Oscillation of Certain Neutral Difference Equations of Mixed Type

JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS ARTICLE NO.

224, 241]254 Ž1998.

AY986001

Oscillation of Certain Neutral Difference Equations of Mixed Type S. R. Grace Department of Engineering Mathematics, Faculty of Engineering, Cairo Uni¨ ersity, Orman, Giza, 12000, Egypt Submitted by Ra¨ i P. Agarwal Received September 16, 1997

Some new criteria for the oscillation of certain neutral difference equations of the form, Di Ž x n q ax nym y bx nqk . s c w qx nyg q px nqh x , are established.

c s "1, i s 1, 2, 3

Q 1998 Academic Press

1. INTRODUCTION This article is concerned with the oscillatory behavior of neutral difference equations of the form, Di Ž x n q ax nym y bx nqk . s c w qx nyg q px nqh x ,

Ž Ei , c .

where c s "1, i s 1, 2, 3 a, b, p, and q are nonnegative real numbers, g, h, m, and k are nonnegative integers and are multiples of i. Let D be the first order forward difference operator D x n s x nq1 y x n and for i G 1, let Di be the ith order forward difference operator Di x n s DŽ Diy1 x n .. A solution  x n4 of Eq. Ž Ei , c . is said to oscillate, if for every n 0 G 0 there exists n G n 0 such that x n x nq1 F 0. Otherwise, the solution is called nonoscillatory. There has been a lot of interest in the oscillations of difference equations of different order. See, for example, w5]8, 10]14x and the references cited therein. For the general theory of difference equations the reader is referred to the monographs w1, 2x. 241 0022-247Xr98 $25.00 Copyright Q 1998 by Academic Press All rights of reproduction in any form reserved.

242

S. R. GRACE

Our aim in this article is to establish some sufficient conditions, involving the coefficients and the arguments only, under which all solutions of Eq. Ž Ei , c . oscillate. The advantage of working with these conditions rather than the characteristic equations associated with the equations under considerations, namely, i Ž l y 1 . w 1 q a lym y bl k x s c q lyg q plh ,

c s "1, i s 1, 2, 3

Ž *. is that they are explicit and therefore, easily verifiable, although determining whether or not a positive root to Eq. Ž*. exists is quite a problem in itself. Furthermore, our technique is given in such a way that is can be extended in a straightforward manner to the case of neutral difference equations with variable coefficients. The analogue of Eq. Ž Ei , c . in the continuous case is the neutral functional differential equations, di dt i

Ž x Ž t . q ax Ž t y m . y bx Ž t q k . . s c qx w t y g x q px w t q h x , Ž Ni , c .

i s 1, 2, 3 and c s "1, a, b, g, h, m, k, p, and q are nonnegative real numbers. The oscillatory behavior of Eq. Ž Ni , c ., i G 1, c s "1, was intensively studied by many authors. We refer to w3, 4, 7x and the references cited therein.

2. MAIN RESULTS The following lemma is needed in the proof of our main results and is extracted from w7, 10x and the discrete analogue of the results in w9x. LEMMA 1. Assume that q is a positi¨ e real number and that k is a positi¨ e integer and is a multiple of i. Then the following statements hold: Ža.

If q)

iiŽ k y i.

ky i

kk

,

for k ) i ,

Ž a1 .

then the difference inequality, Di yn G qynqk ,

iG1

Ž I1 .

OSCILLATION OF MIXED TYPE EQUATIONS

243

has no e¨ entually positi¨ e solution  yn4 which satisfies D j yn G 0 e¨ entually, j s 0, 1, . . . , i. Žb. If q)

iik k

Ž k q i.

kqi

,

k G 1,

Ž b1 .

then the difference inequality, i Ž y1. Di yn G qynyk ,

iG1

Ž I2 .

has no e¨ entually positi¨ e solution  yn4 which satisfies Žy1. jD j yn ) 0 e¨ entually, j s 0, 1, . . . , i. Proof. Ža. Assume for the sake of contradiction that inequality Ž I1 . has an eventually positive solution  yn4 which satisfies D j yn ) 0 eventually, j s 0, 1, . . . , i. Set q s p i, k s t i where t is an integer, and wn s Diy1 yn q pDiy2 ynq t q ??? qp iy1 ynqŽ iy1.t . Then eventually wn ) 0.

