Soluions of hlf-liner differenil equions in he clsses Gmm nd Pi Pvel Řehák Insiue of Mhemics, Acdemy of Sciences CR CZ-6662 Brno, Czech Reublic; Fculy of Educion, Msryk Universiy CZ-60300 Brno, Czech Reublic rehk@mh.cs.cz Vlenin Tddei 2 Di. di Scienze Fisiche, Informiche e Memiche, Universià di Moden e Reggio Emili I-425 Moden, Ily vlenin.ddei@unimore.i Absrc Absrc We sudy symoic behvior of ll) osiive soluions of he nonoscillory hlf-liner differenil equion of he form r) y sgn y ) = ) y sgn y, where, ) nd r, re osiive coninuous funcions on [, ), wih he hel of he Krm heory of regulrly vrying funcions nd he de Hn heory. We show h incresing res. decresing soluions belong o he de Hn clss Γ res. Γ under suible ssumions. Furher we sudy behvior of slowly vrying soluions for which symoic formuls re esblished. Some of our resuls re new even in he liner cse = 2. Keywords: hlf-liner differenil equion; osiive soluion; symoic formul; regulr vriion; clss Gmm; clss Pi 200 Mhemics Subjec Clssificion: 34C; 34C4; 34E05; 26A2 Inroducion Consider he hlf-liner equion r)φy )) = )Φy), ) where r, re osiive coninuous funcions on [, ) nd Φu) = u sgn u wih >. Denoe by he conjuge number of, i.e. =. Equion ) is nonoscillory nd ll is nonrivil soluion re evenully monoone see [7]). Our im is o describe symoic behvior of ll) osiive incresing nd decresing soluions o ) wih he hel of he Krm heory of regulrly vrying funcions nd he de Hn heory. Suored by he grn 20/0/032 of he Czech Science Foundion nd by RVO 67985840. 2 Suored by he rojec PRIN-MIUR 2009 Ordinry Differenil Equions nd Alicions nd by he rojec GNAMPA 203 Toologicl mehods for nonliner differenil roblems nd licions.
We esblish firs condiions gurneeing h ll osiive incresing soluions which end o ) res. osiive decresing soluions which end o zero) of ) belong o he de Hn clss Γ res. Γ wih uxiliry funcions exressed in erms of he coefficiens; hese clsses form roer subses of ridly vrying funcions. The ides of he roofs of hese semens will be uilized o discuss lso regulr vriion of soluions o ). The second r of he er is devoed o soluions of ) in he de Hn clss Π which forms roer subse of slowly vrying funcions. We will disinguish wo cses wih resec o cerin behvior of he coefficiens, where in one cse, ll osiive decresing soluions re shown o be in Π nd o sisfy cerin symoic formuls, while in he oher cse such kind of resuls is esblished for incresing soluions. Theory of regulr vriion hs been shown very useful in he sudy of symoic roeries of differenil equions, see in riculr he monogrh [6] which summrizes he reserch u o 2000. Hlf-liner differenil equions in he frmework of regulr vriion hve been considered in [3, 4, 5, 22, 23], see lso [24, Cher 3] nd [7, Subsecion 4.3]. Our resuls cn be undersood in vrious wys. We give comlemenry informion o sndrd symoic clssificion of nonoscillory soluions see e.g. [2, 3, 5, 7], [7, Cher 4]), nd o some resuls on behvior of soluions o ) mde in he frmework of regulr vriion [3, 4, 23, 24]). Furher, our heory cn be seen s n exension of resuls for he liner equion y = )y see [8, 9, 6, 8, 9, 2, 24]). We however end severl observions nd furher exensions which re new even in he liner cse. The er is orgnized s follows. In he nex secion we recll severl useful fcs bou hlf-liner differenil equions, regulrly vrying funcions, nd de Hn clsses. Secion 3 consiss of hree subsecions. In he firs res. he second one we esblish condiions which gurnee h incresing res. decresing soluions of ) re in Γ res. Γ. In boh cses we give exmles nd simlified versions of sufficien condiions. Subsecion 3.3 discusses reled resuls which re eiher byroducs of he roofs s Hrmn-Winner ye resul) or use modified ides of he roofs sufficien condiions for regulr vriion of soluions). Secion 4 consiss lso of hree subsecions. The firs wo re logiclly disinguished wih resec o he behvior of he coefficiens. In Subsecion 4. we exmine decresing slowly vrying soluions which re in Π) nd derive symoic formuls. The seing of he second subsecion requires o seek for slowly vrying soluions mong incresing soluions nd, gin, symoic formuls for such soluions re esblished. In boh cses we give exmles nd ddiionl observions, see Subsecion 4.3. The er is concluded wih shor secion which indices some direcions for fuure reserch. 2 Preinries I is well known see [7, Cher 4]) h ) wih osiive r, is nonoscillory, i.e. ll is soluions re evenully of consn sign. Thus, wihou loss of generliy, we work jus wih osiive soluions, i.e. wih he clss M = {y : y) is osiive soluion of ) for lrge }. 2
