ARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (1996), 13 { 27. ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION

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1 ARCHIVUM MATHEMATICUM (BRNO) Tomus 32 (996), 3 { 27 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR DIFFERENCE EQUATION P. J. Y. Wong and R. P. Agarwal Abstract. We oer sucient conditions for the oscillation of all solutions of the perturbed dierence equation j m y()j? m y() + Q(; y(? ); y(? ); ; m?2 y(? )) = P (; y(? ); y(? ); ; m? y(? )); 0 where > 0: Examples which dwell upon the importance of our results are also included.. INTRODUCTION The theory of dierence equations and their applications have been and still are receiving intensive attention. In fact, in the last few years several monographs and hundreds of research papers have appeared, e.g., [,8] cover more than 450 articles. In this paper we shall consider the mth order perturbed dierence equation j m y()j? m y() + Q(; y(? ); y(? ); ; m?2 y(? )) = P (; y(? ); y(? ); ; m? y(? )); 0 (:) where > 0 and is the forward dierence operator dened as y() = y( + )?y(): Further, we suppose that 2 Z and lim! (? ) = : Throughout it is also assumed that there exist real sequences fq()g; fp()g and a function f : <! < such that (I) uf(u) > 0 for all u 6= 0; (II) Q(; x(? ); x(? ); ; m?2 x(? )) f(x(? )) q(); P (; x(? ); x(? ); ; m? x(? )) p() for all x 6= 0; f(x(? )) and 99 Mathematics Subject Classication: 39A0. Key words and phrases: oscillatory solutions, dierence equations. Received August 3, 995.

2 4 P. J. Y. WONG AND R. P. AGARWAL (III) lim [q()? p()] 0:! By a solution of (.), we mean a nontrivial sequence fy()g dened for min`0 (`? `); m y() is not identically zero, and y() fullls (.) for 0 : A solution fy()g is said to be oscillatory if it is neither eventually positive nor negative, and nonoscillatory otherwise. Throughout, for i 0 we shall use the usual factorial notation (i) = (? ) (? i + ): In the literature, numerous oscillation criteria for nonlinear dierence and differential equations related to (.) have been established, e.g., see [-4,7,9, 3-20 and the references cited therein]. We refer particularly to [2-4] in which oscillation theorems for higher order nonlinear dierence equations are presented. Thandapani and Sundaram [2] have recently considered a special case of (.) (:2) 2m y() + q()f(y(? )) = 0; 0 where fq()g is an eventually positive sequence. We have extended their wor to general higher order equations. In fact, our results include, as special cases, nown oscillation theorems not only for (.2), but also for several other particular dierence equations considered in []. Further, our results generalize those in [,9]. Finally, we remar that the paper is partly motivated by the analogy between dierential and dierence equations, in fact discrete version of the results in [5,6,0] have been developed. 2. PRELIMINARIES Lemma 2.. [, p.29] Let j m? and y() be dened for 0 : Then, (a) lim inf! j y() > 0 implies lim! i y() = ; 0 i j? ; (b) lim sup! j y() < 0 implies lim! i y() =?; 0 i j? : Lemma 2.2. [, p.29] (Discrete Kneser's Theorem) Let y() be dened for 0 ; and y() > 0 with m y() of constant sign for 0 and not identically zero. Then, there exists an integer p; 0 p m with (m + p) odd for m y() 0 and (m + p) even for m y() 0; such that (a) p m? implies (?) p+i i y() > 0 for all 0 ; p i m? ; (b) p implies i y() > 0 for all large 0 ; i p? : Lemma 2.3. [, p.30] Let y() be dened for 0 ; and y() > 0 with m y() 0 for 0 and not identically zero. Then, there exists a large integer 0 such that y() where p is dened in Lemma 2.2. (m? )! (? ) (m?) m? y(2 m?p? );

