On Third Order Differential Equations with Property A and B

Transcription

1 Journal of Maemaical Analysis Applicaions 3, Aricle ID jmaa , available online a p: on On Tird Order Differenial Equaions wi Propery A B Mariella Cecci Deparmen of Elecronic Engineering, Uniersiy of Florence, Via S. Mara 3, 539 Firenze, Ialy Zuzana Dosla ˇ Deparmen of Maemaics, Masaryk Uniersiy, Janackoo ˇ nam. a, 6695 Brno, Czec Republic Mauro Marini Deparmen of Elecronic Engineering, Uniersiy of Florence, Via S. Mara 3, 539 Firenze, Ialy Submied by Z. S. Aanasson Received Marc 4, 998 INTRODUCTION Tis paper is concerned wi e asympoic beavior of soluions of e linear differenial equaions of e corresponding nonlinear ones were x qž. x rž. x Ž e. x qž. x rž. fž x., Ž n. q, r are coninuous funcions for, qž., rž. Ž. f is a coninuous funcion in suc a fž u. u for u. Ž X99 $3. Copyrig 999 by Academic Press All rigs of reproducion in any form reserved.

2 5 CECCI, DOSLA, ˇ AND MARINI Wen q as coninuous firs derivaive, some ineresing consequences concerning e asympoic beavior of adjoin equaions o e : qž. Ž qž. rž.. Ž e a. are also given. Trougou e paper a soluion of Ž n. will be a ree ime differeniable funcion x saisfying Ž n. for all large sup x Ž.: T4 for every T sufficienly large. For resuls concerning coninuabiliy a infiniy of soluions of Ž n., we refer e reader o 9,, 4. As usual a nonrivial soluion of Ž e. Ž n. is said o be oscillaory or nonoscillaory according o weer i does or does no ave arbirarily large zeros. Equaion Ž e. n is called oscillaory if i as a leas one oscillaory soluion nonoscillaory oerwise, i.e., if all is soluions are nonoscillaory. Many papers ave been devoed o e sudy of e oscillaory asympoic beavior of soluions of Ž e. Ž n., e.g.,, 7, 8,, 5, 8 6, 9,, 3, 7, respecively. According o e classical resuls of V. A. Kondraiev I. T. Kiguradze, is sudy is ofen accomplised by inroducing e conceps of equaion wi propery A or equaion wi propery B. More precisely equaion Ž e. is said o ave e propery A if every soluion of Ž e. eier is oscillaory or saisfies e condiions xž. xž., xž. x Ž. for, Ž. lim xž. lim xž. lim x Ž.. Equaion e is said o ave propery B if every soluion of e eier is oscillaory or saisfies e condiions xž. xž., xž. x Ž. for all large, Ž. lim xž. lim xž. lim x Ž.. We recall a a soluion x of Ž e. wic saisfies Ž. is said o be Kneser soluion of Ž e.. Similarly a soluion x of Ž e. wic saisfies Ž. is said o be srongly monoone soluion of Ž e.. We recall also a Ž e. as always Kneser soluions Ž e. as always srongly monoone soluions Žsee, e.g., 5 Teorem. Lemma... Similar definiions old for e nonlinear equaions Ž n.. In is case condiion Ž. olds for all large. Properies A B ave been discussed by numerous auors by using various ecniques: for recen resuls on is opic we refer e reader o

