Oscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments

Transcription

1 Journal of Mathematical Analysis Applications 6, ) doi: /jmaa , available online at on Oscillation Criteria for Certain nth Order Differential Equations with Deviating Arguments Ravi P. Agarwal Department of Mathematics, National University of Singapore, 10 Kent Ridge Crescent, Singapore matravip@nus.edu.sg Said R. Grace Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 11, Egypt srgrace@alpha1-eng.cairo.eun.eg Donal O Regan Department of Mathematics, National University of Irel, Galway, Irel donal.oregan@nuigalway.ie Submitted by William F. Ames Received February 1, 001 Oscillation criteria for nth order differential equations with deviating arguments of the form x n 1 t ) α 1 x n 1 t + F t x g t = 0 n even are established, where g C t 0 F C t 0, α>0isa constant. 001 Academic Press Key Words: oscillation; nonoscillation; comparison; functional differential equation X/01 $35.00 Copyright 001 by Academic Press All rights of reproduction in any form reserved.

2 60 agarwal, grace, o regan 1. INTRODUCTION In this paper we shall study the oscillatory behavior of the functional differential equation x n 1 t α 1 x n 1 t ) + F t x g t = 0 n even 1.1) where α is a positive constant, g t C t 0 lim t g t =, F t x C t 0, sgn F t x =sgn x t t 0. We shall assume that there exist a constant β>0 a function q t C t 0 + such that F t x sgn x q t x β for x 0 t t 0 1.) By a solution of Eq. 1.1) we mean a function x t C n 1 T x for some T x t 0 which has the property that x n 1 t α 1 x n 1 t C 1 T x satisfies equation 1.1) on T x. A nontrivial solution of Eq. 1.1) is called oscillatory if it has arbitrarily large zeros; otherwise it is said to be nonoscillatory. Equation 1.1) is oscillatory if all of its solutions are oscillatory. The equation 1.1) with n =, namely, the equation x t α 1 x t ) + F t x g t = 0 /or related equations have been the subject of intensive studies in recent years because these equations are natural generalizations of the equation x t +F t x g t = 0 For recent contributions we refer the reader to 5, 15, 19, 0] references therein. As far as we know the equation 1.1) has never been the subject of systematic investigations. In Section, we shall present some oscillation criteria for Eq. 1.1) which extend several known results established in 10, 16, 18 0]. Section 3 contains extensions of some of the results presented in Section to a special case of 1.1), namely, the equation x n 1 t α 1 x n 1 t ) + q t f x g t = 0 1.3) where α>0 is a constant, q t C t 0 + g t C t 0, f x C, lim t g t =, xf x > 0 for x 0. The function f in equation 1.3) need not be a monotonic function. Here, we shall also consider equations of neutral type of the form d x t +p t x τ t n 1 α 1 x t +p t x τ t n 1 ) + F t x g t dt = 0 1.4)

3 oscillation criteria 603 where α F, g are as in Eq. 1.1), p t C t = 0 τ t C 1 t 0, lim t τ t =. The obtained results extend those presented in 10, 1, 16]. In Section 4, we shall consider the more general equation x n 1 t α 1 x n 1 t ) + F t x g t d dt x h t ) = 0 1.5) where α is a positive constant, g t h t C t 0 h t t h t > 0 for t t 0 lim t g t = =lim t h t, F C t 0. We shall assume that there exist a function q t C t 0 + positive constants β µ such that F t x y sgn x q t x β y µ for xy 0 t t 0 1.6) The results presented in this section extend some of our earlier work in 1,, 6]. We shall need the following:. MAIN RESULTS Lemma.1 18]. Let x t C n t 0 +.Ifx n t is eventually of one sign for all large t, say, t 1 t 0, then there exist a t x t 0 an integer l 0 l n, with n + l even for x n t 0 or n + l odd for x n t 0 such that l>0 implies that x k t > 0 for t t x k = 0 1 l 1 l n 1 implies that 1 l+k x k t > 0 for t t x, k = l l + 1 n 1. Lemma. 18]. If the function x t is as in Lemma.1 x n 1 t x n t 0 for t t x, then there exists a constant θ 0 <θ<1, such that x t x t/ θ n 1! tn 1 x n 1 t θ n! tn x n 1 t for all large t for all large t

