Kamenev-Type Oscillation Criteria for Higher-Order Neutral Delay Dynamic Equations

Transcription

1 International Journal of Difference Equations ISSN , Volume 6, Number, pp Kamenev-Type Oscillation Criteria for Higher-Order Neutral Delay Dynamic Equations Lynn H. Erbe Department of Mathematics University of Nebrasa-Lincoln Lincoln, NE , USA lerbe2@math.unl.edu Başa Karpuz Department of Mathematics, Faculty of Science and Arts A.N.S. Campus, Afyon Kocatepe University Afyonarahisar, Turey barpuz@gmail.com Allan C. Peterson Department of Mathematics University of Nebrasa-Lincoln Lincoln, NE , USA apeterso@math.unl.edu Abstract In this paper, we are primarily concerned with the oscillation of solutions to higher-order dynamic equations. We first extend some recent results which have been obtained for second-order dynamic equations to even-order neutral delay dynamic equations. Our technique depends on maing use of the Riccati substitution and the use of properties of the generalized Taylor monomials on time scales. We also extend our results to odd-order neutral delay dynamic equations. Some examples for differential equations, difference equations, and q-difference equations which have important applications in quantum theory are given to illustrate the results obtained. AMS Subject Classifications: 34K, 39A0. Keywords: Delay dynamic equations, even-order, odd-order, oscillation, time scales. Received April 5, 20; Accepted June 20, 20 Communicated by Martin Bohner

2 2 L. H. Erbe, B. Karpuz and A. C. Peterson Introduction The theory of time scales has received a great deal of attention since it was introduced by Hilger in his Ph.D. Thesis in 988. The goal is to unify and extend continuous and discrete analysis see the Ph.D. Thesis of Hilger [9]. Since then, many authors have contributed to various aspects of this new theory; see the survey paper [2] by Agarwal et al. and references cited therein. We refer the readers to the boo [4] by Bohner and Peterson, which summarizes and organizes much of the time scale theory. A time scale T is an arbitrary closed subset of the reals, and the cases when T = R or T = Z represent the classical theories of differential and of difference equations. Many other interesting time scales exist, and they give rise to a number of applications, among them the study of population dynamic models which are discrete in season and may follow a difference scheme with variable step-size or may be modeled by continuous dynamic systems. Then the population dies out, say in winter, while the eggs are incubating or are dormant, and then in season again, the eggs hatch and therefore this model gives rise to a non-overlapping population see [4, Example.39]. Not only does this new theory of so-called dynamic equations unify the existing theories of differential equations and of difference equations, but it also extends these classical cases to cases in between, e.g., to the so-called q-difference equations when T = q N 0 = {q n : n N 0 } where q > which has important applications in quantum theory see the boo [0] by Kac and Cheung. Moreover, these ideas can be applied to many different types of time scales such as T = hz h > 0, T = N 2 and T = { n } / : n N, the set of harmonic numbers. = For a reader not familiar with time scale calculus, we summarize some of the following basic information. A time scale, which inherits the standard topology on R, is a nonempty closed subset of the real line. Here, and later throughout this paper, a time scale will be denoted by the symbol T, and intervals with a subscript are used to denote the intersection of the usual real interval with T. For t T, we define the forward jump operator σ : T T by σt := inft, T while the bacward jump operator ρ : T T is defined by ρt := sup, t T Here we define inf = sup T and sup = inf T. The graininess function µ : T R + 0 is defined by µt := σt t. A point t T is called right-dense if σt = t and t sup T. If σt > t it is called right-scattered. Similarly left-dense and left-scattered points are defined with respect to the bacward jump operator. We recall also that T κ := T\{sup T} if sup T = max T and satisfies ρmax T max T; otherwise, T κ := T. The Hilger derivative of a function f : T R is defined by f σ t ft, µt > 0 f t := µt ft fs lim, µt = 0 s t t s

