Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 DOI 10.1186/s13662-015-0533-4 R E S E A R C H Open Access Non-oscillion of perurbed hlf-liner differenil equions wih sums of periodic coefficiens Per Hsil * nd Michl Veselý * Correspondence: hsil@mil.muni.cz Deprmen of Mhemics nd Sisics, Msryk Universiy, Kolářská 2, Brno, 611 37, Czech Republic Absrc We invesige perurbed second order Euler ype hlf-liner differenil equions wih periodic coefficiens nd wih he perurbions given by he finie sums of periodic funcions which do no need o hve ny common period. Our min ineres is o sudy he oscillory properies of he equions in he cse when he coefficiens give excly he criicl oscillion consn. We prove h ny of he considered equions is non-oscillory in his cse. MSC: 34C10; 34C15 Keywords: hlf-liner equions; oscillion heory; condiionl oscillion; Prüfer ngle; Ricci equion 1 Inroducion The im of his pper is o conribue o he rpidly developing heory of condiionlly oscillory hlf-liner differenil equions. Our pper is orgnized o hree secions. In his secion, we recll he noion of he so-clled condiionl oscillion nd we give hisoricl bckground of he opic. In he second secion, he reder cn find he considered equions ogeher wih he descripion of he used mehods he Ricci nd Prüfer rnsformions. These mehods led o he equion for he Prüfer ngle which is he min ool in our invesigion. Finlly, in he ls secion, we se lemms, resuls, corollries, nd exmples. Le us begin wih he concep of he condiionl oscillion for hlf-liner differenil equions. We consider he equion r x ] γ c x=0, x= x p 1 sgn x, p > 1, 1.1 where γ is given rel consn, coefficiens r nd c re coninuous funcions, nd r is posiive. We sy h 1.1 iscondiionlly oscillory if here exiss posiive consn Ɣ such h 1.1 isoscilloryforγ > Ɣ nd non-oscillory for γ < Ɣ. SuchconsnƔ is clled he criicl oscillion consn of 1.1. Now we collec he milesones in he heory of he condiionl oscillion wih respec o he opic of our pper. I ppers h pproprie hlf-liner equions for he sudy of 2015 Hsil nd Veselý. This ricle is disribued under he erms of he Creive Commons Aribuion 4.0 Inernionl License hp://creivecommons.org/licenses/by/4.0/, which permis unresriced use, disribuion, nd reproducion in ny medium, provided you give pproprie credi o he originl uhors nd he source, provide link o he Creive Commons license, nd indice if chnges were mde.
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 2 of 17 he condiionl oscillion re he Euler ype equions, i.e., he equions wrien in he form r x ] γ s x=0. p The condiionl oscillion s well s mny oher res in he oscillion heory of hlf-liner equions origines from he oscillion heory of liner differenil equions. The firs resul bou he condiionl oscillion of he considered differenil equions wsobinedbykneserin1], where he oscillion consn Ɣ = 1/4 ws found for he equion x γ x =0. 1.2 2 More hn 100 yers ler, in 2, 3], he bove resul concerning 1.2wsexendedforhe liner equions rx ] γ s 2 x =0 1.3 wih posiive α-periodic coefficiens r, s, where he criicl consn is α Ɣ = α2 dτ 1 α 1 sτdτ. 1.4 4 0 rτ 0 Nex, in 4], i ws proved h 1.3 is non-oscillory in he criicl cse γ = Ɣ. For oher reled resuls, we refer o 5 7]. In he field of hlf-liner equions, he bsic criicl consn p 1 p Ɣ = 1.5 p for he equion x ] γ p x=0 comes from 8]see lso9]. Then, in 10 12], he condiionl oscillion ws proved for more generl equions of he form r x ] γ s p x=0. 1.6 Especilly, he criicl consn of 1.6 wih posiive α-periodic funcions r, s ws idenified s cf. 1.4, 1.5 αp 1 p α Ɣ 1 = p 0 1 p α 1 r 1 p 1 τdτ sτdτ 1.7 0 in 10]. For he lierure nd n overview of he heory concerning hlf-liner differenil equions, see 13, 14].