3 TIRD ORDER DIFFERENTIAL EQUATIONS 5 3, 4, 5, 4 e references conained erein. I is wor o noe a many auors sudy ese properies for various ypes of differenial difference equaions wiou explicily inroducing e conceps of propery A B. For linear equaions, e relaionsips beween oscillaory soluions asympoic beavior of nonoscillaory ones are ofen considered. In paricular, in e quoed paper 5, Lazer proved e following: Ž. LAZER TEOREM. Assume q for. Ž. a Equaion Ž e. is oscillaory if only if eery nonoscillaory soluion x of Ž e. is a Kneser soluion lim xž. lim x Ž.. Ž b. If equaion Ž e. is oscillaory, en eery nonoscillaory soluion x of Ž e. is a srongly monoone soluion lim x Ž.lim xž.. According o e case of consan coefficiens, in 5, p. 444 Lazer posed e following quesion: does every nonoscillaory soluion of Ž e. end o zero as ends o infiniy wen Ž e. is oscillaory? In e special case wen q, is conjecure was proved by Gae. Villari in 8 laer by Lazer Ž5, T..5. Gregus Ž8, T. 3.. under addiional assumpions. ere we give a complee posiive answer o e Lazer conjecure, by sowing a i is rue wiou any addiional condiion. Concerning equaion Ž e., e converse of e Lazer resul Ž b. as been proved by Gera Žsee, e.g., 8,.. Te main purpose of is paper is o presen some new resuls concerning relaionsips beween oscillaion propery A or propery B. More precisely in Sec. we will prove, wiou any addiional assumpion, a oscillaion of Ž e. is equivalen o propery A oscillaion of Ž e. is equivalen o propery B. In addiion we will sow an equivalence crierion on propery A for Ž e. on propery B for is adjoin equaion Že a. similarly on propery B for Ž e. on propery A for is adjoin equaion Že a.. Te obained resuls in e linear case are ineresing in emselves by virue of eir necessary sufficien caracer, bu ey are useful also in Sec. 3 were e same problem for e nonlinear case is considered. We remark a our assumpions on e nonlineariy concern only e grow in a neigborood of infiniy: no monooniciy condiions involving e beavior of e nonlineariy on all are assumed.

4 5 CECCI, DOSLA, ˇ AND MARINI. PRELIMINARIES In e following e asympoic beavior of soluions of Ž. qž. Ž. Ž 3. is playing a crucial role. Ten i seems useful o recall e following: PROPOSITION A Žsee, e.g.,, 6.. Assume qž. for. Ten Ž. 3 is nonoscillaory eery nonoscillaory soluion saisfies eier or Ž a. Ž. Ž. for all large lim Ž. Ž 4. Ž b. Ž. Ž. for all. Ž 5. In addiion, Ž. 3 as soluions wic saisfies bo Ž. 4 Ž. 5. Soluions saisfying Ž. 5 are called principal soluions a infiniy. Tey are uniquely deermined up o a consan facor erify e condiions: c d ; Ž d. lim Ž. c if only if qž. d. Le be a posiive soluion of Ž. 3 on,.,. Ten Ž e. can be wrien in e disconjugae form ž ž / / Ž. xž. rž. Ž. xž. Ž.. Ž L. Ž. Equaions L are a special case of e linear equaions were ž ž / / xž. Ž. xž., Ž L. Ž. Ž.,, are posiive coninuous funcions for. Ž. 3 Wen e funcions, do no ave a coninuous firs or second derivaive, en Ž L. may be inerpreed as a firs order differenial sysem for e vecor Ž x, x, x. given by ž / x x, x x, x x Ž x.,

5 TIRD ORDER DIFFERENTIAL EQUATIONS 53 i were x is a soluion of L. Te funcions x are called quasideriaies of x. In, 3 equaions Ž L. were considered ogeer wi adjoin ones: ž ž / / a z Ž. z. Ž L. Ž. Ž. Le us recall some resuls on asympoic beavior of soluions of Ž L. Ž a L. wic will be useful in e sequel. Denoe wi Iu, Ž, w. e riple inegral s IŽ u,, w. už. Ž s. wž. d ds d, were u,, w are coninuous posiive funcions on,.. Te following old: Ž. PROPOSITION B 4. Assume. Ten 3 Ž. a Equaion Ž L. is oscillaory if only if any nonoscillaory soluion saisfies xž. x Ž., xx Ž. Ž. for. Ž b. Equaion Ž L. is oscillaory if only if any nonoscillaory soluion of Ž L. saisfies xž. x Ž., xx Ž. Ž. for large. PROPOSITION C Ž3.. Assume Ž. IŽ,,. 3. Ten any soluion x of Ž L. suc a xž. x Ž., xx Ž. Ž. saisfies lim x Ž.. Ž. PROPOSITION D. Assume. 3 Ž. a Ž a If I,, en L L. are nonoscillaory. Ž b. Ž a If I,, en L L. are nonoscillaory.. OSCILLATION AND PROPERTY AB Te following resul proves e equivalence beween oscillaion propery A for Ž e.. Suc a resul as been already proved by Gregus 8 under e addiional assumpion q C Ž,... TEOREM. Assume Ž.. Ten Ž e. is oscillaory if only if Ž e. as propery A. Proof. By Lazer eorem i is sufficien o prove a if Ž e. is oscillaory en every Kneser soluion of Ž e. ends o zero as.

