- Email: [email protected]
Oscillation of second-order sublinear neutral delay difference equations
Applied Mathematics and Computation 146 (2003) 543–551 www.elsevier.com/locate/amc
Oscillation of second-order sublinear neutral delay difference equations Wan-Tong Li
a,*
, S.H. Saker
b,c
a
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PeopleÕs Republic of China b Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt c Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60–769 Poznan, Poland
Abstract We present new oscillation criteria for the second-order sublinear neutral delay difference equation Dðan Dðxn þ pn xns ÞÞ þ qn xcnr ¼ 0; where 0 < c < 1 is a quotient of odd positive integers and Ó 2002 Elsevier Inc. All rights reserved.
P1
1 n¼0 an
¼ 1:
Keywords: Oscillation; Delay neutral difference equations
1. Introduction In this note we shall consider the second-order sublinear neutral delay difference equation Dðan Dðxn þ pn xns ÞÞ þ qn xcnr ¼ 0;
n ¼ 0; 1; 2 . . . ;
ð1Þ
where 0 < c < 1 is a quotient of odd positive integers, D denotes the forward difference operator Dxn ¼ xnþ1 xn and D2 xn ¼ DðDxn Þ for any sequence fxn g of *
Corresponding author. E-mail addresses: [email protected] (W.-T. Li), [email protected], [email protected] (S.H. Saker). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00604-5
544
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
real numbers, s, r are fixed nonnegative integers, fan g, fpn g and fqn g are sequences of real numbers such that an > 0;
1 X 1 ¼ 1; a n¼n0 n
0 6 pn < 1
for all n P 0 and
qn P 0;
ð2Þ
and qn 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 ¼ 0; 1; 2 . . . Clearly if xn ¼ An
for n ¼ N ; . . . ; 1; 0
ð3Þ
are given, then Eq. (1) has a unique solution satisfying the initial conditions (3). A solution fxn g of (1) is said to be oscillatory if for every n1 > 0 there exists an n P n1 such that xn xnþ1 6 0; otherwise it is nonoscillatory. Eq. (1) is said to be oscillatory if all its solutions are oscillatory. Recently, there has been an increasing interest in the study oscillation and asymptotic behavior of solutions of second-order neutral delay difference equations, for example see [1–18] and the references therein. To the best of our knowledge, nothing is known regarding the qualitative behavior of solutions of Eq. (1) in the sublinear case. Therefore our aim in this paper is to give several oscillation criteria of Eq. (1) when (2) holds.
2. Main results We will assume throughout this paper that Dan P 0. Theorem 2.1. Assume that (2) holds. Furthermore, assume that there exists a positive sequence fqn g such that for every a P 1 " # 1c 2 n X alr ðaðl þ 1 rÞÞ ðDql Þ ql Ql lim sup ¼ 1; ð4Þ n!1 4cql l¼0 c
where Qn ¼ qn ð1 pnr Þ . Then every solution of Eq. (1) oscillates. Proof. Assume for the sake of contradiction that Eq. (1) has a nonoscillatory solution fxn g; we may assume without loss of generality that xnN > 0 for n P n0 > 0: (the case when xn < 0 is similar and hence is omitted). Set zn ¼ xn þ pn xns :
ð5Þ
By, assumption (2) we have zn > 0 for n P n0 and from (1) it follows that Dðan Dzn Þ ¼ qn xcnr 6 0;
n P n0 ;
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
545
