OSCILLATION RESULTS FOR DIFFERENCE EQUATIONSWITH SEVERAL OSCILLATING COEFFICIENTS

G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

Abstract. This paper presents a new su¢ cient condition for the oscillationof all solutions of di¤erence equations with several deviating arguments andoscillating coe¢ cients. Corresponding di¤erence equations of both retardedand advanced type are studied. Examples illustrating the results are alsogiven

Keywords: Oscillating coe¢ cients, retarded argument, advanced argu-ment, oscillatory solutions, nonoscillatory solutions.

2010 Mathematics Subject Classi�cation: 39A10, 39A21.

1. INTRODUCTION

In the present paper, we study the oscillatory behavior of the solutions of theretarded di¤erence equation

�x(n) +

mXi=1

pi(n)x(� i(n)) = 0, n 2 N0, (ER)

where N 3 m � 2, pi, 1 � i � m, are sequences of real numbers and f� i(n)gn2N0 ,1 � i � m, are sequences of integers such that

� i(n) � n� 1, n 2 N0, and limn!1

� i(n) =1, 1 � i � m (1.1)

and the (dual) advanced di¤erence equation

rx(n)�mXi=1

pi(n)x(�i(n)) = 0, n 2 N (EA)

where N 3 m � 2, pi, 1 � i � m, are sequences of real numbers and f�i(n)gn2N,1 � i � m, are sequences of integers such that

�i(n) � n+ 1, n 2 N, 1 � i � m. (1.2)

Here, N0 = f0; 1; 2; : : :g and N = f1; 2; : : :g. Also, as usual, � denotes the forwarddi¤erence operator �x(n) = x(n+1)�x(n) and r denotes the backward di¤erenceoperator rx(n) = x(n)� x(n� 1).By a solution of (ER), we mean a sequence of real numbers fx(n)gn��w which

satis�es (ER) for all n 2 N0: Here,w := � min

n2N01�i�m

� i(n) 2 N0.

HCorresponding author : I. P. Stavroulakis; email address: ipstav@cc.uoi.gr; tel. +30-26510-08283; Greece .

1

2 G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

It is clear that, for each choice of real numbers c�w; c�w+1; : : : ; c�1; c0, thereexists a unique solution fx(n)gn��w of (ER) which satis�es the initial conditionsx(�w) = c�w; x(�w + 1) = c�w+1; : : : ; x(�1) = c�1; x(0) = c0.By a solution of the advanced di¤erence equation (EA), we mean a sequence of

real numbers fx(n)gn2N0 which satis�es (EA) for all n 2 N.A solution fx(n)gn��w

�fx(n)gn2N0

�of (ER) [(EA)] is called oscillatory (around

zero), if for any positive integer n0 � �w [n0 � 0] there exist n1; n2 � n0 such thatx(n1)x(n2) � 0. Otherwise, the solution is said to be nonoscillatory.In the last few decades, the oscillatory behavior of all solutions of di¤erence

equations has been extensively studied when the coe¢ cients pi(n) are nonnegative.However, for the general case when pi(n) are allowed to oscillate, it is di¢ cultto study the oscillation of (ER) [(EA)], since the di¤erence �x(n) [rx(n)] of anynonoscillatory solution of (ER) [(EA)] is in general oscillatory. Therefore, the resultson oscillation of di¤erence and di¤erential equations with oscillating coe¢ cients arerelatively scarce. Thus, a small number of paper are dealing with this case. See,for example, [1�16] and the references cited therein.In 1992 Qian, Ladas and Yan [12], in 2000 Yu and Tang [16] and in 2001 Tang

and Cheng [13] derived oscillation conditions for a special case of equation (ER) thefollowing equation with one oscillating coe¢ cient and constant delay of the form

xn+1 � xn + pnxn�k = 0, n 2 N0,while in 1996 Yan and Yan [15] studied the di¤erence equation with several oscil-lating coe¢ cients of the form

xn+1 � xn +mXi=1

pi(n)xn�ki(n) = 0, n 2 N0,

under some additional conditions on the oscillating coe¢ cients.