6 54 CECCI, DOSLA, ˇ AND MARINI Le be a posiive nonincreasing soluion of Ž. 3 for. Equaion Ž e. can be wrien in e disconjugae form Ž L.. Clearly Ž L. is oscillaory by Proposiion D i olds ž / I r,,. Ž 6. Because is a nonincreasing funcion for, e funcion is nondecreasing for. Ten s s s Ž u. du ds Ž s. du ds ds, Ž s. Ž s. Ž s. s Ž s. du ds Ž s. du ds ds. Ž u. Ž s. Ž s. Consequenly, s s s s u du ds s du ds, Ž s. Ž u. Ž. so, aking ino accoun 6 we ge ž / I r,,. Because Ž L. is oscillaory, by Proposiion B Proposiion C we ge e asserion. From e Lazer resul Teorem, i follows a Ž e. as propery A if only if every nonoscillaory soluion Ž e. is a Kneser soluion, i.e., every Kneser soluion ends o zero as. Tis is no rue in e general case for Ž L. as Example in 3 sows. By virue of Teorem, oscillaion crieria for Ž e. become crieria in order o Ž e. ave propery A vice versa. For insance, from a well-known oscillaion resul by Lazer 5, T..3 we obain e following: COROLLARY. Assume Ten e as propery A. 3 rž. qž. d. Ž 7. 3 ' 3 Now consider equaion Ž e.. As claimed in e inroducion, Gera in 7 as proved e converse of e Lazer eorem Ž b., i.e., if every nonoscilla-

7 TIRD ORDER DIFFERENTIAL EQUATIONS 55 ory soluion of Ž e. is srongly monoone, en Ž e. is oscillaory. Ten, Ž e. is oscillaory if only if every nonoscillaory soluion is srongly monoone. Now we prove a sronger resul wic esablises e equivalency beween oscillaion propery B. Te following olds: TEOREM. Assume Ž.. Ten Ž e. is oscillaory if only if Ž e. as propery B. Proof. By e quoed Gera resul, i is sufficien o sow a if Ž e. is oscillaory en Ž e. as propery B. Le x be a nonoscillaory soluion of Ž e.. By e Lazer resul x is srongly monoone so, wiou loss of generaliy, we can suppose a ere exiss suc a x Ž., xž., x Ž. for. Since xž. for, we obain x Ž. x Ž. or xž. xž. x Ž.Ž. x Ž.Ž., x Ž. x Ž. xž. xž. Ž. Ž. Ž.. Inegraing Ž e. on Ž,. we ge x Ž. x Ž. qž s. xž s. ds rž s. xž s. ds x Ž. Ž s. qž s. ds Ž s. rž s. ds. Ž 8. ž / Since x is an evenually posiive srongly monoone soluion of equaion Ž e., en lim x Ž. lim xž.. ence i is sufficien o sow a xž.. Assume xž.. From Ž 8. we ave rž. d, qž. d. Ž 9. By Proposiion A, ere exiss a posiive nonincreasing soluion of Ž. 3 suc a lim Ž. l. Ten Ž. Ž. l s ds Ž u. du k, Ž s. Ž. were k l. Consequenly, aking ino accoun Ž. 9, we obain s r Ž. Ž. Ž u. du ds d Ž s. k rž. Ž. d kž. rž. d

8 56 CECCI, DOSLA, ˇ AND MARINI, by Proposiion D, e equaion Ž L. is nonoscillaory, wic is a conradicion because Ž L. is e disconjugae form of Ž e.. Now we focus on e sudy of oscillaion asympoic properies of adjoin equaions o Ž e. by assuming a q as coninuous firs derivaive. I is useful o inroduce e following definiions. A funcion g CŽ,.. is said o be lower oscillaory or upper oscillaory if ere exis wo sequences 4, 4,,, suc a k k k k or g Ž., g Ž. k g Ž., g Ž., k k k respecively. Ten, relaing o e sign of q r in e equaion Že a., i is elpful o divide all possibiliies as follows: Ž. a qž. rž. is evenually negaive; Ž. b qž. rž. is evenually nonnegaive; Ž. c qž. r Ž. is lower oscillaory. Similarly i is useful o classify e funcion q r in e equaion Že a. as evenually posiive, evenually nonposiive or upper oscillaory. Concerning propery A propery B, relaionsips beween Ž e. is adjoin Že a. can be easily obained by a well-known resul of anan Teorems. Te following olds: Ž.. TEOREM 3. Assume q C,. Ž. a If q riseenually negaie, en Ž e. as propery A if only if Že a. as propery B. Ž b. If q riseenually nonnegaie, en bo equaions Ž e. Že a. are nonoscillaory. Ž. c If q r is lower oscillaory equaion Ž e. is oscillaory Ži.e. as propery A., en eery soluion of Že a. is srongly monoone. Proof. Ž. a Since Ž. olds, equaion Ž e. is of class I, according o a classical definiion of anan, so Ž e. is oscillaory if only if is adjoin Že a. is oscillaory. Now claim Ž a. follows immediaely from Teorems. Ž b. By e quoed anan resul, a ird order linear equaion is of class I if only if is adjoin Že a. is of class II. I means a Že a. is of class II. Since q r is evenually nonnegaive Ž. 3 is nonoscillaory,

