Elecronic Journl of Differenil Equions, Vol. 06 06), No. 9, pp. 3. ISSN: 07-669. URL: hp://ejde.mh.xse.edu or hp://ejde.mh.un.edu ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS OF FOURTH-ORDER NONLINEAR DIFFERENTIAL EQUATIONS WITH REGULARLY VARYING COEFFICIENTS ALEKSANDRA B. TRAJKOVIĆ, JELENA V. MANOJLOVIĆ Asrc. We sudy he fourh-order nonliner differenil equion `p) x ) x ) + q) x) β x) = 0, > β, wih regulrly vrying coefficien p, q sisfying Z / d <. p) in he frmework of regulr vriion. I is shown h complee informion cn e cquired ou he exisence of ll possile inermedie regulrly vrying soluions nd heir ccure sympoic ehvior infiniy.. Inroducion We sudy he equion p) x ) x ) ) + q) x) β x) = 0, > 0,.) where i) nd β re posiive consns such h > β, ii) p, q : [, ) 0, ) re coninuous funcions nd p sisfies +/) d <..) p) / Equion.) is clled su-hlf-liner if β < nd super-hlf-liner if β >. By soluion of.) we men funcion x : [T, ) R, T, which is wice coninuously differenile ogeher wih p x x on [T, ) nd sisfies he equion.) every poin in [T, ). A soluion x of.) is sid o e nonoscillory if here exiss T such h x) 0 for ll T nd oscillory oherwise. I is cler if x is soluion of.), hen so does x, nd so in sudying nonoscillory soluions of.) i suffices o resric our enion o is evenully) posiive soluions. 00 Mhemics Sujec Clssificion. 34C, 34E05, 6A. Key words nd phrses. Fourh order differenil equion; sympoic ehvior of soluions; posiive soluion, regulrly vrying soluion, slowly vrying soluion. c 06 Texs Se Universiy. Sumied Mrch 6, 06. Pulished June 5, 06.
A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Throughou his pper exensive use is mde of he symol o denoe he sympoic equivlence of wo posiive funcions, i.e., f) g), g) lim f) =. We lso use he symol o denoe he dominnce relion eween wo posiive funcions in he sense h g) f) g), lim f) =. In our nlysis of posiive soluions of.) specil role is plyed y he four funcions s ϕ ) = ps) ds, ϕ ) = s ) s / ps) )/ ds, ψ ) =, ψ ) =, which re he priculr soluions of he unperured differenil equion p) x ) x )) = 0. Noe h he funcions ϕ i nd ψ i, i =, defined ove sisfy he dominnce relion ϕ ) ϕ ) ψ ) ψ ),. Asympoic nd oscillory ehvior of soluions of.) hve een previously considered in [9, 9, 6,, 6, 30, 3]. Kusno nd Tnigw in [9] mde deiled clssificion of ll posiive soluions of he equion.) under he condiion.) nd eslished condiions for he exisence of such soluions. I ws proved h he following four ypes of cominion of he signs of x, x nd p x x ) re possile for n evenully posiive soluion x) of.): p) x ) x )) > 0, x ) > 0, x ) > 0 for ll lrge,.3) p) x ) x )) > 0, x ) > 0, x ) < 0 for ll lrge,.4) p) x ) x )) > 0, x ) < 0, x ) > 0 for ll lrge,.5) p) x ) x )) < 0, x ) < 0, x ) > 0 for ll lrge..6) As resuls of furher nlysis of he four ypes of soluions menioned ove, Kusno nd Tnigw in [9] hve shown h he following six ypes re possile for he sympoic ehvior of posiive soluions of.): P) x) c ϕ ), P) x) c ϕ ) s, P3) x) c 3 s, P4) x) c 4 s, I) ϕ ) x) ϕ ) s, I) x) s, where c i > 0, i =,, 3, 4 re consns. Posiive soluions of.) hving he sympoic ehvior P) P4) re collecively clled primiive posiive soluions of he equion.), while he soluions hving he sympoic ehvior I) nd I) re referred o s inermedie soluions of he equion.). The inerrelion eween he ypes.3)-.6) of he derivives of soluions nd he ypes P) P4), I) nd I) of he sympoic ehvior of soluions is s follows: i) All soluions of ype.3) hve he sympoic ehvior of ype P);
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 3 ii) A soluion of ype.4) hs he sympoic ehvior of one of he ypes P), P), P3) nd I); iii) A soluion of ype.5) hs he sympoic ehvior of one of he ypes P3) nd P4); iv) A soluion of ype.6) hs he sympoic ehvior of one of he ypes P3), P4) nd I). The exisence of four ypes of primiive soluions hs een compleely chrcerized for oh su-hlf-liner nd super-hlf-liner cse of.) wih coninuous coefficiens p nd q s he following heorems proven in [9] show. Theorem.. Le p, q C[, ). Equion.) hs posiive soluion x sisfying P3) if nd only if J = p) s)qs) ds) / d <..7) Theorem.. Le p, q C[, ). Equion.) hs posiive soluion x sisfying P4) if nd only if J = p) s)s β qs) ds) / d <..8) Theorem.3. Le p, q C[, ). Equion.) hs posiive soluion x sisfying P) if nd only if J 3 = q)ϕ ) β d <..9) Theorem.4. Le p, q C[, ). Equion.) hs posiive soluion x sisfying P) if nd only if J 4 = q)ϕ ) β d <..0) Unlike primiive soluions, eslishing necessry nd sufficien condiions for he exisence of he inermedie soluions seems o e much more difficul sk. Thus, only sufficien condiions for he exisence of hese soluions ws oined in [9]. Theorem.5. If.0) holds nd if J 3 = q)ϕ ) β d =, hen equion.) hs posiive soluion x such h ϕ ) x) ϕ ),. Theorem.6. If.8) holds nd J = p) s)qs) ds) / d =, hen.) hs posiive soluion x such h x) s. However, shrp condiions for he oscillion of ll soluions of.) in oh cses su-hlf-liner nd super-hlf-liner) hve een oined in [6].
4 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Theorem.7. Le β <. All soluions of.) re oscillory if nd only if J = p) s)s β qs) ds) / d =. Thus, our sk is o eslish necessry nd sufficien condiions for.) o possess inermedie soluions of ypes I) nd I) nd o deermine precisely heir sympoic ehvior infiniy. Since his prolem is very difficul for equion.) wih generl coninuous coefficiens p nd q, we will mke n emp o solve he prolem in he frmework of regulr vriion, h is, we limi ourselves o he cse where p nd q re regulrly vrying funcions nd focus our enion on regulrly vrying soluions of.). The recen developmen of sympoic nlysis of differenil equions y he mens of regulrly vrying funcions, which ws iniied y he monogrph of Mrić [], hs shown h here exiss vriey of nonliner differenil equions for which he prolem menioned ove cn e solved compleely. The reder is referred o he ppers [8, 0, 3, 4, 8, 0, 8] for he second order differenil equions, o [,, 5, 7, 5] for he fourh order differenil equions nd o [3]-[7], [3, 4, 7] for some sysems of differenil equions. The presen work cn e considered s coninuion of he previous ppers [,, 5], which re he specil cses of.) wih = or p) u hs feures differen from hem in he sense h he generlized regulrly vrying funcions or generlized Krm funcions) inroduced in [] will e used in order o mke cler he dependence of sympoic ehvior of inermedie soluions on he coefficien p. For reder s convenience he definiion of generlized regulrly vrying funcions nd some of heir sic properies re summrized in Secion. In Secions 3 we consider equion.) wih generlized regulrly vrying p nd q, nd fer showing h ech of wo clsses of is inermedie generlized regulrly vrying soluions of ype I) nd I) cn e divided ino hree disjoin suclsses ccording o heir sympoic ehvior infiniy, we eslish necessry nd sufficien condiions for he exisence of soluions nd deermine he sympoic ehvior of soluions conined in ech of he six suclsses explicily nd precisely. In he finl Secion 4 i is shown h our min resuls, when specilized o he cse where p nd q re regulrly vrying funcions in he sense of Krm, provide complee informion ou he exisence nd sympoic ehvior of regulrly vrying soluions in he sense of Krm for h equion.). This informion comined wih h of he primiive soluions of.) cf. Theorems.-.4) enles us o presen full srucure of he se of regulrly vrying soluions for equions of he form.) wih regulrly vrying coefficiens.. Bsic properies of regulrly vrying funcions We recll h he se of regulrly vrying funcions of index ρ R is inroduced y he following definiion. Definiion.. A mesurle funcion f :, ) 0, ) for some > 0 is sid o e regulrly vrying infiniy of index ρ R if fλ) lim f) = λρ for ll λ > 0.
