Oscillation in neutral delay difference equations with variable coefficients

Computers Math. Applic. Vol. 29, No. 3, pp. 5-11, 1995 Copyright@1995 Elsevier Science Ltd Printed in Great Britain. All rights reserved

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Oscillation in N e u t r a l D e l a y Difference E q u a t i o n s w i t h Variable Coefficients M.-P.

CHEN

Institute of Mathematics, Academia Sinica Nankang, Taipei 11529, Taiwan, R.O.C. MAAP0©ccvax. sinica, edu. tw

B. S. LALLI Department of Mathematics and Statistics,University of Saskatchewan Saskatoon, Saskatchewan S7N O W O , Canada Lalli©popserver. Usask. ca

J. S. Y u Department of Applied Mathematics, Hunan University Changsha, Hunan 410082, P.R. China (Received and accepted June 1994)

Abstract--In

this paper, we consider the neutral delay difference equation A(yn -- PnYn-k) + qnYn-t = 0,

n = 0, 1, 2....

(1)

where Pn, qn (n = 0, 1, 2,... ) are real numbers with Pn > O, qn >_ O, k and g are nonnegative integers. We establish some sufficmnt conditions for the oscillation of all solutions of (1) without the condition iq,~

(~)

= oo.

1%=0

K e y w o r d s - - O s c i l l a t i o n , Difference equation. 1.

INTRODUCTION

We consider t h e n e u t r a l delay difference e q u a t i o n

A(yn - P,~Yn-k) + q n Y n - e = O,

n = O, 1, 2 , . . .

(1)

where pn, qn ( n = 0, 1, 2 , . . . ) are real n u m b e r s w i t h Pn >- 0, qn ~- 0, k a n d ~ are n o n n e g a t i v e integers, a n d A d e n o t e s the forward difference o p e r a t o r A x n ----Xn+l - x,~. E q u a t i o n (1) was first c o n s i d e r e d b y B r a y t o n a n d W i l l o u g h b y [1] from the n u m e r i c a l analysis p o i n t of view. O u r a i m is to establish some sufficient c o n d i t i o n s for the oscillation of all solutions of (1) without requiring that oo

(2) n=0

Authors express heartfelt thanks to the referee for his critical review and helpful suggestions for the improvement of the paper. This work was partially supported, to the first author, by the National Science Council of Taiwan under Grant NSC-82-0208-M-l-145, to the second author, by NSERC Grant Number A5290 and a grant from President's Research Fund, and to the third author, by the National Natural Science Foundation of China. Typeset by ¢4A48-TEX

6

M.-P. CHEN et aL

The oscillatory and asymptotic behavior of (1) has been investigated by several authors; see for example [2-7]. However, in most of the papers dealing with (1), hypothesis (2) is required. Therefore, it is useful to study (1) when condition (2) does not hold. In a recent paper [7], the third author and Wang obtained the following result which does not require the condition (2). THEOREM Y W . A s s u m e t h a t Pn =- 1 a n d t h a t oo

oo

nqn n=O

= oo

(3)

j=n

T h e n every s o l u t i o n o f (1) oscillates.

We note that condition (3) is better than (2). For example, for the equation A(yn - yn-3) + (n + 1)-ayn_4 = 0, condition (3) is satisfied when 1 < a < 3/2. Therefore, every solution of this equation is oscillatory. However, condition (2) is not fulfilled. In this paper, we establish several oscillation results for (1) when condition (2) is not satisfied. This is done in Section 3 by using some lemmas which are included in Section 2. For the sake of convenience, we define Oo

Ho,n = E q i .

Let m = max{k, £}. By a solution of (1) we mean a sequence {Yn} of real numbers which is defined for n > - m and satisfies (1) for n = 0, 1, 2 , . . . . It is easy to see under the initial conditions: Yn = A n ,

n = -m, -m

+ 1 . . . . , O.

(4)

Equation (1) has a unique solution satisfying (4). A solution { Y n } of (1) is said to be nonoscillatory if the terms Yn are either eventually positive or eventually negative. Otherwise, the solution is called oscillatory. For the general background on difference equations, one can refer to [8-10]. The results in this paper are discrete analogues of some results in [11] recently obtained for the neutral delay differential equation [x(t) + p ( t ) x ( t - T)]' + q ( t ) x ( t -- 5) = O. 2. S O M E

BASIC

LEMMAS

We make the following assumptions which we need in Section 3. (A1) There exists a positive integer n* such that Pn'+ik ~ 1

for i = 0, 1,2,...

(5)

(A2) There exists a nonnegative integer s such that the sequences oo

H.,

and that

=

iq,

exist for j = 1,2,... ,s

o~

E i=0

iq~Hs,i = oo.

(6)

Neutral Delay Difference Equations

7

Now we establish two lemmas which are needed for establishing our oscillation criteria. LEMMA 1. Assume that (A1) holds. Let xn -- Yn - PnYn-k,

(7)

where {Yn} is an eventually positive solution of the difference inequality A(Yn - PnYn-k) + q,~Yn-e < O.

