Applied Mathematics and Computation 163 (2005) 909–918

www.elsevier.com/locate/amc

Oscillation for second-order nonlinearneutral delay difference equations

Y.G. Sun a,*, S.H. Saker b

a Department of Mathematics, Qufu Normal University, Qufu, Shandong 273165, Chinab Faculty of Science, Mathematics Department, Mansoura University, Mansoura 35516, Egypt

Abstract

Using the Riccati transformation techniques, we will extend some oscillation criteria

of [Appl. Math. Comput. 146 (2003) 791] and [Appl. Math. Comput. 142 (2003) 99] to

the second-order nonlinear neutral delay difference equation

* Co

E-m

0096-3

doi:10.

DðanðDðxn þ pnxn�sÞÞcÞ þ f ðn; xn�rÞ ¼ 0; n ¼ 0; 1; 2; . . .

in the case when 0 < c < 1, which answers a question posed by Saker [Appl. Math.Comput. 142 (2003) 99]. Two examples are considered to illustrate our main results.

� 2004 Elsevier Inc. All rights reserved.

Keywords: Oscillation; Riccati techniques; Second-order neutral difference equations

1. Introduction

We consider the second-order nonlinear neutral delay difference equation

DðanðDðxn þ pnxn�sÞÞcÞ þ f ðn; xn�rÞ ¼ 0; n ¼ 0; 1; 2; . . . ; ð1Þ

where c > 0 is a quotient of odd positive integers, D denotes the forward dif-ference operator Dxn ¼ xnþ1 � xn for any sequence fxng of real numbers, s, r arefixed nonnegative integers, fang and fpng are real sequences satisfying

rresponding author.

ail addresses: yugsun@263.net (Y.G. Sun), shsaker@mum.mans.edu.eg (S.H. Saker).

003/$ - see front matter � 2004 Elsevier Inc. All rights reserved.

1016/j.amc.2004.04.017

910 Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918

ðH1Þ an > 0;X1n¼0

ð1=anÞ1=c ¼ 1; 06 pn < 1;

ðH2Þ f ðn; uÞ : Z R ! R is continuous, and there exists a nonnegative sequencefqng such that f ðn; uÞsgnuP qnuc for n 2 Z, where fqng is not identically zerofor large n.By a solution of (1) we mean a nontrivial sequence fxng which is defined for

nP � N , where N ¼ maxfs; rg, and satisfies Eq. (1) for nP 0. Clearly, if the

initial condition xn ¼ /n for n ¼ �N ; . . . ;�1; 0 is given, then Eq. (1) has aunique solution satisfying the initial condition. A solution fxng of (1) is said tobe oscillatory if for every n0 > 0 there exists an nP n0 such that xnxnþ16 0,otherwise it is nonoscillatory. Eq. (1) is said to be oscillatory if all its solutions

are oscillatory.

In recent years, there has been an increasing interest in studying the oscil-

lation and nonoscillation of solutions of the second-order neutral delay dif-

ference equations. For example, see the monographs [1,2] and the papers

[3,4,6–18] and the references therein. Speaking of oscillation theory of second-

order neutral delay difference equations, most of the previous studies have been

restricted to the linear and nonlinear cases in which c ¼ 1 and f ðn; uÞ ¼ qnf ðuÞ,where f ðuÞ is a continuous function in R.Recently, Jiang [7] and Saker [10] studied the oscillatory behavior of solu-

tions of Eq. (1), respectively. By using the Riccati transformation techniques,

they presented some new oscillation criteria for Eq. (1). However, it is obvious

that Theorems 1–3 in [7] and Theorems 2.2–2.5 in [10] are all focused on the

assumption c > 1. It would be interesting to extend them in the case when

0 < c < 1.The purpose of this paper is to extend the main results of [7] and [10] to Eq.

(1) in the case when 0 < c < 1. Particularly, our results hold also for the casewhen cP 1. At the end of this paper, two examples are considered to illustrate

our main results.

2. Main results

In the sequel, we assume that c > 0. In order to prove our theorems, we needthe following lemma. The similar result can be found in [5].

Lemma. Let

f ðuÞ ¼ bu� auðcþ1Þ=c;

where a > 0 and b are constants, c is a quotient of positive odd integers. Then fassumes its maximal value on R at

Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918 911

u� ¼ bcaðc þ 1Þ

� �c

and

maxu2R

f ðuÞ ¼ f ðu�Þ ¼ cc

ðc þ 1Þcþ1bcþ1

ac: ð2Þ

The proof of the Lemma is evident, and hence is omitted.