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 5 The oliy of ll regulrly vrying funcions of index ρ is denoed y RVρ). In he specil cse when ρ = 0, we use he noion SV insed of RV0) nd refer o memers of SV s slowly vrying funcions. Any funcion f RVρ) is wrien s f) = ρ g) wih g SV, nd so he clss SV of slowly vrying funcions is of fundmenl impornce in he heory of regulr vriion. If f) lim ρ = lim g) = cons > 0 hen f is sid o e rivil regulrly vrying funcion of he index ρ nd i is denoed y f r RVρ). Oherwise, f is sid o e nonrivil regulrly vrying funcion of he index ρ nd i is denoed y f nr RVρ). The reder is referred o Binghm e l. [] nd Sene [9] for complee exposiion of heory of regulr vriion nd is pplicion o vrious rnches of mhemicl nlysis. To properly descrie he possile sympoic ehvior of nonoscillory soluions of he self-djoin second-order liner differenil equion p)x )) +q)x) = 0, which re essenilly ffeced y he funcion p), Jroš nd Kusno inroduced in [] he clss of generlized Krm funcions wih he following definiion. Definiion.. Le R e posiive funcion which is coninuously differenile on, ) nd sisfies R ) > 0, > nd lim R) =. A mesurle funcion f :, ) 0, ) for some > 0 is sid o e regulrly vrying of index ρ R wih respec o R if f R is defined for ll lrge nd is regulrly vrying funcion of index ρ in he sense of Krm, where R denoes he inverse funcion of R. The symol RV R ρ) is used o denoe he oliy of regulrly vrying funcions of index ρ R wih respec o R. The symol SV R is ofen used for RV R 0). I is esy o see h if f RV R ρ), hen f) = R) ρ l), l SV R. If lim f) R) ρ = lim l) = cons > 0 hen f is sid o e rivil regulrly vrying funcion of index ρ wih respec o R nd i is denoed y f r RV R ρ). Oherwise, f is sid o e nonrivil regulrly vrying funcion of index ρ wih respec o R nd i is denoed y f nr RV R ρ). Also, from Definiion. i follows h f RV R ρ) if nd only if i is wrien in he form f) = gr)), g RVρ). I is cler h RVρ) = RV ρ). We emphsize h here exiss funcion which is regulrly vrying in generlized sense, u is no regulrly vrying in he sense of Krm, so h, roughly speking, he clss of generlized Krm funcions is lrger hn h of clssicl Krm funcions. To help he reder we presen here some elemenry properies of generlized regulrly vrying funcions. Proposiion.3. i) If g RV R σ ), hen g RV R σ ) for ny R. ii) If g i RV R σ i ), i =,, hen g + g RV R σ), σ = mxσ, σ ). iii) If g i RV R σ i ), i =,, hen g g RV R σ + σ ). iv) If g i RV R σ i ), i =, nd g ) s, hen g g RV R σ σ ). v) If l SV R, hen for ny ε > 0, lim R)ε l) =, lim R) ε l) = 0.
6 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Nex, we presen fundmenl resul see []), clled generlized Krm inegrion heorem, which will e used hroughou he pper nd ply cenrl role in eslishing our min resuls. Proposiion.4. Le l SV R. Then: i) If >, ii) If <, iii) If =, hen funcions R s)rs) ls) ds R)+ l), ; + R s) Rs) ls) ds R)+ l), ; + R s)rs) ls) ds nd re slowly vrying wih respec o R. R s)rs) ls) ds 3. Asympoic ehvior of inermedie generlized regulrly vrying soluions In wh follows i is lwys ssumed h funcions p nd q re generlized regulrly vrying of index η nd σ wih respec o R, wih R) is defined wih s + ) R) = ps) ds, 3.) / nd expressed s p) = R) η l p ), l p SV R nd q) = R) σ l q ), l q SV R. 3.) From 3.) nd 3.) we hve h Inegring 3.3) from o we hve + + = R )R) η l p ) /. 3.3) + = R s)rs) η l p s) / ds,, 3.4) implying h η. In wh follows we limi ourselves o he cse where η > excluding he oher possiiliies ecuse of compuionl difficuly. Applying he generlized Krm inegrion heorem Proposiion.4) he righ hnd side of 3.4) we oin η ) + R) η + lp ) +, +. 3.5) From 3.3) nd 3.5) we cn express R ) s follows η ) + R + ) R) 3+ η + lp ) +, +, 3.6) which cn e rewrien in he form η ) + + R )R) m,η) l p ) +, +. 3.7)
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 7 The nex lemm, following direcly from he generlized Krm inegrion heorem using 3.7), will e frequenly used in our ler discussions. To h end nd o furher simplify formulion of our min resuls we inroduce he noion: m, η) = η + ), I is cler h m, η) < < 0 < m, η) nd m, η) = m, η) η ; m, η) = η +. 3.8) m, η) η = m, η). 3.9) In proofs of our min resuls consns m i, η), i =,, will e revied s m i, i =,, respecively. Lemm 3.. Le f) = R) µ L f ), L f SV R. Then: i) If µ > m, η), fs) ds m, η) + + µ + m, η) R)µ+m,η) L f )l p ) +, ; ii) If µ < m, η), fs) ds m, η) + + µ + m, η)) R)µ+m,η) L f )l p ) +, ; iii) If µ = m, η), hen funcions fs) ds = fs) ds = re slowly vrying wih respec o R. Rs) m,η) L f s) ds, Rs) m,η) L f s) ds To mke n in deph nlysis of inermedie soluions of ype I) nd I) of.) we need fir knowledge of he srucure of he funcions ψ, ψ, ϕ nd ϕ regrded s generlized regulrly vrying funcions wih respec o R. From 3.5), 3.6) nd 3.7) i is cler h ψ SV R nd ψ RV R m, η)). Using 3.) nd pplying Lemm 3. wice, we oin ϕ ) = s Rr) η/ l p r) / dr ds 3.0) m, η) +) + m, η)m, η) m, η)) R)m,η) l p ) +),, which shows h ϕ RV R m, η)). Furher, y 3.) nd 3.5), in view of 3.9)-ii), noher wo pplicions of Lemm 3. yield ϕ ) m, η) + implying ϕ RV R ). m, η) m, η) + R),, s Rr) m,η) l p r) + dr ds 3.)
