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: [email protected] (Y.G. Sun), [email protected] (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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