and so fan Dzn g is an eventually nonincreasing sequence. We first show that Dzn P 0 for n P n0 : In fact, if there exists an integer n1 P n0 such that an1 Dzn1 ¼ c < 0; then an Dzn 6 c for n P n1 , that is c Dzn 6 ; an and, hence zn 6 zn1 þ c
n1 X 1 ! 1 as a i¼n1 i
n ! 1;
which contradicts the fact that zn > 0 for n P n0 : Also we claim that D2 zn 6 0: If not there exists n1 P n0 such that D2 zn > 0 for n P n1 and this implies that Dznþ1 > Dzn , so that since Dan P 0; anþ1 ðDznþ1 Þ > anþ1 ðDzn Þ P an ðDzn Þ and this contradicts the fact that fan ðDzn Þg is nonincreasing sequence, then D2 zn 6 0 and therefore we have zn > 0;
Dzn P 0;
and
D2 zn 6 0 for n P n0 ;
ð6Þ
and then from (5) and (6) we have xn P ð1 pn Þzn and this implies that for n P n 1 ¼ n0 þ r xnr P ð1 pnr Þznr : From Eq. (1) and the last inequality, we have Dðan Dzn Þ þ Qn zcnr 6 0;
n P n1 :
ð7Þ
Define the function wn ¼ qn
an Dzn : zcnr
ð8Þ
Then wn > 0 and qn q Dðan Dzn Þ Dwn ¼ anþ1 Dznþ1 D c : þ n c znr znr
ð9Þ
From the fact that Dzn P 0
and
Dðan Dzn Þ 6 0
for n P n1 ;
we have anr Dznr P anþ1 Dznþ1
and
znþ1r P znr :
ð10Þ
Then, from Eq. (1), (8) and (9), we get Dwn ¼ qn Qn þ
Dqn q anþ1 Dznþ1 Dzcnr wnþ1 n c : qnþ1 znþ1r zcnr
ð11Þ
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W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
From (10) and (11), we have Dwn 6 qn Qn þ
Dqn q anþ1 Dznþ1 Dzcnr wnþ1 n c : 2 qnþ1 z
ð12Þ
nþ1r
Now, by using the inequality (cf. [5, p. 39]) xc yc P cxc1 ðx yÞ
for all x P y > 0
and
0 < c 6 1;
we find that Dznr c ¼ znþ1r c znr c P cðznþ1r Þc1 ðznþ1r znr Þ ¼ cðznþ1r Þ
c1
ðDznr Þ:
ð13Þ
Substitute from (13) in (12), we have Dwn 6 qn Qn þ
Dqn Dznþ1 ðDznr Þ wnþ1 cðznþ1r Þc1 qn anþ1 c 2 : qnþ1 z
ð14Þ
nþ1r
Again, from (10) in (14), we obtain Dwn 6 qn Qn þ
Dqn ðanþ1 Þ2 ðDznþ1 Þ2 c1 wnþ1 cðznþ1r Þ qn 2 ; qnþ1 ðanr Þ zc nþ1r
ð15Þ
and hence, 2
Dwn 6 qn Qn þ
2
Dqn ðanþ1 Þ ðDznþ1 Þ wnþ1 cqn 2 : 1c c qnþ1 ðanr Þðznþ1r Þ znþ1r
ð16Þ
From (8) and (16), we find Dwn 6 qn Qn þ
Dqn cqn 1 2 wnþ1 : 2 wnþ1 qnþ1 ðznþ1r Þ1c ðanr Þ qnþ1
ð17Þ
From (6), we conclude that zn 6 zn0 þ Dzn0 ðn n0 Þ;
n P n0 ;
and consequently there exists a n1 P n0 and appropriate constant a P 1 such that zn 6 an for n P n1 ; and this implies that znþ1r 6 aðn þ 1 rÞ
for n P n2 ¼ n1 þ r 1;
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
547
and, hence 1 ðznþ1r Þ
1c
P
1 ðaðn þ 1 rÞÞ
1c
:
Substitute from the last inequality in (17), we find Dwn 6 qn Qn þ
Dqn cqn wnþ1 w2 2 1c nþ1 qnþ1 qnþ1 anr ðaðn þ 1 rÞÞ 1c
¼ qn Qn þ 2
anr ðaðn þ 1 rÞÞ 4cqn
ðDqn Þ
2
32 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1c pffiffiffiffiffiffiffi ð aðn þ 1 rÞ Þ Dq a nr cqn n7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 wnþ1 5 pffiffiffiffiffiffiffi 2 cqn 1c qnþ1 ðaðn þ 1 rÞÞ anr " # ðaðn þ 1 rÞÞ1c anr ðDqn Þ2 < qn Qn : ð18Þ 4cqn Then, we have "
# ðanr Þðaðn þ 1 rÞÞ1c ðDqn Þ2 Dwn < qn Qn : 4cqn
ð19Þ
Summing (19) from n2 to n, we obtain " # n X ðalr Þðaðl þ 1 rÞÞ1c ðDql Þ2 ql Ql wn2 < wnþ1 wn2 < ; 4cql l¼n 2
which yields " # 1c 2 n X alr ðaðl þ 1 rÞÞ ðDql Þ ql Ql < c1 ; 4cql l¼n 2
for all large n, which is contrary to (4). The proof is complete.
Remark 2.1. Note that from Theorem 2.1, we can obtain different conditions for oscillation of all solutions of Eq. (1) when (2) holds by different choices of fqn g: Let qn ¼ nk ; n P n0 and k > 1 is a constant. By Theorem 2.2 we have the following result.
