hand, in the above equation

p2n =
8

100
, p2n+1 =

8
100

+
746
1000

sin2
nπ

2
, n = 0, 1, 2, ...,

then it is easy to see that

α = lim inf
n→∞

n−1∑
i=n−3

pi =
24
100

<

(
3
4

)4

and

lim sup
n→∞

n−1∑
i=n−3

pi =
24
100

+
746
1000

> 1 − α
2

4
.

In this case condition (C6) is satisfied and therefore all solutions oscillate. Observe,
however, that condition (C7) is not satisfied.

Example 2.2.([32]) Consider the equation

xn+1 −xn + pnxn−16 = 0, n = 0, 1, 2, ...,
where

p17n = p17n+1 = = p17n+15 =
2

100
, p17n+16 =

2
100

+
655
1000

, n = 0, 1, 2, ....

Here k = 16 and it is easy to see that

α = lim inf
n→∞

n−1∑
i=n−16

pi =
32
100

<

(
16
17

)17

and

lim sup
n→∞

n−1∑
i=n−16

pi =
32
100

+
655
1000

= 0.975 > 1 − α
2

4
.

We see that condition (C6) is satisfied and therefore all solutions oscillate. Observe,
however, that

A = lim sup
n→∞

n∑
i=n−16

pi =
34
100

+
655
1000

= 0.995 < 1;

that is, condition (C3) is not satisfied.
In 1995, Chen and Yu [2], following the above mentioned direction, derived a

condition which formulated in terms of α and A says that all solutions of (1.1) oscillate
if 0 < α ≤ kk+1

(k+1)k+1
and

A > 1 − 1 −α−
√

1 − 2α−α2
2

. (C8)



230 L. K. Kikina and I.P. Stavroulakis
7, 2(2005)

In 1998, Domshlak [9], studied the oscillation of all solutions and the existence
of nonoscillatory solution of (1.1) with r -periodic positive coefficients {pn},pn+r =
pn. It is very important that in the following cases where {r = k},{r = k + 1},
{r = 2},{k = 1,r = 3} and {k = 1,r = 4} the results obtained are stated in terms of
necessary and sufficient conditions and it is very easy to check them.

In 2000, Tang and Yu [38] improved condition (C8) to the condition

A > λk2 (1 −k ln λ2) −
1 −α−

√
1 −α−α2
2

, (C9)

where λ2 is the greater root of the algebraic equation

kλk(1 −λ) = α.
In 2000, Shen and Stavroulakis [30], using new techniques, improved the previous

results.
Lemma 2.3.([30]) Let the number M ≥ 0 be such that

k∑
i=1

pn−i ≥ M for large n.

Assume that (1.1) has an eventually positive solution {xn}. Then
M ≤ kk+1/(k + 1)k+1 and

lim sup
n→∞

xn−k
xn

k∏
i=1

k∑
j=1

pn−i+j ≤ [d(M)]k,

where d(M) is the greater real root of the algebraic equation

dk+1 −dk + Mk = 0, on [0, 1].
Note that from this lemma we obtain a better and perhaps optimal bound which

essentially improves (1.6).
Theorem 2.7.([30]) Assume that 0 ≤ α ≤ kk+1/(k + 1)k+1 and that there exists an
integer l ≥ 1 such that

lim sup
n→∞

⎧⎨
⎩

k∑
i=1

pn−i + [d(α)]
−k

k∏
i=1

k∑
j=1

pn−i+j +
l−1∑
m=0

[d(α/k)]−(m+1)k
k∑

i=1

m+1∏
j=0

pn−jk−i

⎫⎬
⎭ > 1,

(C10)
where d(α) and d(α/k) are the greater real roots of the equations

dk+1 −dk + αk = 0
and

dk+1 −dk + α/k = 0,



7, 2(2005)
A Survey on the Oscillation of Solutions of First Order ... 231

respectively. Then all solutions of (1.1) oscillate.
Notice that when k = 1, d(α) = d(α) = (1+

√
1 − 4α)/2 (see [30]), and so condition

(C10) reduces to

lim sup
n→∞

⎧⎨
⎩Cpn + pn−1 +

l−1∑
m=0

Cm+1
m+1∏
j=0

pn−j−1

⎫⎬
⎭ > 1, (C11)

where C = 2/(1 +
√

1 − 4α), α = lim infn→∞ pn. Therefore, from Theorem 2.7, we
have the following corollary.
Corollary 2.1.([30]) Assume that 0 ≤ α ≤ 1/4 and that (C11) holds. Then all
solutions of the equation

xn+1 −xn + pnxn−1 = 0 (1.7)
oscillate.