4 604 agarwal, grace, o regan Lemma.3. 11]. If X Y are nonnegative numbers, then X λ λxy λ 1 + λ 1 Y λ 0 λ > 1 X λ λxy λ 1 1 λ Y λ 0 0 <λ<1 In the above inequalities the equality holds if only if X=Y. Theorem.1. Let condition 1.) hold with α = β. If there exist σ t ρ t C 1 t 0 +, a constant θ>1 such that σ t inf t g t for T t 0, lim sup t T lim σ t = t σ t > 0 for t t 0.1) ρ s α+1 ] ρ s q s λθ ds =.) ρ s σ n s σ s α where λ = 1/ α + 1 α+1 n 1! α, then Eq. 1.1) is oscillatory. Proof. Suppose to the contrary that Eq. 1.1) has a nonoscillatory solution x t. Without loss of generality, we may assume that x t > 0 for t t 1 t 0 0. Since x n 1 t α 1 x n 1 t ) = F t x g t 0 it follows that the function x n 1 t α 1 x n 1 t is decreasing x n 1 t is eventually of one sign. If x n 1 t < 0 eventually, then since 0 x n 1 t α 1 α 1x x t ) n 1 = α x t ) n 1 n t we find that x n t 0 eventually. But then Lemma.1 implies that x n 1 t > 0 eventually. Further, when x n 1 t > 0 eventually then again from Lemma.1 note n is even) we have x t > 0 eventually. Thus there exists a t t 1 such that Define x t > 0 x n 1 t > 0 for t t.3) x n 1 t ) α w t =ρ t x β σ t / Then, for t t, in view of 1.) we have w t ρ t q t + ρ t ρ t w t βσ t ρ t t t αx x t ) n 1 σ t / x β+1 σ t /.4)

5 oscillation criteria 605 By Lemma. notice since x n 1 t > 0 for t t, we have x n 1 t α 0 for t t, which in turn implies x n t 0 for t t ), there exists a t 3 t a constant θ 1 0 <θ 1 < 1 such that x σ t / since x n 1 σ t x n 1 t for t t 3. Using.5) in.4) with α = β, wefind Fix t t 3, set X = Y = θ 1 n! σn t x n 1 t for t t 3.5) w t ρ t q t + ρ t ρ t w t αθ 1 n! σ n t σ t ρ 1/α t w α+1 /α t ) α/ α+1 αθ1 n! σn t σ w t t λ = α + 1 /α > 1 ρ 1/ α+1 t ) α α ρ ) t α/ α+1 ] α αθ1 α + 1 ρ t ρ1/ α+1 t n! σn t σ t Then, by Lemma.3, we obtain ρ t ρ t w t αθ 1 n! σn t σ t ρ 1/α t w α+1 /α t ) 1 α+1 ρ ) t α+1 ) θ α ] ρ t 1 α + 1 ρ t n! σn t σ t t t 3 Now, inequality.4) reduces to w t ρ t q t λρ t ρ ) t α ] ρ t θ 1 ρ t σ n t σ t for t t 3 Integrating the above inequality from t 3 to t, weget 0 <w t w t 3 t t 3 ρ s q s λρ s ρ s θ 1 ρ s σ n s σ s ) α ] ds.6) Taking lim sup on both sides of.6) as t, we obtain a contradiction to condition.). This completes the proof.

6 606 agarwal, grace, o regan We can apply Theorem.1 to the second order half-linear equation x t α 1 x t ) + q t x g t α 1 x g t = 0.7) where α>0 is a constant, q t C t 0 + g t C t 0, lim t g t =. In fact, we get the following new result. Corollary.1. If there exist two functions ρ t σ t C 1 t 0 + such that condition.1) holds, t ] ρ s q s ds =.8) lim sup t then Eq..7) is oscillatory. t 0 1 ρ s α+1 α + 1 α+1 ρ s σ s α Proof. Let x t be a nonoscillatory solution of Eq..7), say, x t > 0 for t t 1 t 0. It is easy to check that x t > 0 x σ t x t for t t t 1. Next, we define x ) t α w t =ρ t t t x σ t Then, w t ρ t q t + ρ t ρ t αρ 1/α t w α+1 /α t for t t The rest of the proof is similar to that of Theorem.1 hence is omitted. The following example illustrates our theory. Example.1. Consider the second order half-linear differential equation x t α 1 x t ) 1 + t α+1 x t α 1 x t =0 t > 0.9) where α>0 is a constant. Here, we take ρ t =t α. Then, t 1 ρ s α+1 ] ρ s q s ds T α + 1 α+1 ρ α s t ) α α+1 ] 1 = 1 α + 1 s ds = T 1 α α + 1 ) α+1 ] ln t T as t All conditions of Corollary.1 are satisfied hence Eq..9) is oscillatory. We note that the above conclusion do not appear to follow from the known oscillation criteria in the literature.