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W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
Corollary 2.1. Assume that all the assumption of Theorem 2.1 hold, except the condition (4) is replaced by 2 2 3 1c k n asr ðaðs þ 1 rÞÞ ðs þ 1Þ sk X 6k 7 ð20Þ lim sup 4s Qs 5 ¼ 1; k n!1 4cs s¼n0 where Qn ¼ qn ð1 pnr Þc : Then every solution of Eq. (1) oscillates. Remark 2.2. When c ¼ 1, Eq. (1) reduces to the linear delay difference equation Dðan ðDxn þ pn xns ÞÞ þ qn xnr ¼ 0;
n ¼ 0; 1; 2 . . .
and the condition (4) in Theorem 2.1 reduces to " # 2 n X ðalr ÞðDql Þ lim sup ql ql ð1 plr Þ ¼ 1: n!1 4ql l¼0
ð21Þ
ð22Þ
Then Theorem 2.1 and Theorem 1 in [15] are the same in linear case. Also when pn ¼ 0 and c ¼ 1 Theorem 2.1 and Corollary 1 in [16] are the same. 1
Theorem 2.2. Assume that (2) holds, and let fqn gn¼0 be a positive sequence. Furthermore, we assume that there exists a double sequence fHm;n : m P n P 0g such that (i) Hm;m ¼ 0 for m P 0; (ii) Hm;n > 0 for m > n > 0, (iii) D2 Hm;n ¼ Hm;nþ1 Hm;n : If " 2 # m1 pffiffiffiffiffiffiffiffiffi Dqn q2nþ1 1 X lim sup Hm;n qn Qn hm;n Hm;n Hm;n ¼ 1; m!1 Hm;0 n¼n0 4 qn qnþ1 ð23Þ where D2 Hm;n hm;n ¼ pffiffiffiffiffiffiffiffiffi ; Hm;n
1c qn ¼ cqn = ðaðn þ 1 rÞÞ ðanr Þ :
Then every solution of Eq. (1) oscillates. Proof. Proceeding as in Theorem 2.1, we assume that Eq. (1) has a nonoscillatory solution, say xn > 0 and xnr > 0 for all n P n0 . From the proof of Theorem 2.1 we obtain (18) for all n P n2 . From (18), since Dqn 6 0; we have for n P n2 Dwn 6 qn Qn þ
Dqn q wnþ1 n 2 w2nþ1 ; qnþ1 qnþ1
ð24Þ
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
549
or qn Qn 6 Dwn þ
Dqn q wnþ1 n 2 w2nþ1 : qnþ1 qnþ1
ð25Þ
Therefore, we have m1 X
Hm;n qn Qn 6
n¼n2
m1 X
Hm;n Dwn þ
n¼n2
m1 X
Hm;n
n¼n2
m1 X Dqn q wnþ1 Hm;n n 2 w2nþ1 ; qnþ1 qnþ1 n¼n2
ð26Þ which yields, after summing by parts, m1 X
Hm;n qn qnþ1 6 Hm;n2 wn2 þ
n¼n2
m1 X
wnþ1 D2 Hm;n þ
n¼n2
m1 X n¼n2 m1 X
Hm;n
qn qnþ1
2 2 wnþ1 ¼ Hm;n2 wn2
m1 X n¼n2
m1 X
hm;n
Hm;n
Dqn wnþ1 qnþ1
pffiffiffiffiffiffiffiffiffi Hm;n wnþ1
n¼n2
m1 X
Dqn q wnþ1 Hm;n n 2 w2nþ1 ¼ Hm;n2 wn2 q nþ1 qnþ1 n¼n2 n¼n2 " pffiffiffiffiffiffiffiffiffiffiffiffiffi #2 m1 X pffiffiffiffiffiffiffiffiffi Dqn Hm;n qn qnþ1 wnþ1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi hm;n Hm;n Hm;n qnþ1 qnþ1 2 Hm;n qn n¼n2 2 2 m1 qnþ1 1X Dqn pffiffiffiffiffiffiffiffiffi þ hm;n Hm;n : 4 n¼k qn qnþ1 þ
Then, m1 X n¼n2
"
Hm;n
2 # q2nþ1 Dqn pffiffiffiffiffiffiffiffiffi Hm;n Hm;n qn qn hm;n < Hm;n2 wn2 6 Hm;0 wn2 4 qn qnþ1
which implies that " 2 # nX m1 2 1 X q2nþ1 Dqn pffiffiffiffiffiffiffiffiffi Hm;n qn qn hm;n qn qnþ1 þ Hm;0 wn2 : Hm;n < Hm;0 4 qn qnþ1 n¼0 n¼0 Hence
" 2 # m1 q2nþ1 1 X Dqn pffiffiffiffiffiffiffiffiffi lim sup Hm;n Hm;n qn qn hm;n < 1; 4 qn qnþ1 m!1 Hm;0 n¼0
which is contrary to (23). The proof is complete. By choosing the sequence fHm;n g in appropriate manners, we can derive several oscillation criteria for (1.1). For instance, let us consider the double sequence fHm;n g defined by
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W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
Hm;n ¼ ðm nÞk ; or Hm;n ¼
log
mþ1 nþ1
k ;
