Oscillation for second-order nonlinear neutral delay difference equations

Applied Mathematics and Computation 163 (2005) 909–918 www.elsevier.com/locate/amc

Oscillation for second-order nonlinear neutral delay difference equations Y.G. Sun

a,*

, S.H. Saker

b

a

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

b

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 Dðan ðDðxn þ pn xns ÞÞc Þ þ f ðn; xnr Þ 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. Ó 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 c

Dðan ðDðxn þ pn xns ÞÞ Þ þ f ðn; xnr Þ ¼ 0;

n ¼ 0; 1; 2; . . . ;

ð1Þ

where c > 0 is a quotient of odd positive integers, D denotes the forward difference operator Dxn ¼ xnþ1  xn for any sequence fxn g of real numbers, s, r are fixed nonnegative integers, fan g and fpn g are real sequences satisfying

*

Corresponding author. E-mail addresses: [email protected] (Y.G. Sun), [email protected] (S.H. Saker).

0096-3003/$ - see front matter Ó 2004 Elsevier Inc. All rights reserved. doi:10.1016/j.amc.2004.04.017

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ðH1 Þ an > 0;

1 X

ð1=an Þ

1=c

¼ 1;

0 6 pn < 1;

n¼0

ðH2 Þ f ðn; uÞ : Z R ! R is continuous, and there exists a nonnegative sequence fqn g such that f ðn; uÞsgnu P qn uc for n 2 Z, where fqn g is not identically zero for large n. By a solution of (1) we mean a nontrivial sequence fxn g which is defined for n P  N , where N ¼ maxfs; rg, and satisfies Eq. (1) for n P 0. Clearly, if the initial condition xn ¼ /n for n ¼ N ; . . . ; 1; 0 is given, then Eq. (1) has a unique solution satisfying the initial condition. A solution fxn g of (1) is said to be oscillatory if for every n0 > 0 there exists an n P n0 such that xn xnþ1 6 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 oscillation and nonoscillation of solutions of the second-order neutral delay difference equations. For example, see the monographs [1,2] and the papers [3,4,6–18] and the references therein. Speaking of oscillation theory of secondorder 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 qn f ðuÞ, where f ðuÞ is a continuous function in R. Recently, Jiang [7] and Saker [10] studied the oscillatory behavior of solutions 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 case when c P 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 need the 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 f assumes its maximal value on R at

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

u ¼



bc aðc þ 1Þ

911

c

and max f ðuÞ ¼ f ðu Þ ¼ u2R

cc ðc þ 1Þ

cþ1

bcþ1 : ac

ð2Þ

The proof of the Lemma is evident, and hence is omitted. Theorem 2.1. Assume that ðH1 Þ and ðH2 Þ hold. Furthermore, assume that there exist two positive sequences fqn g and f/n g such that 2  cþ1 3 cþ1 /n Dþ qn m1 q a X nr Dþ /n þ q nþ1 7 nþ1 6 c lim sup 5¼1 4qn /n qn ð1  pnr Þ  cþ1 c c m!1 n¼n ðc þ 1Þ q / n n 0 ð3Þ for some n0 > 0, where Dþ /n ¼ maxf0; D/n g and Dþ qn ¼ maxf0; Dqn g, then Eq. (1) is oscillatory. Proof. Suppose to the contrary that fxn g is a positive solution of (1) such that xnN > 0 for n P n0 > 0. Set zn ¼ xn þ pns xns , then zn > 0 for n P n0 . From (1) and ðH2 Þ it follows that c

Dðan ðDzn Þ Þ 6  qn xcnr 6 0;

ð4Þ

n P n0 :

It is not difficult to show that Dzn is eventually positive. In fact, first, we know that Dzn 6 0 for sufficiently large n, since fzn g is nontrivial. Second, if there exists an integer n1 > n0 such that an1 ðDzn1 Þc ¼ c < 0, then an ðDzn Þc 6 c for P 1=c c 1=c n P n1 , i.e., Dzn 6 ðacn Þ , and hence zn 6 zn1 þ n1 ! 1 as n ! 1, i¼n1 ðai Þ which contradicts the fact that zn > 0. Without loss of generality, say Dzn > 0 for n P n0 . Thus, we have xnr P ð1  pnr Þznr ;

n P n1 ¼ n0 þ r:

From (1) and the above inequality, we have Dðan ðDzn Þc Þ þ qn ð1  pnr Þc zcnr 6 0;

n P n1 :

ð5Þ

Using (4), we have c

c

anr ðDznr Þ P anþ1 ðDznþ1 Þ ;

n P n1 ;

i.e., Dznr P Dznþ1



anþ1 anr

1=c ;

n P n1 :

ð6Þ

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Define the sequence fwn g by c

wn ¼ qn

an ðDzn Þ ; zcnr

ð7Þ

n P n1 :

Then wn > 0, and 



q Dwn ¼ anþ1 ðDznþ1 Þ D c n znr c

c

þ

qn Dðan ðDzn Þ Þ : zcnr

ð8Þ

From (5)–(7), we have that c

Dwn 6  qn qn ð1  pnr Þc þ

Dqn q anþ1 ðDznþ1 Þ Dðzcnr Þ wnþ1  n : zcnþ1r zcnr qnþ1

ð9Þ

By the mean value theorem, there exists n 2 ðznr ; znþ1r Þ such that Dðzcnr Þ ¼ cnc1 Dznr :

ð10Þ

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

c

Dwn 6  qn qn ð1  pnr Þ þ

Dqn nc qn anþ1 ðDznþ1 Þc Dznr wnþ1  c qnþ1 nzcnþ1r zcnr ðcþ1Þ=c

c

6  qn qn ð1  pnr Þ þ

Dqn nc qn ðanþ1 Þ ðDznþ1 Þ wnþ1  c 1=c c qnþ1 anr nznþ1r zcnr ðcþ1Þ=c

c

6  qn qn ð1  pnr Þ þ c

¼ qn qn ð1  pnr Þ þ

Dqn q ðanþ1 Þ ðDznþ1 Þ wnþ1  c n 1=c cþ1 qnþ1 anr znþ1r

Dqn cq wnþ1  k nk1 wknþ1 ; qnþ1 qnþ1 anr

cþ1

cþ1

ð11Þ

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

qn /n qn ð1  pnr Þ 6  /n Dwn þ

/n Dqn cq /n k wnþ1  k n k1 w : qnþ1 qnþ1 anr nþ1

ð12Þ

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913

Using the summation by parts we obtain from (12) m1 X

c

qn /n qn ð1  pnr Þ 6 /n1 wn1  /n wn

n¼n1

  m1  X / Dq cq /n k D/n þ n n wnþ1  k n k1 wnþ1 qnþ1 qnþ1 anr n¼n1

   m1 X /n Dþ qn cqn /n k Dþ /n þ 6 /n1 wn1  /n wn þ wnþ1  k k1 wnþ1 qnþ1 qnþ1 anr n¼n1

   m1 X /n Dþ qn cqn /n k Dþ /n þ 6 /n1 wn1 þ wnþ1  k k1 wnþ1 : ð13Þ qnþ1 qnþ1 anr n¼n1 þ

Setting a¼

cqn /n ; k1 qknþ1 anr

b ¼ Dþ /n þ

/n Dþ qn qnþ1

and

u ¼ wnþ1 :

ð14Þ

Using the Lemma, (13) and (14), we have that 2  cþ1 3 cþ1 /n Dþ qn m 1 q a D / þ X6 þ n nþ1 nr qnþ1 7 c 5 6 /n1 wn1 ; 4qn /n qn ð1  pnr Þ  cþ1 c c ðc þ 1Þ q / n n¼n1 n which contradicts the assumption (3). This completes the proof of Theorem 2.1. h If we choose /n ¼ 1 and qn ¼ /n ¼ n, respectively, then we have the following simple criteria. Corollary 2.1. Assume that ðH1 Þ and ðH2 Þ hold. Furthermore, assume that there exists a positive sequence fqn g such that " # cþ1 m1 X anr ðDþ qn Þ c qn qn ð1  pnr Þ  lim sup ¼1 ð15Þ m!1 n¼n ðc þ 1Þcþ1 qcn 0 for some n0 > 0, where Dþ qn is defined as in Theorem 2.1, then Eq. (1) is oscillatory. Corollary 2.2. Assume that ðH1 Þ and ðH2 Þ hold. Furthermore, assume that there exists a positive sequence f/n g such that " # m1 X anr ð2n þ 1Þcþ1 c 2 lim sup n qn ð1  pnr Þ  ¼1 ð16Þ cþ1 m!1 n¼n ðc þ 1Þ n2c 0