Ž a2 .

Observe that Dwn s Di yn q pDiy1 ynq t q ??? qp iy1D ynqŽ iy1.t , and therefore, Dwn y pwnq t s Di yn y p i ynqit G 0. That is Dwn y pwnq t G 0 eventually.

Ž I3 .

But because of Ž a1 . and w7, Theorem 7.6.1x, inequality Ž I3 . cannot have an eventually positive solution. This contradicts Ž a2 . and completes the proof of part Ža.. Žb. Assume that inequality Ž I2 . has an eventually positive solution  yn4 which satisfies Žy1. jD j yn ) 0 eventually, j s 0, 1, . . . , i. Set q s p i, k s s i where s is an integer, and ¨ n s Diy1 yn y pDiy2 yny s q ??? qp iy1 ynyŽ iy1. s .

Then eventually ¨ n ) 0.

Ž b2 .

244

S. R. GRACE

Also, in view of Ž I2 . and Ž b 2 ., D¨ n q p¨ ny s i F 0 eventually.

Ž I4 .

But because of Ž b1 . and w7, Theorem 7.6.1x, inequality Ž I4 . cannot have an eventually positive solution. This contradicts Ž b 2 . and completes the proof of part Žb.. The following two theorems are concerned with the oscillatory behavior of Eqs. Ž E1 , 1. and Ž E1 , y1., respectively. THEOREM 1.

Let b ) 0, h ) 1, and r s g q k G 1. If q b

rr

)

Ž1 q r .

1q r

r G 1,

,

Ž 1.

and p 1qa

)

Ž h y 1.

hy 1

,

hh

h ) 1,

Ž 2.

then Eq. Ž E1 , 1. is oscillatory. Proof. Assume for the sake of contradiction that Eq. Ž E1 , 1. has an eventually positive solution  x n4 , say x n ) 0 for n G n 0 G 0. Set yn s x n q ax nym y bx nqk .

Ž 3.

D yn s qx nyg q px nqh G 0, for n G n1 G n 0 ,

Ž 4.

Then

which implies that  yn4 is eventually of one sign. Therefore, either ŽA. yn - 0 eventually, or ŽB. yn ) 0 eventually. ŽA. Assume yn - 0 for n G n1. Set 0 - ¨ n s yyn s bx nqk y ax nym y x n F bx nqk .

Ž 5.

There exists n 2 G n1 such that xn G

1 b

¨ nyk ,

for n F n 2 .

Ž 6.

Using Ž6. in Ž4., we have D¨ n q

q b

¨ nyŽ gqk . F 0,

for n G n 2 .

Ž 7.

OSCILLATION OF MIXED TYPE EQUATIONS

245

But in view of Lemma 1Žb. and condition Ž1., the inequality Ž7. has no eventually positive solution, a contradiction. ŽB. Assume yn ) 0 for n G n1. Set wn s yn q aynym y bynqk .

Ž 8.

Dwn s qynyg q pynqh ,

Ž 9.

D Ž wn q awnym y bwnqk . s qwnyg q pwnqh .

Ž 10 .

Then and Because yn ) 0 and D yn ) 0 for n G n1 , we have D j wn ) 0, j s 1, 2 for n G N1 G n1 and hence, we see that D j wn ) 0, j s 0, 1, 2, n G N1. Using the fact that  Dwn4 is increasing for n G N1 in Ž10., we have

Ž 1 q a. Dwn G pwnqh , or Dwn G

p

Ž 1 q a.

wnqh , for n G N1 .

Ž 11 .

But in view of Lemma 1Ža. and condition Ž2., the inequality Ž11. has no eventually positive solution, a contradiction. This completes the proof. THEOREM 2.