(8)

Then, we have eventually

=.

(9)

> 0.

PROOF. Let nl be a positive integer such that Yn-m > 0 for all n >_ nl, where m -- max{k,g}. Then, by (7) and (8), we have eventually Ax,~ = - q ~ Y n - t <_ O,

for all n _> nl,

which implies that xn is nonincreasing for n > nl. Hence, if (9) does not hold, then we would have eventually x~ < 0. Therfore, there exist an integer n2 > nl and a constant a > 0 such that for all n > n2.

xn <_ - a , Therefore, we have Yn ~- -oL + PnYn-k,

for all n _> n2.

(10)

Now choose a positive integer n3 such that n* + n3k >_ n2. Thus, by (A1) and (10) we have for j = 0,1,2,... Yn*+nak+jk <-- --(j + 1)a + Y,~*+(na-1)k --* --CO

as j

co,

which contradicts the assumption that Yn is eventually positive. LEMMA 2. Assume that (A2) holds and Pn >- 1,

[orn= 0,1,2,...

(11)

Let {Yn} and {xn} be as in L e m m a 1. Then we have eventually x,~ < O.

(12)

PROOF. It is easy to note that xn is eventually nonincreasing. We assume that xn > 0 eventually. From (11) it follows that y,~ > Yn-k. Hence, there exists a constant M > 0 and a positive integer nl such that Yn-m >_ M for all n > nl. Then, by (8), we have A x n <_ - M q n , for n > nl. And hence, oo

xn >_ M E

qi = MHo,n,

for n > nl,

i=n

which, together with (11) and (7), yields Yn > Yn-k + MHo,n,

(13)

n > nl.

Let ml,n be the greatest integer value of (n - n l ) / k . Then we have Yn > M[Ho,n "4- Ho,n-k -4-'" -4- Ho,n-(m,,,-x)k] + Yn-(ml.,)k

>_M[Ho,,~ + Ho,n-k + ' " + Ho,n-(m,,.-1)k],

n >_nx + k.

8

M.-P. CHEN et al.

In view of the fact that Ho,n is decreasing, we get

n>nl

Yn > Mml,nHo,n,

+k.

Using this inequality in (8), we obtain

A x n <_ -Mqnml,n-~Ho,n-t n>_nl+k+g.

<_ -Mqnml,n-tHo,n,

(14)

Since n/(ml,~-~) ---* k as n ---* co, if follows that

Mqnml,n-~Ho,n nq,~Ho,,~

M k

--..4, - -

a~

n

----~ o o .

Thus, there exists an integer n2 > nl + k + g such that M

Mqnml,,~-~Ho,n >_ ~-~nq,~Ho,n

for n _> n2,

and consequently (14) yields

M Az,~ <_ -~-~nq,~Ho,,~,

for n _> n2.

(15)

In case s = 0, then by directly summing up (15) one gets a contradiction. Therefore, suppose that s ~ 0. Summing (15) from n to oo, we have x n _ > ~M H 1,n,

for n _> n2,

which, together with (8) and (11), yields M

Y,~ >- Yn-k + "~gl,n,

(16)

for n > n2.

Now we let m2,~ denote the greatest integer value of (n - n2)/k. Then

Yn >- -~[Hl,n + Hl,n-k + "" + Hl,n-(m=.~-l)k] + Yn-m2,.~k M [HI,n + Hl,,~-k + " " + Hl,n-(m2,,-1)k],

n>n2+k.

Using the fact that Hl,n is nonincreasing, we have m H 1,n, y,~ >_ ~'-~m2,n

for n > n2 + k.

Thus M

A x n < - ~--km2,,~-tqnHl,n-t M ~__--~;-, m2,n-tqnH1 "zig

for n > n2 + k + g.

Since n/m2,n-~ --4 k as n --* 00, there exists a positive integer n3 _> n2 + k + g such that n

m2,,~-~ _> ~ ,

for n _> n 3.

Neutral Delay Difference Equations

9

Hence, M A x n <_ - ,~.,~nqnHl,n,

n >_ n3,

which yields M

Xn >_ - ~

Thus

H 2,n,

r~ >_ n3.

M Yn >_ Yn-k + ,~,,oH2,n,

07)

n > n3.

By repeating the above procedure, we can choose a positive integer n4 _> n3 + k + ~ such t h a t M A x n <_ - , ~ . , ~ n q n H 3 , n , (za) o

n >_ n4.

In general, by mathematical induction, there exists a sequence {ni} of positive integers such t h a t M A x n <_ - ~ n q n H i , n ,

n >_ ni+l.

M Axn < -,~,,.nqnHs,n,

n > ns+l,

In particular, we have

which, in view of (6), implies t h a t Xn ----4 --(X)

as n ----+Do

is a contradiction and hence (12) holds. The proof is complete.

