This paper present some new criteria for the oscillation of even order dynamic equation on time scale T, where a is the ratio of positive odd integers a and q is a real valued positive rd-continuous functions defined on T.
Journal of Advanced Research (2013) 4, 201–204 Cairo University Journal of Advanced Research SHORT COMMUNICATION On the oscillation of higher order dynamic equations Said R Grace * Department of Engineering Mathematics, Faculty of Engineering, Cairo University, Orman, Giza 12221, Egypt Received March 2012; revised April 2012; accepted 23 April 2012 Available online July 2012 KEYWORDS Abstract Oscillation; Higher Order; Dynamic equations aðtÞðx We present some new criteria for the oscillation of even order dynamic equation DnÀ1 tịịa D ỵ qtịxtịịa ẳ 0; on time scale T, where a is the ratio of positive odd integers a and q is a real valued positive rdcontinuous functions defined on T ª 2012 Cairo University Production and hosting by Elsevier B.V All rights reserved (ii) a, q: T fi R+ = (0, 1) is a real-valued rd-continuous functions, aD(t) P for t [t0, 1)T and Introduction This paper is concerned with the oscillatory behavior of all solutions of the even order dynamic equation nÀ1 aðtÞðxD ðtÞÞa D ỵ qtịxtịịa ẳ 0; 1:1ị on an arbitrary time-scale T ˝ R with Sup T = and n P is an even integer We shall assume that: (i) a P is the ratio of positive odd integers, * Tel.: +20 35876998 E-mail address: saidgrace@yahoo.com Peer review under responsibility of Cairo University Production and hosting by Elsevier Z a1=a sịDs ẳ 1: 1:2ị We recall that a solution x of Eq (1.1) is said to be nonoscillatory if there exists a t0 T Such that x(t)x(r(t)) > for all t [t0, 1)T; otherwise, it is said to be oscillatory Eq (1.1) is said to be oscillatory if all its solutions are oscillatory The study of dynamic equations on time-scales which goes back to its founder Hilger [1] as an area of mathematics that has received a lot of attention It has been created in order to unify the study of differential and difference equations Recently, there has been an increasing interest in studying the oscillatory behavior of first and second order dynamic equations on time-scales, see [2–7] With respect to dynamic equations on time scales it is fairly new topic and for general basic ideas and background, we refer to [8,9] It appears that very little is known regarding the oscillation of higher order dynamic equations [10–15] and our purpose here to establish some new criteria for the oscillation criteria for such equations The obtained results are new even for the special cases when T = R and T = Z 2090-1232 ª 2012 Cairo University Production and hosting by Elsevier B.V All rights reserved http://dx.doi.org/10.1016/j.jare.2012.04.003 202 S.R Grace Theorem 2.2 Let conditions (i), (ii) and (1.2) hold and Main results Z We shall employ the following well-known lemma Z rðsÞ ðaðsÞÞÀ1 t0 Lemma 2.1 [6, Corollary 1] Assume that n N, s, t T and f Crd(T, R) Then Z t s Z t ÁÁÁ gn ỵ1 ẳ 1ịn Z Z Z t!1 t 2:1ị à gðsÞqðsÞ À aðsÞgD ðsÞhÀa nÀ1 ðs; t0 Þ Ds ¼ 1; ð2:4Þ t1 t hn ðs; rðgÞÞfðgÞDg: ð2:3Þ If there exists a positive non-decreasing