9 TIRD ORDER DIFFERENTIAL EQUATIONS 57 Že a. is also of class I so Že a. is nonoscillaory. Te same appens for Ž e.. Ž. c By e same argumen as in claim Ž. a we obain a Že a. is oscillaory. Le be an evenually posiive increasing soluion of Ž. 3 le z be e funcion given by z large, were is a soluion of Že a.. A sard calculaion sows a z is a soluion of ž / Ž Ž. zž.. rž. Ž. zž. Ž large.. Ž. Ž. Ten Ž. is oscillaory, by Proposiion B, every nonoscillaory soluion z of Ž. saisfies zž. z Ž., zž. z Ž. for large. Ž. Le be an evenually posiive soluion of Že a.. To prove claim Ž c., i is sufficien o sow a Ž., Ž. for all large. Consider again e ransformaion Ž.. I is easy o sow a e following ideniies: z Ž. zž. Ž. Ž., Ž. Ž. z Ž. qž. Ž. old for all large. Since z is evenually posiive, from we obain e asserion. Ž a. Similarly relaionsips beween e is adjoin e are given by e following: Ž.. TEOREM 4. Assume q C,. Ž. a If q riseenually posiie, en Ž e. as propery B if only if Že a. as propery A. Ž b. If q riseenually nonposiie, en bo equaions Ž e. Že a. are nonoscillaory. Ž. c If q r is upper oscillaory equaion Ž e. is oscillaory Ži.e. as propery B., en eery soluion of Že a. is Kneser soluion. Proof. Claims Ž. a Ž. b follow by using a similar argumen o is given in Teorem 3. Claim Ž. c follows by using e cange of variable Ž., were now is a nonincreasing soluion of Ž. 3 is a soluion of Ž a e.. Te deails are omied.

10 58 CECCI, DOSLA, ˇ AND MARINI Te following examples illusrae e above given resuls. EXAMPLE. Consider e linear equaion k x x x, Ž 3 3. Ž. Ž. were k is a posiive consan. We will sow a for k sufficienly large is equaion as propery A for k 4 is nonoscillaory. Taking ino accoun a Ž. Ž. is a soluion of e second order associaed equaion Ž., Ž 3. can be wrien in e disconjugae form k Ž. x x. Ž 3 ž / 4. Ž. Ž. Using a Canuria resul 4, Lemmas..3, wic saes a Ž L. is oscillaory if Ž s. ds Ž s. ds s u Ž u. Ž. d du lim sup Ž s. ds Ž s. ds, s Ž u. du we obain a Ž 3. is oscillaory for k sufficienly large. ence, by Teorem, Ž 3. as propery A for k sufficienly large. By Teorem 3Ž. a, e adjoin equaion o Ž 3.: 4 k x x x Ž 4 3. Ž. Ž. as propery B for k sufficienly large. Wen k Ž, 4, i follows from Teorem 3Ž b. a bo Ž 3. Ž 4. are nonoscillaory. Remark a resuls obained by Erbe 6, Teorem 3.4, Gregus 8 Lazer 5 canno be applied in is case. Teorem 3.4 in 6 requires a Ž 3. is oscillaory for k, resuls in 8 need a r Ž. d, as regards resuls in 5, i is sufficien o noe a Ž. 7 fails. EXAMPLE. By Corollary e linear equaion sin 3 x x x 3 Ž. Ž. as propery A. In is case Teorem 3. in 8 is no applicable because e funcion q Ž. sin Ž. 3 does no ave coninuous firs derivaive.