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 3 of 17 Le us urn our enion o he perurbed Euler ype equions. The liner cse of such equions wih periodic coefficiens is sudied in 4, 5]. The hlf-liner cse is reed in 15], where he equions r x ] γ s μd log 2 ] x p =0 1.8 re nlyzed for posiive α-periodic coefficiens r, s,ndd. There i is proved h, in he criicl cse γ = Ɣ 1 see 1.7, 1.8isoscilloryfor μ > Ɣ 2 := αp 2 p 1 p 1 α p 0 1 p α 1 r 1 p 1 τdτ dτdτ 0 nd non-oscillory for μ < Ɣ 2. For furher generlizions, we refer o 16 19] seelso 20]. In his pper, we re ineresed in he cse when he perurbion is lso in he differenil erm nd boh of he perurbions re sums of periodic funcions. In conrs wih he siuion common in he lierure, he funcions in he perurbions do no need o hve ny common period nd cn chnge sign. We prove h ll considered equions re nonoscillory in he criicl cse. According o he bes of our knowledge, his resul is new lso in he liner cse i.e.,forp =2. Concerning he condiionl oscillion of Euler ype liner nd hlf-liner equions, severlresulsreknowninhediscreecseswell.wepoinoulesppers21 23] for difference equions nd 24, 25] for dynmic equions on ime scles. Noe h, in he criicl cse, ny of he discree nd he ime scle counerprs of he bove menioned resuls is no known even for equions wih periodic coefficiens. 2 Preliminries This secion is devoed o he descripion of he considered equions, he corresponding Ricci equions, nd o he modified Prüfer ngle which is he min mehod in our processes. We lso menion bsic definiions nd observions which will be essenilly pplied ler. Throughou he pper, le p > 1 be rbirrily given. We use he sndrd noion R :=, ndhesymbolq denoes he number conjuged wih p i.e., p q = pq. We consider he Euler ype hlf-liner equions expressed s R p q x ] S x p =0, x= x p 1 sgn x, 2.1 where R, S : R e R e snds for he bse of he nurl logrihm log reconinuous funcions such h R is posiive nd bounded nd S is bounded. Noe h he power p/q in he differenil erm does no men ny loss of generliy see lso 17]. Our min objecive is o give non-oscillion crierion for he hlf-liner differenil equions in he form r 0 r p q 1 x ] log 2 s 0 s 1 x log 2 = 0, 2.2 p
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 4 of 17 where r 0, r 1, s 0, s 1 : R R, e, re coninuous funcions such h r 0 is posiive nd α-periodic, s 0 is α-periodic, nd r 1 = R i, s 1 = S i,, 2.3 for rbirrily given periodic coninuous funcions R i nd S i wih periods α i nd β i,respecively. Of course, we cn ssume h ll considered periods α, α i, β i re posiive nd h some of funcions R i, S i re ideniclly zero. A his plce, we recll he definiion of