8 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 3.. Regulrly vrying soluions of ype I). The firs susecion is devoed o he sudy of he exisence nd sympoic ehvior of generlized regulrly vrying soluions wih respec o R of ype I) wih p nd q sisfying 3.). Expressing such soluion x of.) in he form x) = R) ρ l x ), l x SV R, 3.) since ϕ ) x) ϕ ),, he regulriy index ρ of x mus sisfy m, η) ρ. If ρ = m, η), hen since x)/r) m,η) = l x ),, x is memer of nr RV R m, η)), while if ρ =, hen since x)/r) = l x ) 0,, x is memer of nr RV R ). Thus he se of ll generlized regulrly vrying soluions of ype I) is nurlly divided ino he hree disjoin clsses nr RV R m, η)) RV R ρ) wih ρ m, η), ) or nr RV R ). Our im is o eslish necessry nd sufficien condiions for ech of he ove clsses o hve memer nd furhermore o show h he sympoic ehvior of ll memers of ech clss is governed y unique explici formul descriing he decy order infiniy ccurely. Min resuls. Theorem 3.. Le p RV R η), q RV R σ). Equion.) hs inermedie soluions x nr RV R m, η)) sisfying I) if nd only if σ = βm, η) m, η) nd or q)ϕ ) β d =. 3.3) The sympoic ehvior of ny such soluion x is governed y he unique formul x) X ),, where β ) X ) = ϕ ) sqs)ϕ s) β β ds. 3.4) Theorem 3.3. Le p RV R η), q RV R σ). Equion.) hs inermedie soluions x RV R ρ) wih ρ m, η), ) if nd only if in which cse βm, η) m, η) < σ < β m, η), 3.5) ρ = σ + m, η) 3.6) β nd he sympoic ehvior of ny such soluion x is given y he unique formul x) X ),, where X ) = m, η) +) + ) p) + q)r) +) + ) β m, η) ρ)ρ + )ρρ m, η))). 3.7) Theorem 3.4. Le p RV R η), q RV R σ). Equion.) hs inermedie soluions x nr RV R ) sisfying I) if nd only if σ = β m, η) nd q)ϕ ) β d <. 3.8)
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 9 The sympoic ehvior of ny such soluion x is given y he unique formul x) X 3 ),, where β X 3 ) = ϕ ) ) qs) ϕ s) β β ds. 3.9) Preprory resuls. Le x e soluion of.) on [ 0, ) such h ϕ ) x) ϕ ) s. Since lim p)x )) ) = lim x ) = lim x) = 0, lim p)x )) =, 3.0) inegring.) firs on [, ), nd hen on [ 0, ] nd finlly wice on [, ) we oin s s / x) = ξ ps) / + qu)xu) β du dr) ds, 0, 3.) 0 r where ξ = p 0 )x 0 ). To prove he exisence of inermedie soluions of ype I) i is sufficien o prove he exisence of posiive soluion of he inegrl equion 3.) for some consns 0 nd ξ > 0, which is mos commonly chieved y pplicion of Schuder-Tychonoff fixed poin heorem. Denoing y Gx) he righ-hnd side of 3.), o find fixed poin of G i is crucil o choose closed convex suse X C[ 0, ) on which G is self-mp. Since our primry gol is no only proving he exisence of generlized RV inermedie soluions, u eslishing precise sympoic formul for such soluions, choice of such suse X mus e mde ppropriely. I will e shown h such choice of X is possile y solving he inegrl sympoic relion x) s s ps) / r qu)xu) β du dr) / ds,, 3.) for some 0, which cn e considered s n pproximion infiniy) of 3.) in he sense h i is sisfied y ll possile soluions of ype I) of.). Theory of regulr vriion will in fc ensure he solviliy of 3.) in he frmework of generlized Krm funcions. As preprory seps owrd he proofs of Theorems 3.-3.4 we show h he generlized regulrly vrying funcions X i, i =,, 3 defined respecively y 3.4), 3.7) nd 3.9) sisfy he sympoic relion 3.). Lemm 3.5. Suppose h 3.3) holds. Funcion X given y 3.4) sisfies he sympoic relion 3.) for ny nd elongs o nr RV R m, η)). Proof. From 3.), 3.5) nd 3.0), we hve q)ϕ ) β β+) + m m m m )) β R)σ+βm+m l p ) β +) lq ),, nd pplying iii) of Lemm 3., in view of 3.3), we oin sqs)ϕ s) β ds β+) + m m m m )) β Rs) m l p s) β +) lq s) ds SV R, 3.3)
0 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 s, which ogeher wih 3.4) gives where X ) ϕ ) J ) = β+) + m β m m m )) β ) J β ),, Rs) m l p s) β +) lq s) ds. 3.4) Thus, since J SV R, we conclude h X nr RV R m, η)) nd rewrie he previous relion, using 3.0), s X ) R) m l p ) +) m m m m ) ) β ) J β ),. 3.5) To prove h 3.) is sisfied y X, we firs inegre q)x ) β on [, ), pplying Lemm 3. nd using 3.3) we hve m qs) X s) β ds + m ) β ) β β R) m l p ) β +) lq )J ) β β, m m m ) s. Inegring he ove relion on [, ], for ny, we oin s s = m qr) X r) β dr ds m + Rs) m l p s) β +) lq s)j s) β β ds + = m + m m m m ) m m m m ) ) β ) β β m ) β ) β β m m m ) ) β β J s) β β dj s) ) β J ) β,. Inegring he ove relion muliplied y p) nd powered y [, ), pplying Lemm 3. nd using 3.9)-i), we oin r / qω)x ω) dωdu) β dr ds pr) u m m m m ) wice on ) β β ) β m m m m ) R)m l p ) +) J ) β, s, which due o 3.5) proves h X sisfies he desired sympoic relion 3.) for ny. Lemm 3.6. Suppose h 3.5) holds nd le ρ e defined y 3.6). Funcion X given y 3.7) sisfies he sympoic relion 3.) for ny nd elongs o RV R ρ). Proof. Using 3.8) nd 3.6) we oin σ + ρβ + m = ρ + ), σ + ρβ + m = ρ m ). 3.6)
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS The funcion X given y 3.7) cn e expressed in he form where X ) λ ) +) β m +) β) R) ρ l p ) + lq ) λ = ρρ m )) m ρ) ρ + ). ) β,, 3.7) Thus, X RV R ρ). Using 3.6) nd 3.7), pplying Lemm 3. wice, we find +)β++β) +) β) m qs) X s) β ds ) β λ β σ + ρβ + m ) nd for ny, s qr) X r) β dr ds +)β+) +) β) m ) β λ β σ + ρβ + m ))σ + ρβ + m ) +)β+) +) β) m = ) β λ β ρ + ))ρ m ) = ) R) σ+ρβ+m l p ) β + lq ), R) σ+ρβ+m l p ) β + lq ) ) β R) ρ m) l p ) β + lq ) ) β +)β+) +) β) m ) β R) ρ m+ η ) l p ) β + lq ) ) β,, λ β ρ + )m ρ) where we hve used 3.9)-i) in he ls sep. We now muliply he ls relion y p), rise o he exponen / nd inegre he oined relion wice on [, ). As resul of pplicion of Lemm 3., we oin for s /ds qu) X u) β du dr) ps) r +)+β+) β)+) m ) β λ β) m ρ)ρ + )) / ρ m ) nd s pr) r u qω) X ω) β dωdu) / dr ds +) ) R) ρ m l p ) β + β + lq ), β)+) m ) β R) ρ ) l p ) β + lq ),. λ β ρρ m ) m ρ)ρ + )) / This, due o 3.7), complees he proof of Lemm 3.6. Lemm 3.7. Suppose h 3.8) holds. Then he funcion X 3 given y 3.9) sisfies he sympoic relion 3.) for ny nd elongs o nr RV R ). Proof. Using 3.), 3.), 3.8) nd pplying iii) of Lemm 3., we oin qs) ϕ s) β m ) βj3 ds ),, 3.8) m +
A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 where J 3 ) = Rs) m l q s) ds, J 3 SV R, 3.9) implying, from 3.9), m ) β X 3 ) R) β ) m + J β 3),. 3.30) This shows h X 3 RV R ). Nex, we inegre q) X 3 ) β on [, ), using 3.8) we oin qs) X 3 s) β ds m m + m ) β β β ) β β β = m + m ) β β β = m + ) β β ) β β Rs) m l q s) J 3 s) β β ds J 3 s) β β dj3 s)) ) β J 3 ) β SVR,. Furher, inegring previous relion on [, ] for ny fixed, y Lemm 3., we hve s m m + qr)x 3 r) β dr ds ) β β β ) β m + R) m l p ) + J3 ) β,. Muliply he ove y p) nd rise o he exponen /, inegring oined relion wice on [, ), using 3.9)-ii), s resul of pplicion of Lemm 3., we oin s /ds qu)x 3 u) β du dr) ps) nd m ) β m + s m m + r β β pr) ) β ) β r u β β m + m + R) m l p ) qω)x 3 ω) β dωdu) / dr ds +) J3 ) β,, ) β m m + R) J 3 ) β X3 ),, which in view of 3.30), complees he proof of Lemm 3.7. The ove heorems re sis for pplying he Schuder-Tychonoff fixed poin heorem o eslish he exisence of inermedie soluions of he equion.). In fc, inermedie soluions will e consruced y mens of fixed poin echniques, nd ferwrds we confirm h hey re relly generlized regulrly vrying funcions wih he help of he generlized L Hospil rule formuled elow. Lemm 3.8. Le f, g C [T, ). Le lim g) = nd g ) > 0 for ll lrge. 3.3)
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 3 Then f ) f) lim inf g lim inf ) g) If we replce 3.3) wih he condiion lim sup f) f ) lim sup g) g ). lim f) = lim nd g ) < 0 for ll lrge, hen he sme conclusion holds. Proofs of min resuls. Proof of he only if pr of Theorems 3., 3.3 nd 3.4. Suppose h.) hs ype I) inermedie soluion x RV R ρ) on [ 0, ). Clerly, ρ [m, ]. Using 3.) nd 3.), we oin inegring.) on [, ) p)x )) ) = qs)xs) β ds = Rs) σ+βρ l q s)l x s) β ds. 3.3) Noing h he ls inegrl is convergen, we conclude h σ + βρ + m 0 nd disinguish he wo cses: ) σ + βρ + m = 0 nd ) σ + βρ + m < 0. Assume h ) holds. Since y Lemm 3.-iii) funcion S 3 defined wih S 3 ) = Rs) m l q s)l x s) β ds, 3.33) is slowly vrying wih respec o R, inegrion of 3.3) on [ 0, ] shows h p)x )) m + R) m l p ) + S3 ),, 3.34) which is rewrien using 3.9)-ii) s x ) m + R) m l p ) + S3 ) /,. Inegriliy of x ) on [, ), nd m < 0, llows us o inegre he previous relion on [, ), implying x ) m + m + R) m l p ) + S3 ) /,, which we my inegre once more on [, ] o oin x) m m + R) S 3 ) /,. 3.35) This shows h x RV R ). Assume nex h ) holds. From 3.3) we find h p)x )) ) m + + R) σ+βρ+m l p ) + lq )l x ) β,, σ + βρ + m which y inegrion on [ 0, ] implies p)x )) m + + Rs) σ+βρ+m l p s) + lq s)l x s) β ds, 3.36) σ + βρ + m 0 s. In view of 3.0), inegrl on righ-hnd side is divergen, so σ + βρ + m 0. We disinguish he wo cses:.) σ + βρ + m = 0 nd.) σ + βρ + m > 0.