7 oscillation criteria 607 For each t t 0,weletg t t define γ t =sup s t 0 g s t. Clearly, γ t t g γ t =t. Our next result is embodied in the following: Theorem.. then Eq. 1.1) is oscillatory. Let condition 1.) hold with α = β. If lim sup t α n 1 q s ds > n 1! α.10) t γ t Proof. Let x t be an eventually positive solution of Eq. 1.1), say, x t > 0 for t t 1 t 0. As in the proof of Theorem.1, we obtain.3) for t t. Now integrating Eq. 1.1) from t t to u letting u, we get x n 1 t ) α q s x α g s ds By Lemma. there exist a constant θ 0 <θ<1 t 3 t such that Thus, x t t θ n 1! tn 1 x n 1 t for t t 3.11) ) x α θ α t n 1! tn 1 x n 1 t α ) θ α n 1! tn 1 q s x α g s ds t for t t 3 Now by γ t t the fact that x t > 0 g s t for s γ t, it follows that ) x α θ α t n 1! tn 1 q s x α g s ds γ t ) θ α n 1! tn 1 x α t q s ds Dividing both sides of the above inequality by x α t, weget ) θ α n 1! tn 1 q s ds 1 for t t 3.1) γ t Thus, γ t t n 1 ) α t lim sup q s ds = c< t n 1! γ t

8 608 agarwal, grace, o regan Suppose.10) holds. Then there exists a sequence T m m=1, with T m as m such that ) α Tm lim q s ds = c>1 m n 1! γ T m Thus, for ɛ = c 1 / > 0, there exists N>0 such that ) c + 1 α Tm = c ɛ< q s ds for m>n.13) n 1! γ T m Choose K / c + 1 1/α 1. From.1).13), we get ) α 1 K α Tλ q s ds > c + 1 = 1 n 1! γ T λ c + 1 for T λ sufficiently large. This contradiction proves that condition.10) is not satisfied. This completes the proof. In Theorem. if g t t, i.e., g t is an advanced argument, g t 0 for t t 0, we find that Theorem. takes the following form. Theorem.3. Let condition 1.) hold with α = β g t t, g t 0 for t t 0. If. lim sup t then Eq. 1.1) is oscillatory. t α n 1 q s ds > n 1! α.14) t Example.. Consider the half-linear differential equation x n 1 t α 1 x n 1 t ) + ct α n 1 1 x g t α 1 x g t.15) = 0 t > 0 where α c are positive constants, g t C t 0, lim t g t =. We conclude the following: i) If g t =t/, then γ t =t, hence Eq..15) is oscillatory by Theorem. provided that c> α n 1 α n 1 n 1! α ii) If g t t g t 0, then Eq..15) is oscillatory by Theorem.3 provided that c>α n 1 n 1! α Next, we have the following comparison result.

9 oscillation criteria 609 Theorem.4. Let condition 1.) hold assume that there exist a function σ t C 1 t 0 + a constant θ 0 <θ<1 such that σ t inf t g t σ t 0 lim σ t =.16) t for t t 0 If every solution of the delay equation ) y θ α t + σ α n 1 t y σ t β/α sgn y σ t = 0.17) n 1! is oscillatory, then Eq. 1.1) is oscillatory. Proof. Let x t be a nonoscillatory solution of Eq. 1.1), say, x t > 0 for t t 1 t 0. As in the proof of Theorem.1, we see that x n 1 t > 0 for t t t 1. By Lemma. there exist a constant θ, 0<θ<1 t 3 t such that x σ t θ n 1! σn 1 t x n 1 σ t for t t 3.18) Using.18) in Eq. 1.1), for t t 3 we obtain x n 1 t ) ) α) θ β + n 1! σn 1 t q t x n 1 σ t ) β x n 1 t ) ) α + q t x β σ t 0.19) Let y t = x n 1 t ) α t t3 to get y θ ) βq t t + n 1! σn 1 t y β/α σ t ) 0 for t t 3.0) Integrating inequality.0) from t t 3 to u letting u,wefind θ ) βq s y y t n 1! σn 1 s β/α σ s ds for t t 3 t The function y t is obviously decreasing on t 3. Hence, by Theorem 1 in 17], we conclude that there exists a positive solution y t of Eq..17) with lim t y t =0, which contradicts the fact that Eq..17) is oscillatory. This completes the proof. We can apply the results established in 14] to obtain the following corollary.