Bsic clssificion of nonoscillory soluions nd exisence resuls cn be found in [2, 3, 5, 7]. For survey see [7, Cher 4]). Becuse of he sign condiions on he coefficiens, ll osiive soluions of ) re evenully monoone, herefore hey belong o one of he following disjoin clsses: M + = {y M : y ) > 0 for lrge }, M = {y M : y ) < 0 for lrge }. I cn be shown h boh hese clsses re nonemy see [7, Lemm 4..2]). The clsses M +, M cn be furher divided ino four muully disjoin subclsses: We se M + = {y M + : y) = }, M + B = {y M+ : y) = l R}, M B = {y M : y) = l > 0}, M 0 = {y M : y) = 0}. T J = T r) s)ds) d nd T T J 2 = r) s)ds) d. T The convergence or divergence of he bove inegrls fully chrcerize he bove subclsses. In riculr, ccording o [7, Theorem 4..4], M + = M + if nd only if J =, while M + = M + B if nd only if J <. Moreover, M = M B if nd only if J = nd J 2 <, while M = M 0 if nd only if J 2 =. Finlly, if J < nd J 2 <, hen M 0 M B. Le y M nd ke f C wih f) 0 for every. Denoed w = frφ y ), i y sisfies he generlized Ricci equion w f f w f + ) r Φ f) w = 0, 2) where Φ snds for he inverse of Φ, i.e., Φ u) = u sgn u. If f), hen 2) reduces o he usul generlized Ricci equion Dividing 2) by f, we ge w ) + )r ) w = 0. 3) w f = f w )r f 2 + w f. 4) A soluion y of ) is sid o be rincil soluion if for every soluion u of ) such h u λy, λ R, i holds y )/y) < x )/x) for lrge, see [7, Secion 4.2]. According o [4, Corollry ], he se of evenully osiive) rincil soluions is eiher M B if J = nd J 2 < or M 0 oherwise. A soluion of he ssocied generlized Ricci equion which is genered by rincil soluion is clled n evenully miniml soluion. According o [7, Theorem 4.2.2.], if P ) ) for 3
lrge, hen he evenully miniml soluions w = rφy /y) nd z = rφx /x) of he generlized Ricci equions resecively ssocied o ) nd r)φx )) = P )Φx) sisfy w) z) 5) for lrge. In he second r of his secion we recll some bsic informions on he Krm heory of regulrly vrying funcions nd he de Hn heory; for deeer sudy of his oic see he monogrhs [, 9]. Given δ R {± } nd mesurble funcion f : [, ) 0, ) such h for every λ > 0, we sy h fλ) f) = λδ f is regulrly vrying of index δ, we wrie f RV δ), if δ R\{0}; f is slowly vrying, we wrie f SV, if δ = 0; f is ridly vrying, we wrie f RP V ) res. f RP V ), if δ = res. δ =. Here we use he convenion λ = 0, λ = for λ 0, ), nd λ =, λ = 0 for λ > 0. I follows h f RV δ) if nd only if here exiss funcion L SV such h f) = δ L) for every. The slowly vrying comonen of f RV δ) will be denoed by L f, i.e., L f ) = δ f). The Reresenion heorem see e.g. []) sys h f RV δ) if nd only if { f) = φ) δ ex ψs) s ds }, 6), for some > 0, where φ, ψ re mesurble wih φ) = C 0, ) nd ψ) = 0. A regulrly vrying funcion f is sid o be normlized regulrly vrying, we wrie f NRV δ), if φ) C in 6). If 6) holds wih δ = 0 nd φ) C, we sy h f is normlized slowly vrying, we wrie f NSV. Clerly, if f is C f funcion nd ) = δ, hen f NRV δ). f) The following Krm heorem direc-hlf) will be very useful in he sequel. As usul, f) g) s ) mens f)/g) =. Theorem. [, 9]) If L SV hen s. Moreover, if s δ Ls)ds funcion; if cses, L)/ L) 0 s. δ δ+ L) if δ <, s δ Ls)ds δ + δ+ L) if δ > Ls)/s ds converges, hen L) = Ls)/s ds diverges, hen L) = 4 Ls)/s ds is SV Ls)/s ds is SV funcion; in boh
Here re some simle roeries of RV funcions which we frequenly use: If L SV nd ϑ > 0, hen ϑ L) nd ϑ L) 0 s. Le f i RV ϑ i ), i =, 2, ϑ, ϑ 2, γ R. Then f γ RV γϑ ), f + f 2 RV mx{ϑ, ϑ 2 }), f f 2 RV ϑ + ϑ 2 ). Severl furher roeries re sred in he ex, he lces where we need hem. We now recll useful subclsses of slowly nd ridly vrying funcions, which were inroduced by de Hn, see [, 6, 9]. A nondecresing funcion f : R 0, ) is sid o belong o he clss Γ if here exiss funcion v : R 0, ) such h for ll λ R f + λv)) = e λ ; f) we wrie f Γ or f Γv). The funcion v is clled n uxiliry funcion for f. Furher, f Γ v) if /f Γv). Someimes we wrie Γ s Γ +. A mesurble funcion f : [, ) R is sid o belong o he clss Π if here exiss funcion w : 0, ) 0, ) such h for λ > 0 fλ) f) w) = ln λ; we wrie f Π or f Πw). The funcion w is clled n uxiliry funcion for f. I is known h funcions in clss Γ re ridly vrying of index see [6, Theorem.5.]). From Γv) RP V ), i follows h Γ v) RP V ). Moreover, for funcion f in clss Π, here exiss l = f) nd, rovided f is osiive, f is slowly vrying see [9, Corollry.8]). A mesurble funcion f : R 0, ) is Beurling slowly vrying if f + λf)) f) = for ll λ R; 7) we wrie f BSV. If 7) holds loclly uniformly in λ, hen f is clled self-neglecing; we wrie f SN. I is known h if f BSV is coninuous, hen f is self-neglecing see [9, Theorem.34]). Moreover, f SN if nd only here exiss ϕ inegrble such h ϕ) = 0 nd f) ϕs)ds s see [9, Theorem.35]). Trivilly his imlies h f C is BSV if nd only if f is inegrble nd f ) = 0. 3 Soluions in clsses Γ nd Γ 3. Incresing soluions in he clss Γ In his secion we del wih soluions of ) belonging o he de Hn clss Γ where n uxiliry funcion is exressed in erms of he coefficiens of he equion. Theorem 2. Suose h J = nd f C sisfying ) r BSV. If here exiss funcion f) f) := ) ) r) r) 5 nd f ) f 2 )) = 0, 8)
[ ] ) r ) hen M + Γ. Proof. Tke osiive soluion y M +. By definiion, Φy) = y nd Φy ) = y ) for sufficienly lrge. We now show h y ) [ ) ] s. y) r) ) Le us firs ssume h f C f nd h ) = 0 nd define w = f) 2 ) fr y y ). Then w sisfies he generlized Ricci equion 4) nd, since f is osiive nd r f = f, we obin [ ] w f f = + w f 2 )w. 9) We c h w) = ). Consider firs he cse when w ) > 0 for lrge. Then w is definiely osiive, w is incresing nd ends o l 0, ] when f. If l =, ccording o 9) nd f 0, we would ge h f 2 of wo osiive consns M 0, M such h w ) f)) = 0, we would hve h f 2, which leds o conrdicion. Thus l <. If l ), gin by 9) nd f w ) = f)) )l 0, yielding he exisence w ) M 0 0) nd i.e., reclling h f = r ), w ) f)) M ) ) w) w 0 ) + M d r) for > 0 wih 0 sufficienly lrge. By definiion, ) y w = f r ) ) = y hence 0) imlies h 0 ) + r ) ) y ) r) M 0 y) ) y = y ) r y for > 0. Inegring beween 0 nd nd reclling h J = imlies y, we hen would ge ) ) w) w 0 ) + M d w0 ) + M ln y) r) M 0 y 0 ), 0 conrdicion. Resoning similrly, we obin conrdicion lso in he cse when w ) < 0 for sufficienly lrge. In fc, in his cse w is osiive decresing funcion nd ends o l [0, ). Finlly, suose he exisence of sequence { n } n 6 y ),