For equations (ER) and (EA) with several oscillating coe¢ cients, very recently,Bohner, Chatzarakis and Stavroulakis [2,3] and Berezansky, Chatzarakis, Domosh-nitsky and Stavroulakis [1] established the following theorems.

Theorem 1.1 (See [2, Theorem 2.4]). Assume (1.1) and that the sequences� i are increasing for all i 2 f1; : : : ;mg. Suppose also that for each i 2 f1; : : : ;mgthere exists a sequence fni(j)gj2N such that limj!1 ni(j) =1 and

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[�(�(ni(j))); ni(j)] \ N

9=; 6= ;, 1 � k � m (1.3)

where�(n) = max

1�i�m� i(n), n 2 N0. (1.4)

If, moreover

lim supj!1

mXi=1

n(j)Xq=�(n(j))

pi(q) > 1, (1.5)

where n(j) = min fni(j) : 1 � i � mg, then all solutions of (ER) oscillate.

Theorem 1.2 (See [2, Theorem 3.4]). Assume (1.2) and that the sequences�i are increasing for all i 2 f1; : : : ;mg. Suppose also that for each i 2 f1; : : : ;mg

OSCILLATING COEFFICIENTS 3

there exists a sequence fni(j)gj2N such that limj!1 ni(j) =1 and

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[ni(j); �(�(ni(j)))] \ N

9=; 6= ;, 1 � k � m, (1.6)

where�(n) = min

1�i�m�i(n), n 2 N. (1.7)

If, moreover

lim supj!1

mXi=1

�(n(j))Xq=n(j)

pi(q) > 1, (1.8)

where n(j) = max fni(j) : 1 � i � mg, then all solutions of (EA) oscillate.

Theorem 1.3 (See [3, Theorem 2.1]). Assume (1.1) and that the sequences� i are increasing for all i 2 f1; : : : ;mg. Suppose also that for each i 2 f1; : : : ;mgthere exists a sequence fni(j)gj2N such that limj!1 ni(j) =1,

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[� i(� i(ni(j))); ni(j)] \ N

9=; 6= ;, 1 � k � m (1.9)

and

lim supn!1

mXi=1

pi(n) > 0 for all n 2m\i=1

8<:[j2N

[� i(� i(ni(j))); ni(j)] \ N

9=; . (1.10)

If, moreover

lim infj!1

mXi=1

ni(j)�1Xq=� i(ni(j))

pi(q) >1

e, (1.11)

then all solutions of (ER) oscillate.

Theorem 1.4 (See [3, Theorem 3.1]). Assume (1.2) and that the sequences�i are increasing for all i 2 f1; : : : ;mg. Suppose also that for each i 2 f1; : : : ;mgthere exists a sequence fni(j)gj2N such that limj!1 ni(j) =1,

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[ni(j); �i(�i(ni(j)))] \ N

9=; 6= ;, 1 � k � m (1.12)

and

lim supn!1

mXi=1

pi(n) > 0 for all n 2m\i=1

8<:[j2N

[ni(j), �i(�i(ni(j)))] \ N

9=; . (1.13)

If, moreover

lim infj!1

mXi=1

�i(ni(j))Xq=ni(j)+1

pi(q) >1

e, (1.14)

then all solutions of (EA) oscillate.

4 G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

Theorem 1.5 (See [1, Theorem 2.1]). Assume that (1.1) holds, the sequences � iare increasing for all i 2 f1; : : : ;mg and � is de�ned by (1.4). Suppose also that foreach i 2 f1; : : : ;mg there exists a sequence fni(j)gj2N such that limj!1 ni(j) =1,

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[�(�(ni(j))); ni(j)] \ N

9=; 6= ;, 1 � k � m (1.3)

and

� := lim infj!1

mXi=1

n(j)�1Xq=�(n(j))

pi(q), (1.15)

where n(j) = min fni(j) : 1 � i � mg.If 0 < � < 1, and

lim supj!1

mXi=1

n(j)Xq=�(n(j))

pi(q) > 1��2

4 (1� �) , (1.16)

then all solutions of (ER) oscillate.