3 Higher-Order Dynamic Equations 3 for t T κ provided that the limit exists. A function f is said to be rd-continuous provided that it is continuous at right-dense points in T, and has a finite limit at leftdense points. The set of rd-continuous functions will be denoted by C rd T, R. The set of functions C rdt, R denotes those functions f whose derivative is in C rd T, R. For s, t T and a function f C rd T, R, the -integral of f is defined by s fηη = F t F s for s, t T, where F C rdt, R is an anti-derivative of f, i.e., F = f on T κ. In recent years, there have been several papers which have studied oscillation and nonoscillation properties of delay dynamic equations on arbitrary time scales see the papers [2,3,4]. Much of the wor thus far has been motivated by the qualitative theory of second-order equations. There are fewer results dealing with the oscillation and asymptotic behaviour of third and/or higher-order dynamic equations see the papers [7, 2, 3]. In this paper, we shall present some results dealing with the oscillation and asymptotic behaviour of solutions of higher-order neutral delay dynamic equations of the following form: [ xt + Atxαt ] n + Btxβt = 0 for t [t0, T.. Here n N, T is a time scale which satisfies sup T =, and t 0 is a fixed point in T. In addition, we assume A C rd [t 0, T, R and B C rd [t 0, T, R + 0. We suppose that α C rd [t 0, T, T and β C rd[t 0, T, T are strictly increasing and satisfy lim αt =, lim βt =, and αt t, βt t for all t [t 0, T. We also assume that β[t 0, T = [βt 0, T. Our method maes use of the Riccati substitution technique and, after deducing some second-order dynamic inequalities, we relate oscillation of. to second-order dynamic equations, with which we are familiar from well-nown results in the literature. We set t := min { inf t [t0, T {αt}, inf t [t0, T {βt} }. By a solution of. we mean a function x C rd [t, T, R such that x + Atx α C n rd[t 0, T, R and x satisfies. identically on [t 0, T. If x is a solution of., then x is said to be nonoscillatory if x is eventually of one sign. Otherwise, x is said to be oscillatory. Let us briefly recall some classical results for the qualitative theory of the secondorder differential equation x t + Atxt = 0 for t [t 0, R,.2 where A C rd [t 0,, R + 0. In the past two decades, obtaining sufficient conditions for the oscillation and nonoscillation of solutions of second-order differential equations has attracted a great deal of

4 4 L. H. Erbe, B. Karpuz and A. C. Peterson attention, and one of the most powerful tools is the integral averaging technique. These ideas may be traced bac to the classical result due to Wintner in 949 see [7], where lim t t 0 t sasds = is shown to be a sufficient condition for the oscillation of all solutions to.2. Later on, in 952, Hartman see [8] showed that the limit condition given above may not be replaced by a lim sup condition. Specifically, it was shown that < lim inf t t sasds < lim sup t 0 t t 0 t sasds implies that every solution of.2 is oscillatory. In 978, Kamenev see [] improved the result of Wintner by showing that the condition lim t t t s Asds = for some N t 0 is sufficient for the oscillation of all solutions to.2. Analogous results have also been obtained for the second-order difference equation 2 xn + Anxn = 0 for n [n 0, Z,.3 where denotes the forward difference operator and {An} is a nonnegative sequence. It was shown that lim n n n i=n 0 n i Ai = for some N.4 is sufficient for the oscillation of all solutions of.3 see the papers [5,8]. We note also that n lim n i Ai = for some N, n n i=n 0 where t := tt t + is the falling factorial function see the boo [6] by Kelley and Peterson, is equivalent to.4. For additional results on integral averaging see Erbe [6] and the references therein. In order to view our results in the time scale setting as analogues of classical results, we shall use generalized polynomials instead of the usual polynomials. The main motivation behind our paper is the paper [5] by Džurina. Our results extend and generalize the results in [5] to arbitrary time scales. The paper is organized as follows: In Section 2, we give some important results required in the sequel some of which are taen from [4]; in Section 3, we prove some oscillation criteria for. when the order is even; in Section 4, we present some Kamenev-type oscillation criteria for even-order delay dynamic equations with neutral term; and finally in Section 5, we extend our results to neutral delay dynamic equations of odd order.