4 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Assume h.) holds. Denoe y S ) = Then S SV R nd using 3.) we rewrie 3.36) s 0 Rs) m l p s) + lq s)l x s) β ds. 3.37) x ) m + R) η/ l p ) / S ) /,. 3.38) Becuse of inegriliy of x ) on [, ] nd he fc h η +m = m m < 0, vi Lemm 3. we conclude y inegrion of 3.38) on [, ] h x ) m + R) m m l p ) + +) S ) /,. m m which ecuse inegriliy of x ) on [, ) nd m < 0, we my inegre once more on [, ) o ge m x) m m m ) R)m l p ) +) S ) /,. 3.39) implying h x RV R m ). Assume h.) holds. From 3.36), pplicion of Lemm 3. gives +) + m p)x )) l σ + βρ + m )σ + βρ + m ) R)σ+βρ+m p ) + lq )l x ) β, s, which yields +) +) x m ) σ + βρ + m )σ + βρ + m )) / R) σ+βρ+m η l p ) +) lq ) / l x ) β/,. Inegriliy of x ) on [, ] llows us o inegre he previous relion on [, ), implying +) +) x m ) σ + βρ + m )σ + βρ + m )) / where σ+βρ+m η Rs) σ+βρ+m η l p s) +) lq s) / l x s) β/ ds,, 3.40) + m 0, ecuse of he convergence of he ls inegrl. We disinguish wo cses:..) σ+βρ+m η + m = 0 nd..) σ+βρ+m η + m < 0. The cse..) is impossile ecuse he lef-hnd side of 3.40) is inegrle on [ 0, ), while he righ-hnd side is no, ecuse i is in his cse slowly vrying wih respec o R. Assume now h..) holds. Then, pplicion of Lemm 3.) in 3.40) nd inegrion of resuling relion on [, ) leds o +)+) +) m x) σ + βρ + m )σ + βρ + m )) / σ+βρ+m η + m ) Rs) σ+βρ+m η +m l p s) +) lq s) / l x s) β/ ds, 3.4)
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 5 s, which rings us o he oservion of wo possile cses:...) σ+βρ+m η + m = 0 nd...) σ+βρ+m η + m < 0. In he cse...) he inegrl in he righ-hnd side of relion 3.4) is slowly vrying wih respec o R y Proposiion.4 nd so x SV R oo. In he cse...) n pplicion of Lemm 3. gives +) +) x) m σ + βρ + m )σ + βρ + m )) / σ + βρ + m η ) σ + βρ + m η ) ) + m + m 3.4) R) σ+βρ+m η +m l p ) +) lq ) / l x ) β/,, implying h x RV R σ+βρ+m η + m ). Suppose h x is ype I) soluion of.) elonging o nr RV R m ). From he ove oservions his is possile only when.) holds, in which cse 3.39) is sisfied y x). Thus, ρ = m, σ = m β m. Using x) = R) m l x ), 3.39) cn e expressed s where l x ) K l p ) +) S ) /,, 3.43) K = m m m m ), nd S is defined y 3.37). Then 3.43) is rnsformed ino he differenil sympoic relion for S : S ) β S ) K β R) m l p ) β +) lq ),. 3.44) From 3.39), since lim x)/ϕ ) =, we hve lim S ) =. Inegring 3.44) on [ 0, ], since lim S ) β =, in view of noion 3.4) nd 3.3), we find h he second condiion in 3.3) is sisfied nd β ) S ) / Kβ J β ),, implying wih 3.43) h x) R) m l p ) +) β ) K β J ),. 3.45) Noing h in he proof of Lemm 3.5, using 3.), 3.5) nd 3.0), we hve oined expression 3.5) for X given y 3.4), 3.45) in fc proves h x) X ),, compleing he only if pr of he proof of Theorem 3.. Nex, suppose h x is soluion of.) elonging o RV R ρ), ρ m, ). This is possile only when...) holds, in which cse x sisfies he sympoic relion 3.4). Therefore, ρ = σ + βρ + m η which jusifies 3.6). An elemenry clculion shows h + m ρ = σ + m, 3.46) β m < ρ < = βm m < σ < β m,
6 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 which deermines he rnge 3.5) of σ. In view of 3.6) nd 3.46), we conclude from 3.4) h x enjoys he sympoic ehvior x) X ),, where X is given y 3.7). This proves he only if pr of he Theorem 3.3. Finlly, suppose h x is ype-i) inermedie soluion of.) elonging o nr RV R ). Then, he cse ) is he only possiiliy for x, which mens h σ = β m nd 3.35) is sisfied y x, wih S 3 defined y 3.33). Using x) = R) l x ), 3.35) cn e expressed s l x ) K 3 S 3 ) /,, where K 3 = m m +, 3.47) implying he differenil sympoic relion S 3 ) β S 3 ) K β 3 R) m l q ),. 3.48) From 3.35), since lim x)/r) = 0, we hve lim S 3 ) = 0, implying h he lef-hnd side of 3.48) is inegrle over [ 0, ). This, in view of 3.8) nd he noion 3.9), implies he second condiion in 3.8). Inegring 3.48) on [, ) nd comining resul wih 3.47), we find h ) β x) R) β K 3 J 3 ),, which due o he expression 3.30) gives x) X 3 ) s. This proves he only if pr of Theorem 3.4. Proof of he pr if of Theorems 3., 3.3 nd 3.4. Suppose h 3.3) or 3.5) or 3.8) holds. From Lemms 3.5, 3.6 nd 3.7 i is known h X i, i =,, 3, defined y 3.4), 3.7) nd 3.9) sisfy he sympoic relion 3.) for ny. We perform he simulneous proof for X i, i =,, 3 so he suscrips i =,, 3 will e deleed in he res of he proof. By 3.) here exiss T 0 > such h X) s s ps) / T 0 r qu) Xu) β du dr) / ds X), T0. 3.49) Le such T 0 e fixed. Choose posiive consns m nd M such h m β, M β. 3.50) Define he inegrl operor s / Gx) = s ) qu) xu) β du dr) ds, T0, 3.5) ps) nd le i c on he se T 0 r X = {x C[T 0, ) : mx) x) M X), T 0 }. 3.5) I is cler h X is closed, convex suse of he loclly convex spce C[T 0, ) equipped wih he opology of uniform convergence on compc suinervls of [T 0, ). I cn e shown h G is coninuous self-mp on X nd h he se GX ) is relively compc in C[T 0, ). i) GX ) X : Le x) X. Using 3.49), 3.50) nd 3.5) we oin s / Gx) M β/ s ) qu) Xu) β du dr) ds ps) T 0 r
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 7 nd M β/ X) M X), T 0, Gx) m β/ s ) ps) s s T 0 r qu) Xu) β du dr) / ds β/ X) m m X), T 0. This shows h Gx) X ; h is, G mps X ino iself. ii) GX ) is relively compc. The inclusion GX ) X ensures h GX ) is loclly uniformly ounded on [T 0, T ], for ny T > T 0. From r / Gx) = qω) xω) dωdu) β dr ds, pr) we hve Gx) ) = From he inequliy M β/ ps) ps) s T 0 r T 0 s T 0 r u qu) xu) β du dr) / ds, [T0, T ]. qu) Xu) β du dr) / ds Gx) ) 0, [T 0, T ], holding for ll x X i follows h GX ) is loclly equiconinuous on [T 0, T ] [T 0, ). Then, he relive compcness of GX ) follows from he Arzel-Ascoli lemm. iii) G is coninuous on X. Le {x n )} e sequence in X converging o x) in X uniformly on ny compc suinervl of [T 0, ). Le T > T 0 ny fixed rel numer. From 3.5) we hve where G n ) = Gx n ) Gx) T 0 s ps) / G ns) ds, [T 0, T ], ) / qs) x n s) β ds T 0 s qs) xs) β ds) /. Using he inequliy x λ y λ x y λ, x, y R + holding for λ 0, ), we see h if, hen /. G n ) s )qs) x n s) β xs) ds) β On he oher hnd, using he men vlue heorem, if < we oin G n ) ) M β s )qs)xs) β ds s )qs) x n s) β xs) β ds. Thus, using h q) xn ) β x) β 0 s n ech poin [T 0, ) nd q) x n ) β x) β M β q)x) β for T 0, while q)x) β is inegrle on [T 0, ), he uniform convergence G n ) 0 on [T 0, ) follows y he pplicion of he Leesgue domined convergence heorem. We conclude h Gx n ) Gx) uniformly on ny compc suinervl of [T 0, ) s n, which proves he coninuiy of G.