where k P 1, m P n P 0. Then Hm;m ¼ 0 for m P 0 and Hm;n > 0 and D2 Hm;n 6 0 for m > n > 0: Hence we have the following results. Corollary 2.2. Assume that all the assumptions of Theorem 2.2 hold, except the condition (23) is replaced by " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 # m k2 q2nþ1 1 X Dqn k 2 ðm nÞk lim sup k ðm nÞ qn Qn kðm nÞ m!1 m n¼0 4qn qnþ1 ¼ 1; where Qn ¼ qn ð1 pnr Þc . Then every solution of Eq. (1) oscillates. Corollary 2.3. Assume that all the assumptions of Theorem 2.2 hold, except the condition (23) is replaced by " # k m X q2nþ1 1 mþ1 lim sup log qn Qn Bm;n ¼ 1; m!1 nþ1 4qn ð logðm þ 1ÞÞk n¼0 where 0
Bm;n
k ¼@ nþ1
mþ1 ln nþ1
k2 2
Dqn qnþ1
1 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 mþ1 A ln : nþ1
Then every solution of Eq. (1) oscillates. Another Hm;n may be chosen as Hm;n ¼ /ðm nÞ; Hm;n ¼ ðm nÞ
ðkÞ
m P n P 0; k > 2; m P n P 0;
where / : ½0; 1Þ ! ½0; 1Þ is a continuously differentiable function which satisfies /ð0Þ ¼ 0 and /ðuÞ > 0; /0 ðuÞ P 0 for u > 0; and ðm nÞðkÞ ¼ ðm nÞðm n þ 1Þ ðm n þ k 1Þ and D2 ðm nÞ
ðkÞ
¼ ðm n 1Þ
ðkÞ
ðm nÞ
Corresponding corollaries can also be stated.
ðkÞ
¼ kðm nÞ
ðk1Þ
:
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
551
Acknowledgements Supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), the Foundation for University Key Teacher by Ministry of Education of China, and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.
References [1] R.P. Agarwal, Difference Equations and Inequalities, Theory, Methods and Applications, Revised and Expanded, 2nd ed., Marcel Dekker, New York, 2000. [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, Mathl. Comput. Modelling 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, 2nd ed., Cambridge University Press, 1952. [6] H.F. Huo, W.T. Li, Oscillation of the Emden-Fowler difference systems, J. Math. Anal. Appl. 256 (2001) 478–485. [7] J.C. Jiang, Oscillatory criteria for second-order quasilinear neutral delay difference equations, Appl. Math. Comp. 125 (2002) 287–293. [8] B.S. Lalli, B.G. Zhang, On existence of positive solutions and bounded oscillations for neutral difference equations, J. Math. Anal. Appl. 166 (1992) 272–287. [9] W.T. Li, Oscillation theorems for second order nonlinear difference equations, Math. Comput. Modell. 31 (7) (2000) 71–79. [10] W.T. Li, S.S. Cheng, Classifications existence of positive solutions of second order nonlinear neutral difference equations, Funk. Evac. 40 (1997) 371–393. [11] W.T. Li, S.S. Cheng, Asymptotically linear solutions of a discrete Emden-Fowler equation, Far East J. Math. Sci. 6 (4) (1998) 521–542. [12] W.T. Li, S.S. Cheng, Characterizing conditions for the existence of nonoscillatory solutions of a discrete Emden-Fowler equation with summable coefficients, Dyn. Sys. Appl. 9 (2000) 457– 461. [13] W.T. Li, X.L. Fan, C.K. Zhong, On unbounded positive solutions of second order difference equations with singular nonlinear term, J. Math. Anal. Appl. 246 (2000) 80–88. [14] W.T. Li, X.L. Fan, C.K. Zhong, Positive solutions of discrete Emden-Fowler equation with singular nonlinear term, Dyn. Sys. Appl. 9 (2000) 247–254. [15] Z. Szafranski, B. Szmanda, Oscillation of some difference equations, Fasc. Math. 28 (1998) 149–155. [16] Z. Szafranski, B. Szmanda, Oscillation theorems for some nonlinear difference equations, Appl. Math. Comp. 83 (1997) 43–52. [17] B.G. Zhang, G.D. Chen, Oscillation of certain second order nonlinear difference equations, J. Math. Anal. Appl. 199 (1996) 827–841. [18] Z. Zhang, J. Chen, C. Zhang, Oscillation of solutions for second-order nonlinear difference equations with nonlinear neutral term, Comp. Math. Appl. 41 (2001) 1487–1494.