5 Higher-Order Dynamic Equations 5 2 Auxiliary Results The generalized Taylor monomials h : T T R, N, are defined by, = 0 h t, s := t h η, sη, N for all s, t T. Note that these generalized Taylor monomials satisfy s { t h 0, = 0 t, s = h t, s, N for all s, t T and N 0. Any finite linear combination of generalized Taylor monomials is called a generalized polynomial. Since the generalized Taylor monomials play a major role in this paper, we next give some of their important properties. Property 2. [3, Property ]. Using induction and the definition given by 2., it is easy to see that h t, s 0 holds for all N 0 and s, t T with t s and h t, s 0 holds for all N and s, t T with t s. In view of the fact 2.2, for all N, it is evident that h t, s is increasing in t provided that t s, and h t, s is decreasing in t provided that t s. In addition, for all s, t T and all, l N 0 with l, h t, s t s l h l t, s holds when t s, while h t, s l s t l h l t, s when t s. Using L Hôpital s rule see [4, Theorem.20], we have the following corollary. Remar 2.. Let sup T =. Then for any N and fixed r, s T, we have h t, r lim h t, s =. With the following lemma, we are able to give an alternative equivalent formulation of the generalized Taylor monomials. Lemma 2.2. The generalized Taylor monomials h : T T R satisfy, = 0 h t, s = t h t, σηη, N s 2.3 for all s, t T.

6 6 L. H. Erbe, B. Karpuz and A. C. Peterson Proof. By Taylor s formula see [4, Lemma.09, Theorems..3], for any N and fixed r, s T, we have h t, s = h l t, rh l t l=0 r, s + = h l t, rh l r, s + l=0 for all t T, which gives 2.3 by letting r = s. r Corollary 2.3. For all s, t T and N, we have h s t, s = h t, σs. r h t, σηh t η, sη h t, σηη In the sequel, we use the following two important lemmas. Lemma 2.4 Change of order of integration [2, Lemma ]. Assume that s, t T and f C rd T T, R, the s η fη, ξξη = σξ s s fη, ξηξ. Lemma 2.5 Substitution [4, Theorem.93]. Assume that f C rd T, R and g C rdt, R such that g > 0 on T and let T := gt be a time scale. Then, f g = f gg holds on T. The following remar can be extracted from the proof of the one above. Remar 2.6. Assume that g C rdt, R satisfies g > 0 on T κ, and that g[s, t] T = [gs, gt] T holds for some fixed s, t T with t > s. Then for f C rd[gs, gt] T, R, we have f g = f gg on [s, t] T κ. In general the chain rule for the time scale calculus is not the same as in the ordinary calculus case. Remar 2.6 gives us conditions under which the chain rule for time scale calculus coincides with the usual calculus on the real line. One of the most powerful tools for higher-order dynamic equations particularly for difference and differential equations is the following one, which is nown as Kiguradze s theorem. Theorem 2.7 Kiguradze s Theorem [, Theorem 5]. Let sup T =, n N and f C n rdt, R +. Suppose that f n 0 is either nonnegative or nonpositive on T. Then there exists m [0, n Z such that n m f n t 0 holds for all sufficiently large t. Moreover, both of the following conditions hold: i 0 < m implies f t > 0 for all sufficiently large t,

7 Higher-Order Dynamic Equations 7 ii m < n implies m f t > 0 for all sufficiently large t. In what follows, we prove the following lemma, which plays a crucial role in the sequel. Lemma 2.8. Let sup T = and f C n rdt, R + n 2. Moreover, suppose that Kiguradze s theorem holds with m [, n N and f n 0 on T. Then, there exists a sufficiently large t T such that f t h m t, t f m t for all t [t, T. 2.4 Proof. Note that f m is nonincreasing on T. We shall give the proof for the case when m [2, n N, since 2.4 holds trivially by the definition of generalized polynomials for m =. Using the Taylor formula for f centered at t T, Property 2. and the eventually decreasing nature of f m on T, we have m 2 f t = h t, t f t + h m 2 t, σηf m ηη = t h m 2 t, σηη f m t t for all t [t, T see [4, Lemma.09, Theorems..3]. An application of Lemma 2.2 gives 2.4, and completes the proof. The following important result may be found in []. Lemma 2.9 [, Lemma 7]. Let sup T =, n N and f C n rdt, R. Then the following assertions hold: i lim inf f n t > 0 implies lim f t = for all [0, n N0. ii lim sup f n t < 0 implies lim f t = for all [0, n N0. Due to Kiguradze s theorem and Lemma 2.9, we infer the following corollary. Corollary 2.0. Let f be as in Kiguradze s theorem. Then we have lim f t = 0 for all m, n N. 3 Oscillation Criteria for Even-Order Equations In this section, we give some oscillation criteria for. for the even-order case. We first introduce Bt [ Aαt ] B, = n t := n 2 h n 2 t, σηb n ηη, otherwise t 3.