3 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR D. E MAIN RESULTS For clarity the conditions used in the main results are listed as follows : (3:) f is continuous and lim inf f(u) > 0; juj! (3:2) [q()? p()] = = ; (3:3) f is nondecreasing; (3:4) lim! (m?)?m `= 0 (? `? ) (m?) [q(`)? p(`)] = = ; f is nondecreasing; f(uv) Mf(u)f(v) for u; v > 0 (3:5) and some positive constant M; (3:6) (3:7) Z 0 Z du? f(u) < ; du < for all > 0; = f(u) = 0 n f( (m?) )[q()? p()]o = = : Theorem 3.. Suppose (3.) and (3.2) hold. (a) If m is even or m = ; then all solutions of (.) are oscillatory. (b) If m( 3) is odd, then a solution y() of (.) is either oscillatory or y() is oscillatory. Proof. Let fy()g be a nonoscillatory solution of (.), say, y() > 0 for 0 : We shall consider only this case because the proof for the case y() < 0 for 0 is similar. Using (I) - (III), it follows from (.) that (3:8) Hence, we have j m y()j? m y() [p()? q()]f(y(? )) 0; : (3:9) m y() 0; : Case m is even In view of (3.9), from Lemma 2.2 (here p is odd and p m? ; tae i = in (b)) it follows that (3:0) y() > 0; :

4 6 P. J. Y. WONG AND R. P. AGARWAL Let (3:) L = lim! y(? ): Then, since?! and y() is increasing for large (by (3.0)), we have L > 0 and L is nite or innite. (i) Suppose that 0 < L < : Since f is continuous, we get lim! f(y(? )) = f(l) > 0: Thus, there exists an integer 2 such that (3:2) f(y(? )) 2 f(l); 2: Now, from (3.8) we get (3:3) j m y()j? m y() + [q()? p()]f(y(? )) 0; 2 which in view of (III) and (3.2) leads to (3:4) j m y()j? m y() + [q()? p()] 2 f(l) 0; 2: Using (3.9), inequality (3.4) is equivalent to or (3:5)? m y() j m y()j [q()? p()] 2 f(l); 2 Summing (3.5) from 2 to (? ); we obtain (3:6) m? y() m? y( 2 )? [q()? p()] 2 f(l) = ; 2 : =? 2 f(l) `= 2 [q(`)? p(`)] = : By (3.2), the right side of (3.6) tends to? as! : Thus, there exists an integer 3 2 such that m? y() < 0; 3 : It follows from Lemma 2.(b) (j = m? ) that y()!? as! : This contradicts the assumption that fy()g is eventually positive. (ii) Suppose that L = : By (3.), we have lim inf! f(y(? )) > 0: This implies the existence of an integer 2 such that (3:7) f(y(? )) A; 2

5 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR D. E. 7 for some A > 0: In view of (III) and (3.7), it follows from (3.3) that j m y()j? m y() + [q()? p()]a 0; 2 : The rest of the proof is similar to that of Case (i). Case 2 m is odd Here, in view of (3.9), in Lemma 2.2 we have p is even and 0 p m? and hence we cannot conclude that (3.0) is true. Let L be dened as in (3.). We note that L > 0: (i) Suppose that y() > 0 for ; i.e., (3.0) holds. Then, L is nite or innite and the proof follows as in Case. (ii) Suppose that y() < 0 for : Then, L is nite and the proof follows as in Case (i). (iii) Suppose that y() is oscillatory. For the special case m = ; (.) provides (3:8) jy()j? y() [p()? q()]f(y(? )) < 0; where in view of (3.2), we have noted and used in the last inequality (3:9) q()? p() > 0 for suciently large : It follows from (3.8) that y() < 0; which contradicts the assumption that y() is oscillatory. The proof of the theorem is now complete. Example 3.. Consider the dierence equation (3:20) j 4 y()j 2 4 y() + [y( + )] 3 b = b[y( + )] 3 ; 0 where b = b(; y( + ); y( + ); 2 y( + )) is any function. Here, = 3 and m = 4: Tae? ( + ) and f(y) = y 3 : Then, (3.) clearly holds. Further, we have and Q(; y( + ); y( + ); 2 y( + )) f(y( + )) P (; y( + ); y( + ); 2 y( + ); 3 y( + )) f(y( + )) = b q() = b p() and so (3.2) is satised. It follows from Theorem 3.(a) that all solutions of (3.20) are oscillatory. One such solution is given by fy()g = f(?) =2 g: Example 3.2. Consider the dierence equation (3:2) jy()j? y() + y( 2c) (b + 2 ) = b y( 2c); 0

