AbstractSome new oscillation criteria are obtained for the second-order half-linear difference equation (−1)m+1δm yi(n) + σj=1Nqijyi(n−Tji)=0, m ⩾1, i=1,…N where α 0 is a ratio of odd positive integers. The method uses techniques based on a Riccati type difference inequality. Examples are inserted to illustrate the results

Ravi, K.

Thandapani, E.

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Oscillation of second-order half-linear difference equations 

¢) 
PERGAMON Applied Mathematics Letters 13 (2000) 43-49 
Applied 
Mathematics 
Letters 
www.elsevier.nl/locate/aml 
Oscillation of Second-Order Half-Linear 
Difference Equations 
E.  THANDAPANI AND K.  RAVI 
Department of Mathematics, Madras University 
P.G. Centre, Salem-635601, Tamil Nadu, India 
(Received March 1999; accepted April 1999) 
Communicated by R. P. Agarwal 
Abst ract - -Some new oscillation criteria are obtained for the second-order half-linear difference 
equation 
A (pn(Axn) a) + qnx~+ 1 = 0, n = 0, 1, 2 ..... 
where a > 0 is a ratio of odd positive integers. The method uses techniques based on a Pdccati type 
difference inequality. Examples are inserted to illustrate the results. © 2000 Elsevier Science Ltd. 
All rights reserved. 
Keywords--Half-l inear equation, Oscillation, Nonoscillation, Riccati equation. 
1. INTRODUCTION 
Determining oscillation criteria for second-order difference quations has attracted a great deal 
of attention in the last several years, see for example [1-4] and the references cited therein. Many 
of these approaches employ Riccati equations of various types to obtain criteria, which guarantee 
that any nontrivial solution is oscillatory or nonoscillatory. Following this trend, we consider in 
this paper, second-order half-linear difference quations of the form 
X ~ A(pn(AXn)a)+qn +l =0,  n=0,1 ,2 , . . . ,  (1) 
where a > 0 is the ratio of odd positive integers, A is the forward difference operator defined by 
Axn = Xn+l -- xn and {Pn}, {qn} are given sequences of real numbers with Pn > 0 for all n > 0. 
By a solution of equation (1) we mean a real sequence {Xn}, n = 0, 1, 2 , . . . ,  which satisfies 
equation (1) for all n _> 0. We recall that a nontrivial solution of equation (1) is said to be 
oscillatory if for every N > 0 there exists an n _> N such that XnXn+l <_ 0; otherwise it is said to 
be nonoscillatory. Thus, a nonoscillatory solution is eventually positive or negative. 
From [5-7], there exists a striking similarity in the oscillatory behavior between the nonlinear 
equation (1) and the corresponding linear equation 
A(pn(Axn)) + q,~Xn+l ---- 0, n ---- 0, 1, 2, . . . .  (2) 
0893-9659/00/$ - see front matter (~) 2000 Elsevier Science Ltd. All rights reserved. Typeset by ,4A,~-TEX 
PII: S0893-9659(99) 00163-9 
44 E. THANDAPANI AND K. RAVI 
For example, Hille-Wintner type comparison theorem for equation (2) extends in a natural way 
to equation (1). For more recent results of such similarity between equations (1) and (2), we refer 
to [8-10]. 
Our objective here is to proceed further in this direction to obtain some new sufficient condi- 
tions for oscillation of solutions of equation (1) which include and generalize some earlier results 
of [6,8,11]. 
2. PREL IMINARY LEMMAS 
In this section, we state and prove some lemmas which will be needed to prove our main results. 
LEMMA 1, Assume that there exist positive integers No and N1, N1 >_ No such that 
(x) oo 
E qi >- 0, for all N >_ No and E q~ > 0. (3) 
i=N i----N1 
Then there exists an integer N >_ No such that 
E qi >_ O, for all n >_ N. (4) 
i=N 
The proof of the above lemma can be found in [11, Lemma 2.1]. 
