HILLE-KNESER TYPE CRITERIA FOR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES

Transcription

1 HILLE-KNESER TYPE CRITERIA FOR SECOND-ORDER DYNAMIC EQUATIONS ON TIME SCALES L ERBE, A PETERSON AND S H SAKER Abstrat In this paper, we onsider the pair of seond-order dynami equations rt)x ) ) + pt)x t) = 0, and rt)x ) ) + pt)x σ t) = 0, on a time sale T, where > 0 is a quotient of odd positive integers We establish some neessary and suffiient onditions for nonosillation of Hille-Kneser type Our results in the speial ase when T = R involve the well known Hille Kneser type riteria of seond-order linear differential equations established by Hille For the ase of the seond order half-linear differential equation, our results extend and improve some earlier results of Li and Yeh and are related to some work of Došlý and Řehák and some results of Řehák for half-linear equations on time sales Several examples are onsidered to illustrate the main results 1 Introdution The theory of time sales, whih has reently reeived a lot of attention, was introdued by Stefan Hilger in his PhD Thesis in 1988 in order to unify ontinuous and disrete analysis, see [20] This theory of dynami equations unifies the theories of differential equations and differene equations, and also extends these lassial ases to situations in between, eg, to the so-alled q differene equations and an be applied on different types of time sales Many authors have expounded on various aspets of the new theory A book on the subjet of time sales, ie, measure hains, by Bohner and Peterson [5] summarizes and organizes muh of time sale alulus for dynami equations For advanes on dynami equations on time sales, we refer the reader to the book by Bohner and Peterson [6] In reent years, there has been an inreasing interest in studying the osillation of solutions of dynami equations on time sales, whih simultaneously treats the osillation of the ontinuous and the disrete In this way 1991 Mathematis Subjet Classifiation 34K11, 39A10, 39A99 Key words and phrases nonosillation, half-linear dynami equations, time sales, q-differene equations 1

2 2 L ERBE, A PETERSON AND S H SAKER we do not require to write the osillation riteria for differential equations and then write the disrete analogues for differene equations For onveniene we refer the reader to the results given in [1-4, 7-8, 10-12, 14-19, 21-34] In this paper, we present some osillation riteria of Hille-Kneser type for the seond-order dynami equations of the form 11) L 1 x = rt) x t ) ) ) + pt)x t) = 0, and 12) L 2 x = rt) x t ) ) ) + pt)x σ t) = 0, on an arbitrary time sale T, where we assume throughout this paper that r and p are real rd-ontinuous funtions on T with rt) > 0, pt) > 0, and > 0 is a quotient of odd positive integers We denote x σ := x σ, where the forward jump operator σ and the bakward jump operator ρ are defined by σt) := inf{s T : s > t}, ρt) := sup{s T : s < t}, where inf := sup T and sup := inf T A point t T is right-dense provided t < sup T and σt) = t and left-dense if t > inf T and ρt) = t A point t T is right-sattered provided σt) > t and left-sattered if ρt) < t By x : T R is rd-ontinuous we mean x is ontinuous at all right-dense points t T and at all left-dense points t T left hand limits exist finite) The graininess funtion µ : T R + is defined by µt) := σt) t Also T κ := T {m} if T has a left-sattered maximum m, otherwise T κ := T Here the domain of L 1 and L 2 is defined by D = {x : T R : rt)x t)) ) is rd-ontinous} When T = R, equations L 1 x = 0 and L 2 x = 0 are the half-linear differential equation 13) rt)x t)) ) + pt)x t) = 0 See the book by Došlý and Řehák [11] and the referenes there for numerous results onerning 13) When T = Z, L 1 x = 0 is the half-linear differene equation 14) rt) xt)) ) + pt)x t) = 0 in [9] the author studies the fored version of 14)) h > 0, then σt) = t + h, µt) = h, y t) = h yt) = yt + h) yt), h Also, If T = hz,