11 TIRD ORDER DIFFERENTIAL EQUATIONS 59 We conclude is secion by a relaionsip beween oscillaion beavior of e e. COROLLARY. Assume, q C Ž,.., qž. rž. qž.. If Ž e. is oscillaory, en Ž e. is oscillaory. Proof. If Ž e. is oscillaory, en Že a. is oscillaory, by a classical comparison eorem Žsee, e.g., 8, T. 7.5., Ž e. is oscillaory oo. 3. NONLINEAR EQUATIONS In is secion we apply resuls from Sec. o e nonlinear equaions x qž. x rž. fž x. on,.. Ž n. We sar wi some auxiliary resuls on nonoscillaory soluions of n. LEMMA. Le x be a nonoscillaory soluion of n. Ten x is eenually differen from zero. Proof. Te asserion follows by wriing n in e disconjugae form ž ž / / Ž. xž. rž. Ž. fž xž.., Ž N. Ž. were is a soluion of Ž. 3, by using resuls from 4 abou possible classes of nonoscillaory soluions of Ž N.. LEMMA. Le x be a nonoscillaory soluion of Ž n.. Ten eier x is Kneser soluion lim xž. lim x Ž. or ere exiss Tx suc a xž. xž. for T. x Proof. Taking ino accoun Lemma i is sufficien o prove a if x Ž., xž. for, en x Ž. for all large lim xž. lim x Ž.. Assume ere exiss suc a x Ž.. Because xž. for, x becomes negaive for en x is evenually negaive, wic is a conradicion. ence x is evenually posiive. Te remaining par of e proof follows in a rivial way. Now we can sae wo eorems on propery A for equaion n : TEOREM 5. Assume, Ž. i ere exiss a posiie, principal a infiniy, soluion of Ž 3. suc a Ž. d ;

12 5 CECCI, DOSLA, ˇ AND MARINI Ž ii. ere exiss k suc a lim inf fž u. u u k e linear equaion is oscillaory. Ten n as propery A. y qž. y k rž. y Ž 5. Proof. Assume a Ž n. does no ave propery A. By Lemma ere exiss a nonoscillaory soluion x of Ž n. T suc a eier Ž. a xx Ž. Ž. for Tx or Ž. b xx Ž. Ž., xx Ž. Ž. for T xž. xž. x, xž.. Wiou loss of generaliy, we may assume x Ž. for T x. Case Ž. a. Consider in T,. e linearized equaion x f xž. qž. rž.. Ž 6. xž. Ten 6 can be wrien in e disconjugae form ž ž / / f xž. Ž. Ž. rž. Ž.. Ž 6. Ž. xž. x Taking ino accoun Ž. i, Ž 6. is in e canonical form. Ten by 4, Teorem 3.Ž. i every evenually posiive increasing soluion saisfies for all large : or, aking large, ž / Ž. Ž. Ž. Ž. Ž. Ž s. ds, Ž. wic implies Ž.. Because x is an evenually posiive increasing soluion of Ž 6., we ge xž. so ere exiss Tx suc a fžxž.. xž. k for. Ten, aking ino accoun a Ž 5. is oscillaory, we ge by a classical comparison eorem Žsee, e.g., 8, Teorem 7.. a Ž 6. is also oscillaory. Tis conradics e Lazer resul, because x is is nonoscillaory soluion.

13 TIRD ORDER DIFFERENTIAL EQUATIONS 5 Case Ž b.. Te argumen is similar o is given in e proof of Teorem. Equaion Ž n. can be wrien in e disconjugae form Ž N..By 4, Teorem 3. Ž iii. xx Ž. Ž., xx Ž. Ž. so, aking ino ac- Ž coun a x, by 4, Teorem I3 i olds Ir,,.. Now, reasoning as in e proof of Teorem, we ge Ir, Ž,. or Ikr, Ž,.. Ten, by virue of Proposiion D, e equaion ž ž / / Ž. yž. k rž. Ž. y Ž 7. Ž. is nonoscillaory, wic is a conradicion, because Ž 7. is e disconjugae form of Ž 5.. Ž. Remark 3. Condiion i of Teorem 5 is verified in case any of e following assumpions: Ž I. qž. d ; Ž II. qž. for all large is saisfied. Te firs saemen follows by Proposiion A e second one is an easy consequence of a comparison crierion Žsee, e.g.,, Cap. XI, Corollary Te following olds: TEOREM 6. Assume Ž., Ž. fž u. Ž. i lim inf ; ii u u e linear equaion y qž. y kr Ž. y Ž 8. is oscillaory for eery k. Ten n as propery A. Proof. Te asserion follows by using a similar argumen o is given in e proof of Teorem 5. Te deails are omied. Remark 4. Teorem 6 generalizes for Ž n. e resul of 6, Teorem 3.4 were in addiion i is required a lim inf 3 rž..