10 610 agarwal, grace, o regan or Corollary.. lim inf t t σ t Let conditions 1.).16) hold. If σ α n 1 s q s ds > n 1! α e when α = β.1) σ β n 1 s q s ds = when 0 < β/α < 1.) then equation 1.1) is oscillatory. Theorem.5. Let condition 1.) hold with α>1 β>1, assume that there exist two functions σ t ρ t C 1 t 0 + such that condition.1) is satisfied, ρ ρ t ) t 0 0 for t t0.3) σ n t σ t If ρ s q s ds =.4) then Eq. 1.1) is oscillatory. Proof. Let x t be a nonoscillatory solution of Eq. 1.1), say, x t > 0 for t t 1 t 0. As in the proof of Theorem.1 we obtain.3) for t t. Next, we define w t as in the proof of Theorem.1 to obtain.4) which takes the form ) α x n 1 t w t ρ t q t +ρ t x β σ t / for t t.5) Since x n 1 t is nonincreasing on t, there exist a t 3 t positive constants b θ 1 0 <θ 1 < 1 such that x n 1 t α 1 b for t t 3,.5) holds for t t 3. Now.5) takes the form w n! t ρ t q t +b θ 1 ρ t x σ t / σ n t x β σ t / t t 3.6) But, by the Bonnet theorem for a fixed t t 3 for some ξ t 3 t, we have t t 3 ρ s x σ s / σ s / ds σ n s σ s x β σ s / ρ ) t = 3 ξ x σ s / σ s / ds σ n t 3 σ t 3 t 3 x β σ s / ρ ) t = 3 x σ ξ / w β dw σ n t 3 σ t 3 x σ t 3 /

11 hence, since ρ t 3 0 oscillation criteria 611 we find where t x σ ξ / t 3 x σ t 3 / dw w = 1 x 1 β σ t β β 1 3 / x 1 β σ ξ / ) < 1 β 1 x1 β σ t 3 / ρ s x σ s / σ s / ds K for t t σ n s σ s x β σ s / 3.7) ρ t K = 3 1 σ n t 3 σ t 3 β 1 x1 β σ t 3 / Now in view of.7) it follows that t t 3 ρ s q s ds w t +w t 3 +K< This contradicts.4) so the proof is complete. Theorem.6. Let condition.3) in Theorem.5 be replaced by ρ t 0 ρ ) s ds < σ n s σ s then the conclusion of Theorem.5 holds. t 0 The proof is similar to that of Theorem.5 hence is omit- Proof. ted. Theorem.7. Let condition 1.) hold with β>α assume that there exists σ t C 1 t 0 + such that condition.1) is satisfied. If 1/α σ n s σ s q u du) ds =.8) s then Eq. 1.1) is oscillatory. Proof. Let x t be a nonoscillatory solution of Eq. 1.1), say, x t > 0 for t t 1 t 0. As in the proof of Theorem.5 we take ρ t =1 obtain x n 1 t ) α t q s ds x β σ t / <

12 61 agarwal, grace, o regan therefore for t t, x n 1 t ) α q s ds or t x β σ t / ) 1/α q s ds x n 1 t x β/α σ t / t Now by Lemma. there exist a t 3 t a constant θ 1 0 <θ 1 < 1 such that.5) holds for t t 3. Thus, for t t 3, ) θ ) 1/α 1 n! σn t σ t q s ds θ 1 n! σn t σ x n 1 t t x β/α σ t / x σ t / σ t / x β/α σ t / Integrating the above inequality from t 3 to t, weget θ t 1/α t 1 σ n s σ x s q u du) σ s / σ s / ds ds n! t 3 s t 3 x β/α σ s / t = x σ t / x σ t 3 / w β/α dw α β α x α β /α σ t 3 / < which contradicts condition.8). This completes the proof. Example.3. The equation ) x n 1 t α 1 x n 1 t + t n 1 α 1 x γt β sgn x g t = 0 t t 0 > 0 which α β γ are positive constants, β>α γ 1, is oscillatory by Theorem SOME EXTENSIONS Here we shall extend our results of Section to Eqs. 1.3) 1.4). For Eq. 1.3) when the function f need not be monotonic we need the following notations a lemma due to Mahfoud 16], { t0 t t0 = 0 if t 0 > if t 0 = 0,