Let b ) 0, s s g y m G 1, and t s h y k ) 1. If q 1qa

)

ss

Ž1 q s.

1q s

,

s G 1,

Ž 12 .

and p b

)

Ž t y 1. tt

ty1

,

t ) 1,

Ž 13 .

then Eq. Ž E1 , y1. is oscillatory. Proof. Let  x n4 be an eventually positive solution of Eq. Ž E1 , y1., say x n ) 0 for n G n 0 . Define yn by Ž3. and obtain D yn s yqx nyg y px nqh F 0, for n G n1 G n 0 .

Ž 14 .

However, as in the proof of Theorem 1, we consider the two cases ŽA. and ŽB.. ŽA. Assume yn - 0 for n G n1. Set ¨ n as in Ž5. and obtain Ž6.. Using Ž6. in Ž14. we have p D¨ n G ¨ nqŽ hyk . , for n G n 2 G n1 . b

246

S. R. GRACE

The rest of the proof is similar to the proof of Theorem 1ŽB. and hence is omitted. ŽB. Assume yn ) 0 for n G n1. Set wn as in Ž8. and obtain Dwn q qynyg q pynqh s 0, and D Ž wn q awnym y bwnqk . q qwnyg q pwnqh s 0.

Ž 15 .

Clearly Dwn - 0 and D2 wn ) 0 for n G N1 G n1. There are two possibilities to consider: Ži. wn - 0 eventually, and Žii. wn ) 0 eventually. Ži. Suppose that wn - 0 for n G N1. Set 0 - Wn s ywn s bynqk y aynym y yn F bynqk . The rest of the proof is similar to the proof of case ŽA. and hence is omitted. Žii. Suppose that wn ) 0 for n G N1. Using the fact that  yDwn4 is decreasing for n G N1 in Ž15. we have

Ž 1 qa. Dwny m q qwnyg F D Ž wn q awnym y bwnqk . q qwnyg q pwnqh s 0, or q

Dwn q

wnyŽ gym. F 0, for n G N1 .

1qa

The rest of the proof is similar to that of Theorem 1ŽA. and hence is omitted. Next, we consider Eq. Ž E2 , c ., c s "1 and we obtain the following oscillation criteria. THEOREM 3.

Let b ) 0, s s g y m G 1, and h ) 2. If q 1qa

)

4ss

Ž2 q s.

2q s

s G 1,

,

Ž 16 .

and p 1qa

)

4 Ž h y 2. hh

hy 2

,

h ) 2,

Ž 17 .

then Eq. Ž E2 , 1. is oscillatory. Proof. Let  x n4 be an eventually positive solution of Eq. Ž E2 , 1., say x n ) 0 for n G n 0 G 0. Set yn as in Ž3.. Then D2 yn s qx nyg q px nqh G 0, for n G n1 G n 0 ,

Ž 18 .

OSCILLATION OF MIXED TYPE EQUATIONS

247

which implies that Di yn , i s 0, 1 are eventually of one sign. Therefore, the two cases ŽA. and ŽB. in the proof of Theorem 1 are considered. ŽA. Assume yn - 0 for n G n1. Let ¨ n be defined as in Ž5. and obtain Ž6., and hence Ž18., takes the form, D2 ¨ n q

q b

¨ nygyk F 0,

for n G n 2 .

Ž 19 .

Because D¨ n ) 0 for n G n 2 , there exist n 3 G n 2 and c1 ) 0 such that ¨ ny gyk G c1 ,

for n G n 3 .

Ž 20 .

Using Ž20. in Ž19. and summing from n 3 to N y 1 G n 3 we have 0 - D¨ N F D¨ n 3 y

q b

c1 Ž N y n 3 . ª y`, as N ª `,

a contradiction. ŽB. Assume yn ) 0 for n G n1. Let wn be defined as in Ž8.. Then D2 wn s qynyg q pynqh ,

Ž 21 .