8 8 L. H. Erbe, B. Karpuz and A. C. Peterson for t [t 0, T. Here represents an upper index. Our first main result reads as follows. Theorem 3.. Assume that n 2N and A C rd [t 0, T, [0, ] R, and that for every 0, n 2N, there exists a function C rd[t 0, T, R + 0 such that lim sup r B η η η 2 4 ηβ ηh βη, s η = 3.2 for every sufficiently large but fixed r, s [t 0, T with βr > s. Then every solution of. is oscillatory. Proof. Without loss of generality suppose that x is an eventually positive solution of.. Then we may pic t [t 0, T such that xt, xαt, xβt > 0 for all t [t, T. Set y x t := xt + Atxαt for t [t, T. 3.3 Then, y x t > 0 on [t, T. From., we have y n x t = Btxβt 0 for all t [t, T. 3.4 It follows from Kiguradze s theorem that there exist a sufficiently large t 2 [t, T and an odd integer m [, n N0 such that for all t [t 2, T, yx t > 0 for all [0, m N0 and m yx t > 0 for all [m, n N. In particular, we have y x t xt > 0 and yx t > 0 on [t 2, T. Because of α β β on [t 2, T, we have xβt =y x βt Aβtxαβt y x βt Aβty x αβt [ Aβt ] y x βt 3.5 for all t [t 2, T. Using Corollary 2.0, we get from integrating 3.4 over [t, T [t 2, T for a total of n m times and changing the order of integration from the innermost integral to the outermost one, we have n m yx m+ t = t η n m 2 η 2 Bη xβη η η n 2 η n m = n m h n m 2 t, σηbηxβηη 3.6 t for all t [t 2, T. Substituting 3.5 into 3.6, and using the increasing nature of y x β both y x and β are increasing and 3., we obtain y m+ x t + B m ty x βt 0 for all t [t 2, T. 3.7

9 Higher-Order Dynamic Equations 9 We notice that the term n m disappears since n m is even. Now, define Y m t := mtyx m t y x βt > 0 for t [t 2, T. 3.8 Using 3.8, the positivity of m, the increasing nature of y x β and the decreasing nature of yx m, we obtain m tyx m βt y x βt Ym t = mtyx m+ t + y x βt Y m t mty m σ x t y x βσt = mty σ mt σ mt for all t [t 2, T. Hence, from Remar 2.6, 3.7 and 3.8, we get y m σ x t [ mty x βt m t y x βt ] = B m t m t + y x βty x βσt mt mtβ tyx βt y x βt for all t [t 2, T. By Lemma 2.8, for some t 3 [t 2, T, we have Y σ m t σ mt y x t h m t, t 3 y m x t 3. for all t [t 3, T. Using 3.9, 3.0 and 3., for all t [t 4, T, where βt 4 > t 3, we get Ym t B m t m t + mt m tβ th m βt, t 3 ym x βt Y σ m t y x βt σ mt B m t m t + mt m tβ th m βt, t 3 Y mt σ Y σ m t σ mt σ mt m t 2 B m t m t + t 4 4 m tβ th m βt, t 3, 3.2 which yields by integrating over [t 4, T that B m η m η m η 2 η Y 4 m ηβ m t 4. ηh m βη, t 3 This contradicts 3.2, the proof is hence completed. The next result follows from the previous one. Corollary 3.2. Assume that n 2N and A C rd [t 0, T, [0, ] R, and lim sup B ηh η, s h η, s η = 3.3 4h βη, s r for every 0, n 2N and every sufficiently large but fixed r, s [t 0, T with βr > s. Then, every solution of. is oscillatory.