6 8 P. J. Y. WONG AND R. P. AGARWAL where > 0; c is any xed integer, and b = b(; y( 2c)) is any function. Here, m = : Tae? ( 2c) and f(y) = y: Then, it is obvious that (3.) holds. Further, we have Q(; y( 2c)) = b + 2 q() f(y( 2c)) and P (; y( 2c)) = b p() f(y( 2c)) and so (3.2) is satised. Hence, Theorem 3.(a) ensures that all solutions of (3.2) are oscillatory. One such solution is given by fy()g = f(?) g: Example 3.3. Consider the dierence equation (3:22) j 3 y()j? 3 y() + y() b + 4 (2 + 3) = b y(); where 0 < < =2 and b = b(; y(); y()) is any function. Here, m = 3: Taing? and f(y) = y; we note that (3.) holds. Next, and lead to Q(; y(); y()) f(y()) = b + 4 (2 + 3) P (; y(); y(); 2 y()) f(y()) [q()? p()] = = 4 = b p() 2 =? + 3 q() < = and hence (3.2) is not satised. The conditions of Theorem 3. are violated. In fact, (3.22) has a solution given by fy()g = f(?) g; and we observe that both fy()g and fy()g = f(?) + (2 +)g are oscillatory. This illustrates Theorem 3.(b). Theorem 3.2. Suppose (3.3) and (3.4) hold. Then, the conclusion of Theorem 3. follows. Proof. Suppose that fy()g is a nonoscillatory solution of (.), say, y() > 0 for 0 : Using (I) - (III), from (.) we still get (3.8) and (3.9). Case m is even Since (3.9) holds, from Lemma 2.2 (tae i = in (b)) we obtain (3.0). It follows that (3:23) y() y( ) a; : In view of (3.23) and the fact that lim! (? ) = ; there exists an integer 2 such that (3:24) y(? ) a; 2 :

7 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR D. E. 9 Since the function f is nondecreasing (condition (3.3)), (3.24) provides (3:25) f(y(? )) f(a) A; 2 : Now, from (3.8) we get (3.3) which on using (3.25) provides j m y()j? m y() + [q()? p()]a 0; 2 : In view of (3.9), the above inequality is the same as or j m y()j [q()? p()]a; 2 (3:26)? m y() f[q()? p()]ag = ; 2 : By discrete Taylor's formula [, p.26], y() can be expressed as y() = m? i=0 (? 2 ) (i) i! i y( 2 ) + which on rearranging and using (3.26) yields m? i=0 (? 2 ) (i) i!?m (? `? ) (m?) m y(`) (m? )! `= 2 A =?m (? `? ) (m?) [q(`)? p(`)] = (m? )! `= 2 i y( 2 )? y() m? i=0 (? 2 ) (i) Dividing both sides by (m?) ; the above inequality becomes A = (m? )! (3:27)?m (m?) m? i=0 `= 2 (? `? ) (m?) [q(`)? p(`)] = (? 2 ) (i) i! i! (m?) i y( 2 ); 2 : i y( 2 ) ; 2 : By (3.4), the left side of (3.27) tends to as! : However, the right side of (3.27) is nite as! : Case 2 m is odd In this case, taing note of (3.9), in Lemma 2.2 we have p is even and 0 p m? : Therefore, we cannot ensure that (3.0) holds. (i) Suppose that y() > 0 for ; i.e., (3.0) holds. The proof for this case follows from that of Case. (ii) Suppose that y() < 0 for : Then, y() # a (> 0) and so there exists a 2 such that (3.24) holds. The proof then proceeds as in Case.