3 HILLE-KNESER TYPE CRITERIA 3 and L 1 x = 0 beomes the generalized seond-order half-linear differene equation 15) h rt) h xt)) ) + pt)x t) = 0 If T = q N = {t : t = q k, k N, q > 1}, then σt) = q t, µt) = q 1)t, x t) = q xt) = xq t) xt), q 1) t and L 1 x = 0 beomes the seond-order half-linear q-differene equation 16) q rt) q xt)) ) + pt)x t) = 0 If N 2 0 = {t 2 : t N 0 }, then σt) = t + 1) 2 and µt) = t, N yt) = y t + 1) 2 ) yt) 1 + 2, for t [t 2 0, ) t and L 1 x = 0 beomes the seond-order half-linear differene equation 17) N rt) N xt)) ) + pt)x t) = 0, One may also write down the orresponding equations for L 2 x = 0 for the various time sales mentioned above The terminology half-linear arises beause of the fat that the spae of all solutions of L 1 x = 0 or L 2 x = 0 is homogeneous, but not generally additive Thus, it has just half of the properties of a linear spae It is easily seen that if xt) is a solution of L 1 x = 0 or L 2 x = 0, then so also is xt) We note that in some sense muh of the Sturmian theorey is valid for equation 12) but that is not the ase for equation 11) We refer to Řehák [25] and to his Habilitation Thesis [24] in whih some open problems are also mentioned for 12) Sine we are interested in the asymptoti behavior of solutions, we will suppose that the time sale T under onsideration is not bounded above, ie, it is a time sale interval of the form [a, ) T := [a, ) T Solutions vanishing in some neighborhood of infinity will be exluded from our onsideration A solution x of L i x = 0, i = 1, 2, is said to be osillatory if it is neither eventually positive nor eventually negative, otherwise it is nonosillatory The equation L i x = 0, i = 1, 2, is said to be osillatory if all its solutions are osillatory It should be noted that the essentials of Sturmian theory have been extended to the half-linear equation L 2 x = 0 f Řehák [25]) One of the important tehniques used in studying osillations of dynami equations on time sales is the averaging funtion method By means of this tehnique, some osillation riteria for L 2 x = 0 for the ase = 1 have been established in [12] whih involve the behavior of the integral of the oeffiients r and p On the other hand, the osillatory properties an be desribed by the so alled Reid Roundabout Theorem f [5], [11], [25])

4 4 L ERBE, A PETERSON AND S H SAKER This theorem shows the onnetion among the onepts of disonjugay, positive definiteness of the quadrati funtional, and the solvability of the orresponding Riati equation or inequality) whih in turn implies the existene of nonosillatory solutions The Reid Roundabout theorem provides two powerful tools for the investigation of osillatory properties, namely the Riati tehnique and the variational priniple Sun and Li [33] onsidered the half-linear seond order dynami equation L 1 x = 0, where 1 is an odd positive integer, and r and p are positive real-valued rd ontinuous funtions suh that 18) t 0 1 rt) t =, and used the Riati tehnique and Lebesgue s dominated onvergene theorem to establish some neessary and suffiient onditions for existene of positive solutions For the osillation of the seond order differential equation 19) x t) + pt)xt) = 0, t t 0, Hille [21] extended Kneser s theorem and proved the following theorem see also [32, Theorem B] and the referene ited therein) Theorem 1 Hille-Kneser type riteria) Let p = lim t sup t 2 pt) and p = lim t inf t 2 pt) Then 19) is osillatory if p > 1 4, and nonosillatory if p < 1 4 The equation an be either osillatory or nonosillatory if either p or p = 1 4 So the following question arises: an one extend the Hille-Kneser theorem to the half-linear dynami equations L 1 x = 0 and L 2 x = 0 on time sales, and from these dedue the osillation and nonosillation results for halflinear differential and differene equations? The main aim of this paper is to give an affirmative answer to this question onerning the nonosillation result Our results in the speial ase when T = R, involve the results established by Li and Yeh [23], Kusano and Yoshida [22] and Yang [34] for the seondorder half-linear differential equations, and when rt) 1 and = 1 the results involve the riteria of Hille Kneser type for seond-order differential equations established by Hille [21], and are new for equations 14)-16) Also, in the speial ase, = 1, we derive Hille-Kneser type nonosillation riteria for the seond-order linear dynami equation 110) rt) x t) )) + pt)xt) = 0,

