STUDIES OF THE UNIVERSITY OF ŽILINA. Volume 20 Number 1 December Editorial Board

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1 STUDIES OF THE UNIVERSITY OF ŽILINA Volume 20 Number 1 December 2006 Special Issue: Proceedings of Conference on Differenial and Difference Equaions and Applicaions (CDDEA), Rajecké Teplice, Slovakia June 26 30, 2006, Ediors: J. Diblík and M. Růžičková Ediors-in Chief Managing Edior V. Bálin, Žilina P. Grešák, Žilina R. Blaško, Žilina Ediorial Board A. Dekré, B. Bysrica J. Diblík, Brno V. Dolnikov, Yaroslavl F. Fodor, Szeged R. Frič, Košice Á. G. Horváh, Budapes T. Kusano, Fukuoka P. Mihók, Košice M. Růžičková, Žilina S. Palúch, Žilina J. Skokan, Urbana I. P. Savroulakis, Ioannina M. Wozniak, Krakow MATHEMATICAL SERIES

2 Ediorial Board Vojech Bálin (Edior-in Chief), Deparmen of Mahemaics, Faculy of Operaion and Economics of Transpor and Communicaions, Universiy of Žilina, Univerziná 1, Žilina, Slovakia, Anon Dekré, Deparmen of Informaics, Faculy FPB, Universiy of Maej Bell, Tajovského 40, Banská Bysrica, Slovakia, Josef Diblík, Bořeická 9, Brno, Czech republic, Vladimir Dolnikov, Deparmen of Mahemaics, Yaroslavl Sae Universiy, Soveskaya 14, Yaroslavl, Russia, Ferenc Fodor, Geomeriai Tanszék, Bolyai Inéze, Aradi Véranúk ere 1., H-6720 Szeged, Hungary, Roman Frič, MÚ SAV,Grešákova 6, Košice, Slovakia, Pavol Grešák (Edior-in Chief), Deparmen of Mahemaics, Faculy of Operaion and Economics of Transpor and Communicaions, Universiy of Žilina, Univerziná 1, Žilina, Slovakia, Ákos G. Horváh, BME, Műegyeem rkp. 3, H-1111 Budapes, Hungary, Takasi Kusano, Deparmen of Applied Mahemaics, Faculy of Science, Fukuoka Universiy, Fukuoka, , Japan, Peer Mihók, Deparmen of Applied Mahemaics, Faculy of Economics, Technical Universiy Košice, Boženy Němcovej 32, Košice, Slovakia, Sanislav Palúch, Deparmen of Mahemaical Mehods, Faculy of Managemen Science and Informaics, Universiy of Žilina, Univerziná 1, Žilina, Slovakia, Miroslava Růžičková, Deparmen of Mahemaical Analysis and Applied Mahemaics, Faculy of Science, Universiy of Žilina, Hurbanova 15, Žilina, Slovakia, Jozef Skokan, Deparmen of Mahemaics, Universiy of Illinois a Urbana Champaign, 1409 Wes Green Sree, Urbana, IL 61801, USA, jozef@mah.uiuc.edu Ioannis P. Savroulakis, Deparmen of Mahemaics, Universiy of Ioannina, Ioannina, Greece, ipsav@cc.uoi.gr Mariusz Wozniak, Wydzial Maemayki Sosowanej, AGH Krakow, al. Mickiewicza 30, Poland, mwozniak@uci.agh.edu.pl Rudolf Blaško (Managing edior), Deparmen of Mahemaical Mehods, Faculy of Managemen Science and Informaics, Universiy of Žilina, Univerziná 1, Žilina, Slovakia, beerb@frcael.fri.uniza.sk Sudies of he Universiy of Žilina Mahemaical series Faculy of Operaion and Economics of Transpor and Communicaions, Deparmen of Mahemaics, Universiy of Žilina, Univerziná 1, Žilina, Slovakia phone: , fax: , sudies@fpedas.uniza.sk, elecronic ediion: hp://fpedas.uniza.sk or hp://frcael.fri.uniza.sk/sudies Typese using LATEX2e, Prined by: EDIS Žilina Universiy publisher c Universiy of Žilina, Žilina, Slovakia ISSN X

3 Preface This volume conains 8 seleced papers induced by he Conference on Differenial and Difference Equaions and heir Applicaions (CDDEA 2006) held in he represenaive spa own of Rajecké Teplice, Slovak Republic, 26h 30h June This inernaional Conference was he 19h coninuaion of he previous foureen Summer Schools on Differenial Equaions, he firs of which was organized in 1964, and four Inernaional Conferences. The founders of he radiion were universiy professors Pavol Marušiak and Ladislav Berger. The conference was a worhy coninuaion of he radiion and was organized by he Faculy of Science, Universiy of Žilina under he auspices of J. Sloa, mayor of he ciy of Žilina. In he work of he conference paricipaed 76 paricipans from 10 counries. The conference was prepared by Organizing Commiee (B. Bačová, J. Diblík, B. Dorociaková, M. Kúdelčíková, M. Růžičková (chairman), M. Takáč, B. Václavíková (secreary)) and Programme Commiee (A. Boichuk, J. Diblík,Z.Došlá, O. Došlý, J. Džurina, I. Györi, A. Ivanov, J. Jaroš, D. Khusainov, T. Kriszin, T.Kusano,M.Medveď, I. Rachůnková, M. Růžičková, Š. Schwabik, S. Saněk, I. P. Savroulakis, M. Tvrdý). The programme conained 6 plenary lecures, 12 invied lecures, 33 conribued alks an d 13 posers, and covered a broad par of mahemaics conneced wih differenial and difference equaions and heir applicaions and was divided ino five secions (Ordinary differenial equaions, Funcional Differenial Equaions, Difference equaions, Parial differenial equaions and Numerical mehods in differenial and difference equaions). The conference was a successful and fruihful meeing simulaing scienific conacs and collaboraions during nice ime a Rajecké Teplice. Josef Diblík and Miroslava Růžičková Ediors

4 Manuscrip submission Volumes of our journal are prepared using L A TEX2e forma of TEX. Auhors are encouraged o use he following preamble in he L A TEX2e source file \documenclass[woside]{univzil} \usepackage{amsmah} \usepackage{amssymb,amsfons} \usepackage{array} %% for more abulars \usepackage{amscd} %% for commuaive diagrams \usepackage{graphicx} %% for include picures *.eps \usepackage{psfrag} %% for using subsiuion \psfrag Figures in aricles should be prepared eiher wih he TEX package or included as EPS files (recommended). To inser EPS prining files (recommended) use he following commands \begin{figure}[h] \psfrag{c}{new name} \includegraphics[widh=3.5cm]{pic01.eps} \capion{example 2} \label{picure2} \end{figure} In his case add wo following commands o he preamble of he source ex \usepackage{graphicx} and \usepackage{psfrag}. Do no use nonsandard fons. If nonsandard macros or syles are used, corresponding syle files mus be enclosed. See file sample.ex for example. You can find all files (univzil.sy, amsmah.sy, amssymb.sy,..., graphicx.sy, psfrag.sy, psfrag.pro and sample.ex) a hp://fpedas.uniza.sk or hp://frcael.fri.uniza.sk/sudies. To submi a paper, a diskee or CD conaining a L A TEX2e source ex wih corresponding PDF or PS file should be addressed o any member of he ediorial board Summiing by is also accepable. A shor absrac summarizing he aricle, he AMS Mahemaical Subjec Classificaion 2000 ( a lis of key words and phrases, references, and he address of he auhor ( address as well) should be included in he aricle. All papers will be refereed. In special cases we can accep a paper wihou in non-elecronic form. However, in such cases reyping of he aricle could significanly delay is publicaion. The auhor will obain 50 offprins of his aricle. Annual subscripion rae: 50 USD for Libraries or Insiuions and 30 USD for individuals (including posage). Bank: Všeobecnáúverová banka, a.s., Mlynské Nivy 1, Braislava 25. Swif code of bank: SUBA SK BX, bank accoun: , sorcode: Subscripion inquiries should be sen o: Sudies Žilina mahemaical series, Universiy of Žilina, Univerziná 1, Žilina, Slovakia