8 20 P. J. Y. WONG AND R. P. AGARWAL (iii) Suppose that y() is oscillatory. Condition (3.4) implies that (m?) (? (? m)? )(m?) [q(? m)? p(? m)] = > 0 for suciently large ; which ensures that (3.9) holds for suciently large : Hence, for the special case m = ; we get (3.8) and it is seen from the proof of Theorem 3. (Case 2(iii)) that this leads to some contradiction. The proof of the theorem is now complete. Example 3.4. Consider the dierence equation (3:28) j 4 y()j 2 4 y() + [y( + )] 5? b = b[y( + )] 5 ; 0 where b = b(; y(+); y(+); 2 y(+)) is any function. Here, = 3; m = 4; and f(y) = y 5 ; which is nondecreasing. Taing? ( + ); we have Q(; y( + ); y( + ); 2 y( + )) f(y( + )) = b q() and We nd that P (; y( + ); y( + ); 2 y( + ); 3 y( + )) f(y( + )) = b p():?m (? `? ) (m?) [q(`)? p(`)] = (m?) `= 0 =?4 (? `? ) (3) 8 2 4`+ (3) (3) 3(3) 8 2 4(?4)+ = 3 (3) 8 2?323(?4) (3) 3 (3) 8 2?3 ; 7: Hence, (3.4) is satised. By Theorem 3.2(a) all solutions of (3.28) are oscillatory. One such solution is given by fy()g = f(?) =2 g: Remar 3.. It is clear that conditions (3.) and (3.2) are fullled for equation (3.28). Hence, Example 3.4 also illustrates Theorem 3.(a). Remar 3.2. Equation (3.2) also satises conditions (3.3) and (3.4). Hence, Theorem 3.2(a) ensures that all solutions of (3.2) are oscillatory. We have seen that one such solution is given by fy()g = f(?) g: Remar 3.3. In Example 3.3, the condition (3.3) is satised. To chec whether condition (3.4) is fullled, we note that

9 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR D. E. 2?m (? `? ) (m?) [q(`)? p(`)] = (m?) `= 0 =?3 (? `? ) (2) 4(2` + 3) 4 (2) `= `= Letting! ; we get, in view of 0 < < =2; lim!?3 (2) `= (2) 4(2` + 3) (? `? ) 4 `= (? 2)(2) (2) `=? `=? `= : `= 2 `=? + 3 `= < and hence (3.4) is not satised. The conditions of Theorem 3.2 are violated. In fact, it is noted that equation (3.22) has an oscillatory solution given by fy()g = f(?) g where fy()g is also oscillatory. Hence, Example 3.3 also illustrates Theorem 3.2(b). Theorem 3.3. Suppose =?; and (3.5) - (3.7) hold. Then, the conclusion of Theorem 3. follows. Proof. Again suppose that fy()g is a nonoscillatory solution of (.), say, y() > 0 for 0 : Using (I) - (III), from (.) we have (3:29) j m y()j? m y() [p()? q()]f(y( + )) 0; and therefore (3.9) holds. Case m is even Since (3.9) holds, from Lemma 2.2 (tae i = in (b), i = m? in (a)) we get for ; (3:30) y() > 0; m? y() > 0: Using y() > 0 for and Lemma 2.3, we nd that there exists 2 such that y( + ) y() y? 2 p?m+? 2 p?m+ (m?)? 2 m? y() (m? )! It follows that (3:3) y( + ) (m? )! 2(p?m+)(m?) (? 2 m 2 ) (m?) m? y(); 2 : (m? )! 2(p?m+)(m?) 2 m? (m?) m? y() = A (m?) m? y(); 2 m+ 2 + m? 2 3

10 22 P. J. Y. WONG AND R. P. AGARWAL where A = 2 (p?m)(m?) =(m? )!: In view of (3.5), it follows from (3.3) that f(y( + )) f A (m?) m? y() Mf(A)f (3:32) M 2 f(a)f Now, using (3.32) in (3.29) gives (m?) f? m? y() ; 3 : j m y()j? m y() + [q()? p()]m 2 f(a)f (m?) m? y() (m?) f? m? y() 0; which, on noting that m? y() > 0 for 3 and (3.9), is equivalent to M 2 f(a)f (m?) [q()? p()] jm y()j f ( m? y()) ; 3 3 or (3:33) n o = M 2 f(a)f (m?) [q()? p()]? m y() f ( m? y()) = ; 3: Summing (3.33) from 3 to ; we get (3:34) M 2 f(a) = P `= 3 f(`(m?) )[q(`)? p(`)] = P `= 3? m y(`) R m? y( 3) 0 f ( m? y(`)) = du f(u) = : By (3.7), the left side of (3.34) tends to as! ; whereas the right side is nite by (3.6). Case 2 m is odd Here, in view of (3.9), in Lemma 2.2 we have p is even and 0 p m? : Hence, instead of (3.30) we can only conclude that (3:35) m? y() > 0; : (i) Suppose that y() > 0 for : The proof follows as in Case.