Theorem 1.6 (See [1, Theorem 3.1]). Assume (1.2) holds, the sequences �iare increasing for all i 2 f1; : : : ;mg and � is de�ned by (1.7). Suppose also that foreach i 2 f1; : : : ;mg there exists a sequence fni(j)gj2N such that limj!1 ni(j) =1,

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[ni(j), �(�(ni(j)))] \ N

9=; 6= ;, 1 � k � m (1.6)

and

� := lim infj!1

mXi=1

�(n(j))Xq=n(j)+1

pi(q), (1.17)

where n(j) = max fni(j) : 1 � i � mg.If 0 < � < 1, and

lim supj!1

mXi=1

�(n(j))Xq=n(j)

pi(q) > 1��2

4 (1� �) , (1.18)

then all solutions of (EA) oscillate.

In the present paper, the authors study further the equation (ER) [(EA)] and

essentially improve the upper bound of the ratio x(�(n))=x(n+1)hx(n�1)x(�(n))

ifor pos-

sible nonoscillatory solutions x of (ER) [(EA)], when neither (1.5) [(1.8)] nor (1.11)[(1.14)] is satis�ed, and derive a new su¢ cient oscillation condition. Examplesillustrating the results are also given.

2. RETARDED EQUATIONS

In this section, we present a new su¢ cient condition for the oscillation of allsolutions of (ER) when the conditions (1.5) and (1.11) are not satis�ed, under theassumption that the sequences � i are increasing for all i 2 f1; : : : ;mg. To that end,the following lemma provides a useful tool.

OSCILLATING COEFFICIENTS 5

Lemma 2.1. Assume that (1.1) holds, the sequences � i are increasing for alli 2 f1; : : : ;mg, � is de�ned by (1.4), � is de�ned by (1.15) and (x(n))n��w is anonoscillatory solution of (ER). Suppose also that for each i 2 f1; : : : ;mg thereexists a sequence fni(j)gj2N such that limj!1 ni(j) =1, and

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[�(�(ni(j))); ni(j)] \ N

9=; 6= ;, 1 � k � m. (1.3)

If 0 < � � 1=2, then

lim infj!1

x(n(j) + 1)

x(�(n(j)))� 1

2

�1� ��

p1� 2�

�. (2.1)

Proof. Since the solution fx(n)gn��w of (ER) is nonoscillatory, it is eithereventually positive or eventually negative. As f�x(n)gn��w is also a solution of(ER), we may restrict ourselves only to the case where x(n) > 0 eventually.By (1.3), it is obvious that there exists j0 2 N such that

pk(n) � 0 for all n 2m\i=1

[�(�(ni(j0))); ni(j0)] \ N, 1 � k � m (2.2)

and

x(�k(n)) > 0 for all n 2m\i=1

[�(�(ni(j0))); ni(j0)] \ N, 1 � k � m. (2.3)

In view of (2.2) and (2.3), (ER) gives

x(n+ 1)� x(n) = �mXi=1

pi(n)x(� i(n)) � 0,

for every n 2mTi=1

[�(�(ni(j0))); ni(j0)] \ N. This guarantees that the sequence x is

decreasing onmTi=1

[�(�(ni(j0))); ni(j0)] \ N.

Also, by (1.15) we have

mXi=1

n(j0)�1Xq=�(n(j0))

pi(q) � �� ", (2.4)

where " is an arbitrary real number with 0 < " < �. It is clear that there exists a�xed natural number n�(j0) � n(j0) such that

mXi=1

n�(j0)�1Xq=n(j0)

pi (q) <�� "2

andmXi=1

n�(j0)Xq=n(j0)

pi (q) ��� "2

. (2.5)

This is because in the case where pi (q) < ��"2 , there exists n�(j0) > n(j0) such

that (2.5) is satis�ed, while in the case where pi(q) � ��"2 , then n�(j0) = n(j0),

and thereforemXi=1

n�(j0)�1Xq=n(j0)

pi (q) =mXi=1

n(j0)�1Xq=n(j0)

pi (q) (by which we mean) = 0 <�� "2

6 G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

andmXi=1

n�(j0)Xq=n(j0)

pi (q) =mXi=1

n(j0)Xq=n(j0)

pi (q) � pi(n(j0)) ��� "2

.