5 CONTENTS S. Aslega: Small and large ampliude soluions of he second order Neumann boundary value problem... 1 B. Baculíková, D. Lacková: Oscillaion Crieria for Second Order Rearded Differenial Equaions.. 11 O. Došlý: Linearizaion echniques and oscillaion of half-linear differenial equaions S. Dvořáková: On he Asympoics of Superlinear Differenial Equaions wih a Delayed Argumen P. Kundrá: On asympoic properies of a discreizaion of some linear delay differenial equaions Z. Páíková: Hille-Winner ype comparison heorems for half-linear differenial equaions J. Řezníčková: Hille-Nehari nonoscillaion crieria for half-linear differenial equaions 55 I. Yermachenko, F. Sadyrbaev: Mulipliciy resuls for wo-poin nonlinear boundary value problems Sudies of he Universiy of Žilina Mahemaical series Faculy of Operaion and Economics of Transpor and Communicaions, Deparmen of Mahemaics, Universiy of Žilina, Univerziná 1, Žilina, Slovakia phone: , fax: , sudies@fpedas.uniza.sk, elecronic ediion: hp://fpedas.uniza.sk or hp://frcael.fri.uniza.sk/sudies Typese using LATEX2e, Prined by: EDIS Žilina Universiy publisher c Universiy of Žilina, Žilina, Slovakia ISSN X

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7 Sudies of he Universiy of Žilina Mahemaical Series Vol. 20/2006, SMALL AND LARGE AMPLITUDE SOLUTIONS OF THE SECOND ORDER NEUMANN BOUNDARY VALUE PROBLEM S. ATSLEGA Absrac. We prove mulipliciy resuls for he Neumann boundary value problem where he second order differenial equaion is of he form x = f(x). Our esimaes are based on a phase plane analysis and allow f(x) o have muliple simple zeros. We consider equaion Inroducion x = f(x), (1) where f(x) is a coninuously differeniable funcion which has simple zeros, ogeher wih he boundary condiions x (0) = 0, x (1) = 0. (2) Our goal is o obain he mulipliciy resuls for he problem (1), (2). We disinguish beween simple and complicaed cases. We mean by simple cases he cases of he problem o have only small -ampliude soluions in neighborhoods of he cener -ype criical poins. We call he case complicaed if here exis large -ampliude soluions wih rajecories going around more han one cener -ype criical poins. Our resuls can be generalized o he case of f(x) being a funcion wih n simple zeros. 1. Five-zero funcion f(x) 1.1. Simple cases Le us consider equaion (1) ogeher wih he boundary condiions (2). assumpions on a funcion f(x) are: (A1): f C 1 (R); Our 2000 Mahemaics Subjec Classificaion. Primary 34B15. Key words and phrases. Criical poin, phase porrai, Neumann boundary value problem, mulipliciy of soluions, homoclinic soluions. Research suppored by he projec VPD1/ESF/PIAA/04/NP/ /0003/0065.

8 2 S. ATSLEGA (A2): f(x) has simple zeros a p 1 <p 2 <p 3 <p 4 <p 5 ; (A3): f( ) = and respecively f(+ ) =+ as is depiced in figure 1. Figure 1. The funcion f(x). Figure 2. The primiive F (x). Le us consider he primiive F (x) = x f(s) ds, which has exacly 3 local 0 minimums a he poins p 1 <p 3 <p 5 and consequenly 2 local maximums a he poins p 2 <p 4 as is shown in figure 2. The phase porrai of he equivalen sysem x = y, y = f(x) depends on properies of he funcion f(x) andisprimiivef (x). Le us consider he cases: 1. F (p 1 ) <F(p 3 ) <F(p 5 ), 2. F (p 5 ) <F(p 3 ) <F(p 1 ), 3. F (p 3 ) <F(p 1 ) <F(p 5 ), 4. F (p 3 ) <F(p 5 ) <F(p 1 ). The sysem (3) has 3 criical poins of he ype saddle a (p 1 ;0),(p 3 ;0),(p 5 ;0) and 2 criical poins of he ype cener a (p 2 ;0), (p 4 ;0). The following phase porrais describe periodical soluions wih phase rajecories going around of any of he criical poins of he ype cener. There are wo homoclinic soluions (bold lines in figure 3 and figure 4) which go around he poins (p 2 ;0), (p 4 ;0). (3) Figure 3. The phase porrai of he case 1. Figure 4. Thephaseporraiofhecase3.

9 SMALL AND LARGE AMPLITUDE SOLUTIONS 3 Theorem 1.1. Le he condiions m 2 1 π2 < f x (p 2 ) < (m 1 +1) 2 π 2, (4) m 2 2π 2 < f x (p 4 ) < (m 2 +1) 2 π 2 (5) hold. Then he Neumann boundary value problem (1), (2) has a leas 2m 1 +2m 2 nonconsan soluions. Before o prove he heorem le us sae he auxiliary resuls. Four cases are similar. So we consider only he case 1. Lemma 1.2. There exiss he homoclinic soluion emanaing from he poin (p 3 ;0) and going around he poin (p 2 ;0). Proof. Consider he primiive F 3 (x) = x p 3 f(s) ds. Le r be he firs zero of F 3 (x) ohelefofp 3. Consider he rajecory defined by he equaion x 2 () =2F 3 (x()) 2F 3 (p 3 ) (6) and passing hrough he poin (r;0). LeT be he ime needed for he poin (r;0) o move o a posiion (p 3 ; 0) along he rajecory. This ime is given by he formula p3 ds T = (7) r 2F3 (s) 2F 3 (p 3 ) One obains aking ino accoun ha F (s) =f(s), f(p 3 ) = 0 he following relaion F 3 (s) F 3 (p 3 )= 1 2 F 3 (p 3 )(s p 3 ) 2 + ε(s p 3 )= where ε(ξ) 0asξ 0. Then = 1 2 f (p 3 )(s p 3 ) 2 +(s p 3 ) 2 ε(s p 3 ), (8) p3 r ds f (p 3 )(s p 3 ) 2 + ε(ξ) = 1 p3 d(s p 3 ) f (p 3 ) r (s p3 ) 2 + ε 1 (ξ) = = 1 0 f (p 3 ) r p 3 > dξ ξ2 + ξ 2 ε 1 (ξ) = 1 0 f (p 3 ) r p f (p 3 ) δ dξ = ξ f (p 3 ) δ dξ ξ 1+ε 1 (ξ) > dξ =+, (9) ξ where δ>0 is such ha ε 1 (ξ) < 1 2 for ξ [ δ, 0], ε 1 = ε/f (p 3 ). Any rajecory of equaion (6) is symmeric wih respec o he x-axis. Lemma 1.3. There exiss he homoclinic soluion emanaing from he poin (p 5 ;0) and going around he poin (p 4 ;0).

10 4 S. ATSLEGA The proof is similar o ha of Lemma 1.2. Below he proof of he heorem follows. Proof of Theorem 1.1. Le us prove he exisence of muliple soluions going around he criical poin (p 2 ; 0). Le us consider he Cauchy problem (1), x(0) = x 0, x (0) = 0, (10) where x 0 (r; p 2 ). If x 0 is close o p 2, hen zeros of x() end o zeros of y(), where y() is a soluion of he linear problem y = f x (p 2 )y, y(0) = 1, y (0) = 0 (11) π as x 0 p 2 and T (x 0 )endo.if fx(p 2) m 1 T (x 0 ) < 1 < (m 1 +1)T (x 0 ), hen, when x 0 r, T (x 0 ) +. Hence here exis a leas m 1 soluions of he Neumann problem. There are symmerical soluions for he case x 0 (p 2 ; p 3.) Toally a leas 2m 1 soluions. Then we consider he second criical poin (p 4 ; 0). Similar consideraions yield a leas 2m 2 soluions going around he fourh criical poin (p 4 ; 0), he second cener. Remark. Theorem 1 is valid also for he cases 2, 3, 4. The respecive proofs can be carried ou Complicaed cases Le us consider he cases 1. F (p 1 ) <F(p 5 ) <F(p 3 ), 2. F (p 5 ) <F(p 1 ) <F(p 3 ). These cases are symmerical. There exis oher periodical soluions wih he propery ha he respecive phase rajecories go around of boh criical poins of heype cener. Figure 5. The phase porrai for he case F (p 1 ) <F(p 5 ) <F(p 3 ). Figure 6. The phase porrai for he case F (p 5 ) <F(p 1 ) <F(p 3 ).