13 C B t0 oscillation criteria 613 C = f is continuous xf x > 0 for x 0 = { f C f is of bounded variation on any interval a b t0 } Lemma 3.1. Suppose t 0 > 0 f C. Then, f C B t0 if only if f x =H x G x for all x t0, where G t0 + = 0 is nondecreasing on t 0 nonincreasing on t 0, H t0 is nondecreasing on t0. To obtain an extension, we assume that f C t0 t 0 0 let G H be a pair of continuous components of f with H being the nondecreasing one. Also, we assume that H x sgn x x β for x 0 β>0 is a constant 3.1) As in Section, if x t is a nonoscillatory solution of Eq. 1.3), say, x t > 0 for t t 1 t 0, then there exists a t t 1 such that.3) holds for all t t. Next, there exist a t 3 t a constant b>0 such that x n 1 t b for t t 3 3.) Integrating 3.) n 1 times, there exist a t 4 t 3 a positive constant K>0 such that Now it follows from Eq. 1.3) that x g t Kg n 1 t for t t 4 3.3) 0 = d dt x n 1 t ) α + q t G x g t H x g t d dt x n 1 t ) α) + q t G x g t x β g t d x n 1 t ) α) + q t G Kg n 1 t x β σ t dt for t t 4 3.4) where σ C 1 t 0 + σ t inf t g t as t σ t 0 for t t 0 3.5) Integrating the above inequality from t to u t 4 t u letting u, we obtain 1/α x n 1 t q s G K g n 1 s x β σ s ds) t

14 614 agarwal, grace, o regan Following similar steps as in the proof of Lemma.1 in 13], we find that if inequality 3.4) has an eventually positive solution, then so does the equation d y n 1 t ) α + q t G Kg n 1 t y β σ t = 0 3.6) dt Thus, to extend the results of Section, we shall need to apply the following theorem. Theorem 3.1. Assume that f C t0 t 0 0, let G H be a pair of continuous components of f with H being the nondecreasing one. Moreover, assume that conditions 3.1) 3.5) hold. If, for every K>0, the equation x t n 1 α 1 x t ) n 1 + q t G Kg n 1 t x σ t β 1 x σ t = 0 is oscillatory, then Eq. 1.3) is also oscillatory. We note that Theorem 3.1 together with the results of Section can be applied to equations of type 1.3) with f being any of the following functions: i) ii) iii) f x = x β 1 x/ 1 + x γ β γ are positive constants, f x = x β 1 x exp x γ β γ are positive constants, f x = x β 1 x sech x β is a positive constant. However, the results of Section are not applicable to Eq. 1.3) with any one of the above choices of f. Next, we shall extend the results of Section to neutral equations of type 1.4). In fact, if we define z t =x t +p t x τ t, then Eq. 1.4) becomes z n 1 t α 1 z n 1 t ) + F t x g t = 0 3.7) Now if x t is a nonoscillatory solution of Eq. 1.4), say, x t > 0 x τ t > 0 for t t 1 t 0 Then, z t > 0 for t t 1 there exists a t t 1 such that z n 1 t > 0 z t > 0 for t t. In what follows we shall examine the following two cases for τ t p t : i) ii) 0 p t 1 τ t <t p t 1 τ t >t. For case i), we assume that 0 p t 1 τ t <t τ t is strictly increasing for t t 0 p t 1 eventually. 3.8)