That is, in both cases (2.5) is satis�ed.Now, we will show that

�(n�(j0)) � n(j0)� 1:

Indeed, in the case where pi(n(j0)) � ��"2 , since n�(j0) = n(j0), it is obvious

that �(n�(j0)) = �(n(j0)) � n(j0) � 1. In the case where pi(n(j0)) < ��"2 , then

n�(j0) > n(j0). Assume, for the sake of contradiction, that �(n�(j0)) > n(j0) � 1.Hence, n(j0) � �(n�(j0)) � n�(j0)� 1 and then

mXi=1

n�(j0)�1Xq=�(n�(j0))

pi (q) �mXi=1

n�(j0)�1Xq=n(j0)

pi (q) <�� "2

,

which contradicts (2.4). Thus, in both cases, we have �(n�(j0)) � n(j0)� 1.Therefore

mXi=1

n(j0)�1Xq=�(n�(j0))

pi (q) =mXi=1

n�(j0)�1Xq=�(n�(j0))

pi (q)�mXi=1

n�(j0)�1Xq=n(j0)

pi (q)

> (�� ")� �� "2

=�� "2

. (2.6)

Summing up (ER) from n(j0) to n�(j0), and using the fact that the function x isdecreasing and the function � (as de�ned by (1.4)) is increasing, we have

x(n(j0)) = x(n�(j0) + 1) +

mXi=1

n�(j0)Xq=n(j0)

pi (q)x(� i(q))

� x(n�(j0) + 1) +mXi=1

n�(j0)Xq=n(j0)

pi (q)x(�(q)),

or

x(n(j0)) � x(n�(j0) + 1) + x(�(n�(j0)))mXi=1

n�(j0)Xq=n(j0)

pi (q) ,

which, in view of (2.5), gives

x(n(j0)) � x(n�(j0) + 1) +�� "2

x(�(n�(j0))). (2.7)

Summing up (ER) from �(n�(j0)) to n(j0)� 1, and using the same arguments, wehave

x(�(n�(j0))) = x(n(j0)) +mXi=1

n(j0)�1Xq=�(n�(j0))

pi (q)x(� i(q))

� x(n(j0)) +mXi=1

n(j0)�1Xq=�(n�(j0))

pi (q)x(�(q)),

OSCILLATING COEFFICIENTS 7

or

x(�(n�(j0))) � x(n(j0)) + x(�(n(j0)� 1))mXi=1

n(j0)�1Xq=�(n�(j0))

pi (q) ,

which, in view of (2.6), gives

x(�(n�(j0))) > x(n(j0)) +�� "2

x(�(n(j0)� 1)). (2.8)

Combining inequalities (2.7) and (2.8), it is obvious that

x(n(j0)) >

��� "2

�2x(�(n(j0)� 1)) = b1x(�(n(j0)� 1)), (2.9)

where

b1 =

��� "2

�2.

Combining inequalities (2.7),(2.8) and (2.9), we have

x(n(j0)) >

�b1 +

�� "2

�x(�(n�(j0)))

>

�b1 +

�� "2

��x(n(j0)) +

�� "2

x(�(n(j0)� 1))�

>

�b1 +

�� "2

�2x(�(n(j0)� 1)) = b2x(�(n(j0)� 1)), (2.10)

where

b2 =

�b1 +

�� "2

�2.

Following the above procedure, we can inductively construct a sequence of positivereal numbers fb�g��1 with

b�+1 =

�b� +

�� "2

�2,

such thatx(n(j0)) > b�x(�(n(j0)� 1)) (� = 1; 2; ...). (2.11)

It is easy to prove that fb�g��1 is a strictly increasing and bounded sequence ofpositive real numbers. Therefore, it follows that lim�!1 b� exists as a positive realnumber. Set

` = lim�!1

b� .