delta-differentiable function g such that for every t1 [t0, 1)T lim sup gn 1=a quịDu Ds ẳ 1: s t fg1 ịDg1 Dg2 Dgnỵ1 then Eq (1.1) is oscillatory s We shall employ the Kiguradze’s following well-known lemma Theorem 2.1 (Kiguarde’s Lemma [8, Theorem 5]),Let n N; f Cnrd ðT; RÞ and sup T = Suppose that f is either n positive or negative and fD is not identically zero and is either nonnegative or nonpositive on [t0, 1)T for some t0 T Then, there exist t1 [t0, 1)Tm [0, n)Z such that n ðÀ1ÞnÀm fðtÞfD ðtÞ P holds for all t [t0, 1)T with j (I) f ðtÞf D ðtÞ > holds for all t [t0, 1)T and all j [0, m)z, j (II) 1ịmỵj f tịf D ðtÞ > holds for all t [t0, 1)T and all j [m, n)z Proof Let x(t) be a nonoscillatory solution of Eq (1.1) on [t0, 1)T It suffices to discuss the case x is eventually positive (as – x also solves (1.1) if x does), say x(t) > for t P t1 P t0 Now, we see that D nÀ1 aðtÞðxD ðtÞÞa for t P t1 : nÀ1 It is easy to see that xD ðtÞ > for t P t1 for otherwise, and by using condition (1.2) we obtain a contradiction to the fact that x(t) > for t P t1 Now, aD(t) P for t [t0, 1)T We have D nÀ1 D nÀ1 a n1 atịxD tịịa ẳ aD tị xD tị ỵ ar ðtÞ ðxD ðtÞÞa 0: The following result will be used to prove the next corollary Lemma 2.2 [12, Lemma 2.8].Let supT = and f Cnrd ðT; RÞðn P 2Þ Moreover, suppose that Kiguradze’s theon rem holds with m [1, n)N and fD on T Then there exists a sufficiently large t1 T such that m fD ðtÞ P hmÀ1 ðt; t1 ÞfD ðtÞ for all t ẵt1 ; 1ịT : This implies nÀ1 ððxD ðtÞÞa ÞD for t P t1 : nÀ1 Next, we let y ¼ xD rem 1.90] we see that ð2:2Þ Z on [t1, 1)T From Ref [9, Theo- y ỵ hlyD ịa1 dh P ayD Z The proof of the following corollary follows by an integration of (2.2) P ya ịD ẳ ayD Corollary 2.1 Assume that the conditions of Lemma 2.2 hold Then Thus we have yD ¼ xD on [t1, 1)T and from Theorem 2.1, there exists an integer m {1, 3, , n À 1} Such that (I) and (II) hold on [t1, 1)T Clearly xD(t) > for t P t1 and hence, there exists a constant c > such that m fðtÞ P hm ðt; t1 ÞfD ðtÞ for all t ẵt1 ; 1ịT : Next, we need the following lemma see [16] Lemma 2.3 If X and Y are nonnegative and k > 1, then yaÀ1 dh ¼ ayaÀ1 yD : n xðtÞ P c ð2:5Þ for t P t1 : First, we claim that m = n À To this end, we assume that Xk kXYk1 ỵ k 1ịYk P 0; xD tị < where equality holds if and only if X = Y, Integrating Eq (1.1) from t P t1 to u P t, letting u fi we have It will be convenient to employ the Taylor monomials (see [9, Section 1.6]) fhn t; sị1 nẳ0 g which are dened recursively by: Z t hn ðs; sÞDs; t; s T and n P 1; hnỵ1 t; sị ẳ n2 nÀ3 and xD ðtÞ > Z nÀ1 xD ðtÞ P c ðaðtÞÞÀ1 for t P t1 : 1=a for t P t1 : qðsÞDs t s where it follows that h1(t, s) = t À s but simple formulas in general not hold for n P Now we present the following oscillation results for Eq (1.1) Integrating (2.6) from to and letting fi we get nÀ2 < ÀxD ðtÞ À Z t Z ðaðuÞÞÀ1 u 1=a qðsÞDs Du: ð2:6Þ On the oscillation of higher order dynamic equations 203 Integrating this inequality from t and using condition (2.3) after Lemma 2.1 we arrive at the desired contradiction It follows from Lemma 2.2 with m = n À that DnÀ1 D x ðtÞ P hnÀ2 ðt; t1 Þx ðtÞ for t P t1 ; ð2:7Þ and by applying Corollary 2.1 with m = n À instead of Lemma 2.2, we get nÀ1 xðtÞ P hnÀ1 ðt; t1 ÞxD ðtÞ D D g g x wD gq ỵ r wr a r wr : g g x Using Lemma 2.2 with m = n À in (2.15), we find ! D nÀ1 g g xD r wr hnÀ2 w gq ỵ r w a r g g x D n1 Dn1 a ax ị xa on ẵt1 ; 1ÞT : ð2:9Þ g D g nÀ1 axD x 2:10ị Now set 2:11ị a X ẳ aa1=a ịghn2 ịaỵ1 wD tị gtịqtị ỵ atịgD tịhn1 t; t1 ÞÞÀa ; ¼ t P t1 : Integrating this inequality from t2 > t1 to t P t2, we have t ẵgsịqsị asịgD sịha n1 s; t1 ịDs: 2:12ị t2 Taking upper limit of both sides of the inequality (2.12) as t fi and using (2.4) we obtain a contradiction to the fact that w(t) > on [t1, 1)T This completes the proof h Next, we establish the following result t!1  hnÀ2 ðs; t0 ÞgðsÞ ð2:13Þ x ỵ lhxD a1 a a ỵ 1ịaỵ1 ghn2 a gD ịaỵ1 on ẵt2 ; 1ịT : 2:18ị Integrating (2.18) from t2 to t, we get Z t" gsịqsị asị a a ỵ 1ịaỵ1 gsịhn2 s;t1 ị # gD sịịaỵ1 Ds: Taking upper limit of both sides of (2.19) as t fi and using (2.13), we obtain a contradiction to the fact that w(t) > for t P t1 This completes the proof h Finally, we present the following interesting result Proceeding as in the proof of Theorem 2.1, we obtain m = n À and (2.7) and (2.8) Define w as in (2.9) and obtain (2.10) Now from Ref [3, Theorem 1.90], xa ịD ẳ axD ị ; 2:19ị Proof Let x be a nonoscillatory solution of Eq (1.1), say x(t) > for t P t1 P t0 1 a !a !aỵ1 and therefore, we nd t2 then Eq (1.1) is oscillatory Z a1=a aghn2 P 0; asịgD sịịaỵ1 aỵ1 Ds ¼ 1: and Y in Lemma 2.3 with k ẳ aỵ1 > to conclude that a ! D g g a gD ịaỵ1 r 1ỵ1=a w ị r wr ỵ a aỵ1 1ỵ1=a hn2 g ga han2 a ỵ 1ị gr ị wtị wt2 ị a ỵ 1ị !a # t1 r w g a a gD ịa aỵ1 wD gq ỵ Theorem 2.3 Let conditions (i), (ii) (1.2) and (2.3) hold If there exists a a positive non-decreasing delta-differentiable function g such that for every t1 [t0, 1)T ð2:17Þ for t P t2 P t1 ; Using (2.8) in (2.11), we get Z t" lim sup gðsÞqðsÞ on ẵt2 ; 1ịT ; ! D g aị1=a g r hn2 ịwr ị1ỵ1=a w gq ỵ r w a g gr ị1ỵ1=a axD ịa ịr þ a ðaðxD Þa ÞD " x # D a a D DnÀ1 a r g x À gðx Þ ; ẳ gq ỵ ax ị ị xa xr ịa !a Dn1 D x gq ỵ ag : x Z 1=a r 1=a w w P g g D n1 xa wtị wt2 ị ẳ and thus, Then on [t1, 1)T, we have wD ¼ ð2:16Þ where hnÀ2 = hnÀ2(t, t1) Now we see that Now, we let w :¼ g for t P t2 P t1 ; ð2:8Þ for t P t1 : ð2:15Þ dh P axD Z Theorem 2.4 Let conditions (i), (ii), (1.2) and (2.3) hold If there exists a a positive, delta-differentiable function g such that for every t1 [t0, 1)T Z t" lim sup t!1 xaÀ1 dh ¼ axaÀ1 xD : t1 # ðaðsÞÞr ðgD ðsÞÞ2 gðsÞqðsÞ À Ds 4a gðsÞðhnÀ1 ðs; t1 ÞÞaÀ1 hn2 s; t1 ị ẳ 1; 2:20ị 2:14ị Using (2.14) in (2.10), we have then Eq (1.1) is oscillatory 204 S.R Grace Proof Let x be a nonoscillatoy solution of Eq (1.1), say x(t) > for t P t1 P t0 Proceeding as in the proof of Theorem 2.3, we obtain (2.17) which cam be rewritten as ! D g aị1=a g r w gq ỵ r w a hn2 ị g gr ị1ỵ1=a We may also employ other types of the time-scales [8,9] e.g., T = hZ with h > 0; qN o ; q > 1, T ¼ N 20 , etc The detail are left to the reader References D r 1=a1 w ị r w ị on ẵt2 ; 1ÞT : ð2:21Þ Now, using Corollary 2.1 with m = n À we have x nÀ1 xD P hnÀ1 ; implies on [t2, 1)T that (2.22) !1Àa DnÀ1 1=aÀ1 1=aÀ1 1=aÀ1 x w ¼a g x aÀ1 x ¼ a1=aÀ1 g1=aÀ1 DnÀ1 P a1=aÀ1 g1=aÀ1 haÀ1 nÀ1 : x ð2:22Þ Using (2.22) in (2.21) we have on [t2, 1)T that wD gq ỵ D g gr ẳ gq ỵ a1 r gðh Þ h wr À a ðaÞnÀ1r ðgr ÞnÀ2 ðwr Þ2 : r ahÀ1 nÀ2 ðhnÀ1 Þ aÀ1 Á1=2 g À ððaÞr Þ1=2 gr r ðaÞ ðgD Þ2 4aðhrnÀ1 ị gq ỵ a1 hn2 4a !2 ððaÞr Þ1=2 gD aÀ1 2ðahnÀ2 ðhrnÀ1 Þ gÞ1=2 : ðaÞr ðgD Þ2 gðhrnÀ1 Þ aÀ2 hnÀ2 : ð2:23Þ Integrating this inequality from t2 to t, taking upper limit of the resulting inequality as t fi 1, and applying condition (2.20) we obtain a contradiction to the fact that w(t) > for t P t1 This completes the proof h Remarks The results of this paper are presented in a form which is essentially new and of high degree of generality Also, we can easily formulate the above conditions which are new sufficient for the oscillation of Eq (1.1) on different time-scales e.g., T = R and T = Z The details are left to the reader [1] Hilger S Analysis on measure chain-a unified approach to contiguous and discrete calculus Results Math 1990;18:18–56 [2] Braverman E, Karpuz B Nonosillation of first-order dynamic equations with several delays Adv Difference Eq 2010:22 Art ID 873459 [3] Grace SR, Bohner M, Agarwal RP On the oscillation of second order half linear dynamic equations J Differ Eqs Appl 2009;15:451–60 [4] Grace SR, Agarwal RP, Bohner M, O’Regan D Oscillation of second order strongly superlinear and strongly sublinear dynamic equations Commun Nonlinear Sci Numer Simulat 2009;14:3463–71 [5] Sahiner Y Oscillation of second order delay differential equations on time-scales Nonlinear Anal 2005;63(5–7):1073–80 [6] Bohner M Some oscillation criteria for first order delay dynamic equations Far East J Appl Math 2005;18(3):289–304 [7] Braverman E, Karpuz B Nonoscillation of second order dynamic equations with several delays Abstr Appl Anal 2011:34 Art ID 591254 [8] Agarwal RP, Bohner M Basic calculus on time scales and some of its applications Results Math 1999;35(1–2):3–22 [9] Bohner M, Peterson A Dynamic Equations on Time-Scales : An Introduction with Applications Boston: Birkhauser; 2001 [10] Chen DX Oscillation and asymptotic behavior for nth order nonlinear neutral delay dynamic equations on time scales Acta Appl Math 2010;109(3):703–19 [11] Erbe L, Baoguo J, Peterson A Oscillation of nth order superlinear dynamic equations on time scales Rocky Mountain J Math 2011;41(2):471–91 [12] Erbe L, Karpuz B, Peterson A Kamenev-type oscillation criteria for higher order neuyral delay dynamic equations Int J Differ Eq, in press (2012.-IJDE-1106) [13] Karpuz B Asymptotic behavior of bounded solutions of a class of higher-order neutral dynamic equations Appl Math Comput 2009;215(6):2174–2183, [14] Karpuz B Unbounded oscillation of higher-order nonlinear delay dynamic equations of neutral type with oscillating coefficients Electon J Qual Theo Differ Eq 2009;9(34):14 [15] Zhang BG, Deng XH Oscillation of delay differential equations on time scales Math Comput Model 2002;36(11–13):1307–18 [16] Hardy GH, Littlewood IE, Polya G Inequalities Cambridge: University Press; 1959 ... 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