14 5 EXAMPLE 3. CECCI, DOSLA, ˇ AND MARINI Te nonlinear equaion k x x x sgn x Ž 3. Ž. Ž. as propery A for k by Teorem 5. Indeed Ž. i is saisfied, because Ž. Ž. is a soluion of Ž. 3 Ž. ii olds Ž see Example.. In is case Teorem 6 canno be applied, because Ž 8. is no oscillaory for k 4 Ž see again Example.. EXAMPLE 4. Te nonlinear equaion x x x lnx sgn x Ž. as propery A by Teorem 6. Indeed, by Corollary, Ž 8. is oscillaory for every k. In is case Teorem 5 canno be applied because Ž. i is no saisfied. LEMMA 3. Assume condiion Ž. i of Teorem 5. Le x be a nonoscillaory soluion of Ž n.. Ten eier x is srongly monoone soluion lim xž. lim xž. or ere exiss T suc a x xž. xž., xž. x Ž. for T x. Proof. Wiou loss of generaliy we can assume x evenually posiive. Firs we claim a x is evenually posiive. If is is no rue, by Lemma ere exiss suc a x Ž., xž. for. Consider e funcion G given by GŽ. Ž. x Ž. Ž. xž., Ž. Ž. were is a principal soluion of 3 saisfying i. I olds G Ž. rž. Ž. xž. f xž. so G is increasing for. Ten wo cases are possible Ž a. G Ž. for, Ž b. G Ž. for.

15 TIRD ORDER DIFFERENTIAL EQUATIONS 53 Ž. Case a. From we obain d xž. GŽ. Ž. GŽ. d Ž. xž. xž. GŽ. ds Ž. Ž. Ž s. wic gives a conradicion as. Case Ž. b. Since Ž dd. xž. Ž. we obain or xž. xž. Ž. Ž. xž. xž. Ž s. ds xž. Ž. wic gives again a conradicion as. Ten e claim is proved x is evenually posiive. In order o complee e proof i is sufficien o observe a x is evenually posiive us x does no evenually cange e sign. Now we can sae e following eorem for n : TEOREM 7. Assume Ž., Ž., condiion Ž i. of Teorem 5, fž u. Ž ii. lim inf ; iii u u e linear equaion is oscillaory for eery k. Ten n as propery B. y qž. y kr Ž. y Proof. Assume a Ž n. does no ave propery B. Le x be a nonoscillaory soluion of Ž n. defined on T,. x. By Lemma 3, ere exiss a nonoscillaory soluion x of Ž n. T suc a eier Ž. a xx Ž. Ž., xx Ž. Ž. for Tx or Ž. b xx Ž. Ž., xx Ž. Ž. for T xž. xž. x, xž.. x

16 54 CECCI, DOSLA, ˇ AND MARINI Wiou loss of generaliy, we may assume x Ž. for T x. Consider in T,. e linearized equaion x f xž. qž. rž.. Ž 9. xž. Because x is an evenually posiive increasing soluion of Ž 9., ere exis k T suc a fžxž.. xž. x k for. In view of e assumpion Ž iii. we ge from a classical comparison eorem Žsee, e.g., 8, Teorem 7.5. a Ž 9. is oscillaory. Tis conradics Teorem, because x is is nonoscillaory soluion. EXAMPLE 5. Te nonlinear equaion x x x sgn x Ž. Ž. Ž. as propery B for arbirary by Teorem 7. Indeed assumpion Ž. i of Teorem 7 is saisfied because Ž. Ž. is a soluion of Ž. 3 Ž. ii olds. We sow a also Ž iii. olds, i.e., y y y Ž. Ž. Ž. is oscillaory for every. Le k be sufficienly large suc a k Ž. Ž. 3 for every. Using a classical comparison eorem wi Ž 3. Žsee, e.g., 8, Teorem 7.. we ge a y y y Ž. Ž. is oscillaory for every. In view of is Corollary, Ž. is oscillaory for every. CONCLUDING REMARK We conjecure a e ypoesis Ž. i of Teorems 5 7 is also necessary.

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