men vlues for coninuous funcions s ool h helps us o idenify he criicl cse for sudied equions. Le coninuous funcion f : R R be such h he limi 1 Mf := lim b b f sds is finie nd exiss uniformly wih respec o b R.ThenumberMf is clled he men vlue of f.iisseenhfuncionsr 1, s 1 given in 2.3hvemenvlues Mr 1 = MR i, Ms 1 = MS i. 2.4 Concerning he presened resuls, we will ssume h Mr 1, Ms 1 0. In fc, we sudy oscillory properies of he equion r 0 m R i log 2 p q x ] s 0 n S i log 2 x p =0 2.5 wih periodic coefficiens R 1,...,R m, S 1,...,S n on R infiniy i.e.,vlueis lrge enough when m MR i 0, n MS i 0. For simpliciy, we will consider 2.2 onlyinhe criicl cse see he below given Theorem 3.4 nd 12, 16 18] given by i.e., Mr0 ] p q Ms 0 = 1 α p q α r α p 0 τdτ s 0 τdτ = q p, 2.6 Ms 1 Mr 0 ] p q p q Mr 1 Mr p1 0 ] 1 q 1 p = 2, 2.7 1 p q lim s p 1 τdτ r 0 τdτ p r 1 τdτ q p1 r 0 τdτ ] = q1 p 2. Then see he below given Theorem 3.5, we formule he generl resul bou he oscillion nd non-oscillion of 2.5. To sudy 2.2, we will consider he equion Mr 0 Mr p 1 log 2 1 q x ] log 4 Ms 0 Ms 1 log 2 1 x log 4 =0 2.8 p
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 5 of 17 wih consn coefficiens nd we will lso use he noions r 0 := mx{ r 0 ; R }, r 1 := sup { r 1 ; R }, s 0 := mx{ s 0 ; R }, s 1 := sup { s 1 ; R }, 2.9 nd R i := mx { Ri ; R }, S i := mx { Si ; R } 2.10 for ech i. As we hve lredy menioned bove, he min mehod used in his pper is he modified Prüfer ngle. To clrify his mehod, we need some bsic properies of he hlf-liner rigonomeric funcions. We denoe π p := 2π p sin π p 2.11 nd consider he iniil vlue problem x ] p 1 x=0, x0 = 0, x 0 = 1. 2.12 The odd 2π p -periodic exension of he soluion of 2.12 is clled he hlf-liner sine funcion ndisusullydenoedbysin p.thehlf-liner cosine funcion is defined s he derivive of he hlf-liner sine funcion nd i is denoed by cos p. The needed properies of he hlf-liner rigonomeric funcions for our purpose re he vlidiy of he hlf-liner Pyhgoren ideniy sin p y p cos p y p =1, y R, 2.13 nd he boundedness of hese funcions given by see 2.13 cos p y p 1, cos p y sin p y 1, sin p y p 1, y R. For oher properies, we refer, e.g., o14], Secion1.1.2. To inroduce he noion of he modified hlf-liner Prüfer ngle, we consider he concep of he Ricci equion. Using he rnsformion w=r p q x, 2.14 x where x is non-rivil soluionof 2.1, we direcly obin he so-clled Ricci equion w S p p 1R w q =0 2.15 ssocied o 2.1. Hence, we cn inroduce he modified hlf-liner Prüfer rnsformion s follows: x=ρ sin p ϕ, x = Rρ cos p ϕ. 2.16