10 0 L. H. Erbe, B. Karpuz and A. C. Peterson Next, we have an illustrative example for the difference equation case. Example 3.3. Consider the following even-order difference equation: n[ xt + a 0 xt α 0 ] + b 0 h n t, n xt β 0 = 0 for t [, Z, 3.4 where a 0 [0, R, α 0 N 0, b 0 R + and β 0 N 0. For s, t [, Z and N, we have t s h t, s =.! It is easy to show that = h t, s + h + t, s holds, and this implies η=t h η, s = h t, s + for all [2, N. Thus, for t [, Z, we can compute for 0, n 2N that which yields η= B t = λn, b 0 a 0, where λn, := n h + t, +, t B ηh η, h η, nb 0 a 0 t 4h η β 0, 4 η. η= By Corollary 3.2 and Remar 2., every solution of 3.4 oscillates, if { min nb 0 a 0 } = nb 0 n > 0 b 0 a 0 > n {,3,...,n } 4 4 4n since 3.3 holds for every 0, n 2N and every sufficiently large s [, Z. For the following theorem, we need to introduce Bt, B 2 = n t := n 2 h n 2 t, σηb n ηη, 2 otherwise t 3.5 for t [t 0, T. Here 2 is an upper index.

11 Higher-Order Dynamic Equations Theorem 3.4. Assume that n 2N and A C rd [t 0, T, [, 0] R with lim inf At >, and that for every 0, n 2N, there exists a function C rd[t 0, T, R + 0 such that lim sup r B 2 η η η 2 4 ηβ ηh βη, s η = 3.6 for every sufficiently large but fixed r, s [t 0, T with βr > s. Then every solution of. oscillates or tends to zero asymptotically. Proof. Without loss of generality suppose that x is an eventually positive solution of. which does not tend to zero asymptotically. Let t [t 0, T and a 0 [0, R satisfy xt, xαt, xβt > 0 and At a 0 for all t [t, T. Let y x be defined on [t, T by 3.3. From., we have 3.4 on [t, T. Then, for each [0, n N0, yx is of fixed sign on [t 2, T for some t 2 [t, T. Hence, l y := lim y xt exists. Below, we shall show that l y > 0. First consider the case where x is unbounded. Then we may pic an increasing divergent sequence {ξ } N [t 2, T such that xξ = max t [t0,ξ ] T {xt} for all N and {xξ } N is increasing and divergent. Then, we have y x ξ a 0 xξ for all N, which yields l = by letting. Next consider the case where x is bounded. Then we may pic an increasing divergent sequence {ξ } N [t 2, T such that lim xξ = l x, where l x := lim sup xt. It is clear that lim sup xαt l x, and l x > 0 since x does not tend to zero asymptotically. Then, we have y x ξ xξ a 0 xαξ for all N, which yields l y a 0 l x > 0 by letting. In both cases, we obtain y x > 0 on [t 2, T. Then it follows from Kiguradze s theorem that there exists an odd integer m [, n N0 such that for all t [t 2, T, yx t > 0 for all [0, m N0 and m yx t > 0 for all [m, n N. In particular, we have x y x > 0 and yx > 0 on [t 2, T. The remainder of the proof is very similar to that in the proof of Theorem 3.. So we obtain B m 2 η m η m η 2 η Y 4 m ηβ m t 4, ηh m βη, t 3 t 4 where Y m is defined on [t 2, T by 3.8, for some fixed t 4 [t 3, T with βt 4 > t 3 and some fixed sufficiently large t 3 [t 2, T. This contradicts 3.6 and completes the proof. The result below follows from the previous theorem. Corollary 3.5. Assume that A C rd [t 0, T, [, 0] R with lim inf At >, and lim sup B 2 ηh η, s h η, s η = 3.7 r 4h βη, s for every 0, n 2N and every sufficiently large but fixed r, s [t 0, T with βr > s. Then, every solution of. oscillates or tends to zero asymptotically.