D2 Ž wn q awnym y bwnqk . s qwnyg q pwnqh .

Ž 22 .

and

There are two possibilities to consider: Ži. D yn ) 0 eventually, and Žii. D yn - 0 eventually. Suppose that Ži. holds. Then Di wn ) 0, i s 2, 3 for n G N1 G n1 , and hence we obtain Di wn ) 0,

i s 0, 1, 2, 3, and n G N1 .

Ž 23 .

Using the fact that  D2 wn4 is an increasing sequence for n G N1 in Ž22. we have p D2 wn G w , n G N1 . Ž 24 . 1 q a nqh But in view of Lemma 1Ža. and condition Ž17., inequality Ž24. has no eventually positive solution  wn4 satisfying Ž23., a contradiction. Next, suppose that Žii. holds. From Ž21., we see that D2 wn ) 0 and D3 wn - 0 for n G N2 G n1. Therefore, we have three possibilities to consider: ŽI. wn - 0, Dwn - 0 eventually, ŽII. wn ) 0, Dwn ) 0 eventually, and ŽIII. wn ) 0, Dwn - 0 eventually. The proof of case ŽI. is similar to the proof of case ŽA. presented in the previous text hence is omitted.

248

S. R. GRACE

ŽII. Let wn ) 0 and Dwn ) 0 for n G N2 . However, in the proof of case ŽB.Ži. and the fact that the sequence  D2 wn4 is decreasing for n G N2 , we obtain p D2 wny m G w , 1 q a nqh or D2 wn G

p 1qa

wnqhqm , for n G N2 .

Because the sequence  wn4 is increasing for n G N2 , we see that D2 wn G

p 1qa

wnqh , for n G N2 .

The rest of the proof is similar to that of case ŽB.Ži. and hence is omitted. ŽIII. Let wn ) 0 and Dwn - 0 for n G N2 . Then the sequence  wn4 satisfies i Ž y1. Di wn ) 0,

i s 0, 1, 2, 3, and n G N2 .

Ž 25 .

From Ž10., we have D2 wn G

q 1qa

wnyŽ gym. , for n G N2 .

Ž 26 .

But, in view of Lemma 1Žb. and condition Ž16., inequality Ž26. has no eventually positive solution  wn4 satisfying Ž25., a contradiction. This completes the proof. THEOREM 4.

Let b ) 0, t s h y k ) 2, and r s g q k G 1. If q b

)

4r r

Ž2 q r .

2q r

,

r G 1,

Ž 27 .

t ) 2,

Ž 28 .

and p b

)

4 Ž t y 2. tt

ty2

,

then Eq. Ž E2 , y1. is oscillatory. Proof. Let  x n4 be an eventually positive solution of Eq. Ž E2 y1., say x n ) 0 for n G n 0 G 0. Next, define yn by Ž3. and obtain D2 yn s y Ž qx nyg q px nqh . F 0, for n G n1 G n 0 ,

Ž 29 .

OSCILLATION OF MIXED TYPE EQUATIONS

249

which implies that Di yn , i s 0, 1, 2 are eventually of constant sign. The two possibilities ŽA. and ŽB. in the proof of Theorem 3 are considered: ŽA. Let yn - 0 for n G n1. Set ¨ n as in Ž5. and obtain Ž6.. Then Ž29. takes the form, D2 ¨ n G

q b

p

¨ nyŽ gqk . q

b

¨ nqŽ hyk . ,

for n G n 2 G n1 .

Therefore, D¨ n is eventually positive or eventually negative. If D¨ n ) 0 for n G n 2 , then p

D2 ¨ n G

b

¨ nqt ,

n G n2 .

¨ nyr ,

n G n2 .

Also, if D¨ n - 0, n G n 2 , then D2 ¨ n G

q b

The rest of the proof is similar to that of Theorem 3ŽB. and hence is omitted. ŽB. Let yn ) 0 for n G n1. Then D yn ) 0 for n G n 2 G n1. Next, let wn be defined as in Ž8.. Thus, D2 wn q qynyg q pynqh s 0,

n G n2 .