8 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Thus, ll he hypoheses of he Schuder-Tychonoff fixed poin heorem re fulfilled nd so here exiss fixed poin x X of G, which sisfies inegrl equion x) = s ) ps) s T 0 r qu) xu) β du dr) / ds, T0. Differeniing he ove expression four imes shows h x) is soluion of.) on [T 0, ), which due o 3.5) is n inermedie soluion of ype I). Therefore, he proof of our min resuls will e compleed wih he verificion h he inermedie soluions of.) consruced ove re cully regulrly vrying funcions wih respec o R. We define he funcion nd pu χ) = s ) ps) l = lim inf s T 0 r x) χ), qu) Xu) β du dr) / ds, T0, x) L = lim sup χ). By Lemms 3.5, 3.6 nd 3.7 we hve X) χ),. Since, x X, i is cler h 0 < l L <. We firs consider L. Applying Lemm 3.8 four imes, we oin L lim sup x ) χ = lim sup ) lim sup q)x) β lim sup = lim sup ps) ps) s )qs)xs) β ds s )qs)xs) β ds ) / = q)x) β x) ) β/ = L β/, χ) s / T 0 qu) xu) β du dr) r ds s / T 0 qu) Xu) r β du dr) ds ) / qs)xs) β ds lim sup qs)xs) β ds lim sup x) ) β/ X) ) / where we hve used X) χ),, in he ls sep. Since β/ <, he inequliy L L β/ implies h L. Similrly, repeed pplicion of Lemm 3.8 o l leds o l, from which i follows h L = l =, h is, x) lim = = x) χ) X),. χ) Therefore i is concluded h if p RV R η) nd q RV R σ), hen he ype-i) soluion x under considerion is memer of RV R ρ), where ρ = m or ρ = σ + m β m, ) or ρ =, ccording o wheher he pir η, σ) sisfies 3.3), 3.5) or 3.8), respecively. Needless o sy, ny such soluion x RV R ρ) enjoys one nd he sme sympoic ehvior 3.4), 3.7) or 3.9), respecively. This complees he if prs of Theorems 3., 3.3 nd 3.4.
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 9 3.. Regulrly vrying soluions of ype I). Le us urn our enion o he sudy of inermedie soluions of ype I) of equion.); h is, hose soluions x such h x) s. As in he preceding secion use is mde of he expressions 3.) nd 3.) for he coefficiens p, q nd soluions x. Since ψ SV R, ψ ) = nd ψ RV R m, η)), ψ ) = cf. 3.7) nd 3.5)), he regulriy index ρ of x mus sisfy 0 ρ m, η). If ρ = 0, hen since x) = l x ),, x is memer of nr SV R, while if ρ = m, η), hen x)/r) m,η) 0,, nd so x is memer of nr RV R m, η)). If 0 < ρ < m, η), hen x elongs o RV R ρ) nd clerly sisfies x) nd x)/r) m,η) 0 s. Therefore, i is nurl o divide he oliy of ype-i) inermedie soluions of.) ino he following hree clsses nr SV R, RV R ρ), ρ 0, m, η)), nr RV R m, η)). Our purpose is o show h, for ech of he ove clsses, necessry nd sufficien condiions for he memership re eslish nd h he sympoic ehvior infiniy of ll memers of ech clss is deermined precisely y unique explici formul. Min resuls. Theorem 3.9. Le p RV R η), q RV R σ). Then.) hs inermedie soluions x nr SV R sisfying I) if nd only if / σ = m, η) nd s) qs) ds) d =. 3.53) p) The sympoic ehvior of ny such soluion x is governed y he unique formul x) Y ),, where β s / ) β Y ) = s s r)qr) dr) ds. 3.54) ps) Theorem 3.0. Le p RV R η), q RV R σ). Then.) hs inermedie soluions x RV R ρ) wih ρ 0, ) if nd only if m, η) < σ < η + β + )m, η) 3.55) in which cse ρ is given y 3.6) nd he sympoic ehvior of ny such soluion x is governed y he unique formul x) Y ), where Y ) = m, η) +) + ) p) + q)r) +) + ρ m, η) ρ)) ρ m, η)) ρ + ) ) β. 3.56) Theorem 3.. Le p RV R η), q RV R σ). Then.) hs inermedie soluions x nr RV R m, η)) sisfying I) if nd only if / σ = η + β + )m, η), s) s β qs) ds) d <. 3.57) p) The sympoic ehvior of ny such soluion x is governed y he unique formul x) Y 3 ),, where β s ) /ds ) Y 3 ) = s r)r β β qr) dr. 3.58) ps)
0 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Preprory resuls. Le x e ype-i) inermedie soluion of.) defined on [ 0, ). I is known h lim x ) = 0, lim p) x ) x )) = lim p) x ) x ) = lim x) =. 3.59) Inegring.) wice on [ 0, ], hen on [ 0, ) nd finlly on [ 0, ], we oin, for 0, r ) / x) = c 0 + c pr) / +c 3 r 0 )+ r u)qu)xu) β du dr ds, 3.60) 0 0 s where c 0 = x 0 ), c = p 0 ) x 0 )), nd c 3 = p 0 ) x 0 )) ). From 3.60) we esily see h x) sisfies he inegrl sympoic relion r / x) r u)qu)xu) du) β dr ds,, 3.6) pr) s for some, which will ply cenrl role in consrucing generlized RVinermedie soluions of ype I). Lemm 3.. Suppose h 3.53) holds. Then he funcion Y given y 3.54) sisfies he sympoic relion 3.6) for ny nd elongs o nr SV R. Proof. Firs we give n expression for Y ) in erms of R), l p ) nd l q ). Applying Lemm 3. wice we hve s s = qu) du ds Ru) m l q u) du ds Using 3.), 3.5) nd 3.9)-ii), we hve p) + m + l + m ) R)+m p ) + lq ),. + ) / +) m s)qs) ds l + m )) / R) m p ) +) lq ) /. 3.6) Inegring he ove on [, ] for ny, we show h Y ) W β β ) Q β ) where Q ) = Rs) m l p s) +) lq s) / ds SV R, 3.63) W = m + 3.64) + + m ). From 3.63) we conclude h Y nr SV R. To verify he sympoic relion 3.6) for Y, we inegre q)y ) β wice on [, ] nd use Y nr SV R o oin s qr)y r) β dr ds m + + σ + m )σ + m ) R)σ+m l p ) + lq )Y ) β
