CONDITIONAL OSCILLATION OF HALF-LINEAR EQUATIONS

Transcription

1 CONDITIONAL OSCILLATION OF HALF-LINEAR EQUATIONS Per Hasil Habiliaion Thesis

2 c Per Hasil, Masaryk Universiy, 206

3 Preface The naure around us can be described in many ways. One of he mos accurae ways is using equaions. The ype of he invesigaed equaions depends on he analyzed phenomenon iself. There are wo fundamenal ypes, namely, differenial equaions for coninuous ime and difference equaions for discree ime. Of course, during he las decades, he unificaion of boh ypes, he so called ime scale calculus, have been developed. Chaper 2 is devoed o differenial equaions, hen Chaper 3 deals wih difference equaions, and finally he dynamic equaions on ime scales are reaed in Chaper 4. Oscillaion or non-oscillaion is one of he fundamenal opics from he qualiaive heory of differenial and difference equaions. The main idea of his noion is o coun zeros of soluions a infiniy. If here exiss he greaes zero of a soluion, we say ha his soluion is non-oscillaory. On he oher hand, if he zeros of a soluion end o infiniy, he soluion is said o be oscillaory. The main goal of his hesis is o sudy he condiional oscillaion of equaions and o find he so-called criical oscillaion consans. Once we prove ha an equaion is condiionally oscillaory and find he criical consan which usually depends on he coefficiens of he sudied equaion, he oscillaion properies of such an equaion are fully resolved wih no more han one excepion he criical case. This fac means ha condiionally oscillaory equaions are ideal esing equaions. In paricular, many equaions which are no condiionally oscillaory can be compared wih hese esing equaions using comparison heorems and, consequenly, i is possible o specify heir oscillaion properies. This work is based on papers 35, 40, 4, 42, 43, 44, 45, 46] published or submied for publicaion during years 204 and 205. Hence, I am obliged o my coauhors, I hank hem for fruiful collaboraion and look forward o solving open problems which appeared during our conjoin work. Also, I would like o sincerely hank Prof. Ondřej Došlý, who lead my firs seps o mahemaical analysis, for his advice and willingness. Las, bu no leas, I hank my wife, whole family and friends for heir suppor. Per Hasil, Mahemaics Subjec Classificaion: 39A0; 39A2; 39A70; 47B25; 47B36; 47B39.

4 Conens Page Inroducion. The essenials of he used echniques Shor hisory overview Organizaion of he hesis Differenial equaions 3 2. Coefficiens wih mean values Resuls and examples An applicaion Periodic coefficiens Preliminaries Auxiliary resuls Main resuls and examples Sums of periodic coefficiens Preliminaries Lemmaa Resuls and examples Modified Euler equaions Equaion for Prüfer angle Prüfer angle of average funcion Oscillaion consan Criical case Preliminaries Resuls and examples Riemann Weber equaions Modified Prüfer angle and average funcion Preliminaries and resuls Corollaries and examples Difference equaions 3. Coefficiens wih mean values Preliminaries iv

5 CONTENTS v 3..2 Resuls Examples Dynamic equaions on ime scales Equaions wih periodic coefficiens Preliminaries Condiional oscillaion Applicaions and concluding remarks A Bibliography 52 B Curriculum viae 58 C Publicaions 60

6 Chaper Inroducion. The essenials of he used echniques The opic of his hesis belongs o he oscillaion heory of half-linear equaions. The main par Chaper 2 deals wih differenial equaions. Therefore, we recall he basics of half-linear differenial equaions a his place and difference equaions and dynamic equaions on ime scales will be inroduced in Chapers 3 and 4, respecively. Noe ha we will use he sandard noaion R + = 0, and R a = a, for arbirary given a R. Our main ineres is o sudy equaions of he form rφx ] + zφx = 0, Φx = x p sgn x, p >,.. wih coninuous coefficiens r > 0 and z. An equaion of his form appeared for he firs ime in 8] and as he basic pioneering papers in he field of half-linear differenial equaions we menion 24, 60]. During he las decades, hese equaions have been widely sudied in he lieraure. A deailed descripion and a comprehensive lieraure overview concerning he opic can be found in 2] see also 2, Chaper 3]. The name half-linear equaions was inroduced in 7]. This erm is moivaed by he fac ha he soluion space of hese equaions is homogeneous likewise in he linear case, bu i is no addiive. There are several differences beween linear equaions and half-linear equaions. Especially, some ools widely used in he heory of linear equaions are no available for half-linear equaions e.g., see 25] for he Wronskian ideniy and 23] for he Fredholm alernaive. In fac, hese differences are caused, more or less, by he lack of he addiiviy. On he oher hand, many resuls from he heory of linear equaions are exendable o heir half-linear counerpars. Neverheless, according o our bes knowledge, he resuls presened in his hesis are new even for he linear case i.e., for p = 2. Since he main ools in his hesis are based on he Riccai echnique and he Prüfer angle more precisely on heir generalizaions and combinaions, we recall hese noions a his place. To begin wih he Prüfer ransformaion, we have o recall he half-linear rigonomeric funcions as well. For more comprehensive descripion, we refer, e.g., o 2, Secion..2]. The half-linear sine funcion, denoed by sin p, is defined as he odd 2π p -periodic exension of he soluion of he iniial problem Φ x ] + p Φx = 0, x0 = 0, x 0 =,..2

7 .. THE ESSENTIALS OF THE USED TECHNIQUES 2 where π p := 2 p B p, 2Γ = q pγ p Γ p + q In he definiion of π p, we use he Euler bea and gamma funcions Bx, y = τ x τ y dτ, x, y > 0, Γx = q = 2π p sin π...3 p τ x e τ dτ, x > 0, 0 0 and he formula ΓxΓ x = π sinπx, x > 0, ogeher wih he ideniy he conjugacy of he numbers p and q p + q =, i.e., p + q = pq...4 The derivaive of he half-linear sine funcion is called he half-linear cosine funcion and i is denoed by cos p. Noe ha he half-linear sine and cosine funcions saisfy he half-linear Pyhagorean ideniy sin p p + cos p p =, R...5 Especially, he half-linear rigonomeric funcions are bounded. Therefore, here exiss L > 0 such ha cos p y p < L, Φcos p y sin p y < L, sin p y p < L, y R...6 In fac,..6 is valid for any L >. Using he noion of he half-linear rigonomeric funcions, we can inroduce he basic half-linear Prüfer ransformaion x = ρ sin p ϕ,..7 r q x = ρ cos p ϕ,..8 whose modificaions will be used laer, and we apply i o Eq... as follows. We differeniae..7 and combine i wih..8. I leads o r q ρ cos p ϕ = ρ sin p ϕ + ρϕ cos p ϕ...9 Then we apply he funcion Φ o..8, we differeniae he resul, and combine i wih Eq.... This resuls o zρ p Φsin p ϕ = p ϕ ρ p Φsin p ϕ ρ ρ p 2 Φcos p ϕ ]...0