Theorem 2.1. Assume that ðH1Þ and ðH2Þ hold. Furthermore, assume that thereexist two positive sequences fqng and f/ng such that

lim supm!1

Xm�1n¼n0

qn/nqnð1

264 � pn�rÞc �

qcþ1nþ1an�r Dþ/n þ /nDþqn

qnþ1

� �cþ1

ðc þ 1Þcþ1qcn/

cn

375 ¼ 1

ð3Þ

for some n0 > 0, where Dþ/n ¼ maxf0;D/ng and Dþqn ¼ maxf0;Dqng, then Eq.(1) is oscillatory.

Proof. Suppose to the contrary that fxng is a positive solution of (1) such thatxn�N > 0 for nP n0 > 0. Set zn ¼ xn þ pn�sxn�s, then zn > 0 for nP n0. From (1)and ðH2Þ it follows that

DðanðDznÞcÞ6 � qnxcn�r 6 0; nP n0: ð4Þ

It is not difficult to show that Dzn is eventually positive. In fact, first, we knowthat Dzn 6� 0 for sufficiently large n, since fzng is nontrivial. Second, if thereexists an integer n1 > n0 such that an1ðDzn1Þ

c ¼ c < 0, then anðDznÞc 6 c fornP n1, i.e., Dzn 6 ð c

anÞ1=c, and hence zn 6 zn1 þ

Pn�1i¼n1

ð caiÞ1=c ! �1 as n ! 1,

which contradicts the fact that zn > 0. Without loss of generality, say Dzn > 0for nP n0. Thus, we have

xn�r P ð1� pn�rÞzn�r; nP n1 ¼ n0 þ r:

From (1) and the above inequality, we have

DðanðDznÞcÞ þ qnð1� pn�rÞczcn�r 6 0; nP n1: ð5Þ

Using (4), we have

an�rðDzn�rÞc P anþ1ðDznþ1Þc; nP n1;

i.e.,

Dzn�r

Dznþ1P

anþ1an�r

� �1=c; nP n1: ð6Þ

912 Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918

Define the sequence fwng by

wn ¼ qnanðDznÞc

zcn�r; nP n1: ð7Þ

Then wn > 0, and

Dwn ¼ anþ1ðDznþ1ÞcDqn

zcn�r

� �þ qnDðanðDznÞ

cÞzcn�r

: ð8Þ

From (5)–(7), we have that

Dwn 6 � qnqnð1� pn�rÞc þDqn

qnþ1wnþ1 �

qnanþ1ðDznþ1ÞcDðzcn�rÞ

zcnþ1�rzcn�r

: ð9Þ

By the mean value theorem, there exists n 2 ðzn�r; znþ1�rÞ such that

Dðzcn�rÞ ¼ cnc�1Dzn�r: ð10Þ

Thus, from (6), (9) and (10) we have

Dwn 6 � qnqnð1� pn�rÞc þDqn

qnþ1wnþ1 � c

ncqnanþ1ðDznþ1ÞcDzn�r

nzcnþ1�rzcn�r

6 � qnqnð1� pn�rÞc þDqn

qnþ1wnþ1 � c

ncqnðanþ1Þðcþ1Þ=cðDznþ1Þcþ1

a1=cn�rnzcnþ1�rz

cn�r

6 � qnqnð1� pn�rÞc þDqn

qnþ1wnþ1 � c

qnðanþ1Þðcþ1Þ=cðDznþ1Þcþ1

a1=cn�rzcþ1nþ1�r

¼ �qnqnð1� pn�rÞc þDqn

qnþ1wnþ1 �

cqn

qknþ1ak�1

n�r

wknþ1; ð11Þ

where k ¼ ðc þ 1Þ=c. Multiplying (11) by /n, we have that

qn/nqnð1� pn�rÞc 6 � /nDwn þ/nDqn

qnþ1wnþ1 �

cqn/n

qknþ1ak�1

n�r

wknþ1: ð12Þ

Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918 913

Using the summation by parts we obtain from (12)

Xm�1n¼n1

qn/nqnð1� pn�rÞc 6/n1wn1 � /nwn

þXm�1n¼n1

D/n

� þ /nDqn

qnþ1

�wnþ1 �

cqn/n

qknþ1ak�1

n�r

wknþ1

�

6/n1wn1 � /nwn þXm�1n¼n1

Dþ/n

� þ /nDþqn

qnþ1

�wnþ1 �

cqn/n

qknþ1ak�1

n�r

wknþ1

�

6/n1wn1 þXm�1n¼n1

Dþ/n

� þ /nDþqn

qnþ1

�wnþ1 �

cqn/n

qknþ1ak�1

n�r

wknþ1

�: ð13Þ

Setting

a ¼ cqn/n

qknþ1ak�1

n�r

; b ¼ Dþ/n þ/nDþqn

qnþ1and u ¼ wnþ1: ð14Þ

Using the Lemma, (13) and (14), we have that

Xm�1n¼n1

qn/nqnð1

264 � pn�rÞc �

qcþ1nþ1an�r Dþ/n þ /nDþqn

qnþ1

� �cþ1

ðc þ 1Þcþ1qcn/

cn

3756/n1wn1 ;

which contradicts the assumption (3). This completes the proof of Theorem2.1. h

If we choose /n ¼ 1 and qn ¼ /n ¼ n, respectively, then we have thefollowing simple criteria.