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for some n0 > 0, where Dþ /n is defined as in Theorem 2.1, then Eq. (1) is oscillatory. Remark 1. Under the appropriate choices of the sequences fqn g and f/n g, we can 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 ðH1 Þ and ðH2 Þ hold. Furthermore, assume that there exist a positive sequence fqn g and a double sequence fHm;n : m P n P 0g such that Hm;m ¼ 0 for m P 0, Hm;n > 0 for m > n > 0, and D2 Hm;n ¼ Hm;nþ1  Hm;n 6 0 for m P n P 0. If

lim sup m!1

1 Hm;n0

2 m1 X 6 c 4Hm;n qn qn ð1  pnr Þ

n¼n0

 cþ1 3 Dþ qn qcþ1 a D H þ H 2 m;n m;n q nþ1 nr 7 nþ1  5¼1 cþ1 c c ðc þ 1Þ qn Hm;n

ð17Þ

for some n0 > 0, where Dþ qn is defined as in Theorem 2.1, then Eq. (1) is oscillatory. Proof. Proceeding as in Theorem 2.1 we assume that Eq. (1) has a nonoscillatory solution, say xnN > 0 for n P n0 . Similar to the proof of Theorem 2.1 we Pm1 have that (11) holds. Multiplying (11) by Hm;n for n P n1 , we obtain n¼n1 i Pm1 Pm1 h c cqn Hm;n Dqn k Hm;n qn qn ð1  pnr Þ 6  n¼n1 Hm;n Dwn þ n¼n1 Hm;n qnþ1 wnþ1  qk ak1 wnþ1 : nþ1 nr

Using the summation by parts we obtain m1 X

c

 m1  X Dþ qn D2 Hm;n þ Hm;n wnþ1 qnþ1 n¼n1  cqn k  Hm;n k k1 wnþ1 : ð18Þ qnþ1 anr

Hm;n qn qn ð1  pnr Þ 6 Hm;n1 wn1 þ

n¼n1

Setting a ¼ Hm;n

cqn k k1 qnþ1 anr

;

b ¼ D2 Hm;n þ Hm;n

Dþ qn qnþ1

and

u ¼ wnþ1 :

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915

Using the Lemma and (18), we get 2  cþ1 3 m1 qcþ1 anr D2 Hm;n þ Hm;n Dqþnþ1qn X nþ1 7 6 c 5 6 Hm;n1 wn1 ; 4Hm;n qn qn ð1  pnr Þ  cþ1 c c ðc þ 1Þ q H n m;n n¼n1 i.e., 1 Hm;n1

m1 X n¼n1

2 6 c 4Hm;n qn qn ð1  pnr Þ 

qcþ1 nþ1 anr



D2 Hm;n þ

ðc þ 1Þ

Hm;n Dqþnþ1qn

cþ1 c c qn Hm;n

cþ1 3 7 5 6 wn1 ;

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 raised by Saker [10]. When c > 1, Theorems 2.1 and 2.2 are different from the main results 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;n g as the parameter sequence in Theorem 2.2. By choosing specific sequence fHm;n g, we can derive many oscillation criteria for Eq. (1). Let us consider the double sequence fHm;n g defined by k

Hm;n ¼ ðm  nÞ ; k P 1;  k mþ1 ; m P n P 0; k > 1; Hm;n ¼ log nþ1 then Hm;m ¼ 0 for m P 0 and D2 Hm;n 6 0 for m P n P 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 the condition (13) is replaced by 2 m1 X 1 6 k c lim sup 4ðm  nÞ qn qn ð1  pnr Þ k m!1 ðm  n0 Þ n¼n 0   cþ1 3 k k Dþ qn qcþ1  1 nþ1 anr ðm  n  1Þ þ ðm  nÞ qnþ1 7 ð19Þ  5¼1 cþ1 kc ðc þ 1Þ qcn ðm  nÞ for some n0 > 0, where Dþ qn is defined as in Theorem 2.1, then Eq. (1) is oscillatory.