LEMMA 2. In addition to condition (3) assume that 
0¢ 1 
E is -- +co. 
n=O pl/ 
(5) 
If {xn} is a nonosciflatory solution of equation (1) such that xn > 0 for all n >_ N, then there 
exists an integer N1 >_ N such that Axn > 0 for all n >_ N1. 
PROOF. If not, assume first that Axn < 0 for all large n, say n _> N1 _> N. Without loss of 
generality we may assume that (4) holds for n >_ N1 and qN1 >-- O. Define 
n 
Qn = E qi, for n > N1 and QNI-1 = 0. 
i=N1 
We have then 
72 n r$ 
E X~ qi i+l Z Xa÷lAQi-1 = xa+2Qn-  Z QiAx~÷l ~-- O, 
i=N1 i=N1 i=N1 
for n > N1. Since Axi+l < 0 implies that Axe+ 1 < 0. Therefore, 
and so 
X A (pi(Axi) a) ---- Pn+I (n+l )  - PN, (AXN,) a < 0, 
i=N1 
n> N1, 
6 I 1/~ AX,~+I < -- (Pn~l)X/~ , n > N1, where 6 = --PNa AXN1 > O. 
Thus, in view of (5) we have Xn+2 --* - co  as n --* co, a contradiction. 
Next assume that there exists a subsequence nk --* co such that 
Axn~ <_ 0 and AXn~+l >_ O. 
Oscillation of Difference Equations 45 
Letting vn = --pn(Axn/Xn) a, we obtain from (1) 
Pn [ llol °) 
Avn = qn + Vn 
(pll~ 11~\ a 
- -  V n ) 
Further, if 
then 
and (p l /C ' -v l /a )>O.  
Xn.__+llC~_(Xn-#-I i)'::l] <~ O, 
(XnT i~2 (Xn+ 1 a 
\ Xn / Xn \ Xn / ' 
which, in view of a, implies that (Xn+l/Xn - 1) 2 _< 0, a contradiction. 
/ 1 /a  1 /axax  
Thus, VntPn - vn ) ) > 0 also. Now summing the equation (6), we have 
0 )> EAv i  = Vnk+l -- Vn~ = E qi + [ llc~ llv~'~ w " 
i=n~ i=n~ t, p i  - V i ) i~nk 
As nk+ 1 ~ 00, we obtain 
(6) 
Summing (9) we obtain 
Aun + F(un,Pn) + q,~ = 0, n _> N1. 
and hence, 
m m 
Um+l -- ?An "~ E F(tt i 'Pi)  q- E qi = O, 
i=n i=n 
where m > n _> N1. Thus, we have 
O<Um+l <_UN1-- E qi< +CO, 
i=N1 
m m 
un > - Eq i+EF(u i ,p i ) ,  
i=n i=n 
Letting m ~ c~ in (10) we obtain (8). This proves the lemma. 
for all m > Nx 
fo rm >n> N1. 
(9) 
(lO) 
oo 
E qi <0,  
i.~-nk 
which contradicts assumption (3). This proves the lemma. 
REMARK 1. When q~ _> 0 for all i _> N, Lemma 2 is the same as Lemma 5.4 of [9]. 
LEMMA 3. Let conditions of Lemma 2 hold and assume further that 
oo 
i=N 
I f  {xn} is a nonoscillatory solution of (1) such that Xn > 0 for ali n > N, then there exists an 
integer N1 _> N such that 
oo oo 
un >_ E f (u i ,p i )  + E qi' n > Y l ,  (8) 
i=n i=n 
where F(u, p) = u[((u 11~ + plla)a _ p)l(ul la + pl/a)a] and Un = pn(Axnlzn)  a > 0 for n >_ g l .  