11 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR D. E. 23 (ii) Suppose that y() < 0 for : Then, on using Lemma 2.3 we nd that there exists 2 such that y( + ) y(2 p?m+ + + )?? 2 p?m+ (m?) + +? 2 m? y 2 m?p? (2 p?m+ + + ) (m? )! (m? )!? 2 p?m+ + +? 2 (m?) m? y() m? y() m? y? + 2 m? ( + )? 2 p?m+ (m?)? 2 m? y() (m? )! (m? )! 2(p?m+)(m?) (? 2 m 2 ) (m?) m? y(); 2 where we have also used the fact that m? y() is nonincreasing (by (3.9)) and It follows that y( + ) = min m? y( + 2 m? ( + )) > 0: 2 m? y() (m? )! 2(p?m+)(m?) 2 m? (m?) m? y() = A (m?) m? y(); 3 where A and 3 are dened in (3.3). The rest of the proof uses a similar argument as in Case. (iii) Suppose that y() is oscillatory. Condition (3.7) implies that n f( (m?) )[q()? p()]o = > 0 for suciently large : This ensures that (3.9) holds for suciently large : Hence, for the special case m = ; we get (3.8) and we have seen from the proof of Theorem 3. (Case 2(iii)) that this leads to some contradiction. The proof of the theorem is now complete. Example 3.5. Consider the dierence equation (3:36) j 2 y()j 2 y() + y( + ) [b + 6( + )] = b y( + ); 0 where b = b(; y( + )) is any function. Here, = 2 and m = 2: Tae? ( + ) and f(y) = y: Then, (3.5) and (3.6) clearly hold. Further, we have Q(; y( + )) f(y( + )) = b + 6( + ) q()

12 24 P. J. Y. WONG AND R. P. AGARWAL and Hence, P (; y( + ); y( + )) f(y( + )) = b p(): n f( (m?) )[q()? p()]o = = [ 6( + )] =2 = and (3.7) is satised. It follows from Theorem 3.3(a) that all solutions of (3.36) are oscillatory. One such solution is given by fy()g = f(?) g: Remar 3.4. In the above example, it is obvious that the conditions of Theorem 3. are satised. Hence, Example 3.5 also illustrates Theorem 3.(a). Remar 3.5. Equation (3.36) clearly satises condition (3.3). To see that (3.4) is fullled, we note that lim! (m?)?m `= 0 (? `? ) (m?) [q(`)? p(`)] = = lim! = 4 lim! 4 lim! = 4 lim!?2 "?2 "?2 (? `? ) 4 p` + p` +??2 p` +??? `= p` (` + ) 3=2 #?2 p` + # 4 lim! Z p`? d` = 8 3 lim! (? ) 3=2 = : Hence, the conditions of Theorem 3.2 are satised and Example 3.5 also illustrates Theorem 3.2(a). Example 3.6. Consider the dierence equation (3:37) jy()j? y() + y( + 2c) (b + 2 ) = b y( + 2c); 0 where > ; c is any xed positive integer, and b = b(; y( +2c)) is any function. Here, m = : Tae? ( + 2c) and f(y) = y: Then, it is obvious that (3.5)

13 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR D. E. 25 and (3.6) hold. Further, we have Q(; y( + 2c)) f(y( + 2c)) = b + 2 q() and Thus, P (; y( + 2c)) f(y( + 2c)) = b p(): n f( (m?) )[q()? p()]o = = 2 = and (3.7) is fullled. It follows from Theorem 3.3(a) that all solutions of (3.37) are oscillatory. One such solution is given by fy()g = f(?) g: Example 3.7. Consider the dierence equation (3:38) j 3 y()j? 3 y() + [y( + 2)] b + 4 (2 + 3) = b[y( + 2)] ; 0 ( + 2) where > 0; is any odd integer satisfying jj > ; and b = b(; y( + 2); y( + 2)) is any function. We have? ( + 2); f(y) = y ; and Q(; y( + 2); y( + 2)) f(y( + 2)) = b + 4 (2 + 3) ( + 2) q(); P (; y( + 2); y( + 2); 2 y( + 2)) f(y( + 2)) = b p(): Case > It is clear that (3.5) and (3.7) hold whereas (3.6) does not hold. Case 2 <? In this case (3.6) holds but (3.5) and (3.7) are not satised. Hence, the conditions of Theorem 3.3 are violated if jj > : In fact, (3.38) has a solution given by fy()g = f(?) g and both fy()g and fy()g are oscillatory. This example illustrates Theorem 3.3(b). Remar 3.6. Let > 2 in Example 3.7. Condition (3.) clearly holds. However, [q()? p()] = = 4(2 + 3) ( + 2) = < and so (3.2) is not fullled. Hence, the conditions of Theorem 3. are violated. Moreover, we see that (3.3) holds. Since