A condition derived from (C11) and which can be easier verified, is given in the
next corollary.
Corollary 2.2.([30]) Assume that 0 ≤ α ≤ 1/4 and that

lim sup
n→∞

pn >

(
1 +
√

1 − 4α
2

)2
. (C12)

Then all solutions of (1.7) oscillate.
Remark 2.2.([30]) Observe that when α = 1/4, condition (C12) reduces to

lim sup
n→∞

pn > 1/4

which can not be improved in the sense that the lower bound 1/4 can not be replaced
by a smaller number. Indeed, by condition (N1) (Theorem 2.3 in [14]), we see that
(1.7) has a nonoscillatory solution if

sup pn < 1/4.

Note, however, that even in the critical state where limn→∞ pn = 1/4, (1.7) can
be either oscillatory or nonoscillatory. For example, if pn = 14 +

c
n2

then (1.7) will
be oscillatory in case c > 1/4 and nonoscillatory in case c < 1/4 (the Kneser-like
theorem, [8]).
Example 2.2.([30]) Consider the equation

xn+1 −xn +
(

1
4

+ a sin4
nπ

8

)
xn−1 = 0,

where a > 0 is a constant. It is easy to see that

lim inf
n→∞

pn = lim inf
n→∞

(
1
4

+ a sin4
nπ

8

)
=

1
4
,



232 L. K. Kikina and I.P. Stavroulakis
7, 2(2005)

lim sup
n→∞

pn = lim sup
n→∞

(
1
4

+ a sin4
nπ

8

)
=

1
4

+ a.

Therefore, by Corollary 2.2, all solutions oscillate. However, none of the conditions
(C1) − (C9) is satisfied.

The following corollary concerns the case when k > 1.
Corollary 2.3.([30]) Assume that 0 ≤ α ≤ kk+1/(k + 1)k+1 and that

lim sup
n→∞

n−1∑
i=n−k

pi > 1 − [d(α)]−kαk −
k[d(α/k)]−kβ2

1 − [d(α/k)]−kβ, (C13)

where d(α),d(α/k) are as in Theorem 2.7. Then all solutions of (1.1) oscillate.
In 2000, Shen and Luo [29] proved the following theorems.

Theorem 2.8.([29]) Assume that there exists some positive integer l such that

lim sup
n→∞

⎧⎨
⎩

k∑
i=0

pn−i +
k∏

i=0

k∑
j=1

pn−i+j +
l−1∑
m=0

k∑
i=1

m+1∏
j=0

pn−jk−i

⎫⎬
⎭ > 1. (C14)

Then all solutions of (1.1) oscillate.
Theorem 2.9.([29]) Assume that there exists some positive integer l such that

lim sup
n→∞

⎧⎨
⎩

k∑
i=1

pn−i +
k∏

i=1

k∑
j=1

pn−i+j +
l−1∑
m=0

k∑
i=1

m+1∏
j=0

pn−jk−i

⎫⎬
⎭ > 1. (C15)

Then all solutions of (1.1) oscillate.
From Theorem 2.8 and Theorem 2.9 the following corollaries are derived.

Corollary 2.4. ([29]) Assume that

A > 1 −αk+1 − kβ
2

1 −β. (C16)

Then all solutions of (1.1) oscillate.
Corollary 2.5. ([29]) Assume that

lim sup
n→∞

n−1∑
i=n−k

pi > 1 −αk −
kβ2

1 −β. (C17)

Then all solutions of (1.1) oscillate.
Following this historical (and chronological) review we also mention that in the

case where

1
k

n−1∑
i=n−k

pi ≥
kk

(k + 1)k+1
and lim

n→∞
1
k

n−1∑
i=n−k

pi =
kk

(k + 1)k+1
,



7, 2(2005)
A Survey on the Oscillation of Solutions of First Order ... 233

the oscillation of (1.1) has been studied in 1994 by Domshlak [8] and in 1998 by Tang
[33] (see also Tang and Yu [35]). In a case when pn is asymptotically close to one of
the periodic critical states, unimprovable results about oscillation preperties of the
equation

xn+1 −xn + pnxn−1 = 0
were obtained by Domshlak in 1999 [11] and in 2000 [12].