15 oscillation criteria 615 Now, x t =z t p t x τ t = z t p t z τ t p τ t x τ τ t z t p t z τ t 1 p t z t for t t 3.9) Using conditions 1.) 3.9) in Eq. 3.7), we get d α z t ) n 1 + q t 1 p g t β z β g t 0 for t t dt 3 t Now if 3.5) holds, then d dt z n 1 t α + q t 1 p g t β z β σ t 0 for t t ) As in the above discussion, we conclude that if inequality 3.10) has an eventually positive solution, then so does the equation d dt y n 1 t α + q t 1 p g t β y β σ t = ) Thus, we have the following result: Theorem 3.. Let conditions 1.), 3.5), 3.8) hold. If the equation y n 1 t α 1 y n 1 t ) + q t 1 p g t β y σ t β 1 y σ t = 0 is oscillatory, then Eq. 1.4) is also oscillatory. For case ii), we assume that p t 1 p t 1 eventually, τ t >t τ t is strictly increasing for t t 0 3.1) there exists σ t C 1 t 0 + such that σ t inf t τ 1 og t as t σ t 0 for t t ) where τ 1 is the inverse function of τ. We also let ) P 1 1 t = 1 p τ 1 t p τ 1 τ 1 t for all large t

16 616 agarwal, grace, o regan Now, since z t > 0 for t t, we obtain 1 x t = z τ 1 t x τ 1 t ) p τ 1 t = z τ 1 t p τ 1 t 1 p τ 1 t z τ 1 t p τ 1 t 1 p τ 1 t 1 τ τ 1 τ 1 t p τ 1 τ 1 t x τ 1 τ 1 ) t p τ 1 τ 1 t z τ 1 τ 1 t p τ 1 t p τ 1 τ 1 t 1 p τ 1 τ 1 t ] z τ 1 t = P t z τ 1 t for t t 3.14) Using 1.), 3.13), 3.14) in Eq. 3.7), we have 0 d dt z n 1 t ) α + q t P g t β z β τ 1 g t d z n 1 t ) α + q t P g t β z β σ t for t t dt 3 t Thus, similar to Theorem 3. we have the following result: Theorem 3.3. Let conditions 1.), 3.1), 3.13) hold. If the equation y n 1 t α 1 y n 1 t ) + q t P g t β y σ t β 1 y σ t = 0 is oscillatory, then Eq. 1.4) is also oscillatory. Remark 3.1. Further extensions to equations of the form x t +p t x τ t n 1 α 1 x t +p t x τ t 0 n 1 ) + q t f x g t = 0 where f need not be monotonic, can be obtained easily by Theorems Example 3.1. For the equation d x t +px γt n 1 α 1 x t +px γt ) + t n 1 α 1 x λt β sgn x λt dt = 0 for t t 0 > ) where p α β γ, λ are positive constants, β>α, we conclude the following: i) If p<1 γ < 1, λ 1, then Eq. 3.15) is oscillatory by Theorems ii) If p>1 γ > 1, λ γ, then Eq. 3.15) is oscillatory by Theorems 3..7.

17 oscillation criteria FURTHER OSCILLATION CRITERIA Our first oscillatory criterion for the equation 1.5) is embodied in the following theorem. Theorem 4.1. Let condition 1.6) hold, h t g t t for t t 0.If for every θ i 0 <θ i < 1 i= 1 the equations ] y θ t + 1 h t n λ H t q t y h t λ/α sgn y h t n 1 β n! λ = 0 4.1) ] ] z θ t + t h t n λ H t q t t + h t λ/α n n! λ z ] t + h t sgn z = 0 4.) where λ = β + µ α H t = h t β h t λ are oscillatory, then Eq. 1.5) is oscillatory. Proof. Let x t be a nonoscillatory solution of Eq. 1.5), say, x t > 0 for t t 1 t 0. It is easy to check that there exists a t t 1 such that x n 1 t > 0 x t > 0 for t t. We distinguish the following two cases: I) x n 1 t > 0 x t > 0 x t > 0 for t t, II) x n 1 t > 0 x t < 0 x t > 0 for t t. Assume I) holds. By Lemma. there exist a t 3 t b i > 0 0 < b i < 1 i= 1 such that, for t t 3, x g t x h t b 1 n 1! hn 1 t x n 1 h t 4.3) d dt x h t = x h t h b t n! hn t h t x n 1 h t 4.4) Using conditions 1.6), 4.3), 4.4) in Eq. 1.5), we get d x n 1 t ) ) α b β ) + 1 b µ h n λ t H t q t dt n 1! n! x n 1 h t ) λ 0 for t t3