Because of the de�nition of fb�g��1, it holds

` =

�`+

�� "2

�2,

i.e.,

` =1

2

h1� (�� ")�

p1� 2(�� ")

i,

or

` =1

2

h1� (�� ") +

p1� 2(�� ")

i.

8 G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

In both cases, we have

` � 1

2

h1� (�� ")�

p1� 2(�� ")

iand consequently (2.11) yields

x(n(j0))

x(�(n(j0)� 1))� ` � 1

2

h1� (�� ")�

p1� 2(�� ")

i,

orx(n(j0) + 1)

x(�(n(j0)))� 1

2

h1� (�� ")�

p1� 2(�� ")

i.

Thus,

lim infj0!1

x(n(j0) + 1)

x(�(n(j0)))� 1

2

h1� (�� ")�

p1� 2(�� ")

i.

The last inequality holds true for all real numbers " with 0 < " < �. Hence, we canobtain (2.1).The proof of the lemma is complete.

Theorem 2.1. Assume that (1.1) holds, the sequences � i are increasing for alli 2 f1; : : : ;mg, � is de�ned by (1.4) and de�ne � by (1.15). Suppose also that foreach i 2 f1; : : : ;mg there exists a sequence fni(j)gj2N such that limj!1 ni(j) =1,and

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[�(�(ni(j))); ni(j)] \ N

9=; 6= ;, 1 � k � m. (1.3)

If 0 < � � 1=2, and

lim supj!1

mXi=1

n(j)Xq=�(n(j))

pi(q) >1 + �+

p1� 2�

2, (2.12)

then all solutions of (ER) oscillate.

Proof. Assume, for the sake of contradiction, that fx(n)gn��w is an eventuallypositive solution of (ER). Then there exists j0 2 N such that (2.2) and (2.3) hold.Also, as it has already been shown in the proof of Lemma 2.1, the sequence x is

decreasing onmTk=1

[�(�(nk(j0))); nk(j0)] \ N.

Summing up (ER) from �(n(j0)) to n(j0), and using the fact that the functionx is decreasing and the function � (as de�ned by (1.4)) is increasing, we obtain

x(�(n(j0))) = x(n(j0) + 1) +mXi=1

n(j0)Xq=�(n(j0))

pi (q)x(� i(q))

� x(n(j0) + 1) + x(�(n(j0)))mXi=1

n(j0)Xq=�(n(j0))

pi (q) .

Consequently,mXi=1

n(j0)Xq=�(n(j0))

pi (q) � 1�x(n(j0) + 1)

x(�(n(j0))),

OSCILLATING COEFFICIENTS 9

which gives

lim supj0!1

mXi=1

n(j0)Xq=�(n(j0))

pi (q) � 1� lim infj0!1

x(n(j0) + 1)

x(�(n(j0))). (2.13)

Assume that 0 < � � 1=2 and (2.12) holds. Then by Lemma 2.1, inequality (2.1)is ful�lled, and so (2.13) leads to

lim supj0!1

mXi=1

n(j0)Xq=�(n(j0))

pi (q) �1 + �+

p1� 2�

2,

which contradicts condition (2.12). The proof of the theorem is complete.

3. ADVANCED EQUATIONS

In the case of the advanced di¤erence equation (EA) the following Lemma 3.1(analogue to Lemma 2.1) can be easily derived and an easy modi�cation of theproof of Theorem 2.1 leads to the following Theorem 3.1.

Lemma 3.1. Assume that (1.2) holds, the sequences �i are increasing for alli 2 f1; : : : ;mg, � is de�ned by (1.7), � is de�ned by (1.17) and (x(n))n�0 is anonoscillatory solution of (EA). Suppose also that for each i 2 f1; : : : ;mg thereexists a sequence fni(j)gj2N such that limj!1 ni(j) =1, and

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[ni(j); �(�(ni(j)))] \ N

9=; 6= ;, 1 � k � m.

If 0 < � � 1=2, then

lim infj!1

x(n(j)� 1)x(�(n(j)))

� 1

2

�1� ��

p1� 2�

�.