11 SMALL AND LARGE AMPLITUDE SOLUTIONS 5 For he case 1. Define x 0 as he firs zero o he lef of p 5 of he funcion F 5 = x p 5 f(s)ds and x 0 ashefirszeroohelefofp 3 of he funcion F 3 = x p 3 f(s)ds. Obviously p 1 <x 0 <x 0 <p 2. For he case 2. Similarly define x 0 = p 1 and x 0 as he firs zero o he lef of p 3 of he funcion F 3 = x p 3 f(s)ds. Obviously p 1 = x 0 <x 0 <p 2. Consider he funcion T (x 0 )= 1 2 x1(x 0) x 0 ds F (s) F (x0 ), (12) which is defined in he inerval (x 0; x 0 ), where x 1 (x 0 ) is he firs zero o he righ of x 0 of he funcion F (s) F (x 0 ). Theorem 1.4. Le T min =min{t (x 0 ): x 0 (x 0,x 0 )}. Suppose ha here exiss a non-negaive ineger k such ha kt min < 1 < (k +1)T min. (13) Then here are a leas 4k soluions of he Neumann boundary value problem, wih rajecories going around he wo criical poins of he ype cener. Proof. Le z : T (z)=min{t (x 0 ), x 0 <x 0 <x 0 }. Consider he Cauchy problem (1), x(0) = x 0, x (0) = 0, x 0 (x 0; z). When x 0 is close o z, hen he half period T (x 0 ) saisfies he condiion (13). On he oher hand, when x 0 ends o x 0, T (x 0) +. Hence a leas k soluions of he problem. Similarly for he case z<x 0 <x 0. Hence addiionally a leas k soluions. Toally 2k soluions which are monoonically increasing since heir rajecories are in he upper half-plane {(x, x ): x > 0}. Since all rajecories are symmeric wih respec o he x-axis here exis also 2k soluions which are monoonically decreasing since heir rajecories (halves of he closed ones) are in he lower half-plane {(x, x ): x < 0}. Toally a leas 4k soluions. Remark 1.5. There migh be also small-ampliude soluions o he boundary value problem which exis in neighborhoods of he criical poins of he ype cener (he condiions for exisence are given in Theorem 1.4). 2. Seven-zero funcion f(x) 2.1. Simple cases Le us consider equaion (1) ogeher wih he boundary condiions (2). assumpions on a funcion f(x) are(a1), (A3) and (B2): f(x) has simple zeros a p 1 <p 2 <p 3 <p 4 <p 5 <p 6 <p 7 ; Our

12 6 S. ATSLEGA as is depiced in figure Figure 7. The funcion f(x). Figure 8. The primiive F (x). The respecive primiive F (x) = x f(s) ds has exacly 4 local minimums a 0 he poins p 1 <p 3 <p 5 <p 7 and consequenly 3 local maximums a he poins p 2 <p 4 <p 6 as is shown in figure 8. Le us consider he cases: 1. F (p 1 ) <F(p 3 ) <F(p 5 ) <F(p 7 ), 2. F (p 7 ) <F(p 5 ) <F(p 3 ) <F(p 1 ), 3. F (p 3 ) <F(p 1 ) <F(p 5 ) <F(p 7 ), 4. F (p 3 ) <F(p 5 ) <F(p 1 ) <F(p 7 ), 5. F (p 3 ) <F(p 5 ) <F(p 7 ) <F(p 1 ), 6. F (p 5 ) <F(p 7 ) <F(p 3 ) <F(p 1 ), 7. F (p 5 ) <F(p 3 ) <F(p 7 ) <F(p 1 ), 8. F (p 5 ) <F(p 3 ) <F(p 1 ) <F(p 7 ). The sysem (3) has 4 criical poins of he ype saddle a (p 1 ;0), (p 3 ;0), (p 5 ;0), (p 7 ; 0) and 3 criical poins of he ype cener a (p 2 ;0), (p 4 ;0), (p 6 ;0). Theorem 2.1. Le he condiions m 2 1 π2 < f x (p 2 ) < (m 1 +1) 2 π 2, (14) m 2 2 π2 < f x (p 4 ) < (m 2 +1) 2 π 2, (15) m 2 3 π2 < f x (p 6 ) < (m 3 +1) 2 π 2 (16) hold. Then he Neumann boundary value problem (1), (2) has a leas 2m 1 +2m 2 + 2m 3 nonconsan soluions. The proof can be carried ou as he proof of he Theorem Complicaed cases Le us consider he cases: 1. F (p 1 ) <F(p 5 ) <F(p 3 ) <F(p 7 ), 2. F (p 1 ) <F(p 5 ) <F(p 7 ) <F(p 3 ), 3. F (p 3 ) <F(p 1 ) <F(p 7 ) <F(p 5 ), 4. F (p 3 ) <F(p 7 ) <F(p 1 ) <F(p 5 ), 5. F (p 3 ) <F(p 7 ) <F(p 5 ) <F(p 1 ), 6. F (p 5 ) <F(p 7 ) <F(p 1 ) <F(p 3 ), 7. F (p 5 ) <F(p 1 ) <F(p 7 ) <F(p 3 ), 8. F (p 5 ) <F(p 1 ) <F(p 3 ) <F(p 7 ), 9. F (p 7 ) <F(p 3 ) <F(p 1 ) <F(p 5 ), 10. F (p 7 ) <F(p 3 ) <F(p 5 ) <F(p 1 ).

13 SMALL AND LARGE AMPLITUDE SOLUTIONS 7 Here exis oher periodical soluions wih he propery ha he respecive phase rajecories go around of wo criical poins of he ype cener. Figure 9. The phase porrai for he case F (p 1 ) <F(p 5 ) <F(p 3 ) <F(p 7 ). Figure 10. The phase porrai for he case F (p 3 ) <F(p 7 ) <F(p 5 ) <F(p 1 ). Figure 11. The phase porrai for he case F (p 5 ) <F(p 1 ) <F(p 7 ) <F(p 3 ). Figure 12. The phase porrai for he case F (p 7 ) <F(p 3 ) <F(p 1 ) <F(p 5 ). For he case 1 and 2. Define x 0 as he firs zero o he lef of p 5 of he funcion F 5 = x p 5 f(s)ds and x 0 ashefirszeroohelefofp 3 of he funcion F 3 = x p 3 f(s)ds. Obviously p 1 <x 0 <x 0 <p 2. For he case 3, 4 and 5. Define x 0 as he firs zero o he lef of p 7 of he funcion F 7 = x p 7 f(s)ds and x 0 ashefirszeroohelefofp 5 of he funcion F 5 = x p 5 f(s)ds. Obviously p 3 <x 0 <x 0 <p 4. For he case 6, 7 and 8. Define x 0 = p 1 and x 0 ashefirszeroohelefofp 3 of he funcion F 3 = x p 3 f(s)ds. Obviously p 1 = x 0 <x 0 <p 2. For he case 9 and 10. Define x 0 = p 3 and x 0 ashefirszeroohelefofp 5 of he funcion F 5 = x p 5 f(s)ds. Obviously p 3 = x 0 <x 0 <p 4. Theorem 1.4 is valid in his case also.