8 .. THE ESSENTIALS OF THE USED TECHNIQUES 3 Finally, we combine he Φcos p ϕρ ] muliple of..9 wih sin p ϕρ p ] muliple of..0. I leads direcly o he sysem of firs order differenial equaions ϕ = z sin p ϕ p p ρ = ρφsin p ϕ cos p ϕ + r q cos p ϕ p, r q z ]. p.. Remark... The funcion ϕ used above is called he Prüfer angle and he firs equaion from.. is referred as he equaion of he Prüfer angle his equaion will be very imporan laer. The connecion of he Prüfer angle o he oscillaion heory is obvious direcly from..7, i.e., if he Prüfer angle is bounded hen here exiss he greaes zero poin of he soluion x. This is equivalen o he definiion of oscillaion of Eq... given more correcly below. Now, we urn our aenion o he Riccai echnique. We derive he Riccai equaion associaed o Eq... and we show heir muual connecion. To obain he Riccai equaion, we use he ransformaion w = r Φx Φx,..2 where x is a soluion of Eq... which is non-zero on he inerval under consideraion. We simply compue he derivaive of w and use Eq... and..4 as follows w = rφx ] Φx p rφx x p 2 x Φ 2 x = z p r x p x p = z p r q w q. We obained he half-linear Riccai differenial equaion w + z + p r q w q = The connecion of Eq...3 and Eq... is embodied in he below given halflinear Reid roundabou heorem... Neverheless, o formulae i properly, we have o briefly menion he noion of disconjugacy and he energy funcional of Eq.... Definiion... Eq... is said o be disconjugae on he closed inerval a, b] if he soluion x given by he iniial condiion xa = 0, raφx a = has no zero in a, b] by a zero of a soluion x we mean such a 0 ha x 0 = 0. In he opposie case Eq... is said o be conjugae on a, b]. We recall ha he Sobolev space W,p 0 a, b conains absoluely coninuous funcions f such ha f L p a, b = y : a, b R; b a y p + y p d p <, fa = 0 = fb.

9 .. THE ESSENTIALS OF THE USED TECHNIQUES 4 The energy funcional of Eq... is Fy; a, b = b a r y p z y p d, y W,p 0 a, b. Now, we can formulae and prove he half-linear Reid roundabou heorem. Theorem.. Half-linear Reid roundabou heorem. The following saemens are equivalen. i Eq... is disconjugae on he inerval a, b]. ii There exiss a soluion of Eq... having no zero in a, b]. iii There exiss a soluion w of he generalized Riccai equaion..3 which is defined on he whole inerval a, b]. iv The energy funcional Fy; a, b is posiive for every 0 y W,p 0 a, b. Proof. To prove ha all he saemens in he heorem are equivalen, we prove he validiy of he implicaions i ii iii iv i. The firs implicaions i ii is a consequence of he coninuous dependence of soluions of Eq... on iniial condiions. More precisely, we suppose ha Eq... is disconjugae and we consider a soluion ˆx of he iniial value problem given by Eq... and he condiions ˆxa = ε, raφˆx a = wih sufficienly small ε > 0. Then he soluion ˆx is posiive on a, b] The second implicaion ii iii comes direcly from he Riccai subsiuion w = r Φx Φx, i.e., whenever here exiss a soluion of Eq... wih no zero, we obain a soluion w of Eq...3. The hird implicaion iii iv can be proved by a direc compuaion. We suppose ha here exiss a soluion w of Eq...3 which is defined for all a, b]. Then we have for x W,p 0 a, b Fx;a, b = = b = p a b a b a r x p z x p d w x p + pr q p rq x p r q x Φxw + ] q Φxw q d r q p rq x p r q x Φxw + ] q Φxw q d 0.

10 .. THE ESSENTIALS OF THE USED TECHNIQUES 5 In he above compuaion, we used he definiion of w and he Young inequaliy rewrien o x = xφ w r obain A p p AB + Bq q 0 wih A = r q x and B = Φxw where he equaliy holds for ΦA = B. Of course, he ideniy Fx; a, b = 0 holds if and only if Φr q x = Φxw which can be. Immediaely, ogeher wih he fac ha xa = 0, we x = xa exp a ws Φ ds 0. rs Hence, he energy funcional Fy; a, b is posiive for every 0 y W,p 0 a, b and i is equal o zero only if y 0. To prove he final implicaion iv i, we use a conradicion. We suppose ha Eq... is conjugae and, a he same ime, here exiss 0 y W,p 0 a, b such ha he energy funcional Fy; a, b > 0. The fac ha Eq... is conjugae means ha he soluion x given by he iniial condiions xa = 0, raφx a = has a leas one zero in a, b]. We denoe his zero or one from hese zeros by 0 and we inroduce he funcion { x for a, 0 ], y =. 0 for 0, b] Evidenly, y W,p 0 a, b and using inegraion by pars we obain Fy; a, b = Fy; a, 0 = Fx; a, 0 = rxφx ] 0 x { rφx ] + zφx } d = 0. a 0 Which is a conradicion and he proof is complee. a..4 To inroduce properly he definiion of non-oscillaion of Eq..., we formulae and prove he half-linear Surm comparison heorem. Since is usefulness in he upcoming chaper, we sae he half-linear Surm comparison heorem as well. Theorem..2 Surm separaion heorem. Le < 2 be wo consecuive zeros of a nonrivial soluion x of Eq.... Then any oher soluion of his equaion which is no proporional o x has exacly one zero in he inerval, 2. Proof. Since and 2 are consecuive zeros of a soluion x, we can, wihou loss of generaliy, suppose ha x > 0 for, 2. Hence, we can inroduce a soluion w = r Φx of Eq...3 defined on he inerval Φx, 2. Then we have lim w =, + lim 2 w =.

11 .. THE ESSENTIALS OF THE USED TECHNIQUES 6 Now, we suppose by conradicion ha here exiss a soluion ˆx of Eq... wih no zero beween and 2 which is linearly independen of he soluion x. Hence, ˆx 0 ˆx 2. Since he soluion ˆx is non-zero for all, 2 ], here is a soluion ŵ = r Φˆx of Eq...3 which is finie a Φˆx and 2. Especially, ŵ < and ŵ 2 >. Therefore, w 0 = ŵ 0 for some 0, 2 which conradics he uniqueness of soluions of Eq...3. We noe ha he unique solvabiliy of Eq...3 comes from he fac ha i can be rewrien as w = z p r q w q which is a firs order equaion whose righ-hand side has he Lipschiz propery wih respec o w. Definiion..2. We say ha Eq... is non-oscillaory more precisely, non-oscillaory a infiniy, if here exiss 0 R such ha Eq... is disconjugae on any inerval of he form 0, T ], T > 0. Oherwise, Eq... is said o be oscillaory. Theorem..3 Surm comparison heorem. We consider Eq... ogeher wih he equaion RΦx ] + ZΦx = 0,..5 where R > 0, Z are coninuous funcions. Le us denoe < 2 wo consecuive zeros of a nonrivial soluion x of Eq... and le he coefficiens saisfy R r, Z z,, 2 ]...6 Then any soluion of Eq...5 has a zero in, 2 or i is a muliple of he soluion x. The las possibiliy is excluded if a leas one of he inequaliies in..6 is sric on a se of posiive measure. Proof. Le us consider a nonrivial soluion x of Eq... wih wo consecuive zeros < 2. Then, using inegraion by pars analogous o..4, we obain Fx;, 2 = 0. Combined wih..6 i implies F RZ x;, 2 = 2 R x p Z x p d 0. Using Theorem.. equivalency i iv, we know ha he soluion ˆx of Eq...5 given by he iniial condiions ˆx = 0, r Φˆx = has a zero in a, b]. Hence, by Surm separaion heorem..2, any soluion of Eq...5 which is linearly independen on ˆx has a zero in, 2. Nex, we suppose ha he only zero of ˆx in, 2 ] is 2. Hence, here exiss a soluion ŵ = R Φˆx Φˆx,, 2, of he Riccai equaion associaed o Eq...5 and we