Received: June 2003. Revised: October 2003.

References

[1] R. P. AGARWAL AND P. J. Y. WONG, Advanced Topics in Difference
Equations, Kluwer Academic Publishers, 1997.

[2] M. P. CHEN AND Y. S. YU, Oscillations of delay difference equations with
variable coefficients, Proc. First Intl. Conference on Difference Equations,
(Edited by S. N. Elaydi et al), Gordon and Breach 1995, pp. 105-114.

[3] S. S. CHENG, AND B. G. ZHANG, Qualitative theory of partial difference
equations (I): Oscillation of nonlinear partial difference equations, Tamkang
J. Math. 25 (1994), 279-298.

[4] S. S. CHENG, S. T. LIU AND G. ZHANG, A multivariate oscillation theo-
rem, Fasc. Math. 30 (1999), 15-22.

[5] S. S. CHENG, S. L. XI AND B. G. ZHANG, Qualitative theory of partial
difference equations (II): Oscillation criteria for direct control system in
several variables, Tamkang J. Math. 26 (1995), 65-79.

[6] S. S. CHENG AND G. ZHANG, “Virus” in several discrete oscillation the-
orems, Applied Math. Letters, 13 (2000), 9-13.

[7] Y. DOMSHLAK, Discrete version of Sturmian Comparison Theorem for
non-symmetric equations, Doklady Azerb. Acad. Sci. 37 (1981), 12-15 (in
Russian).

[8] Y. DOMSHLAK, Sturmian comparison method in oscillation study for dis-
crete difference equations, I, J. Diff. Integr. Eqs, 7 (1994), 571-582.

[9] Y. DOMSHLAK, Delay-difference equations with periodic coefficients: sharp
results in oscillation theory, Math. Inequal. Appl. , 1 (1998), 403-422.

[10] Y. DOMSHLAK, What should be a discrete version of the Chanturia-
Koplatadze Lemma? Funct. Differ. Equ. , 6 (1999), 299-304.



234 L. K. Kikina and I.P. Stavroulakis
7, 2(2005)

[11] Y. DOMSHLAK, Riccati Difference Equations with almost periodic coeffi-
cients in the critical state, Dynamic Systems Appl. , 8 (1999), 389-399.

[12] Y. DOMSHLAK, The Riccati Difference Equations near ”extremal” critical
states, J. Difference Equations Appl. , 6 (2000), 387-416.

[13] Á. ELBERT AND I. P. STAVROULAKIS, Oscillations of first order differ-
ential equations with deviating arguments, Univ. of Ioannina, T. R. N0 172
1990, Recent trends in differential equations 163-178, World Sci. Ser. Appl.
Anal. , World Sci. Publishing Co. (1992).

[14] L. ERBE AND B. G. ZHANG, Oscillation of discrete analogues of delay
equations, Differential and Integral Equations, 2 (1989), 300-309.

[15] I. GYÖRI AND G. LADAS, Oscillation Theory of Delay Differential Equa-
tions with Applications, Clarendon Press, Oxford, 1991.

[16] J. JAROŠ AND I. P. STAVROULAKIS, Necessary and sufficient conditions
for oscillations of difference equations with several delays, Utilitas Math. ,
45 (1994), 187-195.

[17] J. JAROŠ AND I. P. STAVROULAKIS, Oscillation tests for delay equations,
Rocky Mountain J. Math. , 29 (1999), 197-207.

[18] M. KON, Y. G. SFICAS AND I. P. STAVROULAKIS, Oscillation criteria
for delay equations, Proc. Amer. Math. Soc. , 128 (2000), 2989-2997.

[19] R. KOPLATADZE AND T. CHANTURIA, On oscillatory and monotonic
solutions of first order delay differential equations with deviating arguments,
Differential’nye Uravnenija, 18 (1982), 1463-1465 (Russian).

[20] M. K. KWONG, Oscillation of first-order delay equations, J. Math. Anal.
Appl. , 156 (1991), 274-286.

[21] G. LADAS, Recent developments in the oscillation of delay difference equa-
tions, In International Conference on Differential Equations, Stability and
Control, Dekker, New York, 1990.

[22] G. LADAS, CH. G. PHILOS AND Y. G. SFICAS, Sharp conditions for
the oscillation of delay difference equations, J. Appl. Math. Simulation, 2
(1989), 101-112.