14 8 S. ATSLEGA 2.3. More complicaed cases Le us consider he cases: 1. F (p 1 ) <F(p 7 ) <F(p 3 ) <F(p 5 ), 2. F (p 1 ) <F(p 7 ) <F(p 5 ) <F(p 3 ), 3. F (p 7 ) <F(p 1 ) <F(p 3 ) <F(p 5 ), 4. F (p 7 ) <F(p 1 ) <F(p 5 ) <F(p 3 ). Figure 13. The phase porrai for he case F (p 5 ) <F(p 1 ) < F (p 7 ) <F(p 3 ). For he case 1. Define x 0 as he firs zero o he lef of p 7 of he funcion F 7 = x p 7 f(s)ds and x 0 ashefirszeroohelefofp 3 of he funcion F 3 = x p 3 f(s)ds. Obviously p 1 <x 0 <x 0 <p 2. For he case 2. Define x 0 as he firs zero o he lef of p 7 of he funcion F 7 = x p 7 f(s)ds and x 0 ashefirszeroohelefofp 5 of he funcion F 5 = x p 5 f(s)ds. Obviously p 1 <x 0 <x 0 <p 2. For he case 3. Define x 0 = p 1 and x 0 ashefirszeroohelefofp 3 of he funcion F 3 = x p 3 f(s)ds. Obviously p 1 = x 0 <x 0 <p 2. For he case 4. Define x 0 = p 1 and x 0 ashefirszeroohelefofp 5 of he funcion F 5 = x p 5 f(s)ds. Obviously p 1 = x 0 <x 0 <p 2. Consider he funcion T (x 0 )= 1 2 x1(x 0) x 0 ds F (s) F (x0 ), (17) which is defined in he inerval (x 0 ; x 0 ), where x 1(x 0 ) is he firs zero o he righ of x 0 of he funcion F (s) F (x 0 ).

15 SMALL AND LARGE AMPLITUDE SOLUTIONS 9 Theorem 2.2. Le T min =min{t (x 0 ): x 0 (x 0,x 0 )}. Suppose ha here exiss an ineger k such ha kt min < 1 < (k +1)T min. Then here are a leas 4k soluions of he Neumann boundary value problem, wih rajecories going around he hree criical poins of he ype cener. The proof can be carried ou as he proof of he Theorem 1.4. Remark 2.3. There exis soluions of hree ypes: 1) soluions around criical poin (p 2 ;0), (p 4 ;0), (p 6 ;0); 2) soluion around wo ceners (p 2 ;0),(p 4 ; 0) and, symmerically, (p 4 ;0),(p 6 ;0); 3) soluions around hree ceners. 3. n-zero funcion f(x) 3.1. Simple cases Le us consider equaion (1) ogeher wih he boundary condiions (2). assumpions on a funcion f(x) are(a1), (A3) and (C2): f(x) has simple zeros a p 1 <p 2 < <p n. The sysem (3) has n+1 2 criical poins of he ype saddle a (p 1 ;0), (p 3 ;0),..., (p n ;0) and n 1 2 criical poins of he ype cener a (p 2 ;0), (p 4 ;0),..., (p n 1 ;0). Theorem 3.1. Le he condiions m 2 1 π2 < f x (p 2 ) < (m 1 +1) 2 π 2, (18) m 2 2 π2 < f x (p 4 ) < (m 2 +1) 2 π 2, (19) m 2 n 1 π 2 < f x (p 6 ) < (m n 1 +1) 2 π 2 (20) 2 2 hold. Then he Neumann boundary value problem (1), (2) has a leas 2m 1 +2m mn 1 2 nonconsan soluions. The proof can be carried ou as he proof of he Theorem Complicaed cases One may consider also complicaed cases and prove he exisence of large - ampliude soluions going around 2, 3,..., n 1 2 criical poins of he ype cener. 4. Final remarks There are many papers devoed o mulipliciy resuls for wo-poin boundary value problems for second order nonlinear equaions. We menion [7], [8], [4]. The resuls by A. I. Perov [7, Ch. 15] are obained by comparison of he behavior Our

16 10 S. ATSLEGA of soluions near he rivial soluion (he exisence of which is assumed) and a infiniy. Esimaions of he number of soluions from below can be obained in his way. For discussion abou he mehod by A. I. Perov see [8], [6]. Our approach, alhough for auonomous equaions, was applied for specific equaions which have five or seven or n criical poins and is based on comparison of he behavior of soluions near he criical poins and a boundaries of bounded domains around he criical poins. Besides here are soluions which go around several criical poins, resuling in addiional soluions of he Neumann problem. This is a novel feaure of our work. This approach seems o be applicable for nonauonomous equaions also. Similar mehod was applied boh for auonomous and non-auonomous equaions in he case of he Dirichle boundary condiions in he work [6]. Le us menion also he recen work by Henderson and Thompson [4]. They have proved he mulipliciy resul consrucing pairs of well-ordered (α β) upper and lower funcions β and α. This approach is very resricive when considering rapidly oscillaing soluions. The resricion (wih respec o he Neumann boundary condiions) is formulaed in Erbe [3, Theorem 1]. For discussion one may consul he papers [5], [6]. References [1] Aguirregabiria J. M., Dynamics Solver FreeWare, hp://p.lc.ehu.es/jma/ds/ds.hml. [2] Aslega S., Mulipliciy resuls for he Neumann boundary value problem, Proc.Ins.Mah. of Universiy of Lavia, ser. Mahemaics. Differenial Equaions, 6 (2006), [3] Erbe L., Nonlinear Boundary Value Problems for Second Order Differenial Equaions, Journal of Differenial Equaions, 7 (1970), [4] Henderson J., Thompson H. B., Exisence of Muliple Soluions for Second Order Boundary Value Problems, Journal of Differenial Equaions, 166 (2000), [5] Yermachenko I., Sadyrbaev F., Types of soluions and mulipliciy resuls for wo-poin nonlinear boundary value problems, Proc. of he 4h World Congress of Nonl. Analyss, Orlando, FL, June 30 - July 7, 2004, Nonlinear Analysis 63 (2005), e1725 e1735. [6] Ogorodnikova S., Sadyrbaev F., Muliple Soluions of Nonlinear Boundary Value Problems wih Oscillaory Soluions, Mah. Modelling and Analysis, 11 (2006), N 4, [7] Krasnoselskii M. A. e al., Planar vecor fields, Fizmagiz, Moscow, 1963 (Russian), Acad. Press, New York, 1966 (English ransl.). [8] Sadyrbaev F., Remarks on Mehods of Esimaing henumber of Soluions of Nonlinear Boundary Value Problems for Ordinary Differenial Equaions, Mahemaical Noes, 57 (1995), N 6, Translaed from Maemaicheskie zameki, 57 (1995), N 6, (Russian). [9] Seydel R., Pracical Bifurcaion and Sabiliy Analysis, Springer Verlag, New York, S. Aslega, Universiy of Daugavpils, Curren address: Parades Sr. 1, LV-5400 Daugavpils, Lavia, address: oglana@vne.lv

17 Sudies of he Universiy of Žilina Mahemaical Series Vol. 20/2006, OSCILLATION CRITERIA FOR SECOND ORDER RETARDED DIFFERENTIAL EQUATIONS BLANKA BACULÍKOVÁ and DÁŠA LACKOVÁ Absrac. In his paper we presen some new crieria for oscillaion of he second order rearded differenial equaion [ r() [x()+p()x (τ())] α 1 [x()+p()x (τ())] ] + where τ() and σ() are delayed argumens. + q() x [σ()] α 1 x [σ()] = 0, Inroducion In his paper we are concerned wih he problem of oscillaory properies of he rearded differenial equaion of he form [ r() [x()+p()x (τ())] α 1 [x()+p()x (τ())] ] + + q() x [σ()] α 1 x [σ()] = 0, (E + ) For convenience and furher references, we inroduce he noaion 1 R() = 0 r 1/α (s) ds, 0. We suppose hroughou he paper ha he following hypoheses hold: (H 1 ) α is a posiive consan; (H 2 ) τ(),σ() C 1 [ 0, ), τ(), σ(), lim τ() =, lim σ() =, σ () > 0; (H 3 ) r() C 1 [ 0, ), r() > 0, lim R() = ; (H 4 ) q(),p() C[ 0, ), q() > 0, 0 <p() < 1. We pu z() =x()+p()x(τ()). By a soluion of Eq. (E + ) we mean a funcion x() C 1 [T x, ), T x 0, which has he propery r() z () α 1 z () C 1 [T x, ) and saisfies Eq. (E + )on[t x, ). We consider only hose soluions x() of(e + ) 2000 Mahemaics Subjec Classificaion. Primary 34C10, 34K11. Key words and phrases. Delayed argumen, oscillaory soluion, inegral averaging mehod. This work was suppored by Slovak Scienific Gran Agency, 1/3013/06.