wih n n =, n w n ) = l R {, } nd w n ) = 0 for every n. Then, ccording o 9), i follows h [ ] f n ) + w n ) f n ) 2 n ) )w n) = 0 for every n nd, ssing o he i, we obin )l f = 0, becuse ) 0 f 2 )) s. In conclusion, w) = ), i.e. y ) y) ) ) ) r) ) s. Rising by we hen obin y ) [ ) ]. y) r) ) We reurn now o he generl cse. Le f C be such h 8) is fulfilled. Tke = r f) r. Then ) r) ) r) r) ) =, herefore for every ε 0, ) here exiss ε such h ε) ) ) + ε) ) for ll > ε. Since ε 0, ), ccording o [7, Lemm 4..2], here exis evenully osiive incresing soluions v, u, resecively, of roblems { r)φv )) = ε) )Φv) v ε ) = v 0, v ε ) = v 0, ) nd { r)φu )) = + ε) )Φu) u ε ) = u 0, u ε ) = u 0 2) wih v 0, v 0, u 0, u 0 osiive nd such h v 0 v 0 ) y ε) y ε ) ) u 0 u 0 ). Define w v = r v v ), w y = r y y ), nd w u = r u u ). These funcions sisfy, resecively, he generlized Ricci equions nd w v = ε) )r w v, w y = )r w y, w u = + ε) )r w u. By he heory on differenil inequliies see [, Cher III, Secion 4]), since w v ε ) w y ε ) w u ε ) nd ε) ) ) + ε) ) for every > ε, i follows h w v ) w y ) w u ), yielding v ) v) ) ) ) r) y ) y) ) ) ) r) u ) u) ) ) ) r) 3) for every > ε. Define w v = v fr v ). Then [ f w v f = ε + w v ) w v f 2 herefore, resoning s bove, we ge h w v ) = ε ) ], which imlies v ) v) ) ) ) r) v ) v) ) ) ) r) ) ε 4) 7
s, becuse ) ). From 3) nd 4) we obin h inf for every ε > 0, i.e. h inf y ) y) y ) y) Similrly, from 3) we lso ge h su y ) y) ) ) ) r) ) ε ) ) ) ). r) ) ) ) ), r) hence y ) y) ) ) ) r) ), which imlies gin y ) ) [ ] s. y) r) ) Denoed [ r ) ] = Q, we hve h y ), hence for every ε 0, ) here y) Q) exiss ε such h ε Q) subsiuion, we obin h +λq) y ) +ε y) Q) λ Qs) ds = 0 for ε. For every λ > 0, inegring by Q) Q + ξq)) dξ λ s, becuse Q BSV coninuous imlies Q SN. Thus ε)λ inf +λq) y s) ys) ds su +λq) Hence, in view of he rbirriness of ε, for every λ > 0, +λq) λ = y s) ds = ys) By definiion, y Γ[ r ) ] ) nd he heorem is roved. y + λq)) ln. y) y s) ds + ε)λ. ys) Remrk. i) The revious resul ws obined in [23, Theorem 3] in he secil cse when r) nd hen exended in he sme er o he cse r) d = vi suible rnsformion of deenden vrible. Using differen echniques, we re ble o exend he heorem o he cse when he inegrl cn lso converge. Moreover, we do no need o disinguish wheher he inegrl converges or diverges. Noe h, in conrs o he liner cse, he rnsformion of deenden vrible which cn rnsform convergen cse ino divergen one) is no disosl for equion ). Recll h he sufficien condiion from [23, Theorem 3] reds s r s) ds = nd ) ) R BSV, BSV, 5) r r where R is he inverse of R) = r s) ds. As noed in [23], for, r C, 5) is gurneed by ) r) ) ) 0 nd 8 ) ) r) r ) r) 0 6)
f ) f 2 )) s. Thnks o ideniy 20), condiion 6) imlies lso he condiion 0 s, wih, r C, cf. 8). ii) A comrison wih liner resuls is described in Remrk 3, where lso soluions in Γ re discussed. The nex corollry gives vrious sufficien condiions which gurnee he semen of Theorem 2. ) r Corollry. Assume h J =, BSV. Any of he following ssumions [ ] ) r ) gurnee M + Γ, f being defined in 8): i) is bounded nd here exiss funcion f C such h f) f) s nd f BSV ; ii) iii) ) r is bounded nd here exiss funcion f C such h f) f) s nd ln f BSV ; ) r is bounded nd here exiss funcion f C such h r f) f) s nd f BSV, Proof. i) Since f C, hen f BSV is equivlen o f ) f ) f) 2 ) = 0 from he boundedness of. Similrly, i is ossible o rove he cses ii) nd iii). f) 2 = 0, hence Exmle. Suose h RV δ) nd r RV σ). Then, f RV δ σ ) nd r ) RV σ δ ). According o [9, Proosiion.7], if δ + σ 0 nd δ σ, hen f is symoiclly equivlen o f C, wih f RV δ σ ), while r ) is symoiclly equivlen o h C wih h RV σ δ ). Then f f 2 RV σ δ ). Thus σ δ < imlies f f 2 since r) ) ) 0 nd consequenly f f 2 0. Moreover, h s)ds, he sme condiion imlies h 0, i.e. r ) BSV. Now, if δ <, hen s)ds l R. Hence, J = if nd only if r) d = see [7, ge 36]). Thus σ <, i.e. σ ) >, imlies J =. Noice h σ δ < imlies σ < + δ <. On he oher hnd, if δ >, by Krm s heorem we ge r) ) ) s)ds = σ ) L r ) s δ L s)ds δ+ σ) ) L ) L r )δ+)), 9