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 6 of 17 Ourim is o obin n equionfor he Prüferngle ϕ. Therefore, we skech is derivion his plce. For more deils, see 15]. Le us consider soluion w of 2.15. Using 2.14 nd 2.16, we hve v := p 1 cosp ϕ w = sin p ϕ 2.17 which, ogeher wih he fc h sin p is soluion of he iniil vlue problem in 2.12, leds o v =1 p 1 cos p ϕ p] sin p ϕ ϕ. 2.18 Now we derive v using 2.15nd2.17. We obin v = p 1 w ] =p 1 p 2 w p 1 w = p 1 v S p 1 R cos p ϕ p] sin p ϕ. 2.19 Finlly, using 2.17ndcompring2.18nd2.19, we hve 1 p 1 cos p ϕ p] sin p ϕ ϕ = p 1 cosp ϕ sin p ϕ p 1 R cos p ϕ p] sin p ϕ S which gives consider he Pyhgoren ideniy 2.13 herequiredequion ϕ = 1 R cosp ϕ p cos p ϕ sin p ϕs sin p ϕ p ] p 1 2.20 ssocied o 2.1. In priculr, he Prüfer ngle ϕ ssocied o 2.2vi2.16 sisfies he equion ϕ = 1 r 0 r 1 log 2 cos p ϕ p cos p ϕ sin p ϕ s 0 s 1 log 2 sinp ϕ p p 1 TheequionforhePrüferngleϕ ssocied o 2.8isseegin2.20 ϕ = 1 Mr 0 Mr 1 log 2 1 log 4 cos p ϕ sin p ϕ cos p ϕ p Ms 0 Ms 1 log 2 1 log 4 ]. 2.21 sinp ϕ p p 1 ]. 2.22 For ny soluion ϕ of 2.20onR,wedefinehefuncionψ : R R by he formul ϕτ ψ:= dτ,. 2.23 τ
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 7 of 17 This uxiliry funcion ψ will ply n imporn role in he res of our pper. Noe h 2.21nd2.22respecilcsesof2.20. Thus, he bove funcion ψ is inroduced lso for soluions of 2.21nd2.22. 3 Resuls Now we complee necessry semens which we will use o prove he min resul. We begin wih wo known lemms. Lemm 3.1 If ϕ is soluion of 2.20 on R, hen he funcion ψ : R R defined by 2.23 sisfies ϕ s ψ C log,, s 0, ], for some C >0. Proof See 26], Lemm 3.2. Lemm 3.2 Le ϕ be soluion of 2.20 on R. Then here exis A, c >0such h he funcion ψ : R R defined in 2.23 sisfies he inequliy ψ 1 cosp ψ p Rτdτ cos p ψ sin p ψ sin p ψ p ] p 1 Sτdτ A 1c for ll >. Proof The lemm comes direcly from 26], Lemm 3.4. Nex, we will need he following resuls. Lemm 3.3 Le ϕ be soluion of 2.22 on R. Then he funcion ψ : R R defined in 2.23 sisfies he inequliy ψ 1 Mr 0 Mr 1 log 2 cos p ψ p cos p ψ sin p ψ Ms 0 Ms 1 sinp ψ p ] 1 log 2 p 1 log 5 for ll sufficienly lrge. 3.1 Proof From Lemm 3.2,wehve ψ 1 cosp ψ p Mr 0 Mr 1 log 2 τ 1 log 4 dτ τ cos p ψ sin p ψ
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 8 of 17 1 sin p ψ p p 1 cos p ψ p Mr 0 cos p ψ sin p ψ Ms 0 sin p ψ p p 1 Ms 0 Ms 1 log 2 τ 1 log 4 dτ A ] τ c Mr 1 log 2 1 log 4 Ms 1 log 2 1 log 4 for ll >. Vi he men vlue heorem, one cn direcly compue A c ] 0 lim sup log 2 log 2 log 2 ] lim log 2 2 log =0. 