18 12 BLANKA BACULÍKOVÁ and DÁŠA LACKOVÁ which saisfy sup{ x() : T } > 0 for all T T x. Weassumeha(E + ) possesses such a soluion. Asoluionof(E + ) is called oscillaory if i has arbirarily large zeros on [T x, ) and oherwise i is said o be nonoscillaory. Eq. (E + ) is said o be oscillaory if every is soluion is oscillaory. We need he following lemma. 1. Main resuls Lemma 1.1 (See [6]). If A and B are nonnegaive consans, hen A λ λab λ 1 +(λ 1) B λ 0, λ > 1 and he equaliy holds if and only if A = B. Proof. The case A = 0 holds evidenly, so we can assume ha A 0. Then he lef side of he inequaliy can be wrien in he form 1 λc λ 1 +(λ 1) C λ, (1) where C = B/A. Denoe (1) by f(c). Clearly (1) is saisfied for C =0. Onhe oher hand, if C 0 hen funcion f(c) is decreasing for C (0, 1) and increasing for C (1, ). Furhermore f(1) = 0. Hence he inequaliy holds oo. The proof is complee. The following heorem presens he oscillaory crierion for Eq. (E + ). Theorem 1.2. If [ R α [σ()] q()(1 p[σ()]) α hen Eq. (E + ) is oscillaory. ( ) ] α+1 α σ () d =, (2) α +1 R [σ()] r 1 α [σ()] Proof. Assume o he conrary ha x() is a nonoscillaory soluion of Eq. (E + ). We may assume ha x() > 0. The case of x() < 0 can be proved by he same argumens. Se z() =x()+p()x(τ()). Then z() >x() > 0and [ r() z () α 1 z ()] = q() x [σ()] α 1 x [σ()] < 0. We ge wo possibiliies for z (): (i) z () > 0, (ii) z () < 0for 1 0. The condiion (ii) implies ha for some posiive consan M and 1 0 r() z () α 1 z () M<0. ( ) 1 M Thus z α (). Inegraing he above inequaliy from 1 o, weobain r() ( z() z( 1 ) M 1 α R() R(1 ) ).

19 OSCILLATION CRITERIA FOR SECOND ORDER Leing in he above inequaliy and using (H 3 ), we ge z(). This conradicion proves ha (i) holds. For he case (i), we obain x() =z() p()x(τ()) >z() p()z(τ()) (1 p())z(). Combining he above inequaliy wih Eq. (E + ), we have [ r()(z ()) α] + q()(1 p [σ()]) α z α [σ()] 0 (3) and [ r()(z ()) α] 0. Therefore r()(z ()) α r [σ()] (z [σ()]) α, which implies ha z [σ()] z () ( ) 1 r() α. (4) r [σ()] Define w() =R α [σ()] r()(z ()) α z α > 0 [σ()] for 1. (5) Differeniaing w(), we have w () =αr α 1 σ ()r()[z ()] α [ r()(z [σ()] r 1 α [σ()] z α [σ()] + ()) α] Rα [σ()] z α [σ()] αr α [σ()] r()(z ()) α z [σ()] σ () z α+1. [σ()] (6) Using (3), (4) and (5), we have w () w () ασ () R [σ()] r 1 α [σ()] w() Rα [σ()] q()(1 p[σ()]) α ασ () R [σ()] r 1 α [σ()] Se A = w() andb = λ 1 1 λ,whereλ = α+1 α (7), we obain w () α+1 ασ () R [σ()] r 1 α [σ()] Rα+1 [σ()] r α ()(z ()) α+1 z α+1, [σ()] [w() w α+1 α () ] R α [σ()] q()(1 p[σ()]) α. (7) > 1. Applying he Lemma 1.1 o ( ) α+1 α σ () α +1 R [σ()] r 1 α [σ()] Rα [σ()] q()(1 p [σ()]) α. Inegraing he above inequaliy from 1 o, wege [ w() w( 1 ) R α [σ(s)] q(s)(1 p [σ(s)]) α 1 ( ) ] α+1 α σ (s) ds. (8) α +1 R [σ(s)] r 1 α [σ(s)]

20 14 BLANKA BACULÍKOVÁ and DÁŠA LACKOVÁ Leing in (8), we ge w() on he view of (2). This complees he proof of Theorem 1.2. Now we provide easily verifiable oscillaory crieria for Eq. (E + ). Corollary 1.3. If R α+1 [σ()] r 1 α [σ()] q()(1 p[σ()]) α lim inf σ () hen Eq. (E + ) is oscillaory. > ( ) α+1 α, (9) α +1 Proof. Le (9) holds. Then here exiss ε>0such ha for all large, say 1 R α+1 [σ()] r 1 α [σ()] q()(1 p[σ()]) α ( ) α+1 α σ + ε, () α +1 which follows ha ( ) α+1 R α [σ()] q()(1 p[σ()]) α α σ () α +1 R [σ()] r 1 α [σ()] ε σ () R [σ()] r 1 α [σ()]. Inegraing he above inequaliy from 1 o, weobain [ ( ) ] α+1 R α [σ(s)] q(s)(1 p[σ(s)]) α α σ (s) ds 1 α +1 R [σ(s)] r 1 α [σ(s)] ε [ ln R [σ()] ln R [σ( 1 )] ] as. Now he asserion of Corollary 1.3 follows from Theorem 1.2. Corollary 1.4. If [ [σ()] α q()(1 p [σ()]) α hen he equaion [ [x()+p()x (τ())] α 1 [x()+p()x (τ())] ] + ( ) ] α+1 α σ () d =, (10) α +1 σ() is oscillaory. + q() x [σ()] α 1 x [σ()] = 0 (11) Proof. I is easy o see ha he condiion (2) reduces o (10) for r() 1. Corollary 1.5. If [ σ ] () R [σ()] q()(1 p [σ()]) d =, (12) 4R [σ()] r [σ()] hen he equaion [r()[x()+p()x ] (τ())] + q()x [σ()] = 0 (13) is oscillaory.

21 OSCILLATION CRITERIA FOR SECOND ORDER Proof. I is easy o see ha (2) reduces o (12) for α =1. Remark 1.6. Le us noe σ() =, r() =1, p() =0, α =1. condiion (2) of Theorem 1.2 reduces o ( q() 1 ) d =, 4 Then he which is he well know Kiguradze and Chanuria oscillaion crierion [1] for he corresponding second order differenial equaion x + q()x =0. Remark 1.7. Theorem 1.2 exends resuls presened in [4] and [8], where he differenial equaions of he form [ r() x() α 1 x () ] + q() x [σ()] α 1 x [σ()] = 0 are sudied. Remark 1.8. Theorem 1.2 generalizes Theorem in [5] and resuls presened in [2], [3] and [7]. Now we will use so-called he inegral averaging echnique. Le us consider a funcion H(, s) saisfying he following properies: (i) H(, s) > 0 for >s 0, (ii) H(, ) =0. Denoe for >s h(, s) = H(,s) s, Q(, s) = ασ (s) H(, s) h(, s). H(, s) R[σ(s)]r 1 α [σ(s)] Theorem 1.9. Le α 1. Assume ha for some k (0, 1) lim sup 1 H(, 1 ) 1 [ H(, s)r α [σ(s)] q(s)(1 p[σ(s)]) α R [σ(s)] r 1 α [σ(s)] 4αkσ (s) ] Q 2 (, s) ds =. (14) Then Eq. (E + ) is oscillaory. Proof. Assume o he conrary ha x() is a nonoscillaory soluion of Eq. (E + ). Wihou loss of generaliy we may assume ha x() > 0. Proceeding similarly as in he proof of Theorem 1.2 we have z() > 0, z () > 0 and using he fac ha [ r()(z ()) α] 1 α is nonincreasing, we see ha for any k 1 (0, 1) and for all large ( 1 ) σ() σ() z [σ()] z 1 ( ) (s)ds = r r 1 α (s)z (s) ds α (s) r 1 α [σ()] z [σ()] ( R [σ()] R( 1 ) ) >k 1 R [σ()] r 1 α [σ()] z [σ()]. (15)