Oscillation of second-order sublinear neutral delay difference equations Wan-Tong Li
a,*
, S.H. Saker
b,c
a
Department of Mathematics, Lanzhou University, Lanzhou, Gansu 730000, PeopleÕs Republic of China b Mathematics Department, Faculty of Science, Mansoura University, Mansoura 35516, Egypt c Faculty of Mathematics and Computer Science, Adam Mickiewicz University, Matejki 48/49, 60–769 Poznan, Poland
Abstract We present new oscillation criteria for the second-order sublinear neutral delay difference equation Dðan Dðxn þ pn xns ÞÞ þ qn xcnr ¼ 0; where 0 < c < 1 is a quotient of odd positive integers and Ó 2002 Elsevier Inc. All rights reserved.
P1
1 n¼0 an
¼ 1:
Keywords: Oscillation; Delay neutral difference equations
1. Introduction In this note we shall consider the second-order sublinear neutral delay difference equation Dðan Dðxn þ pn xns ÞÞ þ qn xcnr ¼ 0;
n ¼ 0; 1; 2 . . . ;
ð1Þ
where 0 < c < 1 is a quotient of odd positive integers, D denotes the forward difference operator Dxn ¼ xnþ1 xn and D2 xn ¼ DðDxn Þ for any sequence fxn g of *
Corresponding author. E-mail addresses: [email protected] (W.-T. Li), [email protected], [email protected] (S.H. Saker). 0096-3003/$ - see front matter Ó 2002 Elsevier Inc. All rights reserved. doi:10.1016/S0096-3003(02)00604-5
544
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
real numbers, s, r are fixed nonnegative integers, fan g, fpn g and fqn g are sequences of real numbers such that an > 0;
1 X 1 ¼ 1; a n¼n0 n
0 6 pn < 1
for all n P 0 and
qn P 0;
ð2Þ
and qn 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 ¼ 0; 1; 2 . . . Clearly if xn ¼ An
for n ¼ N ; . . . ; 1; 0
ð3Þ
are given, then Eq. (1) has a unique solution satisfying the initial conditions (3). A solution fxn g of (1) is said to be oscillatory if for every n1 > 0 there exists an n P n1 such that xn xnþ1 6 0; otherwise it is nonoscillatory. Eq. (1) is said to be oscillatory if all its solutions are oscillatory. Recently, there has been an increasing interest in the study oscillation and asymptotic behavior of solutions of second-order neutral delay difference equations, for example see [1–18] and the references therein. To the best of our knowledge, nothing is known regarding the qualitative behavior of solutions of Eq. (1) in the sublinear case. Therefore our aim in this paper is to give several oscillation criteria of Eq. (1) when (2) holds.