Ž 30 .

Clearly D3 wn - 0 and D2 wn - 0 for n G n 2 and hence we conclude that Dwn - 0 and wn - 0 for n G n 2 . Now, set 0 - Wn s ywn s bynqk y aynym y yn F bynqk , and hence we obtain yn G

1 b

Wnyk , for n G N G n 2 .

Ž 31 .

Using Ž31. in Ž30., we have D2 Wn G

p b

Wnqhyk ,

n G N.

The rest of the proof is similar to that of Theorem 3ŽB.Ži. and hence is omitted. Finally, we give the following two criteria for the oscillation of Eq. Ž E3 , c ., c s "1.

250

S. R. GRACE

THEOREM 5.

Let b ) 0, r s g q k G 1, and h ) 3. If q b

)

27r r

Ž3 q r .

3q r

r G 1,

,

Ž 32 .

and p 1qa

)

27 Ž h y 3 . hh

hy 3

,

h ) 3,

Ž 33 .

then Eq. Ž E3 , 1. is oscillatory. Proof. Let  x n4 be an eventually positive solution of Eq. Ž E3 , 1., say x n ) 0 for n G n 0 G 0. Let yn be as in Ž3.. Then D3 yn s qx nyg q px nqh G 0, for n G n1 G n 0 ,

Ž 34 .

which implies that Di yn , i s 0, 1, 2, 3 are eventually of one sign. However, in the proof of Theorem 1, we consider the two cases ŽA. and ŽB.. ŽA. Assume yn - 0 for n G n1. Set ¨ n as in Ž5. and obtain Ž6.. Thus, D3 ¨ n q

q b

¨ nygyk F 0,

for n G n 2 G n1 .

Ž 35 .

There are two possibilities: Ži. D¨ n ) 0 eventually Žii. D¨ n - 0 eventually. If Ži. holds, then there exist n 3 G n 2 and c1 ) 0 such that Ž20. holds. Using Ž20. in Ž35. and summing from n 3 to N y 1 G n 3 we have 0 - D3 ¨ N F D3 ¨ n 3 y

q b

c1 Ž N y n 3 . ª y`, as N ª `,

which is a contradiction. Next, if Žii. holds, then the sequence  ¨ n4 satisfies i Ž y1. Di ¨ n ) 0,

i s 0, 1, 2, 3, and n G n 2 .

Ž 36 .

But, in view of Lemma 1Žb. and condition Ž32., the inequality Ž35. has no eventually positive solution  ¨ n4 satisfying Ž36., a contradiction. ŽB. Assume yn ) 0 for n G n1. Then there exists n 2 G n1 such that D yn ) 0 for n G n 2 . Now let wn be as in Ž8.. Then D3 wn s qynyg q pynqh ,

Ž 37 .

D3 Ž wn q awnym y bwnqk . s qwnyg q pwnqh .

Ž 38 .

and

OSCILLATION OF MIXED TYPE EQUATIONS

251

From Ž37., we see that Di wn ) 0 for i s 3, 4, n G n 2 and hence we have Di wn ) 0,

i s 0, 1, 2, 3, 4, and

n G n2 .

Ž 39 .

Using Ž39. in Ž38., we obtain D3 wn G

p 1qa

n G n2 .

wnqh ,

Ž 40 .

But, in view of Lemma 1Ža. and condition Ž33., the inequality Ž40. has no eventually positive solution satisfying Ž39., a contradiction. This completes the proof. THEOREM 6.

Let b ) 0, s s g y m G 1, and t s h y k ) 3. If q 1qa

)

27s s

Ž3 q s.