22 16 BLANKA BACULÍKOVÁ and DÁŠA LACKOVÁ Taking ino accoun (15) and he monooniciy of r()(z ()) α, we conclude ha z [σ()] z [σ()] = 1 r [σ()] r [σ()] (z [σ()]) α ( ) α 1 z [σ()] (z [σ()]) α z [σ()] r()(z ()) α (z [σ()]) α krα 1 [σ()], (16) r 1 α [σ()] where k = k1 α 1 (0, 1). Using he funcion w() definedin(5),w () in (6) and he inequaliy (16) we obain w ασ () () R [σ()] r 1 α [σ()] w() Rα [σ()] q()(1 p [σ()]) α αr α [σ()] σ () r()(z ()) α (z [σ()]) α z [σ()] z [σ()] ασ () R [σ()] r 1 α [σ()] w() Rα [σ()] q()(1 p [σ()]) α αkσ () R [σ()] r 1 α [σ()] w2 (). Muliplying his inequaliy wih H(, s) > 0 and following inegraing from 1 o we have H(, s)r α [σ(s)] q(s)(1 p [σ(s)]) α ds 1 ασ (s) H(, s) 1 R [σ(s)] r 1 α [σ(s)] w(s)ds αkσ (s) H(, s) 1 R [σ(s)] r 1 α [σ(s)] w2 (s)ds H(, s)w (s)ds. 1 Now inegraing (per pares) from 1 o and using definiion of he funcions h(, s) andq(, s) weareledo H(, s)r α [σ(s)] q(s)(1 p [σ(s)]) α ds = 1 αkσ (s) = H(, 1 )w( 1 ) H(, s) 1 R [σ(s)] r 1 α [σ(s)] w2 (s)ds + [ H(, ασ ] (s) + H(, s) s) h(, s) w(s)ds 1 R [σ(s)] r 1 α [σ(s)] H(, 1 )w( 1 ) αkσ (s) H(, s) 1 R [σ(s)] r 1 α [σ(s)] w2 (s)ds + H(, s)q(, s)w(s)ds. 1

23 Consequenly OSCILLATION CRITERIA FOR SECOND ORDER H(, s)r α [σ(s)] q(s)(1 p [σ(s)]) α ds 1 1 R [σ(s)] r α [σ(s)] H(, 1 )w( 1 )+ 1 4αkσ Q 2 (, s)ds (s) 2 αkσ H(, s) (s) 1 R [σ(s)] r 1 α [σ(s)] w(s) 1 R [σ(s)] r 1 α [σ(s)] 2 αkσ Q(, s) ds. (s) Therefore 1 H(, 1 ) 1 [ H(, s)r α [σ(s)] q(s)(1 p [σ(s)]) α ] R [σ(s)] r 1 α [σ(s)] 4αkσ Q 2 (, s) ds w( 1 ). (s) Leing we ge he conradicion wih (14). This complees he proof of Theorem 1.9. Le us have H(, s) defined by H(, s) =( s) n, n is a posiive ineger. Then Theorem 1.9 provides he following crierion: Theorem Le α 1. Assume ha for some k (0, 1) lim sup 1 ( 0 ) n 0 [ ( s) n R α [σ(s)] q(s)(1 p[σ(s)]) α ] R [σ(s)] r 1 α [σ(s)] 4αkσ Q 2 (, s) (s) ds =, (17) { } where Q(, s) =( s) n 2 ασ (s) R [ σ(s) ] r 1 α [σ(s)] n s. Then Eq. (E + ) is oscillaory. Example We consider differenial equaion ( ( )) α 1 ( ( )) x()+px x()+px + a 2 2 α+1 x(β) α 1 x(β) =0, wih >0, r() =1,τ() = 2, p() =p, 0<p<1, q() = a 0 <β<1. If ( ) α+1 α β α a(1 p) α >, α +1 α+1, a>0, σ() =β,

24 18 BLANKA BACULÍKOVÁ and DÁŠA LACKOVÁ hen by Theorem 1.2 all soluions of his equaion oscillae. References [1] Kiguradze I. T., Chanuria T. A., Asympoic properies of soluions of nonauonomous ordinary differenial equaions, Nauka, Moscow, [2] Džurina J., Oscillaion heorems for neural differenial equaions of higher order, Czechoslovak Ma. Journal 54 (129), No. 1, (2004), [3] Džurina J., Mihalíková B., A noe on unsable neural differenial equaions of he second order, Fasciculi Mahemaici 29 (1999), [4] Džurina J., Savroulakis I. P., Oscillaion crieria for second-order delay differenial equaions, Applied Mah. and Compuaion 140 (2003), [5] Erbe L. H., Kong Q. and Zhang B. G., Oscillaion Theory for Funcional Differenial Equaions, Marcel Dekker, Inc, [6] Hardy G. H., Lilewood J. E. and Polya G., Inequaliies, second ed., Cambridge Universiy Press, Cambridge, [7] Lacková D.,The asympoic properies of he soluions of he n-h order funcional neural differenial equaions, Applied Mah. and Compuaion 146 (2003), [8] Sun Y. G., Meng F. W., Noe on he paper of Džurina and Savroulakis, Applied Mah. and Compuaion 174 (2006), Blanka Baculíková, Deparmen of Mahemaics, Faculy of Elecrical Engineering and Informaics, Technical Universiy of Košice, Curren address: Němcovej 32, Košice, Slovakia, address: blanka.baculikova@uke.sk Dáša Lacková, Deparmen of Applied Mahemaics and Business Informaics, Faculy of Economics, Technical Universiy of Košice, Curren address: Němcovej 32, Košice, Slovakia, address: dasa.lackova@uke.sk

25 Sudies of he Universiy of Žilina Mahemaical Series Vol. 20/2006, LINEARIZATION TECHNIQUES AND OSCILLATION OF HALF-LINEAR DIFFERENTIAL EQUATIONS ONDŘEJ DOŠLÝ Absrac. Oscillaory properies of he half-linear second order differenial equaion (r()φ(x )) + c()φ(x) =0, Φ(x) := x p 2 x, p > 1, (1) are invesigaed using linearizaion echniques, where oscillaory properies of (1) are compared wih hose of a cerain associaed linear equaion. 1. Inroducion Our concern in his conribuion is he second orderhalf-linear differenial equaion (r()φ(x )) + c()φ(x) =0, Φ(x) := x p 2 x, p > 1, (1) where r, c are coninuous funcions and r() > 0. I is well known ha he oscillaion heory of (1) is very similar o ha of he Surm-Liouville linear equaion (r()x ) + c()x =0 (2) which is he special case p = 2 in (1). In paricular, he linear Surmian separaion and comparison heorems exend verbaim o (1) and, herefore, his equaion can be classified as oscillaory or nonoscillaory. Recall ha (1) is said o be nonoscillaory if here exiss T R such ha any nonrivial soluion of (1) has a mos one zero in [T, ), and i is said o be oscillaory in he opposie case. On he oher hand, some mehods of he linear oscillaion heory have no halflinear analogues. Le us menion a leas (non)oscillaion crieria based on he ransformaion heory of (2) or on a suiable applicaion of he elemenary formula (α ± β) 2 = α 2 ± 2αβ + β 2 when dealing wih he Riccai equaion associaed wih (2), see [22]. We will reurn o hese discrepancies beween linear and half-linear oscillaion heories in he nex secion in more deails. The linear oscillaion heory has a long hisory which goes back o he famous Surm s paper from 1836 [20], while he qualiaive heory of (1) is much younger and Elber [12] and Mirzov [18] wih heir papers from he sevenies of he las cenury are usually regarded as pioneers of his heory. From his poin of view, i may be expeced ha a suiable comparison of equaions of he form (1) and (2) can bring new resuls and mehods in he halflinear oscillaion heory. The idea o use he deeply developed linear oscillaion