2. Main results We will assume throughout this paper that Dan P 0. Theorem 2.1. Assume that (2) holds. Furthermore, assume that there exists a positive sequence fqn g such that for every a P 1 " # 1c 2 n X alr ðaðl þ 1 rÞÞ ðDql Þ ql Ql lim sup ¼ 1; ð4Þ n!1 4cql l¼0 c
where Qn ¼ qn ð1 pnr Þ . Then every solution of Eq. (1) oscillates. Proof. Assume for the sake of contradiction that Eq. (1) has a nonoscillatory solution fxn g; we may assume without loss of generality that xnN > 0 for n P n0 > 0: (the case when xn < 0 is similar and hence is omitted). Set zn ¼ xn þ pn xns :
ð5Þ
By, assumption (2) we have zn > 0 for n P n0 and from (1) it follows that Dðan Dzn Þ ¼ qn xcnr 6 0;
n P n0 ;
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
545
and so fan Dzn g is an eventually nonincreasing sequence. We first show that Dzn P 0 for n P n0 : In fact, if there exists an integer n1 P n0 such that an1 Dzn1 ¼ c < 0; then an Dzn 6 c for n P n1 , that is c Dzn 6 ; an and, hence zn 6 zn1 þ c
n1 X 1 ! 1 as a i¼n1 i
n ! 1;
which contradicts the fact that zn > 0 for n P n0 : Also we claim that D2 zn 6 0: If not there exists n1 P n0 such that D2 zn > 0 for n P n1 and this implies that Dznþ1 > Dzn , so that since Dan P 0; anþ1 ðDznþ1 Þ > anþ1 ðDzn Þ P an ðDzn Þ and this contradicts the fact that fan ðDzn Þg is nonincreasing sequence, then D2 zn 6 0 and therefore we have zn > 0;
Dzn P 0;
and
D2 zn 6 0 for n P n0 ;
ð6Þ
and then from (5) and (6) we have xn P ð1 pn Þzn and this implies that for n P n 1 ¼ n0 þ r xnr P ð1 pnr Þznr : From Eq. (1) and the last inequality, we have Dðan Dzn Þ þ Qn zcnr 6 0;
n P n1 :
ð7Þ
Define the function wn ¼ qn
an Dzn : zcnr
ð8Þ
Then wn > 0 and qn q Dðan Dzn Þ Dwn ¼ anþ1 Dznþ1 D c : þ n c znr znr
ð9Þ
From the fact that Dzn P 0
and
Dðan Dzn Þ 6 0
for n P n1 ;
we have anr Dznr P anþ1 Dznþ1
and
znþ1r P znr :
ð10Þ
Then, from Eq. (1), (8) and (9), we get Dwn ¼ qn Qn þ
Dqn q anþ1 Dznþ1 Dzcnr wnþ1 n c : qnþ1 znþ1r zcnr
ð11Þ
546
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
From (10) and (11), we have Dwn 6 qn Qn þ
Dqn q anþ1 Dznþ1 Dzcnr wnþ1 n c : 2 qnþ1 z
ð12Þ
nþ1r
Now, by using the inequality (cf. [5, p. 39]) xc yc P cxc1 ðx yÞ
for all x P y > 0
and
0 < c 6 1;
we find that Dznr c ¼ znþ1r c znr c P cðznþ1r Þc1 ðznþ1r znr Þ ¼ cðznþ1r Þ
c1
ðDznr Þ:
ð13Þ
Substitute from (13) in (12), we have Dwn 6 qn Qn þ
Dqn Dznþ1 ðDznr Þ wnþ1 cðznþ1r Þc1 qn anþ1 c 2 : qnþ1 z
ð14Þ
nþ1r
Again, from (10) in (14), we obtain Dwn 6 qn Qn þ
Dqn ðanþ1 Þ2 ðDznþ1 Þ2 c1 wnþ1 cðznþ1r Þ qn 2 ; qnþ1 ðanr Þ zc nþ1r
ð15Þ
and hence, 2
Dwn 6 qn Qn þ
2
Dqn ðanþ1 Þ ðDznþ1 Þ wnþ1 cqn 2 : 1c c qnþ1 ðanr Þðznþ1r Þ znþ1r
ð16Þ
From (8) and (16), we find Dwn 6 qn Qn þ
Dqn cqn 1 2 wnþ1 : 2 wnþ1 qnþ1 ðznþ1r Þ1c ðanr Þ qnþ1
ð17Þ
From (6), we conclude that zn 6 zn0 þ Dzn0 ðn n0 Þ;
n P n0 ;
and consequently there exists a n1 P n0 and appropriate constant a P 1 such that zn 6 an for n P n1 ; and this implies that znþ1r 6 aðn þ 1 rÞ
for n P n2 ¼ n1 þ r 1;
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547
and, hence 1 ðznþ1r Þ
1c
P
1 ðaðn þ 1 rÞÞ
1c
:
Substitute from the last inequality in (17), we find Dwn 6 qn Qn þ
Dqn cqn wnþ1 w2 2 1c nþ1 qnþ1 qnþ1 anr ðaðn þ 1 rÞÞ 1c
¼ qn Qn þ 2
anr ðaðn þ 1 rÞÞ 4cqn
ðDqn Þ
2
32 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 1c pffiffiffiffiffiffiffi ð aðn þ 1 rÞ Þ Dq a nr cqn n7 6 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi 4 wnþ1 5 pffiffiffiffiffiffiffi 2 cqn 1c qnþ1 ðaðn þ 1 rÞÞ anr " # ðaðn þ 1 rÞÞ1c anr ðDqn Þ2 < qn Qn : ð18Þ 4cqn Then, we have "
# ðanr Þðaðn þ 1 rÞÞ1c ðDqn Þ2 Dwn < qn Qn : 4cqn
ð19Þ
Summing (19) from n2 to n, we obtain " # n X ðalr Þðaðl þ 1 rÞÞ1c ðDql Þ2 ql Ql wn2 < wnþ1 wn2 < ; 4cql l¼n 2
which yields " # 1c 2 n X alr ðaðl þ 1 rÞÞ ðDql Þ ql Ql < c1 ; 4cql l¼n 2
for all large n, which is contrary to (4). The proof is complete.