5 HILLE-KNESER TYPE CRITERIA 5 on a time sale T, whih are essentially new Several examples are onsidered to illustrate the main results 2 Main Results Our interest in this setion is to establish some neessary and suffiient onditions of Hille-Kneser type for nonosillation of L 1 x = 0 and L 2 x = 0 by using the Riati tehnique We searh for a solution of the orresponding Riati equations orresponding to L 1 x = 0 and L 2 x = 0 respetively Assoiated with L 1 x = 0 is the Riati dynami equation 21) where, for u R and t T, 22) F u, t) = R 1 w = w + pt) + w σ F w, t) = 0, 1+µt) u rt) «1 u rt), if µt) > 0 µt), if µt) = 0 Here we take the domain of the operator R 1 to be D := {w : T R : w is rd-ontinuous on T κ and w r R}, where R is the lass of regressive funtions page 58, [5]) defined by R := {x : T R : x is rd-ontinuous on T and 1 + µt)xt) 0} Assoiated with equation L 2 x = 0 is the Riati dynami equation 23) where, for u R and t T, 24) u Su, t) := µt) u R 2 w = w + pt) + Sw, t) = 0, 1+µt) rt)) u 1 «1 1+µt) u rt) ) 1 u rt) «, if µt) > 0,, if µt) = 0 Here we take the domain of the operator R 2 to be D The dynami Riati equation 21) is studied in [33] they assume is an odd positive integer) and the Riati dynami equation 23) is studied extensively in [25] A number of osillation riteria are also given based on the variational tehnique It is easy to show that if w D, then F wt), t) and Swt), t) are rd-ontinuous on T We next state two theorems that relate our seond order half-linear equations to their respetive Riati equations

6 6 L ERBE, A PETERSON AND S H SAKER Theorem 2 Fatorization of L 1 ) If x D with xt) 0 on T and wt) := rt)x t)), t T κ, then w D and x t) 25) L 1 xt) = x t)r 1 wt), t T κ2 Conversely, if w D and xt) := e w r t, t 0), then x D, xt) 0, and 25) holds Furthermore xt)x σ t) > 0 iff w r R + := {x R : 1 + µt)xt) > 0, t T} Proof First we prove the onverse statement Let w D, then sine w r R, we know that xt) = e w r t, t 0) 0 is well defined see [5, page 59]) Let xt) = e w r t, t 0), then x t) = wt) xt) from whih it follows that rt) rt)x t)) = x t)wt) From this last equation and the produt rule we get that 26) L 1 xt) = rt)x t)) ) + pt)x t) = x t)w t) + w σ t)x ) t) + pt)x t) = x t)[w t) + x ) t) w σ t) + pt)] x t) We now show that 27) First if µt) = 0, then x ) t) x t) = F wt), t) x ) t) = x 1 x t) from whih it follows that x ) t) x t) = x t) xt) = wt) rt) = F wt), t)

7 Next assume µt) > 0, then HILLE-KNESER TYPE CRITERIA 7 x ) t) x t) = x σt)) x t) µt)x t) = xσt) xt) ) 1 µt) ) 1 + µt) x t) xt) 1 = µt) 1 + µt) wt) rt) = µt) = F wt), t) ) ) 1 1 Hene in general we get that 27) holds Using 27) and 26) we get the desired fatorization 25) in all ases Next assume x D and xt) 0 Let wt) = rt)x ) t) Using the x t) produt rule 28) w t) = rt)x t)) ) 1 x t) + rt)x t)) ) σ ) 1 x t) Hene 29) x t)w t) = rt)x t)) ) + w σ t)x t)x σ t)x t)) We laim that 210) x σ t)x t)) = F wt), t) If µt) = 0, then x σ t)x t)) = x t) xt) = Next assume that µt) > 0 Then wt) rt) x σ t)x ) t) = x σ t) x ) σ t) x t) µt) = 1 x σ t) x t) x t) µt) = x ) t) x t) = F wt), t) = F wt), t)