26 20 ONDŘEJ DOŠLÝ heory when invesigaing oscillaory properies of various nonlinear differenial equaions has been used in numerous papers, le us menion a leas [2, 3, 17] and he references given herein. These papers deal mosly wih he equaion of he form y + c()f(y) = 0 under various resricions on he nonlineariy f. However, mehods used here are oo general and when applied o such a suble nonlineariy as involved in (1), hey give no subsanially new resuls (comparing wih a direc applicaion of half-linear oscillaion mehods). In his paper we follow he idea inroduced in he paper of Elber and Schneider [14], where equaion (1) is viewed as a perurbaion of he half-linear Euler equaion and insead of a linearizaion of his equaion, a quadraizaion procedure is applied o he associaed Riccai ype equaion, and i enabled o compare oscillaory properies of (1) wih a cerain associaed linear differenial equaion. The paper is organized as follows. In he nex secion we recall elemens of he half-linear oscillaion heory, we also discuss difference beween linear and halflinear. In Secion 3 we provide a brief survey of linearizaion echniques in he half-linear oscillaion heory. Secion 4 is devoed o new (non)oscillaion crieria for (1) based on he linearizaion echnique and he las secion conains some conjecures which sugges possible new research direcions. 2. Elemens of half-linear oscillaion heory The proofs of he resuls given in his secion, as well as a deeper excursion o he heory of half-linear equaions, can be found in he book [10], we also refer o [3] and [6]. The principal mehods of he half-linear oscillaion heory are based on he following saemen which is usually referred o as he Roundabou Theorem. Proposiion 2.1. The following saemens are equivalen: (i) Equaion (1) is disconjugae on an inerval I =[a, b], i.e., any nonrivial soluion of (1) has a mos one zero in I. (ii) There exiss a soluion of (1) havingnozeroin[a, b]. (iii) The p-degree funcional F(y; a, b) := b a [r() y p c() y p ]d is posiive for every 0 y W 1,p (a, b) saisfying y(a) =0=y(b). (iv) There exiss a soluion w of he generalized Riccai equaion R[w] :=w + c()+(p 1)r 1 q () w q =0, q := p p 1, (3) which is defined on he whole inerval [a, b] (soluions of (3) are relaed o soluions of (1) by he subsiuion w = rφ(x )/Φ(x)).

27 LINEARIZATION TECHNIQUES 21 The previous saemen shows ha he Surmian separaion and comparison heorems exend lierally o (1). The separaion heorem is hidden in he equivalence (i) (ii), while he comparison heorem is conained in he equivalence (i) (iii). In fac, he comparison heorem implies (similar o he linear case) ha disconjugacy of (1) is equivalen o solvabiliy of he Riccai inequaliy R[w]() 0. The principal role in he proof of he Roundabou Theorem is played by he half-linear version of Picone s ideniy [16] which (in he modified form suiable for our research) reads for any y C 1 [a, b] as follows b F(y; a, b) =w() y p b a + p r 1 q ()P (r 1 q ()y (), Φ(y())w()) d, where w is a soluion of (3) on [a, b], 1 p + 1 q =1,and a P (α, β) := α p β q αβ + p q. (4) As we have seen in he Roundabou Theorem, he Riccai echnique and he variaional principle are sandard ools of he half-linear oscillaion heory. However, we have no half-linear analogue of he following linear ideniy (suppressing he argumen ) h[(rx ) + cx] =(rh 2 y ) + h[(rh ) + ch]y. (5) This ideniy means, in case h() 0,hax is a soluion of (2) if and only if y is a soluion of he equaion wih (R()y ) + C()y =0 (6) R() =r()h 2 (), C() =h()[(r()h ()) + c()h()]. In paricular, if h = x x2 2, where x 1,x 2 are soluions of (2) saisfying r(x 1 x 2 x 1 x 2 )=±1, he resuling equaion (6) is ( ) 1 q() y + q()y =0, q = 1 rh 2, (7) which has he fundamenal sysem of soluions ) ) y 1 () =sin( q(s) ds, y 2 () =cos( q(s) ds and his means ha (7) (and hence also (2)) is oscillaory if and only if d r()[x 2 =. 1 ()+x2 2 ()]

28 22 ONDŘEJ DOŠLÝ Noe ha he possibiliy o ransform (2) ino an equaion whose soluion can be expressed via he elemenary sine and cosine funcions is he basis of he deeply developed ransformaion heory of linear differenial equaions (see [4, 19]) which is compleely missing in he half-linear case. 3. Linearizaion echniques As we have already menioned in he previous secions, linearizaion echniques applied direcly o he nonlinear funcion Φ in (1) do no give saisfacory resuls. I urns ou ha more efficien is he quadraizaion of some nonlinear erms appearing in Riccai equaion and Picone s ideniy associaed wih (1). In he nex par of his secion we presen he main resuls of he paper [14] which serves as a moivaion for he new (non)oscillaion crieria given in Secion 4. In ha paper, he equaion [ (Φ(x )) γp ] ( ) p p 1 + p + c() Φ(x) =0, γ p =, (8) p is viewed as a perurbaion of he half-linear Euler equaion (Φ(x )) + γ p Φ(x) =0, (9) p wih he so-called criical cnsan γ p. The consan γ p is said o be criical since (9), wih a parameer γ insead of γ p, is oscillaory if and only if γ>γ p (see [10, Sec. 1.4]). Recall also ha x() = p 1 p is a soluion of (9). I is supposed ha he perurbaion funcion c is such ha c() p 1 d is convergen (his assumpion is naural since if c() p 1 d = hen (8) is oscillaory, see [13]) and ha c(s)s p 1 ds 0 for large. Under hese assumpions, half-linear equaion (8) is compared wih he linear equaion ( p ) p 1 p 1 c()y = 0 (10) (y ) p 1 and he following saemens are proved. Theorem 3.1. The following saemens hold: (i) Le p 2 and suppose ha equaion (10) is nonoscillaory, hen (8) is also nonoscillaory. (ii) Le p (1, 2] and suppose ha (8) is nonoscillaory, hen (10) is also nonoscillaory. (iii) Suppose ha p>1 is arbirary and (10) is nonoscillaory. Furher assume ha here exiss a consan θ (0, 1) such ha for a soluion y of (10) he funcions ζ = y /y and μ, defined by

29 LINEARIZATION TECHNIQUES 23 saisfy he relaions μ() = ζ 3 (s) ds, (11) μ(s) ds<, ζ(s)μ(s) ds θ μ() for large. (12) 2 Then (8) is also nonoscillaory and i has a soluion x wih he asympoics x() = p 1 p y 2 p (log )[C + o(1))], x () x() = p p ζ(log )+o (ζ(log )) p where C is a nonzero real consan. as, In he nex saemen, equaion (1) is viewed as a perurbaion of anoher equaion of he same form (r()φ(x )) + c()φ(x) =0. (13) This resul is aken from he paper [9] and we presen i here in a modified form similar o saemens (i) and (ii) of he previous heorem. Theorem 3.2. Le h be a posiive soluion of (13) such ha h () 0for large. Denoe R() =r()h 2 () h () p 2, (i) If p 2 and he linear equaion C() =[c() c()]h p (). (R()y ) + p C()y = 0 (14) 2 is nonoscillaory, hen (1) is also nonoscillaory. (ii) If p (1, 2] and (1) is nonoscillaory, hen linear equaion (14) is also nonoscillaory. If we subsiue r() 1, c() = γp,andh() = p 1 p p equaion (14) reduces o he equaion in he previous heorem, (y ) + p 1 2 ( ) p 1 p p 1 c()y = 0 (15) p 1 where he funcion saying by he erm y is p 1 bigger han in (10). Consequenly, by he Surmian comparison heorem, he assumpion of nonoscillaion of (14) in (i) of Theorem 3.2 is more resricive han he assumpion of nonoscillaion of (10) in Theorem 3.1. Similarly, he conclusion abou nonoscillaion of (14) in Theorem 3.2 (ii) is weaker han he conclusion of Theorem 3.1 (ii).