PROOF. From Lemma 2, there exists an integer N1, such that un > 0 for all n _> N1 and {un} 
satisfies the Riccati equation 
46 E. THANDAPANI AND K. RAVl 
LEMMA 4. Let conditions (5) and (7) hold. Assume further that qn >_ 0 for all n >_ N. Let 
{xn} be a nonoscillatory solution of (1) such that xs > 0 for all n > N. Then u,~ > 0 for a11 
n _> N, un --~ 0 as n --+ oo and 
Un+l <-- ~ . .~ 1/o~ ] ' 
\ i=N ~'i / 
for n_> N. 
PROOF. From Lemma 2 and equation (9), we have us > 0 and 
Aun + F(un,pn) <_ O, for n > N. 
Hence, 
and therefore, 
1/c~ ( ) i/a I/(~ Un+ 1 U 1/°~ -{-pln/°~ < 0 -- Un Pn -- 
1 1 ) ~n+l V ~un --~'n ] -Un  pn 
A - . -Y~/~+~ = 1/~ 1/~ 1/o, <_0, 
\ Un i=1 1~i / Un+lUn Pn 
for n > N. 
It follows that 
and so 
N-1  
I s I 1 
I /c~ -~ E ---~-~ <- E 1/c• ' 
Un+I i=1 Pi i=l Pi 
n>_N, 
lla >- _lla and us -+ 0 by (5). 
Us+I i=N "t)i 
This completes the proof. 
3. OSCILLAT ION RESULTS 
If conditions (3) and (7) hold for n >_ N, then we define the sequence {¢~n} as follows: 
OO 
¢1 = Eq i '  n _> N (11) 
i=n 
and for j = 1 ,2 ,3 . . . ,  
oo 
+¢s, n>_Y. (12) 
i=n 
THEOREM 5. Assume that conditions (3) and (g) hold and let {q~,~} be deigned as in equa- 
tions (11) and (12). Assume further that 
lim sup ¢Jn > 1. 
n---*oo \ i=N Pi / 
(13) 
Then every solution of equation (1) oscillates. 
PROOF. Suppose not and let {xn} be a nonoscillatory solution of equation (1) with x,~ > 0 for 
n > N. Let un = pn(Axn/xn) ~, n >_ N. Then by Lemmas 2 and 3, we have 
oo (3o 
i=n i=n 
fo rn>N1 >_N, 
Osc i l la t ion  of  D i f ference Equat ions  47 
since F(x, c) is nonnegative for x > 0 and c > 0. Since F(x, c) is nondecreasing in x for x > 0 
and c > 0, and therefore, we have 
oo 
an >_ Z F(¢J - l 'p i )  + ¢1 = ¢~n, 
for j = 1, 2, 3 . . . .  Then by Lemma 4, we obtain 
But then we have 
j = 1 ,2 ,3 . . . ,  n>N.  
lim sup 4P~ _< 1, 
n--*cx) \ i=N Pi  f 
which contradicts (13). This completes the proof. 
THEOREM 6. Let the assumptions of Lemma 4 hold and assume further that there exists two 
sequences of positive integers {ink}, {nk} with N + 1 ~_ mk < nk, mk --* co as k ~ c~ such that 
nk 1 1 
Z qi >-- Pnk+l + ~ , for large k. (14) 
i=mk \ i=N l~i / 
Then every solution of equation (1) oscillates. 
PROOF. If not and let {Xn} be a nonoscillatory solution of equation (1) with Xn > 0 for n > N. 
As in the proof of Lemma 4, we may assume that u,~ = pn(AXn/Xn) a > 0 for n >_ N1 _> N and 
{Un} satisfies the Riccati equation (9). Hence, 
Aun + qn < 0, 
which yields 
so that 
nk 
Unk+l -- Urnk + ~_~ qi < O, 
i~-mk 
nk 
Then by Lemma 4, we have 
n> N1, 
N1 <ink < ?Zk, 
Z qi < Umk --  Unk+l  < Pnk+l  + Umk. 
i..~rn k
)o nk 1 1 
Z qi < Pn~+l + -~ , 
i=mk \ i=N 11i ] 
which contradicts (14). This completes the proof. 
REMARK 2. Under the assumptions of Lemma 4, a necessary condition for the existence of a 
nonoscillatory solution is 
f n--1 1 ,~-a qn < [ Z ~ I , for all large n. (15) 
\ i=N Fi / 
Since the inequality Aun + qn < O, n > N implies 0 < Un+l/Un < 1 - (qn/Un). Thus, qn < un 
and so Lemma 4 implies (15). 
Before stating our final result we introduce some notation. Define 
m 
1 
R,~,n = y~ 
1/a  
i=n Pi 
and for a given sequence {17~) with Yn > O, n = 1, 2, 3 . . . ,  we let 
m 
Gm,n = Z ~hqi, n < m. 
48 E. THANDAPANI AND K. RAVI 
THEOREM 7. Let a > 1. In addition to the assumptions of Lemma 4, assume that there exists 
a sequence {~Tn}, ~/,~ > 0, n = 1 ,2 ,3 . . . ,  such that 
m 
lim Grn,N ~-- lim E ~iqi = oo (16) 
m--*oo ?'¢~--400 
i=N 
and 
m 1 vt 
E ( ( i~ i -1 )2 / Ih )  ~Pi [ 1~or + R~_~I,N) 
lira i=N = 0. (17) 
m-*oo Gm,N 
Then every solution of equation (1) is oscillatory. 
PROOF. Suppose the contrary and let {xn} be an eventually position of equation (1). As in 
Lemma 4, we have 
Aun + F(un,pn) + qn = O, n >_ N, (18) 
where un = pn(Axn/xn) '~ > 0. Multiplying equation (18) by Yn and summing we have 
n n n 
Gn,N = E ~?iqi = -- E ~?iAui- E ~hF(ui,pi). 
i=N i~N i=N 
It is easy to see that 
Hence, we have 
-- E ~ iAu i  ~- U iA~i -1  + UN~N" 
i=N i=N 
n n 
c.,N _< .N ,~ + ~. ,a , , _x  - ~ ,,F(u,,p,) 
i=N i=N 
n n 2 
E E UN?~N + uiA?~i-1 -- l]i [ 1/(~ 1/ot~ et' 
where we use the inequality (a + b) a _> a s + b a for c~ _> 1 and a > 0, b > 0. 
Thus, [( )1,2 
n U2 a 
1 (. 1,o 1,o O 
+ "4 ~i '~i + t'i ) 
i=N 
1 ~ (Aria_l) 2 [ x/,~ 1 /~ ~ 
i=N 
- -  ~ U 2 rli i 
1/2 
uiA~i_ 1 
So by Lemma 4, we have 
i ~-~ (a,,-1)~ (_1/~ 1 ) ~ 
i=N 
(19) 
Combing (16), (17), and (19) now yields the desired contradiction. This completes the proof 
of the theorem. 
Oscillation of Difference Equations 49 
EXAMPLE 1. Let pn = 1 for all n and let qn = (1/(n + 1) 2) + ( -1 )n /2 (n  + 1). Then {qn} is not 
of one sign. However, we have 
c~ 1 1 1 
¢~=Zqi -> (n+l )  2 (n+l )  - 2 (n+l ) '  
so that  
A( (A(zn)  a )+qn n+] =0 
is oscil latory for a >_ 1 by Theorem 5. 
EXAMPLE 2. Consider the difference quation 
(1  ) ~n3+(n+l )3 )  3 
a  (axn) 3 zn÷l=0, n > 1 (20) 
All conditions of Theorems 6 are satisfied, and hence, all solutions of equation (20) are oscillatory. 
In fact, {xn} = {( -1 )  n} is such a solution of equation (20). 
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Oscillation of second-order half-linear difference equations

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