12 2 L. H. Erbe, B. Karpuz and A. C. Peterson Another illustrative example for Theorem 3.4 is given below on the quantum set. Example 3.6. Assume q > and consider the following even-order q-difference equation: [ D n q xt + a0 xq α 0 t ] b 0 + h n t, q n xq β 0 t = 0 for t [, q Z {0}, 3.8 where a 0, 0] R, α 0 N 0, b 0 R + and β 0 N 0. Here, for all s, t [, q Z {0} and all N 0, we have xqt xt D, = t q qxt := q t j s and h t, s :=. j D q D q xt, [2, N j=0 i=0 qi For s, t [, q Z {0} and N, it is not difficult to show that q D q = h t, qs q q + h + t, s, which implies t h η, q s d q q qη = q h t, s for all [2, N. Therefore, for t [, q Z {0} and 0, n 2N, we have B 2 t = λn, b nn 0 q 2 q n, where λn, := h + t, q q + 2 q +. For t [, q Z {0}, we have B 2 ηh η, q h η, q d 4h q β 0 η, q q η nn q 2 q n b 0 =q q + 2 q + 2 η d 4q q β 0 η q q η q nn 2 q n b 0 q β 0 q q + 2 q + 2 4q η d qη. for all 0, n 2N. Due to Corollary 3.2 and Remar 2., every solution of 3.8 is oscillatory provided that } {q nn 2 q n b 0 q β 0 min = qn b 0 q β 0 {,3,...,n } q + 2 q + 2 4q q n 2 4q n > 0 or equivalently b 0 q > qn 2 β 0 4q n. 2 In this present case, one can see that 3.3 holds for every 0, n 2N and every sufficiently large s [, q {0}. Z

13 Higher-Order Dynamic Equations 3 4 Kamenev-Type Oscillation Criteria for Even-Order Equations In this section, we give Kamenev type oscillation criteria for. for the even-order case. Theorem 4.. Assume that n 2N and A C rd [t 0, T, [0, ] R. Assume also that for every 0, n 2N, there exist a fixed n N and a function C rd[t 0, T, R + 0 such that lim sup h n t, r h n t, ση B η η η 2 η = r 4 ηh βη, s 4. for all sufficiently large r, s [t 0, T with βr > s. Then, every solution of. is oscillatory. Proof. Proceeding as in the proof of Theorem 3., for all t [t, T [t 0, T, we get 3.0, where x is positive on [t, T and Y m is defined by 3.8 on [t, T. Considering Property 2., Corollary 2.3, and integrating by parts, we get Ym ηh nm t, σηη = h nm t, ηy m η t 2 = h nm t, t 2 Y m t 2 + h nm t, t 2 Y m t 2 η=t η=t 2 t 2 Y m ηh η n m t, ηη t 2 Y m ηh nm t, σηη for all t [t 2, T, where βt 2 > t. Using this, multiplying both sides of 3.2 with t replaced by η by h nm t, ση, and then integrating as η goes from t 2 to t, we get h nm t, ση B m η m η t 2 m η 2 η h 4 m ηβ nm t, t 2 Y m t 2 ηh m βη, t or equivalently, in view of Property 2., we have t h nm t, ση B m η m η h nm t, t 2 t 2 m η 2 η Y 4 m ηβ m t 2, ηh m βη, t which contradicts 4.. The proof is completed.

14 4 L. H. Erbe, B. Karpuz and A. C. Peterson As a particular simple result, we have: Corollary 4.2. Assume that n 2N and A C rd [t 0, T, [0, ] R. And assume for every 0, n 2N that t lim sup h n t, ση B h n t, r ηh η, s r 4.2 h η, s η = 4 m ηβ ηh βη, s for some n N and all sufficiently large but fixed r, s [t 0, T with βr > s. Then, every solution of. is oscillatory. Theorem 4.3. Assume that n 2N and A C rd [t 0, T, [, 0] R with lim inf At >. Assume also that for every 0, n 2N, there exist a fixed n N and a function C rd[t 0, T, R + 0 such that lim sup h n t, r h n t, ση B 2 η η η 2 η = r 4 ηh βη, s 4.3 for all sufficiently large r, s [t 0, T with βr > s. Then, every solution of. oscillates or tends to zero asymptotically. Now we give the following application of Theorem 4.3 to the case T = R. Example 4.4. Consider the following even-order differential equation: [ xt + a0 xα 0 t ] n + b 0 h n t, 0 xβ 0t = 0 for t [, R, 4.4 where n denotes the usual n-th order derivative, a 0, 0] R, α 0 0, ] R, b 0 R + and β 0 0, ] R. For s, t [, R and 0, n 2N. We have and h t, s = t s! h n t, nb 0 4β + and B 2 t = λn, b 0 n, where λn, := h + t, 0 +, h n t, ση t n ηh η, 0 h η, 0 dη 4βh β 0 η, 0 t η n dη η B 2 for all t, R, any n N. Therefore, every solution of 4.4 is oscillatory, by Corollary 4.2 and Remar 2. provided that { min nb 0 } = nb {,3,...,n } 4β0 + 0 n > 0 β n 4β0 n 0 b 0 > n 4n. Notice that 4.3 is true for every 0, n 2N and every sufficiently large r, s [t 0, T with β 0 r > s.