12 .. THE ESSENTIALS OF THE USED TECHNIQUES 7 can calculae he energy funcional F RZ x;, 2 as follows F RZ x;, 2 = = 2 2 R x p Z x p d ] R ŵ x p + pr q q x p R q x Φxŵ + Φxŵ q d 0. p q Indeed, we used Young inequaliy and he facs lim ŵ x p = lim xrφˆx Φx + + Φˆx = 0, lim 2 ŵ x p = 0, which are consequences of he exisence of he limis lim + x ˆx = lim + x ˆx = x ˆx, lim 2 x ˆx = x 2 ˆx 2. Alogeher, we obained ha 0 F RZ x;, 2 0, i.e., F RZ x;, 2 = 0. Hence, ŵ = R Φx see also he proof of he implicaion iii iv of Theorem.., Φx i.e., he soluions x and ˆx are proporional which canno happen if a leas one of he inequaliies in..6 is sric on a se of posiive measure. Remark..2. If inequaliies in..6 hold, Eq... is said o be he minoran equaion of Eq...5 and Eq...5 is said o be he majoran equaion of Eq.... Then we can summarize Theorem..3 as follows. If minoran equaion is oscillaory hen he original equaion is oscillaory as well. If majoran equaion is non-oscillaory hen he original equaion is non-oscillaory as well. The main objecive of his hesis is o sudy he so-called condiional oscillaion of Eq.... I means ha here exiss he so-called criical oscillaion consan, which is a value dependen on coefficiens r and c wih he following propery. Any equaion of he form.. whose coefficiens: indicae a value greaer han he criical one is oscillaory; 2 indicae a value less han he criical one is non-oscillaory. We can reformulae he noion of condiional oscillaion as follows. We say ha he equaion rφx ] + γzφx = 0, γ R,..7 is condiionally oscillaory if here exiss a consan Γ R such ha Eq...7 is nonoscillaory for γ < Γ and oscillaory for γ > Γ. This consan Γ is called he criical oscillaion consan and forms a sharp borderline beween oscillaion and non-oscillaion

13 .2. SHORT HISTORY OVERVIEW 8 of Eq...7. This fac is also one of he reasons why idenifying of condiionally oscillaory equaions and heir criical consans is an imporan par in he field of oscillaion heory. Using many comparison heorems e.g., he Surm comparison heorem..3, we can es equaions which are no condiionally oscillaory owards appropriae condiionally oscillaory equaion. This makes condiionally oscillaory equaion ideal esing equaions. Of course, a naural quesion is a behavior of condiionally oscillaory equaions in he criical case γ = Γ. I urns ou ha some equaions e.g., wih consan or periodic coefficiens, see 9] are non-oscillaory in he border case. Neverheless, he oscillaion properies of more complicaed equaions may be generally unsolvable see Remark In his hesis, we will solve criical cases in heorems 2.2., 2.3., and Anoher imporance of half-linear equaions lies in heir connecion wih he parial differenial equaions wih p-laplacian. Eq... can be considered he equaion wih one dimensional p-laplacian and resuls dealing wih Eq... are helpful ools in he sudy of more general parial differenial equaions for more deails and an example of such use, see Shor hisory overview In his secion, we collec he milesones in he heory of he condiional oscillaion wih respec o he opic of his hesis. I appears ha appropriae half-linear equaions for he sudy of he condiional oscillaion are he Euler ype equaions, i.e., he equaions wrien in he form rφ x ] + γs Φx = 0.2. p wih coninuous coefficien s. The condiional oscillaion as well as many oher areas in he oscillaion heory of half-linear equaions originaes from he oscillaion heory of linear differenial equaions. The firs resul abou he condiional oscillaion of he considered differenial equaions was obained by A. Kneser in 52], where he oscillaion consan Γ = /4 was found for he equaion x + γ x = More han one hundred years laer, in 33, 70], he above resul concerning Eq..2.2 was exended for he linear equaions rx ] + γs 2 x = wih posiive α-periodic coefficiens r, s, where he criical consan is Γ = α2 4 α 0 dτ rτ α 0 sτ dτ..2.4

14 .2. SHORT HISTORY OVERVIEW 9 We should also menion, a leas as references, papers 53, 54, 55, 56, 7] conaining more general resuls see also 3, 32]. Noe ha he criical case γ = Γ of Eq..2.3 was solved as non-oscillaory see 7]. In he field of half-linear equaions, he basic criical consan p p Γ =.2.5 p for he equaion Φ x ] + γ p Φx = 0 comes from 26] see also 27]. Then, in 34, 38], he condiional oscillaion was proved for more general equaions of he form rφ x ] + γs p Φx = Especially, he criical consan of Eq..2.6 wih posiive α-periodic funcions r, s was idenified as cf..2.4,.2.5 p α p α αp Γ = r p τ dτ sτ dτ.2.7 p 0 in 34]. Le us urn our aenion o he perurbed Euler ype equaions. The linear case of such equaions wih periodic coefficiens is sudied in 54, 7]. The half-linear case is reaed in 9], where he equaions rφ x ] + γs + µd ] Φx log 2 = p are analyzed for posiive α-periodic coefficiens r, s, and d. There is proved ha, in he criical case γ = Γ see.2.7, Eq..2.8 is oscillaory for p α p α µ > Γ 2 := αp p r p τ dτ dτ dτ 2 p 0 and non-oscillaory for µ < Γ 2. For furher generalizaions, we refer o 6, 7, 22, 39] see also 28]. Anoher direcion of researches, which is relaed o he one presened here, is based on he oscillaion of Euler ype equaions generalizing Eq..2.2 in a differen way. We poin ou a leas papers 4, 5, 73, 74, 79], where he equaions of he following form and generalizaions of his form x + fgx = 0 are considered and oscillaion heorems are proved. 0 0