nd δ + σ) ) >, i.e. σ δ <, imlies gin J =. In conclusion, he ssumions of Theorem 2 re sisfied if σ δ < nd δ + σ 0 wih δ. The resul holds lso when δ = 0 or σ = 0, i.e. when or r re SV. 3.2 Decresing soluions in he clss Γ In his secion we del wih soluions of ) belonging o he de Hn clss Γ. As fr s we know, he only resul reled wih soluion in his clss ws obined in [9, Corollry 3.2] for he liner equion, i.e. when = 2, in he secil cse when r). Using quie differen roch we exend he semen o quie wider clss of equions nd, moreover, we del wih n enire subclss of decresing soluions. In some secs, he following resul is new even in he liner cse, see Remrk 3. Theorem 3. Suose h J < or J 2 = nd funcions, r C such h, r r nd f := [ r ) ] ) M 0 Γ. r ) r ) r BSV. If here exis sisfies 8), hen Proof. Tke y M 0. By definiion, Φy) = y nd Φy ) = y ) for sufficienly lrge. The min ides of he roof of he heorem re similr o he ones of Theorem 2, bu here re subsnil differences in some ses. Therefore we give jus skech of his roof, oining ou he differences. We firs show he relion y ) ) [ ) ] s. Assuming h, r y) r) ) C f nd h ) y = 0, we define w = fr f) 2 ) y ). Then w is negive nd sisfies he equion w [ f = + w) f f 2 ) w) ]. 7) Le us now show h w) = ). Consider firs he cse when w ) < 0 for sufficienly lrge. Then w is definiely negive, w is decresing nd f ends o l [, 0) when. Suose by conrdicion h l =. Then w, herefore ccording o he definiion of f, i holds r ) [ y ) ]. y Hence, rising by, we ge r ) [ y ) ] ) = y r ) [ y ) ], which imlies y [ ] y 0. 8) r y ) Since, r C, condiion r ) BSV is equivlen o [ r ) ] 0, i.e. ) ) 0. 9) r r 0
Mking now comuions, we obin f f 2 = = ) ) ) r r r ) 2 r r ) ) r r r 2 r 2 = ) ) r r r, ) ) r r ) r r r 20) herefore 9) nd f 0 imly f 2 ) r r r 0 2) hus r r y ) r ) = y r r r y y 0 by 8) nd 2). The condiions, r C yield y C 2 nd, from ), hus, ccording o 8) nd 2), I follows h hence y y r y ) + )r y ) 2 y = y, y y y ) = y + r y ) y = 2 )r y ) y y y )r y ) r y 0. )ry ) = y ) 2 yy y ) 2 = y y y ) 2, which imlies y > 0, conrdicion wih y M. Thus l >. If l ), gin by 7) nd f 0, we would ge h f 2 ) l) 0, yielding he exisence of wo osiive consns M 0, M such h ) ) y ) 22) r) M 0 y) nd 0 w ) f)) = ) ) w) w 0 ) M d 23) r) for > 0 sufficienly lrge. From 22) nd 23), reclling h y 0, we hen would ge w) w 0 ) + M ln y) M 0 y 0 ), conrdicion. Resoning similrly, we obin conrdicion lso in he cse when w ) < 0 for sufficienly lrge or here exiss sequence { n } n wih n n =, n w n ) = l R {, } nd w n ) = 0 for every n. We hve so roved
h w) = ), i.e. y ) y) ) ) ) r) ). Rising by we hen obin y ) [ ) ] s. y) r) ) In he generl cse, given, r C s in he ssumions, le v nd u be soluions resecively of roblems ) nd 2) in clss M 0 nd consider he soluions w v, w y nd w u of he resecively ssocied Ricci equions. Since v, y nd u re rincil soluions nd ϵ) ) ) + ϵ) ) for > ϵ, hen w v ) w y ) w u ) for lrge by 5). Hence i is ossible o reson similrly s in Theorem 2 o rove h y ) y) ) ) ) r) ), which imlies gin y ) [ ) ]. y) r) ) Denoed gin [ r ) ] = Q, from y ) nd Q SN we obin, for every y) Q) λ > 0, +λq) λ = By definiion, y Γ[ r ) y s) y + λq)) ds = ln ys) y) y) = ln y + λq)). ] ), i.e. y Γ [ r ) ] ) nd he heorem is roved. The nex corollry gives condiions which gurnee he semen of Theorem 3. The roof is similr o he one of Corollry. Corollry 2. Assume h J < or J 2 =. If ny of he condiions i) or ii) or iii) of Corollry holds for some, r C wih, r r, hen M 0 [r ] ) ) Γ. Exmle 2. Suose h RV δ) nd r RV σ). According o [9, Proosiion.7], if δ, σ 0, nd r re symoiclly equivlen resecively o nd r C, wih RV δ ) nd r RV σ ). Resoning s in Exmle, i follows h f f 2 0 nd r ) BSV. Now, if δ <, hen δ δ+ L ), hus Krm s heorem we hve s)ds r) s)ds <. According o ) ) s)ds δ+ σ) ) L ). L r ) δ ) Hence, if we furher ssume σ < + δ, i.e. δ + σ) ) >, we obin J 2 =. Noe h while we re ble o gurnee J 2 =, we cnno ssure J < when δ <, since i would yield σ < δ, which is in conrdicion wih σ < + δ; he ler inequliy being required for r ) BSV. If δ >, hen s)ds, which imlies J 2 =. In conclusion, he ssumions of Theorem 2 re sisfied if δ, σ 0 nd σ δ < wih δ. 3.3 Reled observions nd regulrly vrying soluions The following corollry gives recise clssificion of ll soluions of ) in erms of de Hn clsses. Is roof is direc consequence of Theorem 2 nd Theorem 3. 2
) Corollry 3. Assume h J = J 2 = nd h BSV. If here exis r ) funcions, r such h, r r nd f := sisfies 8), hen r r M + Γ [r ) ] ) nd M Γ [r ) ] ). Remrk 2. As by-roduc of Corollry 3, we ge condiions gurneeing h M + = M + RP V ) nd M = M 0 RP V ). Remrk 3. A closer exminion of he roofs of Theorem 2 nd Theorem 3 shows h under he condiions of Corollry 3 we hve gurneed he exisence of soluions y i of ) such h y i) ± ) r) ) ) yi ) 24) s, i =, 2. If = 2 nd r) =, hen 24) reduces o y i) ± )y i ), y i being soluions of y )y = 0. The sme formuls in he liner cse were obined in [2] by Hrmn nd Winner under he ssumions s) ds = nd )/ 3 2 ) 0 s. Omey in [8] rediscovered his semen for n incresing soluion y nd showed h y Γ 2 ) under he ssumion 2 BSV, see lso [9, 9, 2]. A decresing soluion which is in Γ ws found by Omey in [9] wih he hel of reducion of order formul, hving disosl n incresing soluion in Γ. Noe h his ool cnno be used in he hlf-liner cse. We emhsize h in Corollry 3 we work wih ll ossible) decresing soluions nd wih generl r, which mkes his semen new lso in he liner cse. The ides of he roofs of Theorems 2, 3 cn be used o esblish condiions which gurnee regulr vriion of soluions o ). Theorem 4. Assume h here exis, r C such h f ) f 2 ) ) = D R nd ) r) ) ) = C 0, ), 25) where f = ), nd ) ), r) r) s. Le ϱ r r > 0 > ϱ 2 denoe he roos of he equion ϱ i) If J =, hen M + NRV ii) If J < or J 2 =, hen M 0 D ϱ Φ ϱ ) C ). NRV Φ ϱ 2 ) C = 0. 26) ). 3