3.2 Thus, we hve Mr 1 log 2 Mr 1 log 2 Mr 1 log2 log 2 log 4 Ms 1 log 2 Ms 1 log 2 Ms 1 log2 log 2 log 4 for ll lrge.from2.13, i is seen h 1 log 6, 3.3 p 1 log 6 3.4 mx { sin p y p, cos p y p} 1 2, y R. Hence, for lrge,wehve cos p y p log 4 sin p y p p 1log 4 > 2 log 5, y R. 3.5 Alogeher, using 3.3, 3.4, nd 3.5, we obin ψ 1 Mr 0 Mr 1 log 2 Ms 0 Ms 1 sinp ψ p log 2 p 1 cosp ψ p cos p ψ sin p ψ 2 log 5 2 log 6 A ] c for lrge,whichgives3.1. Lemm 3.4 Le ϕ be soluion of 2.21 on R. Then here exiss B >0such h he funcion ψ : R R defined by 2.23 sisfies he inequliy ψ 1 Mr 0 Mr 1 log 2 Ms 0 Ms 1 sinp ψ p log 2 p 1 for ll sufficienly lrge. cos p ψ p cos p ψ sin p ψ B ] log 6
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 9 of 17 Proof From Lemm 3.2, we know h he inequliy ψ 1 cosp ψ p sin p ψ p p 1 r 0 τ r 1τ log 2 τ s 0 τ s 1τ log 2 τ holds for ll >. I mens h i suffices o prove nd 1 1 dτ cos p ψ sin p ψ dτ A c ] r 0 τdτ Mr 0 A 0, 1 s 0 τdτ Ms 0 B 0, 3.6 r 1 τ log 2 τ dτ Mr 1 log 2 A 1 log 6, 3.7 1 s 1 τ log 2 τ dτ Ms 1 log 2 B 1 log 6 3.8 for some A 0, B 0, A 1, B 1 >0,ndforlllrge. Le f : R R be n rbirry coninuous periodic funcion wih period δ >0.Legiven number be sufficienly lrge nd l N be such h lδ,l 1δ. We hve 1 f τdτ Mf 1 f τdτ 1 lδ f τdτ 1 lδ f τdτ Mf 1 f τdτ 1 lδ f τdτ 1 lδ f τdτ lδ lδ δ mx 0,δ f 1 lδ 1 lδmf δ mx 0,δ f δmf. 3.9 Thus, 3.6isvlidforsee2.9 A 0 = α r 0 Mr 0 ], B 0 = α s 0 Ms 0 ]. Since 3.9 is rue for ny periodic coninuous funcion f,weobinsee2.3, 2.4, 2.10 1 r 1 τdτ Mr 1 1 = R i τdτ M R i 1 = R i τdτ MR i
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 10 of 17 1 R i τdτ MR i α i R i α i MR i. 3.10 Anlogously, 1 s 1 τdτ Ms 1 β i S i β i MS i. 3.11 Using 3.10, we hve 1 r 1 τ log 2 τ dτ Mr 1 log 2 1 r 1 τ log 2 τ dτ 1 r 1 τ log 2 dτ 1 r 1 τ log 2 dτ Mr 1 log 2 r 1 1 log 2 τ 1 log 2 dτ 1 1 log 2 r 1 τdτ Mr 1 log r 1 2 log 2 ] 1 α i R i α i MR i log 4 log 2 3.12 for lrge. Considering 3.2, we obin 3.7 from3.12. Anlogously, one cn obin 3.8 pplying 3.11. Hence, he proof is complee. Lemm 3.5 Equion 2.8 is non-oscillory. Proof The non-oscillion of 2.8 follows from 16], Theorem 4.1 see lso 17] nd he Surmin hlf-liner comprison heorem see, e.g.,14], Theorem 1.2.4. More precisely, from 16], Theorem 4.1 i follows h he equion Mr 0 Mr 1 log 2 Ms 0 Ms 1 log 2 p ε q x ] log loglog ] 2 ε log loglog ] 2 x p =0 3.13 is non-oscillory for ny sufficienly smll ε > 0 i is described in 17] nd 3.13 is non-oscillory mjorn of 2.8. Lemm 3.6 For soluion ϕ of 2.22 on R, we hve lim sup ϕ=lim sup ψ<, 3.14 where ψ is inroduced in 2.23. Proof Lemm 3.5 sys h ny considered soluion ϕ is bounded from bove. Indeed, i suffices o consider 2.16 nd2.22 whensin p ϕ = 0. For deils, we cn refer, e.g., o 14], Secion 1.1.3, 4, 15, 19]. Finlly, he equliy in 3.14follows from Lemm 3.1.