3q s

,

s G 1,

Ž 41 .

and p b

)

27 Ž t y 3 . tt

ty3

,

t ) 3,

Ž 42 .

then Eq. Ž E3 , y1. is oscillatory. Proof. Let  x n4 be an eventually positive solution of Eq. Ž E3 , y1., say x n ) 0 for n G n 0 G 0. Define yn by Ž3. and obtain D3 yn s yqx nyg y px nqh F 0, for n G n1 G n 0 ,

Ž 43 .

which implies that Di yn , i s 0, 1, 2, 3 are eventually of constant sign. The two cases ŽA. and ŽB. in the proof of Theorem 5 are also considered: ŽA. Suppose yn - 0 for n G n1. Set ¨ n as in Ž5. and obtain Ž6.. Thus, D3 ¨ n G

p b

¨ nqŽ hyk . ,

for n G n 2 G n1 .

Ž 44 .

There exists n 3 G n 2 such that D¨ n ) 0 for n G n 3 and D2 ¨ n - 0 or D2 ¨ n ) 0 for n G n 3 . If D2 ¨ n - 0, n G n 3 , there exists n 4 G n 3 and c 2 ) 0 such that ¨ nq hyk G c 2 ,

for n G n 4 .

Using Ž45. in Ž44. and summing from n 4 to N y 1 G n 4 we have 0 ) D2 ¨ N G D2 ¨ n 4 q

p b

c 2 Ž N y n 4 . ª `, as N ª `,

Ž 45 .

252

S. R. GRACE

a contradiction. Next, if D2 ¨ n ) 0 for n G n 3 , then the sequence  ¨ n4 satisfies Di ¨ n , i s 0, 1, 2, 3 and n G n 3 . The rest of the proof is similar to that of Theorem 5ŽB. and hence is omitted. ŽB. Suppose yn ) 0 for n G n1. Define wn by Ž8. and obtain D3 wn q qynyg q pynqh s 0,

Ž 46 .

D3 Ž wn q awnym y bwnqk . q qwnyg q pwnqh s 0.

Ž 47 .

and

Now, if D yn ) 0, n G n 2 G n1 , then, D4 wn - 0 and D3 wn - 0 for n G n 2 and hence wn ª y` as n ª `. There exists N G n 2 such that wn - 0 for n G N. Set Wn s ywn and proceed as in case ŽA. in the preceding text, and we arrive at a contradiction. Finally, if D yn - 0 for n G n 2 , one can easily see that the sequence  wn4 satisfies i Ž y1. Di wn ) 0,

i s 0, 1, 2, 3, and

n G n2 .

Ž 48 .

Using Ž48. in Ž47. we have D3 wn q

q 1qa

wnyŽ gym. F 0, for n G n 2 .

The rest of the proof is similar to that of Theorem 5ŽA.Žii. and hence is omitted. The following examples are illustrative: EXAMPLE 1. Each of the following neutral difference equations, Di Ž x n q e m x nym y eyk x nqk . s

Ž e y 1. 2

i

e g x nyg q eyh x nqh , Ž Fi .

where i s 1, 2, 3, g, h, m and k are multiples of i, has a nonoscillatory solution x n s e n ª ` monotonically as n ª `. We observe that for i s 1 condition Ž2. of Theorem 1 is violated, for i s 2, condition Ž17. of Theorem 3 fails and for i s 3 condition Ž38. of Theorem 5 also fails. Remark 1. If p s 0, or conditions involve p in Theorems 1]6 are violated, one can easily observe that the conclusion of Theorems 1]6 may be replaced by: ‘‘Every solution  x n4 of each of the Equations Ž Ei , c ., i s 1, 2, 3, c s "1 is oscillatory or else D j x n ª ` monotonically as n ª `, j s 0, 1, . . . , i y 1.’’