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS s, which ogeher wih 3.63), y ssumpion 3.53) nd 3.9)-ii), yields ) / s)qs)y s) β ds p) + β 3.65) m + ) β R) m l p ) +) lq ) / β ) β + m ) Q β ), s. Inegrion of 3.65) on [, ) gives r / r u)qu)y u) du) β dr pr) + β m + + m ) ) β β ) β β m + R) m l p ) +) lq ) / Q ) β β, s, implying, y inegrion on [, ], r / r u)qu)y u) du) β dr ds s pr) W β β ) β β Rs) m l p s) +) lq s) / Q s) β β ds W β β ) β β Q s) β β dq s) = W β β ) β Q ) β,, eslishing, in view of 3.63), h Y sisfies he sympoic relion 3.6). Lemm 3.3. Suppose h 3.55) holds nd le ρ e defined y 3.6). Then he funcion Y given y 3.56) sisfies he sympoic relion 3.6) for ny nd elongs o RV R ρ). Proof. Using 3.) nd 3.8), since η +) + = m, we cn express Y ) in he form ) Y ) W R) ρ l p ) β + lq ), 3.66) where +) + C = m, ν = ρm ρ) ) ρ m )ρ + ), W = C ) β. 3.67) ν Therefore, Y RV R ρ). Nex we prove h Y sisfies he sympoic relion 3.6) nd o h end we firs inegre q)y ) β wice on [, ] for some wih pplicion of Lemm 3. nd equliies 3.9), 3.6): s W β qr)y r) β dr ds s Rr) σ+ρβ l p ) β + lq ) ) β drds W β + σ + ρβ + m )σ + ρβ + m ) m + R) σ+ρβ+m l p ) β + lq ) ) β W β + + = ρ + )ρ m ) m R) ρ m) l p ) β + lq ) ) β
A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 W β + + = ρ + )ρ m ) m R) ρ m η ) l p ) β + lq ) ) β,, implying furher h r / r u)qu)y u) du) β dr ds s pr) W β + ) / ) ρ + )ρ m ) m + Rr) ρ m l p r) β + β + l q r) dr ds m +) W β/ +) ) ρ + )ρ m )) / m ρ)ρ R)ρ l p ) β) + lq ) = W β/ C ) / ) R) ρ ν l p ) β) + lq ),, s which y 3.66) nd 3.67) proves h Y sisfies he sympoic relion 3.6). Lemm 3.4. Suppose h 3.57) holds. Then he funcion Y 3 given y 3.58) sisfies he sympoic relion 3.6) for ny nd elongs o RV R ). Proof. Using 3.5) nd 3.57), pplicion of Lemm 3. we hve s r β qr) dr ds m β + s s. Since y 3.9)-ii) we hve h from he ls relion, we conclude h s /ds s r)r qr)dr) β ps) +) β + m ) / m + ) + m + m ) s. We denoe y Q 3 ) = Rr) η +)m l p s) β + lq s)ds +) β + m η + )m )η m ) R)η m l p ) β+ + lq ), η + )m = m + ), 3.68) Rs) m l p s) β + +) lq s) / ds, 3.69) Rs) m l p s) β + +) lq s) / ds SV R 3.70) nd comining 3.69) wih 3.58) nd 3.5), we oin he following sympoic represenion for Y 3 ) in erms of R), l p ) nd l q ): Y 3 ) W β 3 R) m l p ) + β ) Q β 3),, 3.7) where m ++ + W 3 = m + )m + m + ). 3.7)
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 3 From 3.7) we conclude h Y 3 RV R m ) nd compue wih he help of Lemm 3., s qr)y 3 r) β dr ds β ) β Q β 3) W β β 3 s. Nex, using 3.57) nd 3.68) we oin s /ds s r)qr)y 3 r) dr) β ps) β β ) β β +) + m R) σ+mβ+m σ + m β + m )σ + m β + m ) l p) β+ + lq ), β β) 3 m +) +) W m + )m + m + )) / Rs) m l p s) β + +) lq s) / Q 3 s) β β ds ) β β β ) = Q β 3) β β) +) +) W3 m m + )m + m + )) / β β) +) +) Q 3 s) β β d Q3 s)) W3 m,. m + )m + m + )) / Noing h he ls expression in he previous relion is slowly vrying wih respec o R, inegrion of his relion over [, ] leds o r / r u)qu)y 3 u) du) β dr ds pr) s β β = ) Q β 3) ) Q β 3) β β) +)+) +) W3 m R) m m + )m + m + )) / W β β) 3 W / 3 R) m l p ) +,, m l p ) + nd in view of 3.7) proves h he desired inegrl sympoic relion 3.6) is sisfied y Y 3. Proof of min resuls. Proof of he only if pr of Theorems 3.9, 3.0 nd 3.. Suppose h.) hs ype-i) inermedie soluion x RV R ρ), ρ [0, m ], defined on [ 0, ). We egin y inegring.) on [ 0, ]. Using 3.), 3.), we hve p) x )) ) 0 qs)xs) β ds = 0 Rs) σ+βρ l q s)l x s) β ds, 3.73) nd conclude y 3.59) h σ + βρ + m 0. Thus, we disinguish he wo cses: ) σ + βρ + m = 0 nd ) σ + βρ + m > 0. Le cse ) hold, so h H 4 ) = 0 Rs) σ+βρ l q s)l x s) β ds = 0 Rs) m l q s)l x s) β ds, 3.74)
4 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 nd H 4 SV R. Inegrion of 3.73) on [ 0, ] wih 3.9)-ii) yields x ) m + R) m η lp ) + H4 ) = m + R) m l p ) + H4 ) /,, Since m < 0 we my inegre previous relion on [, ) nd oin vi Lemm 3. h x ) m + m + R) m H 4 ) /,. The righ hnd side in he ls relion is inegrle on [, ), ecuse m < m, u on he oher hnd in view of 3.59) he lef hnd side of ls relion isn inegrle on [, ), so we conclude h his cse is impossile. Le cse ) hold. Then, from 3.73) i follows h p) x )) ) m + + σ + βρ + m R) σ+βρ+m l p ) + lq )l x ) β which, inegred on [ 0, ] nd he fc h σ + βρ + m > 0, gives +) x + m ) σ + βρ + m )σ + βρ + m ) R) σ+βρ+m η ) / l p ) +) lq ) / l x ) β/, s, implying in view of 3.59) y inegrion on [, ), +) x + m ) σ + βρ + m )σ + βρ + m ) Rs) σ+βρ+m η ) / l p s) + lq s)l x s) β) / ds. 3.75) Thus, we furher consider he following wo possile cses:.) σ+βρ+m η + m = 0 nd.) σ+βρ+m η + m < 0. Suppose h.) holds, nd le H 3 ) = Rs) m l p s) +) lq s) / l x s) β/ ds. 3.76) Using 3.68) nd 3.9)-ii), since we hve σ + ρβ + m = m + ), inegrion of 3.75) on [ 0, ] implies x) ++) + m m + )m + ) + m ) ) /R) m l p ) + H3 ),. 3.77) Since H 3 SV R, we conclude h x RV R m ). Suppose h.) holds. Applicion of Lemm 3. in 3.75) implies x ) +)+) +) m σ + βρ + m )σ + βρ + m )) / σ+βρ+m η + m ) R) σ+βρ+m η +m l p ) +) lq ) / l x ) β/,. 3.78)
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 5 Inegring 3.78) on [ 0, ] using 3.59) we oin +)+) +) m x) σ + βρ + m )σ + βρ + m )) / σ+βρ+m η + m )) 0 Rs) σ+βρ+m η +m l p s) +) lq s) / l x s) β/ ds,. 3.79) Thus, since x) s, from he previous relion we conclude h wo possiiliies my hold:..) σ+βρ+m η + m = 0 nd..) σ+βρ+m η + m > 0. In he cse..), using 3.9)-ii) we oin σ + βρ + m =. Applicion of Lemm 3. in 3.79) leds us o where H ) = x) ++) m + ) /H ),, 3.80) + m ) Rs) m l p s) +) lq s) / l x s) β/ ds, H SV R. 3.8) 0 Thus, x SV R. Applicion of Lemm 3. in 3.79) in he cse..) gives +) ) +) x) m σ + βρ + m )σ + βρ + m )) / σ + βρ + m η ) σ + βρ + m η ) ) + m ) + m R) σ+βρ+m η +m l p ) +) lq ) / l x ) β/ ds,. 3.8) This implies h x RV σ+βρ+m η + m ). Now, le x e ype-i) inermedie soluion of.) elonging o nr SV R. Then, from he ove oservions i is cler h only he cse..) is dmissile, in which cse σ = m, nd 3.80) is sisfied y x). Using x) = l x ), from 3.80) we hve l x ) W / H ),, 3.83) where W is given y 3.64) nd H is defined y 3.8). Then, 3.83) is rnsformed ino he following differenil sympoic relion for H, H ) β H ) W β/ R) m l p ) +) lq ) /,. 3.84) From 3.59), since lim x) =, we hve lim H ) =. Inegring 3.84) on [ 0, ], using h lim H ) β =, in view of noion 3.64) nd 3.6), we find h he second condiion in 3.53) is sisfied nd H ) W β/ β ) Q β ), which wih 3.83) implies x) W β β ) Q β ),. 3.85)