15 .3. ORGANIZATION OF THE THESIS 0.3 Organizaion of he hesis In his secion, we briefly menion only he main resul of each upcoming secion. As we already menioned above, he main par of his hesis is he analysis of half-linear differenial equaions which is he opic of Chaper 2. In Secion 2., we sudy Euler-ype equaion.2. wih coefficiens r, s having mean values. From echnical reasons, we rewrie Eq..2. ino he form ] r p q Φx s + Φx = 0.3. p Since he coefficien r is considered posiive, i does no mean any loss of generaliy. We prove ha his equaion is condiionally oscillaory and we idenify he borderline of oscillaion and non-oscillaion. Of course, his resul cover all previously known resuls concerning Euler-ype equaions.2.3 and.2.6 whose coefficiens are consan, periodic, asympoically almos periodic ec. This secion is based on paper 35]. The criical case remains unsolved in Secion 2. and for such general equaions i is no possible o solve i in general for deails see Remark Therefore, Secions 2.2 and 2.3 are devoed o his problem. Secion 2.2 based on paper 40] deals wih he sudy of he criical case of Eq..3. wih periodic coefficiens r, s whose periods do no have o coincide e.g., one may be raional and he oher irraional. In Secion 2.3, we solve he criical case for he equaion m i= r + R p ] q i Φ x log 2 + s + n i= S i log 2 Φx p = wih α-periodic coefficiens r, s and general periodic coefficiens R,..., R m, S,..., S n defined on some inerval a,, a R + see 4]. Moreover, we are able o combine resuls of Secion 2.3 o prove ha Eq..3.2 is non-oscillaory if and only if p lim a+α q a+α rτ dτ sτ dτ q p α p a + p a+ a a n S i τ dτ i= a+ a rτ dτ p q p + a+ a m R i τ dτ q p 2, rτ dτ i= q p+ a+ see Theorem Then we urn our aenion o anoher ype of condiionally oscillaory equaions r p Φx ] s + log p Φx = 0,.3.3 a

16 .3. ORGANIZATION OF THE THESIS where r > 0 and s are coninuous. Eq..3.3 is discussed in Secion 2.4 see paper 42]. We prove ha his equaion remains condiionally oscillaory and we idenify he sharp borderline. Remarkable is he fac ha he obained resuls cover any equaion of he form.3.3 wih coefficiens r > 0, s saisfying lim +α r q τ dτ log = 0, lim +α sτ dτ log = 0 for some posiive α. Unforunaely, as well as above, he criical case is complicaed and remains unsolved in general. Hence, we dedicae Secion 2.5 o he criical case of Eq..3.3 wih periodic coefficiens r, s whose period may differ, i.e., wihou any common period see 44]. The las secion in Chaper 2 deals wih he behavior of he perurbed Eq The objecive of Secion 2.6 is o idenify he form of perurbaions which preserve he condiional oscillaion of he original equaion and o solve is behavior see 45]. More precisely, we perurb Eq..3.3 in he criical case and we analyze non-oscillaion of he resuling equaion r + p r 2 q p loglog ] 2 Φx ] + log p s + s 2 loglog ] 2 Φx = 0, where r > 0 and s are α-periodic coninuous funcions and r 2, s 2 are coninuous funcions saisfying r 2 r + 2 > 0, > e, loglog ] lim +α log r 2 u du = 0, lim +α s 2 u du = 0. log Chapers 3 and 4 are devoed o a presenaion of he siuaion in he discree case and he case of dynamic equaions on ime scales. Since he discree and ime scale mehods are more complicaed han in he coninuous case, he resuls are in general no as advanced as in he coninuous case. Neverheless, we succeeded o obain a discree version of he resul from Secion 2.. This resul is conained in Chaper 3 which is based on paper 43]. We find he oscillaion consan of he half-linear difference equaion r k Φ x k ] + s k k + p Φx k+ = 0, where k p sands for he generalized power funcion, he sequence {r k } is a posiive bounded sequence such ha here exiss a posiive mean value of {r q k } and he sequence {s k } has a mean value.

17 .3. ORGANIZATION OF THE THESIS 2 Finally, in Chaper 4 see 46] are reaed dynamic half-linear equaions on ime scales rφy ] + s p σ Φyσ = 0,.3.4 where σ, f, p sand for he forward jump operaor, -derivaive, and generalized power funcion, respecively. The considered coefficiens r, s are rd-coninuous, posiive, and periodic. The obained resul shows ha Eq..3.4 is condiionally oscillaory and reveals is criical consan. In spie of he fac ha his resul is much weaker han he resuls in coninuous and discree case, as far as we know, i is he mos general resul abou condiional oscillaion on ime scales and i could be a sep o his rich field.

18 Chaper 2 Differenial equaions 2. Equaions wih coefficiens having mean values In his secion, we will sudy Eq... in he form ] r p q Φx s + Φx = 0, 2.. p where r : R a R is a coninuous funcion having mean value Mr = and saisfying 0 < r := inf R a r r + := sup R a r < 2..2 and s : R a R is a coninuous funcion having mean value Ms > 0. The Riccai equaion associaed o Eq. 2.. has he form see..3 w + s p + p r w q = Finally, using he subsiuion w = ζ p, we obain he adaped Riccai equaion ζ = p ζ + s + p r ζ q ], 2..4 which will play a crucial role in he proof of he announced resul see he below given Theorem To prove he main resuls, we will apply he Riccai echnique for Eq.... The fundamenal connecion beween he non-oscillaion of Eq... and he solvabiliy of Eq...3 is described by he following heorem. Theorem 2... Eq... is non-oscillaory if and only if here exiss a funcion w which solves Eq...3 on some inerval T,. Proof. The heorem is a consequence of he Reid roundabou heorem... We will also use he Surm comparison heorem in he form given below. Theorem Le z, Z : R a R be coninuous funcions saisfying Z z for all sufficienly large. Le us consider Eq... and he equaion rφx ] + ZΦx =

19 2.. COEFFICIENTS WITH MEAN VALUES 4 i If Eq... is oscillaory, hen Eq is oscillaory as well. ii If Eq is non-oscillaory, hen Eq... is non-oscillaory as well. Proof. The heorem is a weaker reformulaion of Theorem..3 see Remark..2. Now, we recall he concep of mean values which is necessary o find an explici oscillaion consan for general half-linear equaions. Definiion 2... Le a coninuous funcion f : R a R be such ha he limi Mf := lim b+ b fs ds 2..6 is finie and exiss uniformly wih respec o b R a. The number Mf is called he mean value of f. 2.. Resuls and examples To prove he announced resul, we need he following lemmaa. Lemma 2... If here exiss a soluion of Eq on some inerval T,, hen Eq. 2.. is non-oscillaory. Proof. A soluion ζ of Eq on an inerval T, gives he soluion w = ζ p of Eq on he same inerval. Thus, he lemma follows from Theorem 2... Lemma Le Eq. 2.. be non-oscillaory and le here exiss M > 0 such ha c sτ dτ τ p < M, a b < c For any soluion w of Eq on T,, i holds b T rτ wτ q dτ < Proof. The lemma follows, e.g., from 2, Theorem 2.2.3], where i suffices o use Lemma If Eq. 2.. is non-oscillaory, hen here exiss a soluion ζ of Eq on some inerval T, wih he propery ha ζ < A for all T and for some A > 0.