18 618 agarwal, grace, o regan Setting w t = x n 1 t α t t 3 we have w t + θ β 1 θµ n 1 β n! λ h n λ t H t q t w λ/α h t 0 for t t 3 4.5) Integrating 4.5) from t t 3 to u letting u,wefind ] θ β 1 w t θµ h n λ s H s q s w λ/α h s ds n 1 β n! λ t The function w t = x n 1 t α is clearly strictly decreasing for t t 3. Hence, by Theorem 1 in 17], there exists a positive solution y t of Eq. 1.5) with y t 0ast. But this contradicts the assumption that Eq. 4.1) is oscillatory. Assume II) holds. By Lemma. there exists a T 1 t 1 a constant a 0 <a<1 such that x g t x h t ah t x h t for t T 1 4.6) Using conditions 1.6) 4.6) in Eq. 1.5) setting v t =x t for t T 1, we obtain d v n t ) α + a β H t q t v λ h t 0 for t T dt 1 4.7) It is clear that function v t satisfies 1 i v i t > 0 i = 0 1 n t T 1 4.8) Now by Lemma..4 in ], there exists a T T 1 such that t h t n ] ] t + h t v h t v n for n n! Thus, 4.7) takes the form w t + T h t t + h t 4.9) a β n n! λ t h t n λ H t q t w λ/α h t 0 t T 4.10) where w t = v n t ) α t T. The rest of the proof is similar to that of case I) hence is omitted.

19 oscillation criteria 619 Now applying the results established in 14] to Theorem 4.1, we obtain Corollary4.1. t t 0.If Let condition 1.6) hold, let h t g t t for I 1 t lim inf h n α s H s q s ds > n 1 β n! α t h t e t lim inf s h s n α H s q s ds > n n! α t t+h t / e are satisfied when λ = β + µ = α, I h n λ s H s q s ds = s h s n λ H s q s ds = hold when λ<α, then Eq. 1.5) is oscillatory. Next we shall provide sufficient conditions for the oscillation of Eq. 1.5) when β α µ α. Theorem 4.. Let condition 1.6) hold, let g t t g t 0 for t t 0. If for every positive constant θ i i= 1 the equations ] y θ t + 1 h t µ g n 1 β t q t y g t β/α n 1! β sgn y g t = ) ] z θ t + n n! t µ h t n µ h t µ q t t + h t µ/α z ] t + h t sgn z = 0 4.1) are oscillatory, then Eq. 1.5) is oscillatory.

20 60 agarwal, grace, o regan Proof. Let x t be a nonoscillatory solution of Eq. 1.5), say x t > 0 for t t 1 t 0. As in Theorem 4.1, we have cases I) II) for t t. Assume I) holds. Then there exist a t 3 t positive constants a b such that x g t d dt x h t ah t for t t ) b n 1! gn 1 t x n 1 g t for t t ) Using conditions 1.6), 4.13), 4.14) in Eq. 1.5), we obtain w a µ b β ] t + h t µ g n 1 β t q t w β/α g t 0 for t t n 1! β 3 where w t = x n 1 t α t t 3. Now proceeding as in the proof of Theorem 4.1I), we arrive at the desired contradiction. Assume II) holds. Then there exist a T t 1 a positive constant a 1 such that 4.9) holds, Thus 4.10) takes the form w a β 1 t + n n! µ 0 x g t a 1 for t T 4.15) ] for t T t h t n µ h t µ q t w µ/α t + h t The rest of the proof is similar to that of Theorem 4.1II) hence is omitted. The following result provides sufficient conditions for the oscillation of Eq. 1.5) when β µ are arbitrary positive constants. Theorem 4.3. Let condition 1.6) hold, let g t t g t 0 for t t 0. If for every positive constant θ 1 θ the equation y t n 1 α 1 y t ) n 1 +θ1 h t µ q t y g t β sgn y g t =0 4.16) is oscillatory, every bounded solution of the equation z t n α 1 z t ) n +θ h t µ q t z h t µ sgn z h t =0 4.17) is oscillatory, then Eq. 1.5) is oscillatory. ]