Corollary 2.1. Assume that ðH1Þ and ðH2Þ hold. Furthermore, assume that thereexists a positive sequence fqng such that

lim supm!1

Xm�1n¼n0

qnqnð1"

� pn�rÞc �an�rðDþqnÞ

cþ1

ðc þ 1Þcþ1qcn

#¼ 1 ð15Þ

for some n0 > 0, where Dþqn is defined as in Theorem 2.1, then Eq. (1) isoscillatory.

Corollary 2.2. Assume that ðH1Þ and ðH2Þ hold. Furthermore, assume that thereexists a positive sequence f/ng such that

lim supm!1

Xm�1n¼n0

n2qnð1"

� pn�rÞc �an�rð2nþ 1Þcþ1

ðc þ 1Þcþ1n2c

#¼ 1 ð16Þ

914 Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918

for some n0 > 0, where Dþ/n is defined as in Theorem 2.1, then Eq. (1) isoscillatory.

Remark 1. Under the appropriate choices of the sequences fqng and f/ng, wecan obtain many new criteria for the oscillation of (1) from Theorem 2.1.

Because of the limited space, we omit them here.

Theorem 2.2. Assume that ðH1Þ and ðH2Þ hold. Furthermore, assume that thereexist a positive sequence fqng and a double sequence fHm;n : mP nP 0g such thatHm;m ¼ 0 for mP 0, Hm;n > 0 for m > n > 0, and D2Hm;n ¼ Hm;nþ1 � Hm;n 6 0 formP nP 0. If

lim supm!1

1

Hm;n0

Xm�1n¼n0

Hm;nqnqnð1

264 � pn�rÞc

�qcþ1nþ1an�r D2Hm;n þ Hm;n

Dþqnqnþ1

� �cþ1

ðc þ 1Þcþ1qcnH

cm;n

375 ¼ 1 ð17Þ

for some n0 > 0, where Dþqn is defined as in Theorem 2.1, then Eq. (1) isoscillatory.

Proof. Proceeding as in Theorem 2.1 we assume that Eq. (1) has a nonoscil-

latory solution, say xn�N > 0 for nP n0. Similar to the proof of Theorem 2.1 wehave that (11) holds. Multiplying (11) by Hm;n for nP n1, we obtain

Pm�1n¼n1

Hm;nqnqnð1� pn�rÞc 6 �Pm�1

n¼n1Hm;nDwn þ

Pm�1n¼n1

Hm;nDqnqnþ1

wnþ1

h� cqnHm;n

qknþ1a

k�1n�r

wknþ1

i:

Using the summation by parts we obtain

Xm�1n¼n1

Hm;nqnqnð1� pn�rÞc 6Hm;n1wn1 þXm�1n¼n1

D2Hm;n

� þ Hm;n

Dþqn

qnþ1

�wnþ1

� Hm;ncqn

qknþ1ak�1

n�r

wknþ1

�: ð18Þ

Setting

a ¼ Hm;ncqn

qknþ1ak�1

n�r

; b ¼ D2Hm;n þ Hm;nDþqn

qnþ1and u ¼ wnþ1:

Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918 915

Using the Lemma and (18), we get

Xm�1n¼n1

Hm;nqnqnð1

264 � pn�rÞc �

qcþ1nþ1an�r D2Hm;n þ Hm;n

Dþqnqnþ1

� �cþ1

ðc þ 1Þcþ1qcnH

cm;n

3756Hm;n1wn1 ;

i.e.,

1

Hm;n1

Xm�1n¼n1

Hm;nqnqnð1

264 � pn�rÞc �

qcþ1nþ1an�r D2Hm;n þ Hm;n

Dþqnqnþ1

� �cþ1

ðc þ 1Þcþ1qcnH

cm;n

3756wn1 ;

which contradicts the assumption (17). This completes the proof of Theorem

2.2. h

Remark 2. When 0 < c < 1, Theorems 2.1 and 2.2 answer the question raisedby Saker [10]. When c > 1, Theorems 2.1 and 2.2 are different from the mainresults in [7,10] and are sharper than Theorems 2.2–2.5 of [10] for some cases

(see the following two examples).