8 8 L ERBE, A PETERSON AND S H SAKER by 27) Now by 27) and 29) we get 25) Finally note that if xt) 0 and wt) := rt)x t)), then x t) = 1 + µt) x t) wt) xt) = 1 + µt) rt) x σ t) xt) = xt) + µt)x t) xt) It follows that wt) rt) R Also we get xt)x σ t) > 0 iff In a similar manner, we may obtain w r R + Theorem 3 Fatorization of L 2 ) If x D with xt) 0 and wt) :=, then w D and rt)x t)) x t) 211) Conversely, if w D and L 2 xt) = x σ t)r 2 wt), t T κ xt) := e w r t, t 0), then x D and 211) holds Furthermore xt)x σ t) > 0 iff w r R + The following orollary follows easily from the fatorizations given in Theorems 2 and 3 respetively and the fat that if xt) 0 and wt) := rt)x t)), then x t) ) x σ 1 t) wt) xt) = 1 + µt) rt) Corollary 1 For i = 1, 2 the following hold: a) The dynami equation L i x = 0 has a solution xt) with xt) 0 on T iff the Riati equation R i w = 0 has a solution wt) on T κ with w R r b) The dynami equation L i x = 0 has a solution xt) with xt)x σ t) > 0 on T iff the Riati equation R i w = 0 has a solution wt) on T κ with w R + r ) The dynami inequality L i x 0 has a positive solution xt) on T iff the Riati inequality R i z 0 has a solution zt) on T κ with z r R + We state for onveniene the following theorem involving the Riati tehnique for equations L 1 x = 0 and L 2 x = 0 This theorem follows immediately from Theorems 2 and 3 Part B) is proven in Řehák [25] Part A) is onsidered in Sun and Li [33] when is an odd positive integer The proof

9 HILLE-KNESER TYPE CRITERIA 9 of A) is quite straightforward and is based on an iterative tehnique We omit the details Theorem 4 Assume sup T = and 18) holds A) The Riati inequality R 1 z 0 has a positive solution on [t 0, ) T iff the dynami equation L 1 x = 0 has a positive solution on [t 0, ) T B) The Riati inequality R 2 z 0 has a positive solution on [t 0, ) T iff the dynami equation L 2 x = 0 has a positive solution on [t 0, ) T Theorem 5 Assume sup T = and 18) holds A) If 1 and there is a t 0 [a, ) T suh that the inequality 212) z + pt) + ) ) 1 1 z 1 + µt) z +1 r 1 0 t) rt) has a positive solution on [t 0, ) T, then L 1 x = 0 is nonosillatory on [a, ) T B) If 1 and there exists a t 0 [a, ) T suh that the inequality 213) z + pt) + ) ) 1 1 z 1 + µt) z +1 r 1 0 t) rt) has a positive solution on [t 0, ) T, then L 2 x = 0 is nonosillatory on [a, ) T Â) If 0 < 1 and there is a t 0 [a, ) T suh that the inequality 214) z + pt) + r 1 t) z +1 0 has a positive solution on [t 0, ) T, then L 1 x = 0 is nonosillatory on [a, ) T ˆB) If 0 < 1 and there exists a t 0 [a, ) T suh that the inequality 215) z + pt) + ) ) 1 z 1 + µt) z +1 r 1 0 t) rt) has a positive solution on [t 0, ) T, then L 2 x = 0 is nonosillatory on [a, ) T Proof Assume 1 Using the mean value theorem one an easily prove that if x y 0 and 1, then the inequality 216) y 1 x y) x y x 1 x y)

10 10 L ERBE, A PETERSON AND S H SAKER holds We will use 216) to show that if u 0 and t T, then ) ) 1 1 u u 217) F u, t) 1 + µt) rt) rt) For those values of t T, where µt) = 0 it is easy to see that 217) is an equality Now assume µt) > 0, then using 216) we obtain for u 0 ) ) 1 u 1 + µt) 1 rt) F u, t) = µt) ) ) 1 1 u u 1 + µt), rt) rt) and hene 217) holds To prove A) assume z is a positive solution of 212) on [T, ) T Now onsider R 1 zt) = z t) + pt) + z σ t)f zt), t) ) ) 1 1 zt) z t) + pt) + z σ zt) t + µt) rt) rt) ) ) 1 1 zt) z zt) t) + pt) + zt + µt) rt) rt) = z t) + pt) + 0 by 212) 1 + µt) z rt) ) 1 z +1 t) r 1 t) by 217) by z t) 0 The proof of Part B) of this theorem is very similar, where instead of the inequality 217) one uses the inequality Su, t) ) r 1 1 )u t) µt) for 1, u 0, t T Now assume 0 < 1, then using the mean value theorem one an show that if 0 < y x, then 218) u rt) x 1 x y) x y y 1 x y) Using 218) we have that for u 0, t T, u 219) F u, t) rt)