14 26 P. J. Y. WONG AND R. P. AGARWAL?m (? `? ) (m?) [q(`)? p(`)] = (m?) `= 0 =?3 (? `? ) (2) 4(2` + 3) (2) (` + 2) = 4 (? )(2) (2)?3 4(2` + 3) (` + 2) 4 = 4(2` + 3) < ; (` + 2) = the condition (3.4) is not satised. Therefore, the conditions of Theorem 3.2 are violated. Hence, when > 2 Example 3.7 also illustrates both Theorems 3.(b) and 3.2(b). References [] Agarwal, R. P., Dierence Equations and Inequalities, Marcel Deer, New Yor, 992. [2] Agarwal, R. P., Properties of solutions of higher order nonlinear dierence equations I, An. St. Univ. Iasi 29(983), [3] Agarwal, R. P., Dierence calculus with applications to dierence equations, in General Inequalities, ed. W. Walter, ISNM 7, Birhaver Verlag, Basel (984), [4] Agarwal, R. P., Properties of solutions of higher order nonlinear dierence equations II, An. St. Univ. Iasi 29(985), [5] Grace, S. R., Lalli, B. S., Oscillation theorems for nth order delay equations, J. Math. Anal. Appl. 9(983), [6] Grace, S. R., Lalli, B. S., Oscillation theorems for damped dierential equations of even order with deviating arguments, SIAM J. Math. Anal. 5(984), [7] Hooer. J. W., Patula, W. T., A second order nonlinear dierence equation: oscillation and asymptotic behavior, J. Math. Anal. Appl. 9(983), [8] Lashmiantham, V., Trigiante, D., Dierence Equations with Applications to Numerical Analysis, Academic Press, New Yor, 988. [9] Popenda, J. Oscillation and nonoscillation theorems for second order dierence equations, J. Math. Anal. Appl. 23(987), [0] Staios, V. A., Basic results on oscillation for dierential equations, Hiroshima Math. J. 0(980), [] Thandapani, E., Oscillatory behavior of solutions of second order nonlinear dierence equations, J. Math. Phys. Sci. 25(99), [2] Thandapani, E., Sundaram, P., Oscillation theorems for some even order nonlinear dierence equations, preprint. [3] Wong, P. J. Y., Agarwal, R. P., Oscillation theorems and existence of positive monotone solutions for second order nonlinear dierence equations, Math. Comp. Modelling 2(995), [4] Wong, P. J. Y., Agarwal, R. P., Oscillation theorems and existence criteria of asymptotically monotone solutions for second order dierential equations, Dynamic Systems and Applications, to appear. [5] Wong, P. J. Y., Agarwal, R. P., Oscillatory behavior of solutions of certain second order nonlinear dierential equations, J. Math. Anal. Applic., 98 (996), [6] Wong, P. J. Y., Agarwal, R. P., Oscillation theorems for certain second order nonlinear dierence equations, to appear.

15 ON THE OSCILLATION OF AN mth ORDER PERTURBED NONLINEAR D. E. 27 [7] Wong, P. J. Y., Agarwal, R. P., Oscillation and monotone solutions of a second order quasilinear dierence equation, to appear. [8] Wong, P. J. Y., Agarwal, R. P., On the oscillation and asymptotically monotone solutions of second order quasilinear dierential equations, Appl. Math. Comp., to appear. [9] Wong, P. J. Y., Agarwal, R. P., Oscillations and nonoscillations of half-linear dierence equations generated by deviating arguments, Advances in Dierence Equations II, Computers Math. Applic., to appear. [20] Wong, Z., Yu, J., Oscillation criteria for second order nonlinear dierence equations, Fun. Ev. 34(99), Patricia J. Y. Wong Division of Mathematics Nanyang Technological University Singapore Ravi P. Agarwal Department of Mathematics National University of Singapore Singapore