6 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Noe h in Lemm 3. we hve oined expression 3.63) for Y ) given y 3.54). Therefore, 3.85) in fc proves h x) Y ),, compleing he only if pr of Theorem 3.9. Nex, le x e ype-i) inermedie soluion of.) elonging o RV R ρ) for some ρ 0, m ). Clerly, only cse..) cn hold nd hence x sisfies he sympoic relion 3.8). This mens h ρ = σ + βρ + m η + m ρ = σ + m, 3.86) β verifying h he regulriy index ρ is given y 3.6). An elemenry compuion shows h 0 < ρ < m m < σ < + m β ), showing h he rnge of σ is given y 3.55). In view of 3.6) nd 3.86), we conclude from 3.8) h x enjoys he sympoic ehvior x) Y ),, where Y is given y 3.56). This proves he only if pr of he Theorem 3.0. Finlly, le x is ype-i) inermedie soluion of.) elonging o RV R m ). Since only he cse.) is possile for x, i sisfies 3.77), where H 3 is defined y 3.76), implying ρ = m nd σ = + m β ). Using x) = R) m l x ), 3.77) cn e expressed s l x ) W / 3 l p ) + H3 ),, 3.87) where W 3 is defined y 3.7), implying he differenil sympoic relion β H 3 ) β H 3 ) W3 R) m l p ) β+ +) lq ) /,. 3.88) From 3.77), since lim R) m x) = 0, we hve h lim H 3 ) = 0, implying h he lef-hnd side od 3.88) is inegrle over [, ). This, in view of 3.69) nd noion 3.70) implies he second condiion in 3.57). Inegring 3.88) on [, ) nd comining resul wih 3.87), using he expression 3.7), we find h x) W β 3 R) m l p ) + ) β Q β 3) Y 3 ),, where Q 3 is defined wih 3.70). Thus he only if pr of he Theorem 3. hs een proved. Proof of he if pr of Theorem 3.9, 3.0 nd 3.. Suppose h 3.53) or 3.55) or 3.57) holds. From Lemms 3., 3.3 nd 3.4 i is known h Y i, i =,, 3, defined y 3.54), 3.56) nd 3.58) sisfy he sympoic relion 3.6). We perform he simulneous proof for Y i, i =,, 3 so he suscrips i =,, 3 will e deleed in he res of he proof. By 3.6) here exiss T 0 > such h r ) / r u)qu)y u) β du dr ds Y ), T0. s pr) T 0 T 0 Le such T 0 e fixed. We my ssume h Y is incresing on [T 0, ). Since 3.6) holds wih = T 0, here exiss T > T 0 such h r ) / r u)qu)y u) β Y ) du dr ds s pr) T 0, T. T 0
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 7 Choose posiive consns k nd K such h k β, β K 4, ky T ) KY T 0 ). Considering he inegrl operor r ) / Hy) = y 0 + r u)qu) yu) β du dr ds, T0, s pr) T 0 T 0 where y 0 is consn such h ky T ) y 0 K Y T 0), we my verify h H is coninuous self-mp on he se Y = {y C[T 0, ) : ky ) y) KY ), T 0 }, nd h H sends Y ino relively compc suse of C[T 0, ). Thus, H hs fixed poin y Y, which generes soluion of equion.) of ype I) sisfying he ove inequliies nd hus yields h Denoing L) = 0 < lim inf s pr) y) y) lim sup Y ) Y ) <. r nd using Y ) L) s we oin 0 < lim inf r u)qu)y u) β du) / dr ds y) y) lim sup L) L) <. Then, proceeding excly s in he proof of he if pr of Theorems 3. 3.4, wih pplicion of Lemm 3.8, we conclude h y) L) Y ),. Therefore, y is generlized regulrly vrying soluion wih respec o R wih requesed regulriy index nd he sympoic ehvior 3.54), 3.56), 3.58) depending on if q RV R σ) sisfies, respecively, 3.53) or 3.55) or 3.57). Thus, he if pr of Theorems 3.9, 3.0 nd 3. hs een proved. 4. Corollries The finl secion is concerned wih equion.) whose coefficiens p) nd q) re regulrly vrying funcions in he sense of Krm). I is nurl o expec h such equion my possess. Our purpose here is o show h he prolem of geing necessry nd sufficien condiions for he exisence of inermedie soluions which re regulrly vrying in he sense of Krm, cn e emedded in he frmework of generlized regulrly vrying funcions, so h he resuls of he preceding secion provide full informion ou he exisence nd he precise sympoic ehvior of inermedie regulrly vrying soluions of.). We ssume h p) nd q) re regulrly vrying funcions of indices η nd σ, respecively, i.e., p) = η l p ), q) = σ l q ), l p, l q SV, 4.) nd seek regulrly vrying soluions x) of.) expressed in he from x) = ρ l x ), l x SV. 4.)
8 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 We egin y noicing h in order h he condiion.) e sisfied we hve o ssume h η +. Since R) defined y 3.) due o 4.) kes he form, R) = s + η lp s) ds) / i is esy o see h η ) R RV. 4.3) An imporn remrk is h he possiiliy η = + should e excluded. If his equliy holds, hen R) is slowly vrying y 4.3), nd his fc prevens p) from eing generlized regulrly vrying funcion wih respec o R. In fc, if p RV R η ) for some η, hen here exiss f RVη ) such h p) = fr)), which implies h p SV. Bu his conrdics he hypohesis h p RVη) = RV + ). Thus, he cse η = + is impossile, nd so η mus e resriced o η > +, 4.4) in which cse R sisfies R) η η l p ) /,, 4.5) implying h R RV ) η. Since R is monoone incresing, is inverse funcion R ) is regulrly vrying of index /η ). Therefore, ny regulrly vrying funcion of index λ is considered s generlized regulrly vrying funcion wih respec o R which regulriy index is λ/η ), nd conversely ny generlized regulrly vrying funcion wih respec o R of index λ is regrded s regulrly vrying funcion in he sense of Krm of index λ = λ η )/. I follows form 4.) nd 4.) h η σ ρ ) p RV R ), q RV R ), x RV R. η η η Pu η η = η, σ σ = η, ρ ρ = η. Noe h 4.4) implies η > ecuse > 0 nd h he wo consns given y 3.8) re reduced o m, η ) = η η, m, η ) = η. I urns ou herefore h ny ype-i) inermedie regulrly vrying soluion of.) is memer of one of he hree clsses η ) η nr RV, RVρ), ρ, + η ) + η ), nr RV, while ny ype-i) inermedie regulrly vrying soluion elongs o one of he hree clsses nr SV, RVρ), ρ 0, ), nr RV). Bsed on he ove oservions we re le o pply our min resuls in Secion 3, eslishing necessry nd sufficien condiions for he exisence of inermedie regulrly vrying soluions of.) nd deermining he sympoic ehvior of ll such soluions explicily.