Theorem 3.1. Assume (1.2) holds, the sequences �i are increasing for alli 2 f1; : : : ;mg, � is de�ned by (1.7) and de�ne � by (1.17). Suppose also that foreach i 2 f1; : : : ;mg there exists a sequence fni(j)gj2N such that limj!1 ni(j) =1,and

pk(n) � 0 for all n 2m\i=1

8<:[j2N

[ni(j), �(�(ni(j)))] \ N

9=; 6= ;, 1 � k � m. (1.6)

If 0 < � � 1=2, and

lim supj!1

mXi=1

�(n(j))Xq=n(j)

pi(q) >1 + �+

p1� 2�

2, (3.1)

then all solutions of (EA) oscillate.

Remark 3.1. When � ! 0, then the conditions (2.12) and (3.1) reduce tothe condition (1.5) and (1.8) respectively. However the improvement is clear when0 < � � 1=2. It is easy to see that

1

2

�1� ��

p1� 2�

�>

�2

4 (1� �) ;

10 G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

which means that the condition (2.12) is weaker than the condition (1.16) and thecondition (3.1) is weaker than the condition (1.18).

4. EXAMPLES

The signi�cance of the results is illustrated in the following examples.

Example 4.1. Consider the retarded di¤erence equation

�x(n) + p1(n)x(n� 3) + p2(n)x(n� 4) = 0; n 2 N0, (4.1)

where p1(n) and p2(n) are oscillating coe¢ cients, as shown in Figure 1.

n0 11

1216

23

0n

)n(p1

)n(p2

23

4.0

02.0

3.0

12

Figure 1.

11

07.0

635.0

16

In view of (1.4), it is obvious that �(n) = n� 3: Observe that forn1(j) = 24j + 11; j 2 N,

we have p1(n) � 0 for every n 2 A, where

A =[j2N

[�(�(n1(j))); n1(j)] \ N =[j2N

[24j + 5; 24j + 11] \ N.

Also, forn2(j) = 24j + 9; j 2 N,

we have p2(n) � 0 for every n 2 B, where

B =[j2N

[�(�(n2(j))); n2(j)] \ N =[j2N

[24j + 3; 24j + 9] \ N.

Therefore

p1(n) � 0 and p2(n) > 0 for all n 2 A \B =[j2N

[24j + 5; 24j + 9] \ N 6= ;.

OSCILLATING COEFFICIENTS 11

Observe that

n(j) = min fni(j) : 1 � i � 2g = 24j + 9, j 2 N.

Now,

� = lim infj!1

2Xi=1

n(j)�1Xq=�(n(j))

pi(q)

= lim infj!1

24 24j+8Xq=24j+6

p1(q) +

24j+8Xq=24j+6

p2(q)

35 = 3 � 7100

+ 3 � 2100

= 0:27

lim supj!1

2Xi=1

n(j)Xq=�(n(j))

pi(q)

= lim supj!1

24 24j+9Xq=24j+6

p1(q) +

24j+9Xq=24j+6

p2(q)

35 = 4 � 7100

+ 3 � 2100

+635

1000= 0:975

Observe that

0:975 >1 + �+

p1� 2�

2' 0:974116499;

that is, condition (2.12) of Theorem 2.1 is satis�ed and therefore all solutions ofequation (4.1) oscillate.On the other hand,

0:975 < 1,

0:975 < 1� �2

4 (1� �) ' 0:975034246.

Observe that p1(n) � 0 for every n 2 A0 = A and p2(n) � 0 for every n 2 B0, where

B0 =[j2N

[�2(�2(n2(j))); n2(j)] \ N =[j2N

[24j + 1; 24j + 9] \ N.

Therefore

p1(n) � 0 and p2(n) > 0 for all n 2 A0 \B0 =[j2N

[24j + 5; 24j + 9] \ N 6= ;.