Proof. i) Tke y M +. Similrly s in he roof of Theorem 2 we ge he relion ) f)r) y ) y) ϱ s wih f = ). Since C > 0, ccording o he r r l Hosil rule, ) r) ) [ ) r) ) ] C, hence from he second condiion in 25) we ge ) r) r) C s. Thus, ϱ ) r) ) y ) y) ) C) y ) y) ) y = C ) y) ) Φ ϱ ). ) y s. This imlies ) = ϱ y) C, or y NRV C ii) Tke y M 0. Similrly s in he roof of Theorem 3 we ge he relion ) f)r) y ) y) ϱ2 s. We only noe h o show h w) is excluded for negive soluion w of he ssocied Ricci ye equion we cn use he sme rgumens s in he roof Theorem 3, since r boundedness follows from he ideniy ) r r is bounded. The f ) f 2 ) ) = ) ) r) ) ) ) ) r) r ) r) 27) nd condiion 25). Similrly s in i), we obin y ) y) = Φ ϱ 2 ) C. Remrk 4. Jroš, Kusno, nd Tnigw in [4], ssuming r s) ds =, showed h equion ) wih no sign condiion on ) ossesses ir of soluions y i NRV R Φ λ i )), i =, 2, if nd only if R ) s) ds = A ), ), 28) where R) = r s) ds nd λ, λ 2 re he rel roos of he equion λ λ A = 0. 29) The noion f NRV R ϑ) mens f R NRV ϑ); we sek bou generlized regulr vriion wih resec o R. A similr semen is roved for he cse r s) ds <. To mke comrison wih our resuls simler, le r) =. Then he i in condiion 28) reduces o s) ds nd generlized regulr vriion becomes usul regulr vriion, hus y i NRV Φ λ i )), i =, 2. The second condiion in 25) hen imlies ) ) C s nd so, by he L Hosil rule, s) ds =, which yields )C = /A. )C Furher, from he ideniy 27) nd he firs condiion in 25), we hve D = )C. The relion beween he corresonding rel roos λ of 29) nd ϱ of 26) reds s λ = ϱa )). Hence, Φ λ i ) = Φ ϱ i )/C, i =, 2, nd so, s execed, he corresonding indices of regulr vriion in boh resuls re he sme. Noe h he inegrl condiion from [4] is more generl hn our condiions in Theorem 4 nd, moreover, i is shown o be necessry. On he oher hnd, he fixed oin roch used in [4] gurnees he exisence of les one osiive incresing decresing) RV soluion, while Theorem 4 sys h ll osiive incresing decresing) soluions re regulrly vrying. 4
Remrk 5. I is well known h J < imlies M + = M + B SV. Consequenly, ssuming 25), no mer wh ddiionl inegrl condiion holds, M + RV SV. The sme conclusion holds for M, reclling h, if J = nd J 2 <, hen M = M B SV. Remrk 6. Noe h, wih r C, hving obined ) ) ) y ) = f)r)φ ϱ 30) r) y) s, where y is soluion of ) nd ϱ is roo of 26) which cnno be zero), D being rel number, he semens of he roofs of Theorems 2, 3, nd 4 cn be roved in n lernive nd unified wy; comre wih he ler rs of heir roofs. The number C in 25) is ssumed o be in [0, ). We show h here exiss he i y )y) = K R. 3) y 2 ) We hve y y y = 2 )r y y r )r y y. Relion 30) imlies ) y ). From ideniy 27), in view of r) = r), 30), r) y ) ϱ nd ) ) s, we ge r ) r) y) ) y ) = r ) ) r) ) y) r) ) r) y ) )C D ) Φ ϱ) s. Since ϱ is roo of 26), for K from 3) i holds K = ϱ Cϱ + ) Dϱ = ϱ ϱ Cϱ) = C Φ ϱ). If C = 0, hen K =, which imlies y Γ ± y /y ) deending on wheher y is incresing or decresing, see [20, Proosiion 2.]. Since for he uxiliry funcion i holds y ) y) ) )r) ) s, we ge he semens of Theorems 2 nd 3. If C > 0 s in Theorem 4), hen K nd y NRV / K)) = NRV Φ ϱ)/c) by [20, Proosiion 2.]. Noe h in [20], he semen which we jus lied is formuled for regulrly vrying funcions, bu closer exminion of he roof in h er shows h regulr vriion is normlized. Corollry 4. Assume h here exis, r C such h ) ) ) r NRV γ), γ R, nd = C 0, ). 32) r) Then he semen of Theorem 4 holds, where D in 26) sisfies D = C γ). Proof. The second condiion in 32) imlies r) ) C s. Hence, in view of he firs condiion in 32), we hve ) r) ) r ) r) C r ) r) Cγ s. For D from condiion 25), in view of 27), we ge D = )C Cγ. The resul now follows from Theorem 4. 5
4 Soluions in he clss Π 4. Decresing soluions in he cse δ < We sr wih showing h ny osiive decresing soluion of ) is normlized slowly vrying nd in he de Hn clss Π) nd sisfies n symoic formul. Theorem 5. Le RV δ) nd r RV δ + ) wih δ <. If L ) L r ) 0 s, hen y Π y )) for every y M. Moreover, for every y M, i) if ii) if h ss) rs) wih y M 0. h ss) rs) ) ds =, hen here exiss ε) wih ε) 0 s such y) = ex { + εs)) ss) ) } ds δ + )rs) 33) ) ds <, hen here exiss ε) wih ε) 0 s such { y) = l ex + εs)) ss) δ + )rs) where l = y) 0, ), wih y M B. ) ds }, 34) Proof. Firs noe h s)ds < hnks o δ <. Furher, r RV )δ + )). Since )δ + ) = )δ + + ) = )δ + ), >, nd δ <, we hve )δ + ) >, nd so r s) ds =. Tke y M. We firs rove h y NSV. By definiion rφy ) = r y ) for sufficienly lrge nd i is negive incresing. Therefore here exiss r) y )) = M, 0]. If M < 0, hen, ccording o he monooniciy, r) y )) M for every, hus y ) M r). Inegring now from o we would ge y) y) M rs) ds s, conrdicion. Hence M = 0. Inegring now ) from o, we ge r) y )) = hus, dividing by r) nd reclling h y is decresing, y )) = r) s)ys) ds, 35) s)ys) ds y) r) s)ds. Therefore, dividing by y) nd mulilying by, we obin ) 0 < y ) s)ds 36) y) r) 6