20 2.. COEFFICIENTS WITH MEAN VALUES 5 Proof. Considering Theorem 2.., he non-oscillaion of Eq. 2.. implies ha here exiss a soluion w of Eq on some inerval T, which gives he soluion ζ = w p of Eq on he inerval. We show ha his soluion ζ is bounded above. A firs, we prove he convergence of he inegral T sτ τ p dτ R 2..9 and he inequaliy sup T p sτ τ p dτ < L for some L > Evidenly, i suffices o prove 2..9 and lim sup p sτ τ p dτ <. 2.. Le b > 0 be such ha +b sτ dτ > 0, +b sτ dτ bms < b, T, 2..2 where we use direcly Definiion 2.. he exisence of Ms > 0. The symbol f ] + and f ] will denoe he posiive and negaive par of funcion f, respecively. We choose 0 T. We can express 0 +kb 0 +k b sτ τ p dτ = 0 +kb 0 +k b sτ] + dτ τ p 0 +kb 0 +k b sτ] τ p dτ, k N. For an arbirarily given posiive ineger k, we have I k := 0 +kb 0 +k b sτ τ p dτ 0 + k b] p 0 +kb 0 +k b 0 + kb] p sτ] + dτ 0 +kb 0 +k b sτ] dτ 2..3

21 2.. COEFFICIENTS WITH MEAN VALUES 6 if I k > 0, and 0 +kb 0 +k b sτ τ p dτ 0 + kb] p 0 +kb 0 +k b + sτ] + dτ 0 + k b] p 0 +kb 0 +k b sτ] dτ 2..4 if I k < 0. Using 0 + k b] p lim = k 0 + kb] p and using 2..2, 2..3, 2..4, we obain he exisence of n 0 N such ha i holds 0 +kb sτ 2bMs + ] dτ < τ p 0 + k b], k n 0, k N. p 0 +k b Since 0 T is arbirary, i also holds +b sτ dτ τ p < 2bMs + ] p 2..5 for all sufficienly large. Hence, he inegral sττ p dτ is convergen. Especially, T sup sτ dτ T τ p < K for some K > Moreover, we have see 2..5 lim sup sτ p dτ τ p T = lim sup lim sup p p k= k= +kb +k b sτ τ p 2bMs + ] + k b] p Thus, 2.. is valid, i.e., here exiss L > 0 for which 2..0 is valid. Inegraing Eq. 2..3, we obain dτ <. w = wt T sτ dτ p rτ wτ q dτ, T τ p T

22 2.. COEFFICIENTS WITH MEAN VALUES 7 We know ha rτ wτ q dτ <, i.e. see 2..2, wτ q dτ < T Indeed, considering 2..7 ogeher wih 2..6, one can ge 2..8 from Lemma From 2..9 and 2..7 i follows ha here exiss he limi lim w R. In addiion, he convergence of he inegral in 2..8 gives T lim w q = 0, i.e., lim w = Again, we consider arbirarily given 0 T. We can rewrie Eq ino or see direcly Eq w = w 0 0 sτ dτ p τ p 0 rτ wτ q dτ, T Puing 0, from 2..9, 2..8, 2..9, and 2..20, we obain w = sτ τ p dτ + p Finally, le us denoe w = f + f 2, where f := sτ τ p dτ, f 2 := p We know ha see 2..0 and 2..2 sup T rτ wτ q dτ, T rτ wτ q dτ, T. p w f 2 < L We denoe S := { T : w < 0}. If w is posiive, hen he saemen of he lemma is rue for all A > 0. Therefore, we can assume ha S. Since f 2 is non-increasing and lim f 2 = 0, funcion f 2 is nonnegaive. From i follows Hence, we have sup S p w] + f 2 = sup S p w f 2 < L. sup S p w = sup S p w] + 0 sup S p w] + f 2 < L and, consequenly, we obain ha ζ = p w < L, T. saemen of he lemma is valid for A = L. I means ha he

23 2.. COEFFICIENTS WITH MEAN VALUES 8 Remark 2... Le Eq. 2.. be non-oscillaory. If he considered funcion s is posiive for all a, hen he saemen of Lemma 2..3 is rue for a negaive soluion ζ of Eq See, e.g., 2, Lemma 2.2.5]. Theorem Eq. 2.. is oscillaory for Ms > q p, and non-oscillaory for Ms < q p. Proof. The proof is organized as follows. In he firs par, we derive upper bounds for wo inegrals involving funcion s. Then we prove he oscillaory par and, finally, he non-oscillaory par. A firs, we use he exisence of Ms and he coninuiy of funcion s. There exiss β wih he propery ha b+β sτ dτ < β Ms + ], b a,, β b+β b sτ dτ β + ξ b b+β+ξ b sτ dτ <, b a,, ξ 0, ], and, consequenly, here exiss R > 0 wih he propery ha c+ξ sτ dτ < R, c a, a + β], ξ 0, ] We can rewrie ino he form ξ β + ξ b+β b c sτ dτ Using 2..23, we obain β β + ξ b+β+ξ b+β β β + ξ b+β+ξ b+β sτ dτ < β, b a,, ξ 0, ]. sτ dτ < β + βms + ], b a,, ξ 0, ], i.e., b+β+ξ b+β sτ dτ < 2β Ms + 2], b a,, ξ 0, ]

24 2.. COEFFICIENTS WITH MEAN VALUES 9 Combining and 2..26, we have b+ξ b sτ dτ < S, b a,, ξ 0, ], where S := max {R, 2βMs + 2]}. posiive on R a, i holds Since he funcion y = / is decreasing and 2 sτ τ 3 dτ = sτ dτ sτ dτ for all 2 a and for some 3, 2 ]. Analogously, for any 2 a, here exiss 4, 2 ] such ha 2 sτ τ 4 dτ = sτ dτ. Hence, from i follows b+ξ b sτ τ dτ < S, b a,, ξ 0, ] b Now, we prove he oscillaory par. Le Ms > q p. By conradicion, in his par of he proof, we will suppose ha Eq. 2.. is non-oscillaory. Lemma 2..3 says ha here exiss a soluion ζ of Eq on some inerval T, and ha ζ < A for all T and for a cerain number A > 0. Evidenly, we can assume ha T >. We show ha here exiss K < saisfying ζ > K, T On conrary, le us assume ha lim inf ζ =. Le ζ P for all from some inerval, 2 ], where 2 + j, + j], j N, and le P > 0 be such ha see 2..2 hx := p r x q S p x > 0, x P

25 2.. COEFFICIENTS WITH MEAN VALUES 20 Indeed, lim x ± hx =. We can assume ha h is increasing for x P. Using 2..4, 2..28, 2..30, i holds 2 2 ζ 2 ζ = ζ p ζτ + sτ + p rτ ζτ q τ dτ = dτ τ 2 p P + p r P q 2 dτ sτ τ dτ τ + p P + p r P q + dτ sτ τ dτ τ + +j p P + p r P q +j + dτ sτ τ dτ τ +j 2 +j 2 2 p P + p r P q 2 + dτ sτ τ dτ τ +j > S + S + + S + j S + j j S + j = S. Thus, ζ P ST for all T which proves Indeed, i suffices o consider ζ = P. In addiion, we can assume ha K > A, i.e., K < ζ < K, T Thus see direcly 2..4 and 2..28, we have 2 2 ζ 2 ζ = ζ τ dτ p ζτ + rτ ζτ q ] dτ + τ 2 sτ τ dτ p K 2 + S + p r + K q 2 ] for all 2 > T, where 2 +. The previous inequaliy implies ζ 2 ζ p K + S + p r+ K q, T, 2, + ] Considering Definiion 2.. and Ms > q p, here exis n N and ε > 0 such ha n b+n b sτ dτ q p > ε, b a,, 2..33

26 2.. COEFFICIENTS WITH MEAN VALUES 2 and, a he same ime, such ha ε 4p K q < n For such an ineger n, we define b+n b rτ dτ < 2, b a, ϑ := n +n ζτ dτ, T Since we have n +n ζτ dτ < n +n A dτ = A, T, ϑ < A, T Hence, o prove he firs implicaion in he saemen of he heorem, i suffices o show ha is no rue. From i follows where Especially see 2..35, gives Nex, we consider he funcion ζ + τ ζ C, T, τ 0, n], C = n p K + S + p r + K q]. ϑ ζ C, T g := p ζ + q p + p n +n rτ ζτ q dτ, T. If ζ 0 for some T, hen g > 0. Henceforh in his paragraph, we consider he case when ζ < 0, T. Le us define fx := x + q p p + xq, x 0.