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 11 of 17 Now we cn prove he nnounced resul. Theorem 3.1 Equion 2.2 wih 2.6 nd 2.7 is non-oscillory. Proof We recll h he non-oscillion of 2.2 is equivlen o he boundedness of soluion ϕ of 2.21 onr see gin ech one of ppers 4, 15] or19]. In ddiion, soluion ϕ of 2.21 onr is bounded if nd only if lim sup ϕ<. Iisseenfrom he righ-hnd side of 2.21whensin p ϕ=0. Le sufficienly lrge T > be given. Le us consider n rbirry soluion ϕ of 2.21on R T nd he corresponding funcion ψ : R T R given by 2.23. Lemm 3.4 ensures ψ 1 Mr 0 Mr 1 log 2 Ms 0 Ms 1 sinp ψ p log 2 p 1 Thus, we hve ψ < 1 Mr 0 Mr 1 log 2 Ms 0 Ms 1 sinp ψ p log 2 p 1 cosp ψ p cos p ψ sin p ψ B ] log 6, > T. cosp ψ p cos p ψ sin p ψ 1 log 5 ], > T, 3.15 becuse T cnbechosenrbirrily. We consider he soluion ϕ of 2.22 given by he iniil condiion see 2.11 ϕt=mx { ϕt ; 0, T] } π p 3.16 nd he corresponding funcion ψ given by 2.23. Considering he form of 2.22 nd 3.16, one cn show h ψt< ψt. 3.17 Lemm 3.6 sys h 3.14isvlidfor ϕ nd ψ, i.e.,wehve lim sup ϕ=lim sup ψ<. 3.18 Lemm 3.3 gives ψ 1 Mr 0 Mr 1 log 2 Ms 0 Ms 1 sinp ψ p log 2 p 1 cos p ψ p cos p ψ sin p ψ 1 log 5 Considering 3.15, 3.17, 3.18, nd 3.19, we obin ], > T. 3.19 lim sup ψ lim sup ψ<.
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 12 of 17 Indeed, i suffices o consider he cse when ψ 0 = ψ 0 for ny 0 > T. Using Lemm 3.1, we know h ϕ is bounded from bove which implies he non-oscillion of 2.2. To illusre our resuls, we menion exmples. We remrk h ll given exmples re no generlly solvble using ny previously known non-oscillion crieri. Exmple 1 Immediely, Theorem 3.1 gives he non-oscillion of severl equions. For exmple, he equions 1 sin p q2 sin 2 2p log 2 1rcnsin3 p q x ] q p πq sin 1 4 log 2 re non-oscillory. p q x ] q p sin5 x =0, p x p =0 Theorem 3.1 implies new resuls in mny specil cses. We obin new resul even for liner equions wih consn nd periodic coefficiens which is formuled s he corollry below. Corollry 3.1 Le f, g be periodic nd coninuous funcions such h Mf, Mg 0 nd Mf Mg=1.The equion 1 f 1 ] log 2 x 1 1 g 4 2 log 2 x =0 3.20 is non-oscillory. Exmple 2 Le 0, 1 nd ϱ, σ > 1 be rbirry. For he liner equions x ] 1 sin ϱ /log 2 11 sin σ / log 2 x =0, 4 2 x ] 1 sin ϱ /log 2 11 cos σ / log 2 x =0, 4 2 x ] 1 cos ϱ /log 2 11 sin σ / log 2 x =0, 4 2 x ] 1 cos ϱ /log 2 11 cos σ / log 2 x =0, 4 2 we cn pply Corollry 3.1. Thus, he bove equions re non-oscillory. Now we menion wo relevn resuls. Theorem 3.2 Le c : R R be coninuous funcion, for which men vlue Mc 1 q exiss nd for which 0< inf R c sup R c<,
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 13 of 17 nd le d : R R be coninuous funcion hving men vlue Md. Le us consider he equion c x ] d x=0 3.21 p nd denoe := q p M c 1 q] 1 p. The following semens hold. i Equion 3.21 is oscillory if Md>. ii Equion 3.21 is non-oscillory if Md<. Proof See 12], Theorem 9. Theorem 3.3 Le c 1 be posiive α-periodic coninuous funcion, le d 1 be n α-periodic coninuous funcion, nd le c 2, d 2 : R R be rbirry coninuous funcions for which men vlues Mc 2, M c 2, Md 2, M d 2 exis. Le us consider he equion c 1 c p q 2 x ] log 2 d 1 d 2 x log 2 =0 3.22 p nd denoe Ɣ := 2q p 1 Md 2 Mc 1 ] p q 2q 2 pmc 2 Mc 1 ] 1. Le c 1 c 2 log 2 >0,, qp Md 1 Mc 1 ] p q =1. The following semens hold. i Equion 3.22 is oscillory if Ɣ >1. ii Equion 3.22 is non-oscillory if Ɣ <1. Proof See 18], Theorem 5.1, where i suffices o pu n =1. Combining Theorems 3.2 nd 3.3, we obin he following one. Theorem 3.4 The following semens hold. i If Mr 0 ] p q Ms 0 >q p, hen 2.2 is oscillory. ii If Mr 0 ] p q Ms 0 <q p, hen 2.2 is non-oscillory. iii If Mr 0 ] p q Ms 0 =q p nd Ms 1 Mr 0 ] p q p q Mr 1 Mr p1 0 ] 1 q 1 p > 2, hen 2.2 is oscillory.