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 9 Firs, we se he resuls on ype-i) inermedie soluions h cn e derived s corollries of Theorems 3., 3.3 nd 3.4. Theorem 4.. Assume h p RVη) nd q RVσ). Equion.) possess inermedie soluions elonging o nr RV η ) if nd only if σ = β η β nd q)ϕ ) β d =. Any such soluion x enjoys one nd he sme sympoic ehvior x) X ) s, where X ) is given y 3.4). Theorem 4.. Assume h p RVη) nd q RVσ). Equion.) possess inermedie regulrly vrying soluions of index ρ wih ρ η, + η ) if nd only if β η β < σ < β η ) β, in which cse ρ is given y ρ = η + σ + 4.6) β nd ny such soluion x enjoys one nd he sme sympoic ehvior p) q) ) β x) ρρ )),. η) ρ + η ) Theorem 4.3. Assume h p RVη) nd q RVσ). Equion.) possess inermedie soluions elonging o nr RV ) + η if nd only if σ = β η ) β nd q) ϕ ) β d <. Any such soluion x enjoys one nd he sme sympoic ehvior x) X 3 ) s, where X 3 ) is given y 3.9). Proof. To prove Theorem 4. nd 4.3 we need only o check h σ = m, η )β m, η ) σ = β η β, σ = β m, η ) σ = β η ) β, nd o prove Theorem 4. i suffices o noe h ρ = σ + m, η ) β ρ = + σ η +, β nd o comine he relion 4.5) wih he equliy m, η ) +) + [m, η ) ρ )ρ + ) ρ m, η )ρ ) ] = η)ρ + η )ρρ )). Similrly, we re le o gin hrough knowledge of ype-i) inermedie regulrly vrying soluions of.) from Theorems 3.9, 3.0 nd 3..
30 A. B. TRAJKOVIĆ, J. V. MANOJLOVIĆ EJDE-06/9 Theorem 4.4. Assume h p RVη) nd q RVσ). Equion.) possess inermedie nonrivil slowly vrying soluions if nd only if / σ = η nd s) qs) ds) d =. p) The sympoic ehvior of ny such soluion x is governed y he unique formul x) Y ),, where Y ) is given y 3.54). Theorem 4.5. Assume h p RVη) nd q RVσ). Equion.) possess inermedie regulrly vrying soluions of index ρ wih ρ 0, ) if nd only if η < σ < η β, in which cse ρ is given y 4.6) nd he sympoic ehvior of ny such soluion x is governed y he unique formul p) q) ) β x) ),. ρ ρ) η ) ρ + η ) Theorem 4.6. Assume h p) RVη) nd q) RVσ). Equion.) possess inermedie nonrivil regulrly vrying soluions of index if nd only if /d σ = η β nd s)s qs)ds) β <. p) The sympoic ehvior of ny such soluion x is governed y he unique formul x) Y 3 ),, where Y 3 ) is given y 3.58). The ove corollries comined wih Theorems..4 enle us o descrie in full deils he srucure of RV-soluions of equion.) wih RV-coefficiens. Denoe wih R he se of ll regulrly vrying soluions of.) nd define he suses Rρ) = R RVρ), r Rρ) = R r RVρ), nr Rρ) = R nr RVρ). Corollry 4.7. Le p RVη), q RVσ). i) If σ < β η β, or σ = β η β nd J 3 <, hen R = r R η ) + η ) r R r R0) r R). ii) If σ = β η β nd J 3 =, hen R = nr R η ) + η ) r R r R0) r R). iii) If σ β η β, β η ) β ), hen R = R σ + + η ) + η ) r R r R0) r R). β iv) If σ = β η ) β nd J 4 <, hen R = r R + η ) + η ) nr R r R0) r R). v) If σ = β η ) β nd J 4 =, or σ β η ) β, η ), or σ = η nd J <, hen R = r R0) r R).
EJDE-06/9 ASYMPTOTIC BEHAVIOR OF INTERMEDIATE SOLUTIONS 3 vi) If σ = η nd J =, hen R = nr R0) r R). vii) If σ η, η β ), hen R = R σ + + η ) r R). β viii) If σ = η β nd J <, hen R = r R) nr R). ix) If σ = η β nd J =, or σ > η β, hen R =. Acknowledgemens. The uhors re greful o he nonymous referee who mde numer of useful suggesions which improved he quliy of his pper. Boh uhors cknowledge finncil suppor hrough he Reserch projec OI- 74007 of he Minisry of Educion nd Science of Repulic of Seri References [] N. H. Binghm, C. M. Goldie, J. L. Teugels; Regulr Vriion, Encyclopedi of Mhemics nd is Applicions, 7, Cmridge Universiy Press, 987. [] J. Jroš, T. Kusno; Self-djoin differenil equions nd generlized Krm funcions, Bull. Clsse Sci. M. N., Sci. Mh., Acd. Sere Sci. Ars, CXXIX, No. 9 004), 5 60. [3] J. Jroš, T. Kusno; Slowly vrying soluions of clss of firs order sysems of nonliner differenil equions, Ac Mh. Univ. Comenine, vol. LXXXII, 03), no., 65 84. [4] J. Jroš, T. Kusno; Exisence nd precise sympoic ehvior of srongly monoone soluions of sysems of nonliner differenil equions, Diff. Equ. Applic., 5 03), no., 85 04. [5] J. Jroš, T. Kusno; Asympoic Behvior of Posiive Soluions of Clss of Sysems of Second Order Nonliner Differenil Equions, Elecronic Journl of Quliive Theory of Differenil Equions 03, no. 3, 3. [6] J. Jroš, T. Kusno; On srongly monoone soluions of clss of cyclic sysems of nonliner differenil equions, J. Mh. Anl. Appl. 47 04), 996 07. [7] J. Jroš, T. Kusno; Srongly incresing soluions of cyclic sysems of second order differenil equions wih power-ype nonlineriies, Opuscul Mh. exf35 05), no., 47 69 [8] J. Jroš, T. Kusno, J. Mnojlović; Asympoic nlysis of posiive soluions of generlized Emden-Fowler differenil equions in he frmework of regulr vriion, Cen. Eur. J. Mh., ), 03) 5 33 [9] K. Kmo, H. Usmi; Nonliner oscillions of fourh order qusiliner ordinry differenil equions, Ac Mh. Hungr., 3 3) 0), 07. [0] T. Kusno, J. Mnojlović; Asympoic ehvior of posiive soluions of suliner differenil equions of Emden-Fowler ype, Compu. Mh. Appl. 6 0), 55 565. [] T. Kusno, J. Mnojlović; Posiive soluions of fourh order Emden-Fowler ype differenil equions in he frmework of regulr vriion, Appl. Mh. Compu., 8 0), 6684 670. [] T. Kusno, J. Mnojlović; Posiive soluions of fourh order Thoms-Fermi ype differenil equions in he frmework of regulr vriion, Ac Applicnde Mhemice, 0), 8 03. [3] T. Kusno, J. Mnojlović, V. Mrić; Incresing soluions of Thoms-Fermi ype differenil equions - he suliner cse, Bull. T. de Acd. Sere Sci. Ars, Clsse Sci. M. N., Sci. Mh., CXLIII, No.36, 0), 36. [4] T. Kusno, J. Mnojlović, J. Milošević; Inermedie soluions of second order qusiliner differenil equion in he frmework of regulr vriion, Appl. Mh. Compu., 9 03), 878-89.