27 2.. COEFFICIENTS WITH MEAN VALUES 22 I can be direcly verified ha funcion f has he global minimum f q q = q q = q q q q p + p + q +q + q +q q q q = q q q = q q q q + q q q ] + q q q+ = q q q + q I means ha fx 0, x 0. Especially, i gives he inequaliy q q q q q q + q q ] = 0. p ζ + q p + p ζ q 0, T Considering and 2..39, we have g 2 := p ζ + q p + p n +n rτ ζ q dτ ε, T Applying and he inequaliies K < ζ < A, T, here exiss T such ha ζ q ζ + τ q < Hence see also 2..34, we ge ε 8p,, τ, + n]. g g 2 p n p n +n +n From and 2..4, we know ha rτ ζ q ζτ q dτ ε rτ 8p dτ < 2 p ε 8p = ε 4, Of course, remains rue for ζ 0 as well. Le us consider for which g ε 2, ζ ϑ < ε, p Noe ha he exisence of such a number follows from I is seen ha and imply p ϑ + q p + p n +n rτ ζτ q dτ 3ε,

28 2.. COEFFICIENTS WITH MEAN VALUES 23 Evidenly, we can consider he soluion ζ in an arbirarily given neighborhood of +. Hence, we can assume ha From and 2..45, we see n +n sτ τ dτ n +n sτ from 2..3 and 2..46, we have Sn Knp r + K q np dτ = n n +n +n < ε 24, < ε 24, < ε sτ τ sτ dτ dτ n < Sn 2 < ε 2 24,, p n +n ζτ τ dτ p n +n p 2 ζτ +n dτ = p n ζτ dτ < +n ζτ τ dτ Knp 2 < ε 24,, and, analogously, from 2..2, 2..3, and i follows p n +n rτ ζτ q τ = p n p 2 dτ p n +n +n +n rτ ζτ q rτ ζτ q τ dτ dτ rτ ζτ q dτ < r+ K q np 2 < ε 24,

29 2.. COEFFICIENTS WITH MEAN VALUES 24 For all, using 2..33, 2..44, 2..48, 2..49, and 2..50, we obain ϑ = n = n +n +n +n > n = n Thus, i holds Since ζ τ dτ = n sτ τ sτ +n dτ q p dτ q p +n p ζτ + sτ + p rτ ζτ q + p n + p n +n +n ζτ τ ζτ τ dτ + q p dτ + q p sτ dτ q p + p ϑ + q p + p n ϑ ϑ = lim ϑ τ dτ ε 8τ ε 8τ dτ =, + p n + p n +n +n +n dτ,. dτ rτ ζτ q τ rτ ζτ q dτ dτ ε 8 rτ ζτ q dτ ε > ε 8 8. we obain ha lim ϑ =. The conradicion wih proves he firs implicaion. In he non-oscillaory par of he proof, we consider Ms < q p. Le n N and ε > 0 saisfy n b+n b sτ dτ q p < ε, b a, Le us consider soluion ζ of Eq given by ζt 0 = for some sufficienly large T 0 a. Since he righ-hand side of Eq is coninuous, he considered soluion ζ can be defined on an inerval T 0, T, where T 0 < T. In addiion, if T <, we can assume ha lim sup ζ = T If T =, hen he considered soluion of Eq saisfies he condiion of Lemma 2... I means ha i suffices o find B, C R for which B ζ C, T 0, T As in he oscillaory par of he proof see 2..29, we can prove ha ζ > K for some K < and for all T 0, T. Indeed, we can analogously show ha he inequaliy

30 2.. COEFFICIENTS WITH MEAN VALUES 25 ζ < P ST0 canno be valid for any T 0, T, where S is aken from and P from We wan o prove ha T =. On conrary, le be valid for some T R. Especially, soluion ζ has o be posiive on some inerval T 2, T 3 ] T 0, T in his case. We denoe p p := = q p p and we compue p + q p + p q = pq p + pq p = q p pq + p] = We know ha ζ is negaive on an inerval T 0, T T 0, T. Le T have he propery ha ζ T = 0. For all, 2 T 0, T ], 2 +, we have see ζ 2 ζ = ζ τ dτ ζτ + r + ζτ q 2 p dτ + sτ τ dτ τ p K + r+ K q + S = p K + r+ K q ] + S. Thus, for general, 2 T 0, T ] saisfying 2 + 2n, we have We can assume ha T 0 is so large ha ζ 2 ζ 2n p ] K + r+ K q ] + S ζτ < 0, τ, + 2n] T 0, T, if ζ see Namely ζt 0 = <, we can define he funcion ϑ := n +n ζτ dτ for all T 0, T 0 + n and for all T 0 + n when ζ n. Especially, le T 0 be so large ha ϑt 0 <. We repea ha we assume he posiiviy of ζ which implies he inequaliy ϑ > for from some inerval. The coninuiy of ϑ gives he exisence of > T 0 such ha ϑ =, ϑ From i follows ha, for any δ > 0, one can choose T 0 such ha ζ 2 ζ < δ, T n T.

31 2.. COEFFICIENTS WITH MEAN VALUES 26 Thus, we can assume ha Consequenly, le ϑ ζτ < δ, τ, + n ] ϑ q ζτ q < ε 8p, τ, + n ] A he same ime, we can assume ha n N was chosen in such a way ha i is valid b+n { } rτ dτ n < min ε 4p,, b a, b Using and 2..60, we have p ϑ + q p + p +n rτ ζτ q dτ p ϑ + q p + p ϑ q] n +n = p rτ ζτ q dτ +n ϑ q + rτ +n ϑ q dτ rτ ϑ q dτ n n n +n p rτ ζτ q dτ +n rτ ϑ q dτ n n +n + p ϑ q rτ ϑ q dτ n +n p rτ ζτ q ϑ q dτ n + p +n rτ dτ n < ε 4 + ε 4 = ε 2. Since see 2..55, we have p ϑ + q p + p ϑ q = p + q p + p q = 0, p ϑ + q p + p n +n rτ ζτ q dτ < ε Le T 0 be so large ha see and also n sτ dτ +n sτ n τ n + n dτ < ε 8 + n, T 0, 2..62