21 oscillation criteria 61 Proof. Let x t be a nonoscillatory solution of Eq. 1.5), say, x t > 0 for t t 1 t 0. As in the proof of Theorem 4.1 we consider cases I) II). In case I) inequality 4.13) holds for t t 3. Thus, Eq. 1.5) leads to d x n 1 t ) α + a µ h t µ q t x β g t 0 for t t dt 3 Using an argument presented in Section 3, we find that the equation d x n 1 t ) α + a µ h t µ q t x β g t = 0 dt has a positive solution, which is a contradiction. If II) holds, then 4.15) is satisfied for t T t 1, hence we have d v n t ) α β + a 1 dt t µ q t v β h t 0 for t T 4.18) where v t =x t 4.8) holds for t T. Integrating inequality 4.18) n 1 times from t T to u, using 4.8), letting u,wefind v t a β s t n 3 1/α 1 h τ µ q τ v h τ dτ) β ds t n 3! s Now following similar steps of the proof of Theorem 1 in 17], we conclude that Eq. 4.17) has a solution z t with lim t z t =0, which is a contradiction. This completes the proof. Example 4.1. Consider the equation x n 1 t ) α 1 x n 1 x t ] β d t ] t + q t dt x µ t ] sgn x = ) where q t C t 0 + α β, µ are positive constants. Let β α µ α. Then, by Theorem 4., Eq. 4.19) is oscillatory if for every positive constant θ 1 θ the equations ] y θ t + 1 t n 1 β t ] ] β/α t q t y sgn y = 0 µ n 1 n 1! β ] z θ t + t n µ q t z n 3 n! µ 3t 4 ] ] µ/α t sgn z = 0 are oscillatory. We also note that Eq. 4.19) is oscillatory if we take q t = t n k 0 <k<1 when α = β = µ, when β<α µ<α. q t = 1 t min t n 1 β t n µ t > 1

22 6 agarwal, grace, o regan REFERENCES 1. R. P. Agarwal S. R. Grace, Oscillation of certain functional differential equations, Comput. Math. Appl ), R. P. Agarwal, S. R. Grace, D. O Regan, Oscillation Theory for Difference Functional Differential Equations, Kluwer, Dordrecht, R. P. Agarwal, S.-H. Shieh, C. C. Yeh, Oscillation criteria for second-order retarded differential equations, Math. Comput. Model ), A. Elbert, A half-linear second order differential equation, in Proceedings of the Colloquia Math. Soc. János Bolyai 30: Qualitative Theory of Differential Equations, Szeged, 1979, pp A. Elbert T. Kusano, Oscillation nonoscillation theorems for a class of second order quasilinear differential equations, Acta Math. Hungar ), S. R. Grace, Oscillatory asymptotic behavior of delay differential equations with a nonlinear damping term, J. Math. Anal. Appl ), S. R. Grace, Oscillation theorems for damped functional differential equations, Funkcial. Ekvac ), S. R. Grace, Oscillation theorems for certain functional differential equations, J. Math. Anal. Appl ), S. R. Grace, Oscillation criteria of comparison type for nonlinear functional differential equations, Math. Nachr ), S. R. Grace B. S. Lalli, Oscillation theorems for nth order delay differential equations, J. Math. Anal. Appl ), G. H. Hardy, J. E. Littlewood, G. Polya, Inequalities, nd ed., Cambridge Univ. Press, Cambridge, UK, J. Jeroš T. Kusano, Oscillation properties of first order nonlinear functional differential equations of neutral type, Differential Integral Equations ), A. G. Kartsatos, On nth order differential inequalities, J. Math. Anal. Appl ), R. G. Koplatadze T. A. Chanturia, On oscillatory monotone solutions of first order differential equations with deviating arguments, Differencial nye Uravnenija ), T. Kusano B. S. Lalli, On oscillation of half-linear functional differential equations with deviating arguments, Hiroshima Math. J ), W. E. Mahfoud, Remarks on some oscillation theorems for nth order differential equations with a retarded argument, J. Math. Anal. Appl ), Ch. G. Philos, On the existence of nonoscillatory solutions tending to zero at for differential equations with positive delays, Arch. Math ), Ch. G. Philos, A new criterion for the oscillatory asymptotic behavior of delay differential equations, Bull. Acad. Pol. Sci. Ser. Sci. Mat ), P. J. Y. Wong R. P. Agarwal, Oscillation theorems existence criteria of asymptotically monotone solutions for second order differential equations, Dynam. Systems Appl ), P. J. Y. Wong R. P. Agarwal, Oscillatory behaviour of solutions of certain second order nonlinear differential equations, J. Math. Anal. Appl ),