Remark 3.We used a general class of double sequence fHm;ng as the parametersequence in Theorem 2.2. By choosing specific sequence fHm;ng, we can derivemany oscillation criteria for Eq. (1). Let us consider the double sequence fHm;ngdefined by

Hm;n ¼ ðm� nÞk; kP 1;

Hm;n ¼ logmþ 1nþ 1

� �k

; mP nP 0; k > 1;

then Hm;m ¼ 0 for mP 0 and D2Hm;n 6 0 for mP nP 0. Hence, we have the

following corollaries by Theorem 2.2.

Corollary 2.3. Assume that all the assumptions of Theorem 2.2 hold, except thecondition (13) is replaced by

lim supm!1

1

ðm� n0ÞkXm�1n¼n0

ðm

264 � nÞkqnqnð1� pn�rÞc

�qcþ1nþ1an�r ðm� n� 1Þk þ ðm� nÞk Dþqn

qnþ1� 1

� �� �cþ1

ðc þ 1Þcþ1qcnðm� nÞkc

375 ¼ 1 ð19Þ

for some n0 > 0, where Dþqn is defined as in Theorem 2.1, then Eq. (1) isoscillatory.

916 Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918

Corollary 2.4. Assume that all the assumptions of Theorem 2.2 hold, except thecondition (13) is replaced by

lim supm!1

1

log mþ1n0þ1

� �k

Xm�1n¼n0

logmþ 1nþ 1

� �k

qnqnð1

26664 � pn�rÞc

�qcþ1nþ1an�r log mþ1

nþ2

� �kþ log mþ1

nþ1

� �kDþqnqnþ1

� 1� �� �cþ1

ðc þ 1Þcþ1qcn log mþ1

nþ1

� �kc

37775 ¼ 1 ð20Þ

for some n0 > 0, where Dþqn is defined as in Theorem 2.1, then Eq. (1) isoscillatory.

3. Some applications

In this section, we will consider the following two examples.

Example 1. Consider the following difference equation

D D xn

��þ nþ r � 1

nþ rxn�s

�c�þ naxc

n�r ¼ 0; nP 1; ð21Þ

where a and c are constants with c > 0, s and r are nonnegative integers. In(21), an � 1, pn ¼ nþr�1

nþr and qn ¼ na. It is easy to see that assumptions ðH1Þ andðH2Þ hold. For the case when c > 1, we choose qn ¼ n. By Corollary 2.1, wehave

lim supm!1

Xm�1n¼1

n1þa�c

"� 1

ðc þ 1Þcþ1nc

#¼ 1

when a � cP � 2. Therefore, Eq. (21) with c > 1 is oscillatory if a � cP � 2.For the case when 0 < c6 1, if we choose qn ¼ n�k ð0 < k < 1Þ and /n ¼ n

such that k þ c > 1, then Dþqn � 0 and D/n � 1. By Theorem 2.1, we have

lim supm!1

Xm�1n¼1

n1�kþa�c

"� 1

ðc þ 1Þcþ1ðnþ 1Þkð1þcÞnð1�kÞc

#

P lim supm!1

Xm�1n¼1

n1�kþa�c

"� 1

ðc þ 1Þcþ1ðnþ 1Þkþc

#¼ 1

Y.G. Sun, S.H. Saker / Appl. Math. Comput. 163 (2005) 909–918 917

when a � k � cP � 2. Therefore, Eq. (21) with 0 < c6 1 is oscillatory if thereexists a k 2 ð0; 1Þ such that a � k � c P � 2.

Example 2. Consider the following difference equation

D ðn�

þ rÞc�1D xn

�þ nþ r � 1

nþ rxn�s

�c�þ bnc�2xc

n�r ¼ 0; nP 1; ð22Þ

where b > 0 and cP 1 are constants, s and r are nonnegative integers. In (22),an ¼ ðnþ rÞc�1, pn ¼ nþr�1

nþr and qn ¼ bnc�2. It is easy to see that assumptions

ðH1Þ and ðH2Þ hold. Let qn ¼ n, by Corollary 2.1, we have

lim supm!1

Xm�1n¼1

bn

"� 1

ðc þ 1Þcþ1n

#¼ 1

when b > 1

ðcþ1Þcþ1. Thus, Eq. (22) is oscillatory when b > 1

ðcþ1Þcþ1. However, by

Theorem 2.4 in [10], the left-hand side of (2.41) takes the form

lim supm!1

Xn

l¼1

bl

� 1

23�cl2�c

�:

We can easily show that

lim supm!1

Xn

l¼1

bl

� 1

23�cl2�c

�¼ 1;

only holds for the case when c ¼ 1 and b > 1=4.

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