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Corollary 2.4. Assume that all the assumptions of Theorem 2.2 hold, except the condition (13) is replaced by 2 k m1 6 X 1 mþ1 6 c lim sup  log qn qn ð1  pnr Þ 6 k 4 n þ 1 m!1 n¼n0 log nmþ1 0 þ1 qcþ1 nþ1 anr





cþ1 3  k  Dþ qn þ log mþ1  1 7 nþ1 qnþ1 7 7¼1  kc 5 cþ1 ðc þ 1Þ qcn log mþ1 nþ1

log mþ1 nþ2

k

ð20Þ

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

3. Some applications In this section, we will consider the following two examples. Example 1. Consider the following difference equation   c  nþr1 D D xn þ xns þ na xcnr ¼ 0; n P 1; nþr

ð21Þ

where a and c are constants with c > 0, s and r are nonnegative integers. In (21), an  1, pn ¼ nþr1 and qn ¼ na . It is easy to see that assumptions ðH1 Þ and nþr ðH2 Þ hold. For the case when c > 1, we choose qn ¼ n. By Corollary 2.1, we have " # m1 X 1 1þac n lim sup  ¼1 cþ1 m!1 n¼1 ðc þ 1Þ nc when a  c P  2. Therefore, Eq. (21) with c > 1 is oscillatory if a  c P  2. For the case when 0 < c 6 1, if we choose qn ¼ nk ð0 < k < 1Þ and /n ¼ n such that k þ c > 1, then Dþ qn  0 and D/n  1. By Theorem 2.1, we have " # m1 X 1 1kþac n lim sup  cþ1 kð1þcÞ ð1kÞc m!1 n¼1 ðc þ 1Þ ðn þ 1Þ n " # m1 X 1 P lim sup n1kþac  ¼1 cþ1 kþc m!1 n¼1 ðc þ 1Þ ðn þ 1Þ

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917

when a  k  c P  2. Therefore, Eq. (21) with 0 < c 6 1 is oscillatory if there exists a k 2 ð0; 1Þ such that a  k  c P  2. Example 2. Consider the following difference equation   c  nþr1 c1 D ðn þ rÞ D xn þ xns þ bnc2 xcnr ¼ 0; nþr

n P 1;

ð22Þ

where b > 0 and c P 1 are constants, s and r are nonnegative integers. In (22), c1 and qn ¼ bnc2 . It is easy to see that assumptions an ¼ ðn þ rÞ , pn ¼ nþr1 nþr ðH1 Þ and ðH2 Þ hold. Let qn ¼ n, by Corollary 2.1, we have # " m1 X b 1 lim sup ¼1  n ðc þ 1Þcþ1 n m!1 n¼1 1 1 . However, by when b > ðcþ1Þ cþ1 . Thus, Eq. (22) is oscillatory when b > ðcþ1Þcþ1

Theorem 2.4 in [10], the left-hand side of (2.41) takes the form  n X b 1 lim sup  3c 2c : l 2 l m!1 l¼1 We can easily show that  n X b 1  3c 2c ¼ 1; lim sup l 2 l m!1 l¼1 only holds for the case when c ¼ 1 and b > 1=4. References [1] R.P. Agarwal, Difference Equations and Inequalities, Theory, Methods and Applications, second ed., Marcel Dekker, New York, 2000, revised and expanded. [2] R.P. Agarwal, P.J.Y. Wong, Advanced Topics in Difference Equations, Kluwer Academic Publishers, 1997. [3] R.P. Agarwal, M.M.S. Manuel, E. Thandapani, Oscillatory and nonoscillatory behavior of second-order neutral delay difference equations, Math. Comput. Model. 24 (1996) 5–11. [4] R.P. Agarwal, M.M.S. Manuel, E. Thandapani, Oscillatory and nonoscillatory behavior of second-order neutral delay difference equations II, Appl. Math. Lett. 10 (1997) 103–109. [5] G.H. Hardy, J.E. Littlewood, G. Polya, Inequalities, second ed., Cambridge University Press, Cambridge, 1952. [6] J. Jianchu, Oscillatory criteria for second-order quasilinear neutral delay difference equations, Appl. Math. Comput. 125 (2002) 287–293. [7] J. Jiang, Oscillation of second order nonlinear neutral delay difference equations, Appl. Math. Comput. 146 (2003) 791–801. [8] B.S. Lalli, B.G. Zhang, On existence of positive and bounded oscillations for neutral difference equations, J. Math. Anal. Appl. 166 (1992) 272–287. [9] J.W. Lou, D.D. Bainov, Oscillatory and asymptotic behavior of second-order neutral difference equations with maximum, J. Comput. Appl. Math. 131 (2001) 331–341.

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