[23] G. S. LADDE, V. LAKSHMIKANTHAM AND B. G. ZHANG, Oscillation
Theory of Differential Equations with Deviating Arguments, Marcel Dekker,
New York, 1987.

[24] B. LALLI AND B. G. ZHANG, Oscillation of difference equations, Collo-
quium Math. , 65 (1993), 25-32.



7, 2(2005)
A Survey on the Oscillation of Solutions of First Order ... 235

[25] ZHIGUO LUO AND J. H. SHEN, New results for oscillation of delay dif-
ference equations, Comput. Math. Appl. 41 (2001), 553-561.

[26] ZHIGUO LUO AND J. H. SHEN, New oscillation criteria for delay differ-
ence equations, J. Math. Anal. Appl. 264 (2001), 85-95.

[27] CH. G. PHILOS AND Y. G. SFICAS, An oscillation criterion for first
order linear delay differential equations, Canad. Math. Bull. , 41 (1998),
207-213.

[28] Y. G. SFICAS AND I. P. STAVROULAKIS, Oscillation criteria for first-
order delay differential equations, Bull. London Math. Soc. 35 (2003), no.
2, 239-246.

[29] J. H. SHEN AND ZHIGUO LUO, Some oscillation criteria for difference
equations, Comput. Math. Applic. , 40 (2000), 713-719.

[30] J. H. SHEN AND I. P. STAVROULAKIS, Oscillation criteria for delay
difference equations, Univ. of Ioannina, T. R. N0 4, 2000, Electron. J. Diff.
Eqns. Vol. 2001 (2001), no. 10, pp. 1-15.

[31] I. P. STAVROULAKIS, Oscillations of delay difference equations, Comput.
Math. Applic. , 29 (1995), 83-88.

[32] I. P. STAVROULAKIS, Oscillation Criteria for First Order delay difference
equations, Mediterr. J. Math. 1 (2004), 231-240.

[33] X. H. TANG, Oscillations of delay difference equations with variable coef-
ficients, (Chinese), J. Central So. Univ. of Technology, 29 (1998), 287-288.

[34] X. H. TANG AND S. S. CHENG, An oscillation criterion for linear differ-
ence equations with oscillating coefficients, J. Comput. Appl. Math., 132
(2001), 319-329.

[35] X. H. TANG AND J. S. YU, Oscillation of delay difference equations,
Comput. Math. Applic. , 37 (1999), 11-20.

[36] X. H. TANG AND J. S. YU, A further result on the oscillation of delay
difference equations, Comput. Math. Applic. , 38 (1999), 229-237.

[37] X. H. TANG AND J. S. YU, Oscillations of delay difference equations in a
critical state, Appl. Math. Letters, 13 (2000), 9-15.

[38] X. H. TANG AND J. S. YU, Oscillation of delay difference equations,
Hokkaido Math. J. 29 (2000), 213-228.

[39] X. H. TANG AND J. S. YU, New oscillation criteria for delay difference
equations, Comput. Math. Applic. , 42 (2001), 1319-1330.



236 L. K. Kikina and I.P. Stavroulakis
7, 2(2005)

[40] P. J. Y. WONG AND R. P. AGARWAL, Oscillation criteria for nonlinear
partial difference equations with delays, Comput. Math. Applic. , 32 (6)
(1996), 57-86.

[41] WEIPING YAN AND JURANG YAN, Comparison and oscillation results
for delay difference equations with oscillating coefficients, Internat. J. Math.
& Math. Sci. , 19 (1996), 171-176.

[42] J. S. YU, B. G. ZHANG AND X. Z. QIAN, Oscillations of delay difference
equations with oscillating coefficients, J. Math. Anal. Appl. , 177 (1993),
432-444.

[43] J. S. YU, B. G. ZHANG AND Z. C. WANG, Oscillation of delay difference
equations, Appl. Anal. , 53 (1994), 117-124.

[44] B. G. ZHANG, S. T. LIU AND S. S. CHENG, Oscillation of a class of delay
partial difference equations, J. Differ. Eqns Appl. , 1 (1995), 215-226.

[45] B. G. ZHANG AND YONG ZHOU, The semicycles of solutions of delay
difference equations, Comput. Math. Applic. , 38 (1999), 31-38.

[46] B. G. ZHANG AND YONG ZHOU, Comparison theorems and oscillation
criteria for difference equations, J. Math. Anal. Appl. , 247 (2000), 397-409.