OSCILLATION OF MIXED TYPE EQUATIONS

253

EXAMPLE 2. Each of the following neutral difference equations, Di Ž x n q eym x nym y e k x nqk . s

1 1 2

ž

e

i

y1

/

eyg x nyg q e h x nqh , Ž Hi .

where i s 1, 2, 3, g, h, m, and k are multiples of i, has a nonoscillatory solution x n s eyn ª 0 monotonically as n ª `. We see that for i s 1, condition Ž2. of Theorem 1 fails, for i s 2, condition Ž16. of Theorem 3 is violated and for i s 3 condition Ž41. of Theorem 6 is also violated. Remark 2. When q s 0, or conditions involve q in Theorems 1]6 are violated, one can easily see that the conclusions of these theorems may be replaced by: ‘‘Every solution  x n4 of each of the Equations Ž Ei , c .,i s 1, 2, 3 and c s "1 is oscillatory or else D j x n ª 0 monotonically as n ª `, j s 0, 1, . . . , i y 1.’’ EXAMPLE 3. Each of the following neutral difference equations, Di Ž x n q ax nym y ax nqm . s 2 iy1 Ž y1 .

gq i

Ž x nyg q x nqg . ,

Ž Oi .

where i s 1, 2, 3, a is a positive real number, m and g are positive integers, has an oscillatory solution x n s Žy1. n. All conditions of Theorems 1, 4, and 5 are satisfied for g is odd and also all conditions of Theorems 2, 3, and 6 are satisfied for g is even. Remark 3. 1. The results of this article are presented in a form which is essentially new. We note that the results of this article are not applicable to Eq. Ž Ei , c ., i s 1, 2, 3, c s "1 when p s 0 or q s 0. It is remarkable that our results are possibly valid if p s 0 or q s 0 Žbut not p s q s 0. provided that either a s 0, b s 0 or a s b s 0. Here we omit the details. 2. By using the same technique presented in this article, one can easily obtain similar criteria for Eq. Ž Ei , c . for a different sign of m and k and also for a different sign of a and b. The details are left to the reader. 3. It would be interesting to obtain results similar to these presented here for Ž Ei , c ., i ) 3, c s "1. REFERENCES 1. R. P. Agarwal, ‘‘Difference Equations and Inequalities,’’ Dekker, New York, 1992. 2. R. P. Agarwal and P. J. Y. Wong, ‘‘Advanced Topics in Difference Equations,’’ Kluwer, Dordrecht, 1997. 3. S. R. Grace, Oscillation criteria for nth order neutral functional differential equations, J. Math. Anal. Appl. 184 Ž1994., 44]55.

254

S. R. GRACE

4. S. R. Grace and B. S. Lalli, Oscillation theorems for second order neutral functional differential equations, Appl. Math. Comput. 51 Ž1992., 119]133. 5. S. R. Grace and B. S. Lalli, Oscillation theorems for second order delay and neutral difference equations, Utilitas Math. 45 Ž1994., 197]211. 6. S. R. Grace and B. S. Lalli, Oscillation theorems for forced neutral difference equations, J. Math. Anal. Appl. 187 Ž1994., 91]106. 7. I. Gyori and G. Ladas, ‘‘Oscillation Theory of Delay Differential Equations with Applications,’’ Oxford Univ. Press, Oxford, U.K., 1991. 8. J. W. Hooker and W. T. Patula, Growth and oscillation properties of solutions of a fourth order linear difference equations, J. Austral. Math. Soc. Ser. B 26 Ž1985., 310]328. 9. T. Kusano, On even order functional differential equations with advanced arguments and retarded arguments, J. Differential Equations 45 Ž1982., 75]84. 10. G. Ladas and C. Qian, Comparison results and linearized oscillations for higher order difference equations, Internat. J. Math. Math. Sci. 15 Ž1992., 129]142. 11. B. S. Lalli, B. G. Zhang, and J. Zhao, On oscillation and existence of positive solutions of neutral difference equations, J. Math. Anal. Appl. 158 Ž1991., 213]233. 12. B. S. Lalli and S. R. Grace, Oscillation theorems for second order neutral difference equations, Appl. Math. Comput. 62 Ž1994., 47]60. 13. Ch. G. Philos, Oscillation in a class of difference equations, Appl. Math. Comput. 48 Ž1992., 45]57. 14. Ch. G. Philos and Y. G. Sficas, Positive solutions of difference equations, Proc. Amer. Math. Soc. 108 Ž1990., 107]115.