3. M A I N R E S U L T S In this section, we apply the lemmas in Section 2 to establish several new oscillation criteria for (1) without the condition (2). Theorem 1 is an immediate consequence of L e m m a s 1 and 2, which contains T h e o r e m 3 in [7] as a special when case s = 0. THEOREM 1. Assume that (A2) holds with Pn - 1. Then every solution of (1) oscillates. EXAMPLE 1. Consider the neutral delay difference equation 1

A(x~ - xn_k) + (n - - - +- - -1) - ~ xn-~ ----0.

(El)

Since 1/((1 + n)l'6)[n -°'6 - (n + 1)-°"6] -1 --* 5/3 as n -* co, it follows t h a t there exists an integer no such t h a t n -0"6 - (n + 1) -0.8 <

and hence

( l + n ) 1.6 oo

n -°'6

< Ho,. = ~

< 2In - ° ' s - (n + 1)-°"61,

n _>no,

1

(1 +

n ) 1.6 < 2n -°'6,

n _> no.

Thus we have n n_O. 0 n .2n_O.6, (1 + n) 1"6 < nqnHo,n < (1 + n ) 1"6

T~ ~ nO,

10

M.-P. CHEN et al.

and

oo

oo

n=0

j=n

ZnqoZq,< Since

nO.4 (1 + n) 1-6 [n-°'2 - (7"/.+ 1)-°"2] -1 --4 5

as?Z--*

OO,

there exists another integer nl > no such that n0.4

n -°'2 - (n + 1) -0.2 < (1 + n) 1.6 < 6[n-°'2 - (n + 1)-°'2],

n ~ 7~1,

which implies oo

Hl,n = ~

iqiHo,i > n -°'2,

n>_nl.

i=n

Since

n n q n H l , n > (1 + n) 1.6

it follows that

• Tt - 0 . 2

n 0s (1 -[- n) 1"6'

_

n ~ hi,

(2O

~-~ nqnHl,n -~ oo. n=0

Thus with s = 2 the condition (A2) is satisfied. Thus, by Theorem 1, every solution of (El) is oscillatory. THEOREM 2. Assume that both (A1) and (A2) hold and that Pn-eq,~ >_ qn-k

for n _> m.

(18)

T h e n every solution o f (1) oscillates.

PROOF. Otherwise, (1) would have an eventually positive solution {Yn}. Let Xn he as defined by (7). By Lemma 1, there exists an integer no _> 0 such that xn >_ 0 for n _> no. Using (1) and (18), we have A X n = --qnYn-e = --qn[Xn-e + P n - e Y n - e - k ] = --qnXn-e -- qnPn-eYn-~-k <__--qnXn-e -- q n - k Y n - e - k = --qnXn-e q- A x n - k ,

n >_ no + k + g.

That is, A [ x n - Xn-k] + q,~xn-e < O,

n > no + k + e.

(19)

By Lemmas 1 and 2, assumption (A2) implies that (19) cannot have an eventually positive solution, which is a contradiction. With this contradiction the proof is complete. THEOREM 3. In addition to (A2) and (11), suppose that Pn-eqn <- qn-k

for n >_ m.

(20)

T h e n every solution o f (1) oscillates.

PROOF. Otherwise, (1) would have an eventually positive solution {Yn}. We take {Xn} as defined by (7). By Lemma 2, there exists a positive integer nl such that Xn < 0 for n > nz. Thus, by (1) and (20), we have A x n = --qnXn-e = --qn[Xn-e + Pn-~Yn-t-k] > --qnXn-e -- q n - k Y n - e - k = --qnXn-e q-

mxn-k,

Neutral Delay Difference Equations

11

which shows t h a t { - x n } is an eventually positive solution of the inequality

A(zn -

z n - k ) + q~z~_e <_ O.

But, by using L e m m a s I and 2, (21) can not have any eventually positive solution, a contradiction, and thus the proof of the t h e o r e m is complete. EXAMPLE 2. T h e neutral delay difference equation A

(

n+2 ) Yn-- n + l yn-1 + ( n + l ) - S / a y n _

1 =0

(E2)

satisfies the conditions of T h e o r e m 3. Therefore, every solution of (E2) oscillates. We believe t h a t none of the criteria in literature is applicable to (E2). THEOaEM 4. A s s u m e t h a t ( A 2 ) and (11) hold a n d t h a t there exists a positive integer n* such t h a t Pn*+ik = 1 for i = 0, 1 , 2 , . . . . T h e n every solution o f (1) oscillates. PROOF. T h e p r o o f of this t h e o r e m is obvious from L e m m a s 1 and 2. EXAMPLE 3. T h e neutral difference equation A ( x n - p,~Xn-2) q- (n ÷ 1)-l"Sx,~_2 = 0

(E3)

satisfies all the conditions of T h e o r e m 4, where {Pn} is defined as P0 -- 2, Pl = 1, Pn+2 -~ Pn, n = 0, 1, 2 , . . . . Therefore, by T h e o r e m 4, every solution of (E3) oscillates.

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