for lrge. By Krm s heorem, r) s)ds δ+ L r ) Hence, ccording o he ssumion, r) δ δ+ L ) = L ) L r ) δ. s)ds 0, nd from 36) we ge y )/y) 0, which imlies y NSV. Now, since RV δ), i follows h r y ) ) = y RV δ). From 35) nd Krm s heorem, we obin r y ) RV δ + ). Since r RV δ + ), we ge y ) RV ), i.e. y RV ). Finlly, inegring by subsiuion, we ge, for every λ > 0, yλ) + y) y ) = λ y u) λ y ) du = y s) λ y ) ds s, i.e. y Π y )), becuse y s) y ) s [min{, λ}, mx{, λ}]. Se h) = δ r) y )) δ + ) ds = ln λ, 37) s uniformly in he inervl s δ 2 rs) y s)) ds. Le us show h h Πh )) nd h Π δ + ) δ r) y )) ). Indeed, h ) = δ ) δ 2 r) y )) + δ )y) δ + ) δ 2 r) y )) = δ )y), hence, reclling h RV δ) nd y SV, we hve h RV δ + δ + 0) = RV ) nd, resoning like in 37), we obin h h Πh )). Moreover, inegring by subsiuion, we ge hλ) h) δ + ) δ r) y )) = λ δ δ rλ) y λ)) δ + ) δ r) y )) + δ r) y )) δ + ) δ r) y )) δ + ) λ s δ 2 rs) y s)) ds δ + ) δ r) y )) = λ δ rλ) y λ)) δ + )r) y )) λ δ + δ 2 u δ 2 ru) y u)) δ r) y )) = λ δ rλ) y λ)) δ + )r) y )) δ + λ Since r y ) RV δ + ), i follows h λ δ rλ) y λ)) δ + )r) y )) nd he uniform convergence of y u) y ) [ λ o u u δ 2 ru) y u)) du r) y )) 7 u δ 2 ru) y u)) r) y )) = λ δ δ + λδ+ = δ + du. du in [min{, λ}, mx{, λ}] imlies ] = λ u δ 2 u δ+ du = ln λ, 38)
hus hλ) h) = ln λ, δ + ) δ r) y )) i.e. h Π δ + ) δ r) y )) ). Becuse of he uniqueness of he uxiliry funcion u o symoic equivlence, δ + ) δ r) y )) h ) = δ )y), which imlies [ y ) y) ] ), i.e. y ) [ ) ] δ+)r) y) δ+)r). Therefore here exiss funcion ε), wih ε) = 0, such h Assume now h y ) y) ss) [ ] rs) ln y) ln y) = [ = + ε)) ) δ + )r) ] ds =. Inegring 39) from o, i follows y s) ys) ds = [ + εs)) ss) ] ds, δ + )rs) which yields 33). In fc, i esily follows h here exiss ε) 0 such h ln y) + εs))[ ss) ] δ+)rs) ds = + εs))[ ss) ] δ+)rs) ds. I is cler h y) 0 s. On he conrry, ssuming ss) [ ] rs) ds < nd inegring 39) from o, i follows ln l ln y) = y s) ys) ds = [ + εs)) ss) ] ds, δ + )rs) wih l = y), which imlies 34). I is cler h l mus be osiive. The nex remrk revels h he condiion gurneeing normlized slow vriion of decresing soluions cn be relxed. Moreover, his condiion is shown o be necessry for he exisence of decresing slowly vrying soluion of ). In ddiion, in Remrk 8, we rove h slowly vrying soluions necessrily decrese. Remrk 7. i) From he roof of Theorem 5 i cn be deduced h M NSV follows from he weker condiions nd )d <, r) r) d = 39) s)ds = 0. 40) We oin ou h, in his cse, i is no necessry o ssume he regulr vriion of or r. ii) We now show he necessiy of 40). More recisely, we rove he following semen. Assume r RV δ + ) wih δ <. If here exiss y M NSV, hen 40) holds. Indeed, se w = rφy /y) = r y /y). Then w sisfies he generlized Ricci equion 3) for lrge, nd 0 < w) = y ) r) y) 8 ) 0 s
, becuse y NSV. Hence, here exiss M > 0 such h w) Mr) RV δ + ), nd so w) 0 s 0, becuse δ <. Furher, since y NSV, here exiss N > 0 such h r ) w) Nr)/ RV δ), which imlies r s) ws) ds <. Inegring 3) from o nd mulilying by /r) we obin w) = r) r) s) ds+ )z), z) := r) r s) ws) ds. 4) We c h z) 0 s. Wihou loss of generliy we my ssume r NRV δ + ) C. Indeed, if r is no normlized or is no in C, hen we cn ke r NRV δ + ) C wih r) r) when, nd we hve By he L Hosil rule, z) r) r s) rs)φy s)/ys)) ds. r ) ws) z) = r ) + )r) = r ) w) r )/r) + ) y )/y) = r )/r) + ) = 0 δ + = 0. Condiion 40) hen follows from 4). If, in ddiion RV δ), hen he necessry condiion my red s L )/L r ) 0 s. A closer exminion of he roofs shows h he condiion r RV δ + ) cn be relxed o he exisence of r i RV δ i +), i =, 2, wih r ) r) r 2 ) for lrge, nd δ, δ 2 <. The nex remrk shows h SV soluions cnno increse. Hence, in Theorem 5 we re deling wih ll SV soluions of ). Remrk 8. Assume RV δ), r RV δ + ), wih δ <, nd ke y M +. Then Φy ) = y ) nd Φy) = y. Since y is osiive, hen ry ) is osiive incresing, hence here exiss osiive consn M such h r)y ) M for sufficienly lrge. Dividing by r) nd rising by, i follows h y ) M ) r), which imlies y) y) + M Since r RV δ + ), i holds r ) δ >, i.e. δ rs) ) ds. 42) rs) RV δ ). From hyohesis, δ <, hus >. Alying Krm s heorem, we hen ge h ) ds RV δ + ) = RV ) δ. Since δ <, i follows h δ > 0, herefore 42) imlies h y is greer hn or equl o RV δ ) funcion, nd herefore cnno be SV. We hve so roved h if δ <, hen M SV M. Observe h regulr vriion of cully ws no used. 9
Remrk 9. i) If = 2 nd r) =, hen Theorem 5 reduces o [8, Theorem 0.-A]. ii) Under he condiions of Theorem 5-i), i does no follow h { ) } ss) y) ex ds δ + )rs) s. This fc ws observed lredy in he liner cse nd wih r) = ), see [8, Remrk 2]. 4.2 Incresing soluions in he cse δ > The nex heorem dels wih SV soluions in he comlemenry cse δ >. As i follows from subsequen Remrk, we mus sough for SV soluions mong elemens of M +. The resul is new lso in he liner cse. Theorem 6. Le RV δ) nd r RV δ + ) wih δ >. If L ) L r ) 0 s, hen y Πy )) for every y M +. Moreover, for every y M +, i) if ii) if h ss) rs) wih y M +. h ss) rs) ) ds =, hen here exiss ε) wih ε) 0 s such { ) } ss) y) = ex + εs)) ds δ + )rs) 43) ) ds <, hen here exiss ε) wih ε) 0 s such y) = ex { where l = y) 0, ), wih y M + B. ) } ss) + εs)) ds, 44) δ + )rs) Proof. Firs noe h, since δ >, we hve s)ds =. Tke y M+. We firs rove h y NSV. By definiion, Φy ) = y ) for > 0 sufficienly lrge nd hence, inegring ) from 0 o nd reclling h y is incresing we ge r)y ) = r 0 )y 0 ) + 0 s)ys) ds r 0 )y 0 ) + y 0 ) 0 s)ds s. Moreover, i is ossible o find osiive consn A such h 45) r)y ) A 0 s)ys) ds for lrge. Now, dividing he ls inequliy by r) nd reclling h y is incresing, we ge y ) Ay) r) 20 0 s)ds.