Remark 2.1. Note that from Theorem 2.1, we can obtain different conditions for oscillation of all solutions of Eq. (1) when (2) holds by different choices of fqn g: Let qn ¼ nk ; n P n0 and k > 1 is a constant. By Theorem 2.2 we have the following result.
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Corollary 2.1. Assume that all the assumption of Theorem 2.1 hold, except the condition (4) is replaced by 2 2 3 1c k n asr ðaðs þ 1 rÞÞ ðs þ 1Þ sk X 6k 7 ð20Þ lim sup 4s Qs 5 ¼ 1; k n!1 4cs s¼n0 where Qn ¼ qn ð1 pnr Þc : Then every solution of Eq. (1) oscillates. Remark 2.2. When c ¼ 1, Eq. (1) reduces to the linear delay difference equation Dðan ðDxn þ pn xns ÞÞ þ qn xnr ¼ 0;
n ¼ 0; 1; 2 . . .
and the condition (4) in Theorem 2.1 reduces to " # 2 n X ðalr ÞðDql Þ lim sup ql ql ð1 plr Þ ¼ 1: n!1 4ql l¼0
ð21Þ
ð22Þ
Then Theorem 2.1 and Theorem 1 in [15] are the same in linear case. Also when pn ¼ 0 and c ¼ 1 Theorem 2.1 and Corollary 1 in [16] are the same. 1
Theorem 2.2. Assume that (2) holds, and let fqn gn¼0 be a positive sequence. Furthermore, we assume that there exists a double sequence fHm;n : m P n P 0g such that (i) Hm;m ¼ 0 for m P 0; (ii) Hm;n > 0 for m > n > 0, (iii) D2 Hm;n ¼ Hm;nþ1 Hm;n : If " 2 # m1 pffiffiffiffiffiffiffiffiffi Dqn q2nþ1 1 X lim sup Hm;n qn Qn hm;n Hm;n Hm;n ¼ 1; m!1 Hm;0 n¼n0 4 qn qnþ1 ð23Þ where D2 Hm;n hm;n ¼ pffiffiffiffiffiffiffiffiffi ; Hm;n
1c qn ¼ cqn = ðaðn þ 1 rÞÞ ðanr Þ :
Then every solution of Eq. (1) oscillates. Proof. Proceeding as in Theorem 2.1, we assume that Eq. (1) has a nonoscillatory solution, say xn > 0 and xnr > 0 for all n P n0 . From the proof of Theorem 2.1 we obtain (18) for all n P n2 . From (18), since Dqn 6 0; we have for n P n2 Dwn 6 qn Qn þ
Dqn q wnþ1 n 2 w2nþ1 ; qnþ1 qnþ1
ð24Þ
W.-T. Li, S.H. Saker / Appl. Math. Comput. 146 (2003) 543–551
549
or qn Qn 6 Dwn þ
Dqn q wnþ1 n 2 w2nþ1 : qnþ1 qnþ1
ð25Þ
Therefore, we have m1 X
Hm;n qn Qn 6
n¼n2
m1 X
Hm;n Dwn þ
n¼n2
m1 X
Hm;n
n¼n2
m1 X Dqn q wnþ1 Hm;n n 2 w2nþ1 ; qnþ1 qnþ1 n¼n2
ð26Þ which yields, after summing by parts, m1 X
Hm;n qn qnþ1 6 Hm;n2 wn2 þ
n¼n2
m1 X
wnþ1 D2 Hm;n þ
n¼n2
m1 X n¼n2 m1 X
Hm;n
qn qnþ1
2 2 wnþ1 ¼ Hm;n2 wn2
m1 X n¼n2
m1 X
hm;n
Hm;n
Dqn wnþ1 qnþ1
pffiffiffiffiffiffiffiffiffi Hm;n wnþ1
n¼n2
m1 X
Dqn q wnþ1 Hm;n n 2 w2nþ1 ¼ Hm;n2 wn2 q nþ1 qnþ1 n¼n2 n¼n2 " pffiffiffiffiffiffiffiffiffiffiffiffiffi #2 m1 X pffiffiffiffiffiffiffiffiffi Dqn Hm;n qn qnþ1 wnþ1 þ pffiffiffiffiffiffiffiffiffiffiffiffiffi hm;n Hm;n Hm;n qnþ1 qnþ1 2 Hm;n qn n¼n2 2 2 m1 qnþ1 1X Dqn pffiffiffiffiffiffiffiffiffi þ hm;n Hm;n : 4 n¼k qn qnþ1 þ