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 14 of 17 iv If Mr 0 ] p q Ms 0 =q p nd Ms 1 Mr 0 ] p q p q Mr 1 Mr p1 0 ] 1 q 1 p < 2, hen 2.2 is non-oscillory. Proof The heorem follows immediely from Theorem 3.2 prs i, ii nd Theorem 3.3 prs iii, iv. I suffices o consider he ideniies p 1=p/q, 1 q p/q=1. ApplyingTheorem 3.4, we cn improve Theorem 3.1ndCorollry 3.1 ino he following more convenien forms. We give illusring exmples s well. Theorem 3.5 Equion 2.5 is non-oscillory if nd only if lim α p α 1 p p q α r 0 τdτ s 0 τdτ q p p q p m S i τdτ r 0 τdτ R ] iτdτ q p1 r 0 τdτ q1 p 2. Proof I suffices o consider Theorems 3.1 nd 3.4. Exmple 3 Le, b, c, d >0, 1, 2, 3, b 1 0,p = 3/2. Le us consider he hlf-liner equion 1 ccos1 sin 1 cos 2 sin 3 /log 2 x ] x b d cosb1 sinb 1 log ] 2 x =0. 3.23 3 x Theorem 3.5 gurnees he oscillion of 3.23 ifb 2 > 1/27; nd is non-oscillion if b 2 < 1/27. We pu b 2 = 1/27. Since M cosα sinα = M cosα = M sinα =0, α 0, nd M cosα sinα ] 2 = 1 8, α 0, we obin he oscillion of 3.23 ford 2 > 16/3 nd he non-oscillion in he opposie cse d 2 16/3. Corollry 3.2 Le f, g be periodic nd coninuous funcions such h Mf, Mg 0. Equion 3.20 is oscillory if nd only if Mf Mg>1.
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 15 of 17 Exmple 4 Using Corollry 3.2 nd Theorem 3.5, we cn generlize Exmple 2. For ny 1, 2, b 1, b 2 >0,ndϱ, σ > 1, he liner equions x ] 1 b 1 sin ϱ /log 2 2 b 2 sin σ / log 2 x =0, 4 2 x ] 1 b 1 sin ϱ /log 2 2 b 2 cos σ / log 2 x =0, 4 2 x ] 1 b 1 cos ϱ /log 2 2 b 2 sin σ / log 2 x =0, 4 2 x ] 1 b 1 cos ϱ /log 2 2 b 2 cos σ / log 2 x =0 4 2 re oscillory for 1 2 > 1 nd non-oscillory for 1 2 < 1. In he limiing cse 1 2 =1, one cn esily rewrie he considered equions in he form of 3.20, where Mf =b 1 / 1 nd Mg= 1 b 2. Therefore, in he cse 1 2 = 1, he bove equions re oscillory if nd only if b 1 > 1 1 1 b 2. If we know h n equion is condiionlly oscillory, hen we cn use i s esing equion for mny oher equions. For exmple, using he Surmin comprison heorem see 14], Theorem 1.2.4, we cn proceed for perurbed Euler ype hlf-liner equions s follows. Le us consider nd r p q 0 x ] s0 x p g x=0 3.24 r0 f ] p q x ] s0 x p =0, 3.25 where f, g re rbirry coninuous funcions nd r 0, s 0 re α-periodic coninuous funcions such h r 0, f re posiive nd Mr 0 =1,Ms 0 =q p. Equion 3.24 is non-oscillory if here exis β i -periodic coninuous funcions S i, i {1,...,n},suchh M S i =1, S i >0, R, 3.26 nd lim sup g p log 2 n S i < q1 p 2. 3.27 Equion 3.24 is oscillory if he funcions S i sisfy 3.26nd lim inf g p log 2 n S i > q1 p 2. 3.28