15 Higher-Order Dynamic Equations 5 5 Kamenev-Type Oscillation Criteria for Odd-Order Equations In this section, we extend our results to odd-order neutral delay dynamic equations. Theorem 5.. Assume that n 2N and A C rd [t 0, T, [0, ] R with lim sup At <. Assume also that n t 0 h n t 0, σηbηη = 5. holds, and for every 0, n 2N, there exist a fixed n N and a function C rd[t 0, T, R + 0 such that 4. holds for all sufficiently large r, s [t 0, T with βr > s. Then, every solution of. oscillates or tends to zero asymptotically. Proof. The proof maes use of similar arguments if Kiguradze s theorem holds for m 0, n 2N and for y x defined by 3.3. And if m = 0, then we apply [3, Theorem 3.] since y x is bounded, and this implies boundedness of x. Then, we see that x tends to zero asymptotically. Theorem 5.2. Assume that n 2N and A C rd [t 0, T, [, 0] R with lim inf At >. Assume also that 5. holds, and for every 0, n 2N, there exist a fixed n N and a function C rd[t 0, T, R + 0 such that 4.3 holds for all sufficiently large r, s [t 0, T with βr > s. Then, every solution of. oscillates or tends to zero asymptotically. Proof. The proof is similar to that of Theorem 5., and hence we omit it. References [] R. P. Agarwal and M. Bohner, Basic calculus on time scales and some of its applications, Results Math., 35-2: 3 22, 999. [2] R. P. Agarwal, M. Bohner, D. O Regan, and A. C. Peterson, Dynamic equations on time scales: a survey, J. Comput. Appl. Math., 4-2: 26, [3] R. P. Agarwal, M. Bohner and S. H. Saer, Oscillation of second order delay dynamic equations, Can. Appl. Math. Q., 3: 7, [4] M. Bohner and A. Peterson, Dynamic Equations on Time Scales: An Introduction with Applications, Birhäuser, Boston, 200. [5] J. Džurina, Oscillation theorems for neutral differential equations of higher order, Czechoslova Math. J., 29:85 95, 2004.

16 6 L. H. Erbe, B. Karpuz and A. C. Peterson [6] L. H. Erbe, Oscillation criteria for second order linear equations on a time scale, Canad. Appl. Math. Q., 94: , 200. [7] L. H. Erbe, A. C. Peterson and S. H. Saer, Hille and Nehari type criteria for thirdorder dynamic equations, J. Math. Anal. Appl., 329:2 3, [8] P. Hartman, On nonoscillatory linear differential equations of second order, Amer. J. Math., 74: , 952. [9] S. Hilger, Ein Maßettenalül mit Anwendung auf Zentrumsmannigfaltigeiten, Ph.D. Thesis, Universität Würzburg, 988. [0] V. Kac and P. Cheung, Quantum Calculus, Universitext, Springer, 200. [] I. V. Kamenev, Certain specifically nonlinear oscillation theorems, Mat. Zameti, 0:29 34, 97. [2] B. Karpuz, Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients, Electron. J. Qual. Theory Differ. Equ., :4 pp., [3] B. Karpuz, Asymptotic behaviour of bounded solutions of a class of higher-order neutral dynamic equations, Appl. Math. Comput., 256: , [4] S. H. Saer, Oscillation of second-order delay and neutral delay dynamic equations on time scales, Dynam. Systems Appl., 62: , [5] S. H. Saer and S. S. Cheng, Oscillation criteria for difference equations with damping terms, Appl. Math. Comput., 482:42 442, [6] W. Kelley and A. C. Peterson, Difference Equations: An Introduction With Applications, Academic Press, Second Edition, 200. [7] A. Wintner, A criterion of oscillatory stability, Quart. Appl. Math., 7:5 7, 949. [8] B. G. Zhang and S. H. Saer, Kamenev-type oscillation criteria for nonlinear neutral delay difference equations, Indian J. Pure Appl. Math., 34:57 584, 2003.