32 2.. COEFFICIENTS WITH MEAN VALUES 27 and see 2..2 ogeher wih and also n p rτ ζτ q dτ p +n rτ ζτ q dτ n τ n + n < Considering 2..5, 2..6, 2..62, and 2..63, we obain ϑ = n +n < p n = + n ζ τ dτ = n +n +n ζτ + n dτ + n p ζτ + sτ + p rτ ζτ q +n p ϑ + q p + p n τ sτ + n dτ + p n +n + n +n rτ ζτ q dτ +n ε 8 + n dτ rτ ζτ q + n dτ + ε 4 + n sτ dτ q p + ε ε < n. This conradicion see means ha is rue for B = K and C = 0. Since canno be valid for any T <, he considered soluion ζ exiss on inerval T 0,. We repea ha he non-oscillaion of Eq. 2.. acually follows from Lemma 2... The following heorem is a version of Theorem 2..3 which is ready for applicaions o he half-linear equaions wrien in he form common in he lieraure. Theorem Le c : R a R be a coninuous funcion, for which mean value M c q exiss and for which i holds 0 < inf R a c sup R a c <, and le d : R a R be a coninuous funcion having mean value Md. Le Γ := q p M a+ p c q] p = q p lim c q τ dτ. Consider he equaion cφx ] + d Φx = p Eq is oscillaory if Md > Γ, and non-oscillaory if Md < Γ. a

33 2.. COEFFICIENTS WITH MEAN VALUES 28 Proof. Le Md > 0. Eq can be rewrien ino he form i.e., c q ] ] p q Φx d + Φx = 0, p ] c q p q Φx ] + Mc q ] p q d Φx = Mc q p Eq has he form of Eq. 2.. for r = c q Mc q, s = Mc q ] p q d, a. Noe ha Mr = and Ms > 0 and ha 2..2 follows from Thus see Theorem 2..3, Eq is oscillaory for Ms = Mc q ] p q Md = Mc q ] p Md > q p, i.e., Md > Γ, and non-oscillaory if he opposie inequaliy Md < Γ holds. I remains o consider he case when Md 0. Of course, here exiss k > 0 such ha 0 < Md + k = Md + k < Γ. We know ha he equaion cφx ] + d + k p Φx = 0 is non-oscillaory. Now, i suffices o use Theorem 2..2, ii. Remark For reader s convenience, we consider Eq insead of Eq. 2.. in Theorem The form of Eq shows how he presened resul improves he known ones. Especially, we ge new resuls in wo imporan cases, when funcion s changes sign and when i is unbounded. For deails, we refer o paper 38]. Remark For Md = Γ, i is no possible do decide wheher Eq is oscillaory or non-oscillaory for general funcions c, d saisfying he condiions from he saemen of Theorem I follows, e.g., from he main resuls of 7, 9]. One of he mos sudied classes of funcions which have mean values is formed by almos periodic funcions. Based on he consrucions from 76], i is conjecured in 38] ha he case Md = Γ is no generally solvable in he sense wheher i is oscillaory or non-oscillaory even for almos periodic coefficiens of Eq I means ha here exis almos periodic funcions c, d such ha Md = Γ and Eq is oscillaory. A he same ime, here exis anoher se of almos periodic funcions c, d saisfying Md = Γ such ha Eq is non-oscillaory. Noe ha he case of periodic funcions c, d wih he same period was proved o be non-oscillaory see 7, 9]. To illusrae Theorem 2..4, we menion a leas wo examples.

34 2.. COEFFICIENTS WITH MEAN VALUES 29 Example 2... For A > /2, B, C > 0, and q = 3, le us consider he equaion ] Φx arcsincos + cosc sinc]2 + Φx = A + cosb sinb Eq has he form of Eq for c = A + cosb sinb, I can be direcly verified ha arcsincos + cosc sinc]2 d = and ha M c q = M cosb sinb + A] 2 = + 8A2 8 Md = M cosc sinc] 2 = 8. Hence, if 2 9 < A 2 hen Eq is oscillaory and if 2 9 > A 2 hen i is non-oscillaory; i.e., Eq is oscillaory for A κ, and non-oscillaory for A /2, κ, where κ = 8 3 /3 3 /8. = Since d is oscillaory, he oher relaed resuls in he lieraure give no conclusion for Eq Example Le K R and k 0 be arbirarily given. We define he funcion d : R R by d := K + k n 3 n, n, n + 2, n N; n d := K + k n n, n + 2, n + 22, n N; n n n d := K, n + 22, n +, n N. n I is seen ha lim sup d = and ha Md = K. Analogously, for L > 0 and l > L, we define c : R R as n, n +, n N; 2n c := L + l n 2n, c := L + l n + n 2n, c := L, n + n, n +, n N. n + 2n, n + n, n N; We have and Mc q = L q. Thus, Eq for he above given funcions c, d is oscillaory if K > q p L, and non-oscillaory if K < q p L. Since d is no bounded, he oher relaed resuls in he lieraure give no conclusion for his equaion.

35 2.. COEFFICIENTS WITH MEAN VALUES 30 Theorem 2..4 gives a new resul for linear equaions as well. Therefore, we menion he following corollary. Corollary 2... Le r, s : R a R be coninuous funcions having mean values Mr, Ms and le 2..2 hold. Then, he equaion x r ] + s 2 x = 0 is oscillaory if 4MrMs >, and non-oscillaory if 4MrMs <. Example Using Corollary 2.., we can decide abou he oscillaion and nonoscillaion of several equaions. Le a, b R and m, n N be relaive prime. Le f, g : R R be arbirary posiive coninuous funcions wih he propery ha The equaions lim f =, lim g =. 2 fx ] + γ sin m + a + sin n + b x = 0, g fx ] + γ cos m + a + sin n + b x = 0, g fx ] + γ sin m + a + cos n + b x = 0, g fx ] + γ cos m + a + cos n + b x = 0 g are oscillaory for 8γ > π and non-oscillaory for 8γ < π. Noe ha his simple crierion does no follow from known ones. In he following corollary, we menion he case of negaive coefficien r. Corollary Le r, s : R a R be coninuous funcions such ha funcion s has mean value Ms and ha here exiss mean value R := M r q. Le be fulfilled. The equaion < inf R a r sup R a r < 0 rφx ] + s p Φx = 0 is oscillaory if Ms < q p R p, and non-oscillaory if Ms > q p R p. Proof. To prove he corollary, i suffices o use Theorem 2..4 and he fac ha he soluion space of half-linear equaions is homogeneous.

36 2.. COEFFICIENTS WITH MEAN VALUES 3 One can prove a number of consequences using differen ypes of comparison heorems. For example, applying Theorem 2..4 and 2, Theorem 2.3.], we immediaely obain he nex new resul. Corollary Le r, s : R a R be coninuous funcions having mean values Mr, Ms. Le 2..2 hold and le a+ p Γ := q p M r] p = q p lim rτ dτ. Le us consider he equaion a r p q Φx ] + yφx = 0, where y : R a R is a coninuous funcion saisfying yτ dτ <. a i If here exiss 0 a for which sτ dτ τ p and if Ms > Γ, hen Eq is oscillaory. ii If here exiss 0 a for which sτ dτ τ p yτ dτ, 0, yτ dτ, 0, and if Ms < Γ, hen Eq is non-oscillaory. In he following example, we demonsrae, ha he resuls are applicable even if Ms does no exiss. Example Le p > and X, Y, Z, Z 2 > 0. Puing s := X + Y 2 n 2 n, 2 n, 2 n +, n N; 4 s := X + Y 2 n n, 2 n + 4, 2n +, n N; 2 s := X Y 2 n 2 n, 2 n + 2 2, 2n + 3, n N; 4 s := X Y 2 n + 2 n, 2 n + 3 ] 4, 2n +, n N; s := X, R 2 \ n N 2 n, 2 n + ],