11 HILLE-KNESER TYPE CRITERIA 11 and Su, t) u +1 r 1 t) 1 + µt) u rt) ) The rest of the proof for parts Â) and ˆB) is similar to the proofs for A) and B) respetively We note that as a speial ase when T = R, Theorem 5, is related to some results of Li and Yeh [23, Theorem 32], Yang [34, Theorem 2] and Yang [34, Corollary 2] for the seond-order half-linear differential equation 13) Now, we are ready to establish our main osillation and nonosillation results Theorem 6 Hille-Kneser type nonosillation riteria for L 1 x = 0) Assume sup T = and 18) holds Assume 1 Suppose there exist a t 0 [a, ) T, and onstants 0 and d 1 suh that for t [t 0, ) T, 220) pt) µt) t d rt) ) ) 1 1 d t d ) +1 r 1 t) tσt)) d Then L 1 x = 0 is nonosillatory on [a, ) T In partiular, if for t t 0 suffiiently large there is a 0 suh that [ 221) pt) ) ] 2 1 σt) 1 tσt)) rt) t then L 1 x = 0 is nonosillatory on [a, ) T Now assume 0 < 1 Suppose there exist a t 0 [a, ) T, and onstants 0 and 0 < d 1 suh that for t [t 0, ) T, ) pt) + d t d ) +1 r 1 t) t d σt) Then L 1 x = 0 is nonosillatory on [a, ) T In partiular, if for t t 0 suffiiently large there is a 0 suh that [ 223) pt) ) ] σt) 1 t σt) rt) t then L 1 x = 0 is nonosillatory on [a, ) T

12 12 L ERBE, A PETERSON AND S H SAKER Proof First assume 1 From Theorem 5 we see that if the inequality 212) has a positive solution in a neighborhood of, then L 1 x = 0 is nonosillatory Let zt) := for t t t d 0, where > 0 and d 1 We laim that 224) z t) d tσt)) d If µt) = 0 it is easy to see that 224) is an equality Now assume µt) > 0, then [ z 1 t) = µt) σt)) ] [ 1 = d t d σt) t σt)) 1 ] d t d ) σt)) d t d 225) = σt)) d t d σt) t Applying inequality 216) to 225) we get that z t) σt)) d t d dtd 1 ) = d tσt)) d Hene 224) holds in general It follows from 224) that 226) z t) + pt) µt) r 1 t) d +1 tσt)) + pt) + d 0 by 220)) 1 + µt) ) ) 1 1 zt) z +1 t) rt) t d ) +1 t d rt) r 1 t) ) ) 1 1 It then follows from Theorem 5 that L 1 x = 0 is nonosillatory on [a, ) T Letting d = in 220) and simplifying, we have 227) pt) tσt)) +1 t +1 r 1 t) 1 + µt) tr 1 t) 1 ) 1 Hene, if for some 0, pt) satisfies 227) for t [t 0, ) T, then L 1 x = 0 is nonosillatory on [a, ) T Note that sine we are assuming pt) satisfies 221) and pt) > 0) we have that 1 whih we use in the next hain rt) of inequalites

13 By 221) pt) = = HILLE-KNESER TYPE CRITERIA 13 [ 1 tσt)) rt) tσt)) tσt)) tσt)) +1 ) ] 2 1 σt) t ) σt) ) 1 t +1 r 1 t) t ) µt) t +1 r 1 t) t ) µt) t +1 r 1 t) t ) 1 rt) ) ) 1 1 Hene 227) holds and thus L 1 x = 0 is nonosillatory on [a, ) T To prove the seond half of this theorem the ase 0 < 1) note that from 218) in this ase sine 0 < d 1) one gets the inequality z t) d t d σt) instead of 224) The proof of the result onerning 222) follows diretly from Â) in Theorem 5 and the result onerning 223) follows easily by letting d = in 222) Similar to the proof of Theorem 6, one an establish the following result Theorem 7 Hille-Kneser type nonosillation riteria for L 2 x = 0) Assume sup T = and 18) holds Assume 1 If for t t 0 suffiiently large, there exist positive onstants and d 1 suh that 228) pt) + +1 t d ) +1 r 1 t) 1 + µt) t d rt) ) ) 1 d tσt)) d Then L 2 x = 0 is nonosillatory on [a, ) T In partiular, if for t t 0 suffiiently large there is a 0 suh that 229) pt) ) σt) rt) 1 tσt)) t 1 + µt) t rt) then L 2 x = 0 is nonosillatory on [a, ) T