Also,

lim infj!1

2Xi=1

ni(j)�1Xq=� i(ni(j))

pi(q)

= lim infj!1

24 24j+10Xq=24j+8

p1(q) +

24j+8Xq=24j+5

p2(q)

35 = 3 � 7100

+ 4 � 2100

= 0:29 <1

e.

Therefore none of the conditions (1.5), (1.16) and (1.11) are satis�ed.

Example 4.2. Consider the advanced di¤erence equation

rx(n)� p1(n)x(n+ 2)� p2(n)x(n+ 3) = 0; n 2 N, (4.2)

12 G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

where p1(n) and p2(n) are oscillating coe¢ cients, as shown in Figure 2.

n0 11

12 23

0n

)n(p1

)n(p2

23

5.0

06.012

Figure 2.

11

08.0

612.0

In view of (1.7), it is obvious that �(n) = n+ 2: Observe that for

n1(j) = 24j + 3; j 2 N,

we have p1(n) � 0 for every n 2 A, where

A =[j2N

[n1(j); �(�(n1(j)))] \ N =[j2N

[24j + 3; 24j + 7] \ N.

Also, for

n2(j) = 24j + 4; j 2 N,

we have p2(n) � 0 for every n 2 B, where

B =[j2N

[n2(j); �(�(n2(j)))] \ N =[j2N

[24j + 4; 24j + 8] \ N.

Therefore

p1(n) � 0 and p2(n) > 0 for all n 2 A \B =[j2N

[24j + 4; 24j + 7] \ N 6= ;.

Observe that

n(j) = max fni(j) : 1 � i � 2g = 24j + 4, j 2 N.

OSCILLATING COEFFICIENTS 13

Now

� = lim infj!1

2Xi=1

�(n(j))Xq=n(j)+1

pi(q)

= lim infj!1

24 24j+6Xq=24j+5

p1(q) +

24j+6Xq=24j+5

p2(q)

35 = 2 � 8100

+ 2 � 6100

= 0:28.

Also

lim supj!1

2Xi=1

�(n(j))Xq=n(j)

pi(q)

= lim supj!1

24 24j+6Xq=24j+4

p1(q) +

24j+6Xq=24j+4

p2(q)

35 = 3 � 8100

+ 2 � 6100

+612

1000= 0:972

Observe that

0:972 >1 + �+

p1� 2�

2' 0:971662479;

that is, condition (3.1) of Theorem 3.1 is satis�ed and therefore all solutions ofequation (4.2) oscillate.On the other hand,

0:972 < 1,

0:972 < 1� �2

4 (1� �) ' 0:972777777.

Observe that p1(n) � 0 for every n 2 A0 = A and p2(n) � 0 for every n 2 B0, where

B0 =[j2N

[n2(j); �2(�2(n2(j)))] \ N =[j2N

[24j + 4; 24j + 10] \ N.

Therefore

p1(n) � 0 and p2(n) > 0 for all n 2 A0 \B0 =[j2N

[24j + 4; 24j + 7] \ N 6= ;.

Also,

lim infj!1

2Xi=1

�i(ni(j))Xq=ni(j)+1

pi(q)

= lim infj!1

24 24j+5Xq=24j+4

p1(q) +

24j+7Xq=24j+5

p2(q)

35 = 2 � 8100

+ 3 � 6100

= 0:34 <1

e.

Therefore none of the conditions (1.8), (1.18) and (1.14) are satis�ed.

Acknowledgement The authors would like to thank both referees for the con-structive remarks which improved the presentation of the paper.

14 G. E. CHATZARAKIS, M. LAFCI, AND I. P. STAVROULAKISH

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Department of Electrical and Electronic Engineering Educators, School of Peda-gogical and Technological Education (ASPETE), 14121, N. Heraklio, Athens, Greece

E-mail address : geaxatz@otenet.gr, gea.xatz@aspete.gr

Department of Mathematics, Faculty of Sciences, Ankara University, Tando¼Gan,Ankara, 06100, Turkey

E-mail address : mlafci@ankara.edu.tr

Department of Mathematics, University of Ioannina, 451 10 Ioannina, GreeceE-mail address : ipstav@cc.uoi.gr