37 2.. COEFFICIENTS WITH MEAN VALUES 32 we define he coninuous funcion s : R 2 R. Evidenly, mean value Ms does no exis. Since 0 < 2 n+ 2 n sτ X τ p dτ = 2 n + 2 n = Y 2n 6 sτ X dτ < Y ] 2 n τ p 6 2 n 2 n p 2 n + p ] 2 n + p 2 n p < Y 2 n + p 2 n p 6, n N, 2 n p we have Considering 0 < 2 sτ τ p dτ < Y 6 lim n n= 2 p + X dτ < n τ p 2 n + p 2 n p 2 n + p 2 n p 2 n+]p = 0 and 2..69, here exiss a posiive coninuous funcion h : R 2 R wih he propery ha 2 n+ 2 hτ n+ dτ > sτ X τ p dτ τ p, 2 n, 2 n+, n N, 2..7 and ha lim h = 0. Using 2..7, we obain X hτ τ p dτ < sτ τ p dτ < In addiion, MX + h = MX h = X and I is also seen ha 2 for all sufficienly large. The equaions 2 X + hτ τ p dτ, hτ τ p dτ < X hτ τ p dτ > ] sin Z ] + cos Z ] + sin Z 2 ] + cos Z 2 ] p q Φx X h + Φx = 0, p ] sin Z ] + cos Z ] + sin Z 2 ] + cos Z 2 ] p q Φx X + h + Φx = 0 p

38 2.. COEFFICIENTS WITH MEAN VALUES 33 are oscillaory for X > q p π/8 p and non-oscillaory for X < q p π/8 p see Theorem Indeed, M sin Z + cos Z + sin Z 2 + cos Z 2 ] pq q = 8 π. Therefore see Corollary 2..3 ogeher wih 2..70, 2..72, 2..73, 2..74, we know ha he equaion ] sin Z ] + cos Z ] + sin Z 2 ] + cos Z 2 ] p q Φx s + Φx = 0 p is oscillaory if X > q p π/8 p, and non-oscillaory if X < q p π/8 p An applicaion In his paragraph, we use our main resul o derive a heorem relaed o ellipic parial differenial equaions wih p-laplacian and he power ype non-lineariy div Ax u p 2 u + CxΦu = 0, where x = x i n i= R n, A is an ellipic n n marix funcion wih differeniable componens, and C is a Hölder coninuous funcion. As a soluion of Eq in Ω R n, we undersand a differeniable funcion u such ha Ax ux p 2 ux is also differeniable and u saisfies in Ω. The following noaion is used. We consider he usual Euclidean norm he induced marix norm n b = i= b 2 i 2, A A = sup b b =0 b, and λ min x, λ max x sands for he smalles and larges eigenvalue of marix Ax, respecively. From he fac ha Ax is posiive definie, i follows ha Ax = λ max x. Denoe Ωr 0 := {x R n : x r 0 }. We say ha a soluion u of Eq is oscillaory if i has a zero in Ωr for every r r 0. Eq is said o be oscillaory if every soluion of his equaion is oscillaory. Oherwise, Eq is said o be non-oscillaory. Recall ha in general we disinguish wo ypes of oscillaion in he heory of 2..75: he weak oscillaion defined in he previous paragraph and he so called srong nodal oscillaion, which is based on nodal domains i.e., bounded domains such ha he equaion possesses a nonrivial soluion which vanishes on he boundary of his domain. Concerning his concep of oscillaion, Eq is said o be nodally oscillaory if every

39 2.. COEFFICIENTS WITH MEAN VALUES 34 soluion has a nodal domain ouside of any ball in R n, and o be nodally non-oscillaory in he opposie case. I is known ha he nodal oscillaion implies oscillaion. The opposie implicaion has been proved only in he linear case p = 2 see 6] and remains an open quesion in he half-linear mulidimensional case. We also refer o 72] which relaes he weak oscillaion of linear PDE s and hus nodal oscillaion wih he finieness of negaive specrum of he Laplace operaor, which is of ineres in physical applicaions. Since he resuls of his secion are based on he mehod of Riccai ype equaion, in he oscillaion crieria we essenially prove a nonexisence of evenually posiive soluion, i.e., we deal wih weak oscillaion. In 20, 57], here is proved a heorem which allows o deduce he oscillaion of cerain half-linear parial differenial equaions from he oscillaion of ordinary differenial equaions if Ax is eiher he ideniy marix or a scalar muliple of he ideniy marix. This heorem has been laer exended in 30] see also 59] as follows. Theorem Le us define he funcions p λmax x λ maxx ds for p > 2; r := x = λ min x λ max x ds for < p 2, x = c := Cx ds. If he equaion x = rφu + cφu = 0 is oscillaory, hen Eq is oscillaory as well We can easily apply he oscillaion par of Theorem 2..4 and Theorem 2..5 o obain he following resul. Theorem Le r and c be defined by Suppose ha 0 < lim inf and ha M r q and M c p exis. If hen Eq is oscillaory. r lim sup r < M c p > q p M r q ] p, Remark Noe ha in conras o Theorem 2..4, we lack he non-oscillaion par in Theorem 2..5 and consequenly also in Theorem 2..6, because here is a principal problem wih non-oscillaion crieria for parial differenial equaions via he Riccai mehod. A deailed discussion relaed o he relaionship of he Riccai equaion and he non-oscillaion of second order equaions in he mulidimensional case can be found in 5].

40 2.2. PERIODIC COEFFICIENTS Non-oscillaion of equaions wih periodic coefficiens in criical case In he previous secion, we found he criical oscillaion consan for Eq. 2.. wih coefficiens r and s having mean values. Despie ha he criical case canno be resolved in full generaliy, here remains sill a solvable open problem. I is no known wheher Eq. 2.. wih posiive α-periodic coefficien r and β-periodic coefficien s is oscillaory or no in he criical case r and s do no need o have any common period, e.g., α =, β = 2. In his secion, we prove ha Eq. 2.. is non-oscillaory in his case. We poin ou ha coefficien s can change is sign in conras wih he siuaion common in he lieraure and we remark ha, according o our bes knowledge, he resul presened in his secion is new in he half-linear case as well as in he linear one i.e., for p = 2. To prove his resul, we have o use anoher mehod han in Secion Preliminaries In his paragraph, we menion he used form of sudied equaions ogeher wih he corresponding Riccai equaion, and he concep of he modified Prüfer angle. These ools will be applied in and We sudy he equaion ] r p q Φ x s + Φx = 0, 2.2. p where r, s : R a R, R a := a,, a e e denoes he base of he naural logarihm log. Henceforh, le funcion r be bounded and posiive and s be such ha lim sup s <. For furher use, we denoe r + := sup{r; R a }, s + := sup{ s ; R a } Le us recall ha via he Riccai ransformaion w = r p q Φ x x where x is a nonrivial soluion of Eq and funcion w is well defined whenever x 0, we obain he Riccai equaion, w + s p + p r w q = associaed o Eq Using he noion of he half-linear rigonomeric funcions recalled in Secion., we can inroduce he modified half-linear Prüfer ransformaion x = ρ sin p ϕ, x = rρ cos p ϕ