14 14 L ERBE, A PETERSON AND S H SAKER Now assume 0 < 1 If for t t 0 suffiiently large, there exist positive onstants and 0 < d 1 suh that 230) pt) + +1 t d ) +1 r 1 t) 1 + µt) t d rt) ) ) 1 d t d σt) Then L 2 x = 0 is nonosillatory on [a, ) T In partiular, if for t t 0 suffiiently large there is a 0 suh that ) 1 231) pt) ) σt) t σt) 1 rt) t ) ) µt) t rt) then L 2 x = 0 is nonosillatory on [a, ) T We now give some interesting examples Example 1 If T = R, then L 1 x = 0 and L 2 x = 0 are the same If = 1 the onditions 221) and 229) both redue to pt) 1 ) t 2 rt) In the speial ase rt) 1, this redues taking = 1 2 ) to pt) 1 4t, 2 whih is the Hille Kneser riterion mentioned in Theorem 1 More generally, if > 0 and rt) 1, then 221) and 223) with = +1) both redue to ) +1 1 pt) + 1 t 1+ Moreover, in the ase > 1, Došlý and Řehák [11] have improved this riterion Example 2 If T = N, = 1 and rt) 1, then the ondition 221) for L 1 x = 0 redues to pt) 1 t + 1 ) tt + 1) t Letting = 1, it is easily seen that if there is a k < 1, suh that 2 4 k pt) tt + 1),

15 HILLE-KNESER TYPE CRITERIA 15 for large t, then L 1 x = 0 is nonosillatory on N If T = N, rt) 1, = 1, ondition 229) redues to pt) 1 tt t Letting = 1 one an argue that if there is a k < 1 suh that 2 4 k pt) tt + 1) for large t, then L 2 x = 0 is nonosillatory on N If 1, rt) 1 then using 229) it is not diffiult to see that if there +1 exists k < +1) suh that pt) k tt + 1) for large t, then L 2 x = 0 is nonosillatory on N On the other hand if ) 0 < +1 1, then using 231) it is easily shown that if there exists k < +1 suh that k pt) t t + 1) for large t, then L 2 x = 0 is nonosillatory on N Combining these results +1 we see that if > 0, rt) 1, and there is a k < +1) suh that pt) k t +1 for large t, then L 2 x = 0 is nonosillatory on N See Agarwal et al [1] for additional results Example 3 If T = q N 0, q > 1, then 221) beomes in the ase = 1 and rt) 1), pt) 1 q) qt2 Taking = 1 2q 232) we get pt) 1 4q 2 t 2, for large t implies L 1 x = 0 is nonosillatory on q N 0 With the same assumptions T = q N 0, q > 1, rt) 1, = 1) ondition 229) beomes pt) 1 qt q 1)

16 16 L ERBE, A PETERSON AND S H SAKER and with = 1 1+, we get q 233) pt) 1 q1 + q) 2 t 2, for large t implies the nonosillation of L 2 x = 0 on q N 0 We see therefore that the riteria for nonosillation of the linear = 1) equations L 1 x = 0 and L 2 x = 0 as onsequenes of Theorems 6 and 7, are different in general Solving the Euler Cauhy equations 234) x + a tσt) x = 0 and 235) x + a tσt) xσ = 0 one an show that if a 1, then 234) is nonosillatory on 4 qn 0, and if 1 a, then 235) is nonosillatory on q N 0 We note that the result q+1) 2 232) is not sharp; however, the result 233) is sharp as an be seen by a more detailed analysis See also Řehák [25] If 1, rt) 1, then applying 221) we get that if pt) 1 ) +1 1 q t, 1+ for large t, then L 1 x = 0 is nonosillatory On the other hand if 0 < 1, then applying 223) we get that if pt) 1 ) +1 1 q t, 1+ for large t, then L 1 x = 0 is nonosillatory If 1, rt) 1, we get using 229) that if )] pt) [1 q 1 t 1+ q 1 + q 1) 1 for large t, then L 2 x = 0 is nonosillatory On the other hand if 0 < 1, then using 231) we get that if pt) 1 q 1 [ ] t 1+ q 1 + q 1) 1 for large t, then L 2 x = 0 is nonosillatory