Then, m1 X n¼n2
"
Hm;n
2 # q2nþ1 Dqn pffiffiffiffiffiffiffiffiffi Hm;n Hm;n qn qn hm;n < Hm;n2 wn2 6 Hm;0 wn2 4 qn qnþ1
which implies that " 2 # nX m1 2 1 X q2nþ1 Dqn pffiffiffiffiffiffiffiffiffi Hm;n qn qn hm;n qn qnþ1 þ Hm;0 wn2 : Hm;n < Hm;0 4 qn qnþ1 n¼0 n¼0 Hence
" 2 # m1 q2nþ1 1 X Dqn pffiffiffiffiffiffiffiffiffi lim sup Hm;n Hm;n qn qn hm;n < 1; 4 qn qnþ1 m!1 Hm;0 n¼0
which is contrary to (23). The proof is complete. By choosing the sequence fHm;n g in appropriate manners, we can derive several oscillation criteria for (1.1). For instance, let us consider the double sequence fHm;n g defined by
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Hm;n ¼ ðm nÞk ; or Hm;n ¼
log
mþ1 nþ1
k ;
where k P 1, m P n P 0. Then Hm;m ¼ 0 for m P 0 and Hm;n > 0 and D2 Hm;n 6 0 for m > n > 0: Hence we have the following results. Corollary 2.2. Assume that all the assumptions of Theorem 2.2 hold, except the condition (23) is replaced by " qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2 # m k2 q2nþ1 1 X Dqn k 2 ðm nÞk lim sup k ðm nÞ qn Qn kðm nÞ m!1 m n¼0 4qn qnþ1 ¼ 1; where Qn ¼ qn ð1 pnr Þc . Then every solution of Eq. (1) oscillates. Corollary 2.3. Assume that all the assumptions of Theorem 2.2 hold, except the condition (23) is replaced by " # k m X q2nþ1 1 mþ1 lim sup log qn Qn Bm;n ¼ 1; m!1 nþ1 4qn ð logðm þ 1ÞÞk n¼0 where 0
Bm;n
k ¼@ nþ1
mþ1 ln nþ1
k2 2
Dqn qnþ1
1 s ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi k 2 mþ1 A ln : nþ1
Then every solution of Eq. (1) oscillates. Another Hm;n may be chosen as Hm;n ¼ /ðm nÞ; Hm;n ¼ ðm nÞ
ðkÞ
m P n P 0; k > 2; m P n P 0;
where / : ½0; 1Þ ! ½0; 1Þ is a continuously differentiable function which satisfies /ð0Þ ¼ 0 and /ðuÞ > 0; /0 ðuÞ P 0 for u > 0; and ðm nÞðkÞ ¼ ðm nÞðm n þ 1Þ ðm n þ k 1Þ and D2 ðm nÞ
ðkÞ
¼ ðm n 1Þ
ðkÞ
ðm nÞ
Corresponding corollaries can also be stated.
ðkÞ
¼ kðm nÞ
ðk1Þ
:
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551
Acknowledgements Supported by the NNSF of China (10171040), the NSF of Gansu Province of China (ZS011-A25-007-Z), the Foundation for University Key Teacher by Ministry of Education of China, and the Teaching and Research Award Program for Outstanding Young Teachers in Higher Education Institutions of Ministry of Education of China.
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