Hsil nd Veselý Advnces in Difference Equions 2015 2015:190 Pge 16 of 17 Indeed, from inequliy 3.27, we obin ε > 0 wih he propery h q 1 p n g< 2 ε S i p log 2 for ll sufficienly lrge.thus,isufficesousetheorem3.5 nd he Surmin comprison heorem. Anlogously, we ge he semen concerning inequliy 3.28. Similrly, 3.25 is non-oscillory if here exis α i -periodic coninuous funcions R i for i {1,...,m} such h m M R i =1, m R i >0, R, 3.29 nd we hve lim sup f log 2 m R i < q2 2p. On he oher hnd, if he funcions R i sisfy 3.29nd lim inf f log 2 m R i > q2 2p, hen 3.25isoscillory. Compeing ineress The uhors declre h hey hve no compeing ineress. Auhors conribuions The uhors declre h he reserch ws relized in collborion wih he sme responsibiliy nd conribuions. All uhors red nd pproved he finl mnuscrip. Acknowledgemens The firs uhor is suppored by Grn P201/10/1032 of he Czech Science Foundion. The second uhor is suppored by he projec Employmen of Bes Young Scieniss for Inernionl Cooperion Empowermen CZ.1.07/2.3.00/30.0037 co-finnced from Europen Socil Fund nd he se budge of he Czech Republic. The uhors would like o hnk he referees for heir commens which improved he presenion of he resuls. Received: 25 Februry 2015 Acceped: 8 June 2015 References 1. Kneser, A: Unersuchungen über die reellen Nullsellen der Inegrle linerer Differenilgleichungen. Mh. Ann. 423, 409-435 1893. doi:10.1007/bf01444165 2. Geszesy, F, Ünl, M: Perurbive oscillion crieri nd Hrdy-ype inequliies. Mh. Nchr. 189, 121-144 1998. doi:10.1002/mn.19981890108 3. Schmid, KM: Oscillion of perurbed Hill equion nd lower specrum of rdilly periodic Schrödinger operors in he plne. Proc. Am. Mh. Soc. 127, 2367-2374 1999. doi:10.1090/s0002-9939-99-05069-8 4. Schmid, KM: Criicl coupling consn nd eigenvlue sympoics of perurbed periodic Surm-Liouville operors. Commun. Mh. Phys. 211, 465-485 2000.doi:10.1007/s002200050822 5. Krüger, H, Teschl, G: Effecive Prüfer ngles nd relive oscillion crieri. J. Differ. Equ. 24512, 3823-3848 2008. doi:10.1016/j.jde.2008.06.004 6. Krüger, H, Teschl, G: Relive oscillion heory for Surm-Liouville operors exended. J. Func. Anl. 2546, 1702-1720 2008. doi:10.1016/j.jf.2007.10.007 7. Krüger, H, Teschl, G: Relive oscillion heory, weighed zeros of he Wronskin, nd he specrl shif funcion. Commun. Mh. Phys. 2872, 613-640 2009.doi:10.1007/s00220-008-0600-8 8. Elber, Á: Oscillion nd nonoscillion heorems for some nonliner ordinry differenil equions. In: Ordinry nd Pril Differenil Equions, Dundee, 1982. Lecure Noes in Mh., vol. 964, pp. 187-212. Springer, Berlin 1982 9. Elber, Á: Asympoic behviour of uonomous hlf-liner differenil sysems on he plne. Sudi Sci. Mh. Hung. 192-4, 447-464 1984
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