30 24 ONDŘEJ DOŠLÝ The reason is ha equaion (13) is quie general, while he generalized Euler equaion (9) has many nice properies which enable o formulae sronger saemen as given in Theorem 3.1. In he nex secion we presen a sronger version of Theorem 3.2 (conaining addiional resricions on equaion (13)) which in case r() 1, c() = γp,andh() = p 1 p p reduces exacly o (i) and (ii) of Theorem Main resuls Suppose ha (13) is nonoscillaory and h is a posiive coninuously differeniable funcion such ha Denoe lim h() =. (16) and R[w h ]() :=w h ()+ c()+(p 1)r1 q () w h q, w h () = r()φ(h ()) Φ(h()) (17) R() :=r()h 2 () h () p 2, C() :=[c() c()]h p () R[w h ()]. (18) Observe ha in conras o Theorem 3.2, he funcion h need no be a soluion of (13) and in case when i is a soluion of (13), hen we have (by a shor compuaion) R[w h ]() 0 andhence C() = C(). Theorem 4.1. In addiion o he assumpions ha (13) is nonoscillaory and (16) holds, suppose ha he inegral c() d is convergen, c(s) ds 0 for large, and r 1 q () d =. Furher suppose ha lim r()h()φ(h ()) =: L (0, ) (19) exiss finie, he inegral C() d is convergen, and for large. (i) If p 2 and he linear equaion C(s) ds 0 (20) (R()y ) + q 2 C()y = 0 (21) is nonoscillaory, hen (1) is also nonoscillaory. (ii) If p (1, 2] and (1) is nonoscillaory, hen linear equaion (21) is also nonoscillaory.

31 LINEARIZATION TECHNIQUES 25 Proof. (i) Equaion (21) can be wrien in he form ( ) 2 q R()y + C()y =0 and nonoscillaion of his equaion implies he exisence (for large ) ofasoluion v of he associaed Riccai equaion v + C()+ q =0. (22) 2 R() We will show ha he funcion w =w h +h p v saisfies he inequaliy R[w]() 0 for large (wih he Riccai operaor given by (3)) and his means, by he remark given below Proposiion 2.1, ha (1) is nonoscillaory. To his end, denoe G() :=r()h()φ(h ()) and consider he funcion v 2 H(, v) :=(p 1)r 1 q ()h q () { v + G() q qg q 1 ()v G q () }. (23) Then we have H(, 0) = 0 = H v (, 0) and by he second degree Taylor formula (aking ino accoun ha (p 1)(q 1) = 1) [ q ] H(, v) =r 1 q ()h q () 2 G() q 2 v 2 + R 2 (, v), where q(q 2) R 2 (, v) = ξ()+g() q 3 sgn(ξ()+g()) 3! wih ξ() beween 0 and v(). We have he ideniy (suppressing he argumen ) r 1 q h q G q 2 = r 1 q h q rhφ(h ) q 2 = 1 R Therefore, using (19) we obain and hence 1 R() = 1 r()h 2 () h () p 2 h () Lh() 1 lim R(s) ds 1 L lim h (s) h(s) where f() g() for a pair of funcions f,g means 1 ds = lim L log h() =, f() lim g() =1. This implies, by he classical Harman-Winner heorem (see, e.g. [15, Chap XI.]), ha Riccai equaion (22) can be wrien in he inegral form v() = C(s) ds + q 2 v 2 (s) R(s) ds

32 26 ONDŘEJ DOŠLÝ and his means, in view of (20), ha v() 0forlarge and hence ξ() 0in R 2 (, v). Consequenly, sgn R 2 (, v) =sgn(q 2) for large, i.e., v 2 H(, v) q for q 2 and H(, v) q for q 2. (24) 2 R() 2 R() This implies he inequaliy (we suppose ha p 2 in he par (i)) for large v + C()+H(, v) 0. (25) Subsiuing for v = h p (w w h )wehave v 2 H(, v)= { =(p 1)r 1 q h q h p (w w h )+rhφ(h ) q q(rhφ(h )) q 1 v r q h q (h ) q(p 1)} = =(p 1)r 1 q h q+pq w q ph p 1 h w + ph p 1 h (h /h) p 1 (p 1)r(h ) p = =(p 1)r 1 q h p w q ph p 1 h w + r(h ) p and v + C = pφ(h)h (w w h )+ + h p { w R[w h ]+ c +(p 1)r 1 q w q} +(c c)h p + h p R[w h ]= = ph p 1 h w r(h ) p + h p (w + c) which yields v + C()+H(, v) =h p () [ w + c()+(p 1)r 1 q () w q], so from (25) we obain he inequaliy w + c()+(p 1)r 1 q () w q 0 which means ha (1) is nonoscillaory. (ii) Nonoscillaion of (1) implies he exisence of a differeniable funcion w saisfying (3) for large. Puv = h p (w w h ). Then v =pφ(h)h (w w h )+h p { c (p 1)r 1 q w q R[w h ]+ c+(p 1)r 1 q w h q} = and hence = C pr 1 q h p P (Φ 1 (w h ),w), (26) T v(s) T + C(s) ds + p r 1 q (s)h p (s)p (Φ 1 (w h (s)),w(s)) ds =0, (27) T where he funcion P is given by (4) and Φ 1 (s) := s q 2 s is he inverse funcion of Φ.

33 LINEARIZATION TECHNIQUES 27 Since r 1 q () d = and 0 c(s) ds<, w solves also he inegral Riccai equaion (see [6, p. 207] or [10, p. 57]) w() = c(s) ds +(p 1) and herefore w() 0forlarge. Hence r 1 q (s) w(s) q ds, v(t ) v() =h p (w h w) T hp w h ()+h p (w(t ) w h (T )) and leing in (27) we have (wih L given by (19)) L + h p (w(t ) w h (T )) C(s) ds + p T T r 1 q (s)h p (s)p (Φ 1 (w h (s)),w(s)) ds. Since we suppose ha C() d is convergen, his means ha r 1 q ()h p ()P (Φ 1 (w h ()),w()) d <. (28) and i follows by (27) ha here exiss a finie limi lim v() = lim hp ()(w() w h ()) =: L (29) and hence also he limi lim w() w h () = lim h p ()w() h p ()w h () = L + L L. (30) Therefore, leing in (27) and hen replacing T by, wegeheequaion v() L = C(s) ds + p r 1 q (s)h p (s)p (Φ 1 (w h (s)),w(s)) ds. Now we use he inequaliy for he funcion P (α, β) (see [11]) which claims ha β in he region where Φ(α) <T, T being a posiive consan, here exiss a consan K = K(T ) such ha P (α, β) K α p 2 (β Φ(α)) 2 (31) and hence for large (suppressing his argumen) Kr 1 q h p w q 2 h (w w h ) 2 r 1 q h p P (Φ 1 (w h ),w). for some posiive consan K, since he raio w/w h is bounded by (30). Consequenly, since we have r 1 q h p w q 2 h (w w h ) 2 = v2 R,

34 28 ONDŘEJ DOŠLÝ K R() v2 () r 1 q ()h p ()P (Φ 1 (w h ()),w()) (32) for large. By (29) we have v() L as.if L 0, i.e. v() 0, we have v 2 (s) K lim R(s) ds = K L 2 ds lim R(s) < < L 2 lim r 1 q (s)h p (s)p (Φ 1 (w h (s)),w(s)) ds < a conradicion, since similarly as in he par (i) of he proof ds lim R(s) = lim 1 h (s) L h(s) ds = 1 L lim log h() =. Therefore, L = 0 and hence v() 0forlarge. Concerning he funcion pr 1 q h p P (Φ 1 (w h ),w), subsiuing for v =h p (w w h ), similarly as in he par (i) of he proof we have pr 1 q h p P (Φ 1 (w h ),w)=h(, v). Now we use again inequaliy (24). This inequaliy copuled wih (26) implies ha v saisfies he inequaliy v + C()+ q 2 R() 0 for large and his is he Riccai inequaliy associaed wih (21), which means ha his equaion is nonoscillaory. v 2 5. Open problems A he end of his paper we formulae some open problems and conjecures which sugges he research direcions for he nex fuure. (i) In Theorem 4.1 we have exended he resuls given in pars (i) and (ii) of Theorem 3.1 which disinguish beween he cases p 2andp 2. A naural quesion is how he par (iii) of ha heorem exends o he general siuaion, in paricular, wha modificaion of (12) wih he funcion μ given by (11) is sufficien for he relaionship beween nonoscillaion of (1) and (21) for any p>1. (ii) The principal role in he proof of Theorem 4.1 is played by he assumpion (19). This assumpion is saisfied for he ypical model r() 1, c() = γp p when (13) reduces o he generalized Euler equaion (9) wih h() = p 1 p. However, if we ake c() = γp ( ) + μp p p log 2, μ p := 1 p 1 p 1, 2 p i.e., (13) is he generalized Euler-Weber equaion (an alernaive erminology is he Euler-Riemann equaion)