Thus, dividing by y) nd mulilying by, we obin By Krm s heorem, 0 < ) y ) A y) r) r) s)ds L ) L r ) δ + s)ds. 46) s. Hence, from 46) nd he hyohesis, we ge y )/y) 0, which imlies y NSV. From y RV δ) nd 45), we obin ry RV mx{0, δ+}) = RV δ+), becuse δ >. Consequenly, y ) RV ), i.e. y RV ), nd concluding s in 37) we obin y Πy ). Since y RV δ), wih δ >, from 45) nd Krm s heorem we ge r)y ) 0 s)ys) ds sδ L s)l y s)ds δ+ δ+ L )L y ) = δ+ )y). Thus, dividing by r)y), we ge y ) y) ) ) y ) ) y). δ + )r) ), hence, rising by, δ+)r) Therefore here exiss funcion ε), wih ε) = 0, such h y ) y) ) ) = + ε)). 47) δ + )r) Assume now h ) δ+)r) ) d =. Inegring 47) from o, we ge ln y) ln y) = ) ss) + εs)) ds, δ + )rs) nd 43) follows wih y) s. Oherwise, if ) δ+)r) ) d <, hen, inegring 47) from o, we obin ln y) l = ) ss) + εs)) ds, δ + )rs) wih l = y), which yields 44). Clerly, l 0, ). The nex remrk cs h sufficien condiions gurneeing M + NSV cn be relxed. Moreover, he condiion 48) which cn be undersood s counerr o 40) is necessry for he exisence of n incresing SV soluion. 2
Remrk 0. The roof of Theorem 6 esily imlies M + NSV when ssuming, insed of he regulr vriion of nd r, he weker condiions nd r) )d = s)ds = 0. 48) Similrly s in Remrk 7, i is ossible o rove he following semen. Assume r RV δ + ) wih δ >. If here exiss y M + NSV, hen 48) holds. Noe h in his cse, insed of 4), we work wih he Ricci ye inegrl equion of he form w) r) r) = r) s) ds ) r) r s) ws) ds. Here, /r) 0 nd w)/r) 0 s. If, in ddiion RV δ), hen necessry condiion reds s L )/L r ) 0 s. Moreover, he condiion r RV δ + ) cn be relxed o he exisence of r i RV δ i + ), i =, 2, wih r ) r) r 2 ) for lrge, nd δ, δ 2 >. As we will see nex, slowly vrying soluions cnno decrese in our curren seing. Thus, in Theorem 6 we re deling wih ll SV soluions. Remrk. Assume RV δ), r RV δ + ), wih δ >, nd le y M. Then Φy ) = y ) nd Φy) = y. Inegring ) from o, we ge r) y )) = r) y )) + s)ys) ds. 49) From he fc h r y ) is negive incresing, we obin h here exiss r) y )) = M, 0]. Suose now h y SV. Then y RV δ) hus s)ys) ds, becuse δ >, conrdicion. We hve so roved h if δ >, hen M SV M +. Observe h regulr vriion of r cully ws no used. 4.3 Reled observions nd exmle We sr wih jusifying he fc h he relion bou he indices of regulr vriion of he coefficiens is quie nurl when looking for slowly vrying soluions. Remrk 2. Assume h r RV γ), RV δ), nd h y SV is soluion of equion ). Then rφy )) = y RV δ). Suose firs δ >. Then inegring ) we hve r)φy )) = r)φy )) + s)y s) ds +, hus, wihou loss of generliy, we my ssume y ) > 0 for. Hence, r)y )) = r)y )) + s)y s) ds RV mx{0, δ + }) = RV δ + ) nd so y ) δ+ γ ). Now, if, + ), herefore, since + = 0, conrdicion. Thus δ+ γ =, i.e. γ = δ +. I RV δ+ γ), becuse r RV γ), which imlies y RV δ+ γ similrly s before i is ossible o rove h y RV δ+ γ y SV, we ge δ+ γ 22
urns ou h δ > is equivlen o γ >, so now le γ <. This imlies γ ) >, consequenly r s) ds =. Resoning like in Remrk 8 we obin h y M, hen r) y )) = s)ys) ds RV δ+), ccording o 35) nd Krm s heorem. Similrly s before we ge γ = δ +. The resuls from Theorems 5 nd 6 cn be unified o obin he following corollry. Corollry 5. Assume h RV δ) nd r RV δ + ), wih δ nd L )/L r ) 0 s. Then, for every y M here exiss ε) wih ε) 0 s such h i) if ss) ) rs) ds =, hen { y) = ex sgnδ + ) ii) if ss) ) rs) ds <, hen { y) = l ex sgnδ + ) where l = y) 0, ). ) } ss) + εs)) ds δ + rs) ) } ss) + εs)) ds, δ + rs) Remrks 8 nd in combinion wih Theorems 5 nd 6 yield he following corollry. Corollry 6. Assume h RV δ), r RV δ + ) wih L ) L r) 0, i) Le δ <. Then ) if ss) ) rs) ds = hen M SV = M = M 0 ; b) if ss) ) rs) ds < hen M SV = M = M B. ii) Le δ >. Then ) if ss) ) rs) ds = hen M SV = M + = M + ; b) if ss) ) rs) ds < hen M SV = M + = M + B. In he nex remrk we discuss relions beween he condiions from Theorems 5 nd 6 which involve he inegrl ss) ) rs) ds nd he inegrl condiions involving J nd J 2 from he generl exisence heory see Preinries) under our seing. Remrk 3. Assume h RV δ) nd r RV γ). We recll, firs of ll, h M + = M + B if nd only if J <, while M + = M + if nd only if J =. Noice h, if δ >, hen s)ds =, nd, ccording o Krm s heorem, s)ds L ) ) δ+ γ) ) r) δ+)l ) r). Under our nurl ssumion γ = δ +, we obin h J = if nd only if L ) ) L r ) d =, which recisely corresonds o our condiions in seing ii) of Corollry 6. 23