17 HILLE-KNESER TYPE CRITERIA 17 Example 4 For a general time sale T, where sup T = and 18) holds, it follows from 221) that if > 1, and ) 2 1 σt) mr 1 t) t for large t, for some onstant m > 0, then L 1 x = 0 is nonosillatory on [a, ) T provided ) +1 tσt)) 236) pt) m, m + 1) for large t To see this, observe that in 221) the right hand side is bounded above by tσt)) h), where h) := +1 m, whih has its maximum at = m+1)) For T = R, = 1, rt) 1, we an take m = 1 and this again redues to the Hille Kneser riterion) For the ase of L 2 x = 0, we first observe that in 229), the expression 1 + µt) t rt) rt), rt) so that the right side of 229) is bounded below by [ ) σt) 1 tσt)) t rt) Therefore, if there exists an m > 0 suh that ) σt) m r 1 t) t for large t, then L 2 x = 0 is nonosillatory provided 236) holds Notie that if = 1, T = R, rt) 1, then 236) redues to the Hille Kneser riterion pt) 1 One an also give additional speial ases We leave 4t 2 this to the interested reader Aknowledgement: The authors gratefully aknowledge the referee s detailed omments and orretions on an earlier version ]

18 18 L ERBE, A PETERSON AND S H SAKER Referenes [1] R P Agarwal, M Bohner, S Grae, and D O Regan, Disrete Osillation Theory, Hindawi, New York, 2005 [2] R P Agarwal, M Bohner and S H Saker, Osillation of seond order delay dynami equations, Canad Appl Math Quart, to appear [3] E Akin-Bohner, M Bohner and S H Saker, Osillation riteria for a ertain lass of seond order Emden-Fowler dynami equations, Elet Trans Numer Anal, to appear [4] E A Bohner and J Hoffaker, Osillation properties of an Emden-Fowler type equations on disrete time sales, J Differene Eqns Appl, ) [5] M Bohner and A Peterson, Dynami Equations on Time Sales: An Introdution with Appliations, Birkhäuser, Boston, 2001 [6] M Bohner, A Peterson, Advanes in Dynami Equations on Time Sales, Birkhäuser, Boston, 2003 [7] M Bohner and S H Saker, Osillation of seond order nonlinear dynami equations on time sales, Roky Mountain J Math, [8] M Bohner and S H Saker, Osillation riteria for perturbed nonlinear dynami equations, Math Comp Modelling, ) [9] X Cai, An existene theorem for seond order disrete boundary value problems, Mathematis in Eonomis, 22:2 2005) [10] O Došlý and S Hilger, A neessary and suffiient ondition for osillation of the Sturm Liouville dynami equation on time sales, Speial Issue on Dynami Equations on Time Sales, edited by R P Agarwal, M Bohner, and D O Regan, J Comp Appl Math, [11] O Došlý and P Řehák, Half Linear Differential Equations, North-Holland, Mathematis Studies 202, Amsterdam, 2005 [12] L Erbe, Osillation riteria for seond order linear equations on a time sale, Canad Appl Math Quart, [13] L Erbe and A Peterson, Positive solutions for a nonlinear differential equation on a measure hain, Math Comput Modelling, Boundary Value Problems and Related Topis, ) [14] L Erbe and A Peterson, Riati equations on a measure hain, In G S Ladde, N G Medhin, and M Sambandham, editors, Proeedings of Dynami Systems and Appliations, , Atlanta Dynami publishers [15] L Erbe and A Peterson, Osillation riteria for seond order matrix dynami equations on a time sale, Speial Issue on Dynami Equations on Time Sales, edited by R P Agarwal, M Bohner, and D O Regan, J Comput Appl Math, ) [16] L Erbe and A Peterson, Boundedness and osillation for nonlinear dynami equations on a time sale, Pro Amer Math So ) [17] L Erbe, A Peterson and S H Saker, Osillation riteria for seond-order nonlinear dynami equations on time sales, J London Math So, ) [18] L Erbe, A Peterson and S H Saker, Asymptoti behavior of solutions of a thirdorder nonlinear dynami equation on time sales, J Comp Appl Math, ) [19] G Sh Guseinov and B Kaymakçalan, On a disonjugay riterion for seond order dynami equations on time sales, Speial Issue on Dynami Equations on Time

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