Recll now h M = M + B if nd only if J = nd J 2 <, while M = M 0 if nd only if J 2 =. Suose δ <. Then s)ds <, hus J = if γ < =, which is fulfilled when γ = + δ. Moreover, gin from Krm s heorem, s)ds r) ) δ+ γ) ) L ) δ + )L r ) ) = L ) δ + )L r ) ) reclling h we ssume γ = δ +. Therefore, lso in his cse, we obin h he inegrl condiions in seing i) of Corollry 6 re equivlen o he known generl exisence condiions involving J nd J 2. Exmle 3. Consider equion ) wih ) = δ L ) nd r) = δ+, where L ) = ln ) γ + h ), L r ) = ln ) γ 2 + h 2 ), wih h i ) = oln ) γ i ), i =, 2, for some γ < γ 2. For exmle, boh h i ) = cos or h i ) = lnln)) sisfy he revious condiion. Trivilly, for every λ > 0, we hve h i.e. L, L r SV. Now, ln λ) γ i + h i λ) ln ) γ i + hi ) lnλ) ln = lnλ) ln = ) γ i + h iλ) ln ) γ i + h i) ln ) γ i ) γ i + h iλ) ln λ) γ i + h i) ln ) γ i ln λ ln )γ i =, L ) L r ) = ln ) γ + h ) ln ) γ 2 + h2 ) = = + h) ln ) γ ln ) γ 2 γ [ + h 2 ) ln ) γ 2 ] = 0, + h ) ln ) γ ln ) γ 2 γ + h 2 ) ln ) γ 50) becuse γ 2 > γ. Finlly, ccording o 50), i follows h ) ) r) ) δ+ L ) = = δ+ L r ) { + h ) ln ) γ ln ) γ 2 γ [ + h 2 ) ln ) γ 2 ] } γ γ2 ln ) s. Thus, since ln s s)λ ds < if nd only if λ <, we hve h ss) ) rs) ds < if nd only if ξ := γ γ 2 <. In his cse, ccording o Corollry 5, every soluion hs finie non zero i l nd { } y) = l ex sgnδ + ) + o))ln ) ξ+ ξ + δ + s. Furher, if ξ >, hen incresing soluions re unbounded, while decresing soluions hve zero i, nd { } y) = ex sgnδ + ) + o))ln ) ξ+ ξ + δ + 24
s. Finlly, if ξ =, hen s. y) = ln ) sgnδ+)+o)) δ+ Remrk 4. For reled resuls sufficien nd necessry condiions for he exisence of RV soluions of )) see [4] where generlized regulr vriion nd fixed oin heorem ly imorn roles. Comre lso wih Remrk 4 nd he discussion on condiions 40) nd 48). 5 Some oen roblems In his ls rgrh we indice some direcions for ossible fuure reserch reled o he bove oics: To esblish second order resul ssocied o Theorems 2 nd 3, i.e., o ke ) closer look he behvior of y ) y), y being soluions of ), ) r) ) cf. 24), in he sense of [9, Secion 3.2] nd [2, Secion 5.], where he liner equion y = )y is considered. To give n imrovemen of Theorems 5 nd 6 in he sense of [8, Theorem 0.- B], where he liner equion y = )y is considered nd he clss ΠR 2 u, v) is uilized. To del wih RV ϑ) soluions ϑ being cerin osiive number) in he siuion of Theorem 5, in riculr, o show M + RV ϑ) nd derive n symoic formul. Similrly for Theorem 6. Noe h in conrs o he liner cse, he reducion of order formul or some usul rnsformion ricks re no our disosl. To esblish symoic formuls for RV ϱ) soluions wih ϱ differen from 0 nd ϑ here we men he ϑ from he revious iem), nd ossibly under more generl inegrl condiion, in he sense of [0, Theorem.2], where he liner equion y = )y is considered. To exmine he borderline cse for δ in Theorems 5 nd 6), nmely δ =. To exend some of) he bove resuls o he so-clled nerly hlf-liner equion, i.e., r)gy )) = )F y), where F, G re regulrly vrying funcions wih he sme osiive index. Some observions long his line more recisely, exension of Theorem 5 o he equion where F ), G ) re regulrly vrying zero of index ) cn be found in [25]. References [] N. H. Binghm, C. M. Goldie, J. L. Teugels, Regulr Vriion, Encycloedi of Mhemics nd is Alicions, Vol. 27, Cmbridge Universiy Press, 987. [2] M. Cecchi, Z. Došlá, M. Mrini, On he dynmics of he generlized Emden- Fowler equion, Georgin Mh. J. 7 2000), 269 282. 25
[3] M. Cecchi, Z. Došlá, M. Mrini, On nonoscillory soluions of differenil equions wih -Llcin, Adv. Mh. Sci. Al. 200), 49 436. [4] M. Cecchi, Z. Došlá, M. Mrini, Princil soluions nd miniml ses of qusiliner differenil equions, Dyn. Sys. Al. 3 2004), 223-234. [5] T. A. Chnuriy, Monoone soluions of sysem of nonliner differenil equions, Ann. Polon. Mh. 37 980), 59 70, in Russin). [6] L. de Hn, On Regulr Vriion nd is Alicions o he Wek Convergence of Smle Exremes, Mhemisch Cenrum Amserdm, 970. [7] O. Došlý, P. 2005. Řehák, Hlf-liner differenil equions, Elsevier, Norh Hollnd, [8] J. L.Geluk, On slowly vrying soluions of he liner second order differenil equion, Publ. Ins. Mh. 48 990), 52 60. [9] J. L. Geluk, L. de Hn, Regulr vriion, exensions nd Tuberin heorems, CWI Trc, 40. Siching Mhemisch Cenrum, Cenrum voor Wiskunde en Informic, Amserdm, 987. [0] J. L. Geluk, V. Mrić, M. Tomić, On regulrly vrying soluions of second order liner differenil equions, Differenil Inegrl Equions 6 993), 329 336. [] P. Hrmn, Ordinry differenil equions, SIAM, 2002. [2] P. Hrmn, A. Winner, Asymoic inegrions of liner differenil equions, Amer. J. Mh. 77 955), 45 86. [3] J. Jroš, T. Kusno, T. Tnigw, Nonoscillion heory for second order hlfliner differenil equions in he frmework of regulr vriion, Resuls Mh. 43 2003), 29 49. [4] J. Jroš, T. Kusno, T. Tnigw, Nonoscillory hlf-liner differenil equions nd generlized Krm funcions, Nonliner Anl. 64 2006), 762 787. [5] T. Kusno, V. Mrić, Nonoscillory liner nd hlf-liner differenil equions hving regulrly vrying soluions, Adv. Mh. Sci. Al. 4 2004), 35 357. [6] V. Mrić, Regulr Vriion nd Differenil Equions, Lecure Noes in Mhemics 726, Sringer-Verlg, Berlin-Heidelberg-New York, 2000. [7] J. D. Mirzov, Asymoic roeries of nonliner sysems of nonuonomous ordinry differenil equions, Mjko, 993, in Russin). [8] E. Omey, Regulr vriion nd is licions o second order liner differenil equions, Bull. Soc. Mh. Belg. Sér. B 33 98), 207 229. [9] E. Omey, Ridly vrying behviour of he soluions o second order liner differenil equion, Proc. of he 7h in. coll. on differenil equions, Plovdiv, Augus 8-23, 996. Urech: VSP 997), 295 303. 26
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