DIFFERENIAL EQUAIONS AND CONROL PROCESSES 4, 8 Elecroic Joural, reg. P375 a 7.3.97 ISSN 87-7 hp://www.ewa.ru/joural hp://www.mah.spbu.ru/user/diffjoural e-mail: jodiff@mail.ru Oscillaio, Secod order, Half-liear differeial equaios, Dampig. OSCILLAION CRIERIA FOR SECOND ORDER HALF-LINEAR DIFFERENIAL EQUAIONS WIH DAMPING ERM E. M. Elabbasy¹, A. A. S. Zaghrou² ad H. M. Elshebay³ ¹Deparme of Mahemaics, Faculy of Sciece, Masoura Uiversiy, Masoura, 3556, Egyp. E-mail: emelabbasy@mas.edu.eg.,3 Deparme of Mahemaics, Faculy of Sciece, AI-Azhar Uiversiy, Nasr-ciy, P.O. Box 99. 765 Cairo, Egyp. E-mail: afaf@yahoo.com. Absrac By usig averagig fucios, several ew oscillaio crieria are esablished for he half-liear damped differeial equaio - - dx dx - dx dx - r() ψ(x) +p() ϕ x x,r() ψ(x) +q() x x=, d d d d ad he more geeral equaio - - d dx dx dx dx r() ψ(x) +p() ϕ g(x),r() ψ(x) +q()g(x)=, d d d d d where p,q,r:[, ) R ad ψ,g: R R are coiuous, r()>, p() ad ψ(x)>, xg(x)> for x, > a fixed real umber. Our resuls geeralize ad exed some kow oscillaio crieria i he lieraure.
. INRODUCION Differeial Equaios ad Corol Processes, 4, 8 We are cocered wih he oscillaio of soluios of secod order differeial equaios wih dampig erms of he followig form - - dx dx - dx dx - r() ψ(x) +p() ϕ x x,r() ψ(x) +q() x x=, (.) d d d d ad he more geeral equaio - - d dx dx dx dx r() ψ(x) +p() ϕ g(x),r() ψ(x) +q()g(x)=, d d d d d (.) where r C[[, ),R + ], p C[[, ),[, )], q C[[, ),R], ψ C[R,R + ] ad g C¹[R,R] such ha xg(x)> for x ad g (x)> for x. φ is defied ad coiuous o R R-{} wih uφ(u,v)> for uv ad φ(λu, λv)= λφ(u,v) for <λ< ad (u,v) R R-{}. By oscillaio of equaio (.)[(.)], we mea a fucio x C¹([ x, ),R) for some - x, which has he propery ha dx dx ad saisfies equaio r() ψ (x) C ([ x, ), R) d d (.)[(.)] o [ x, ). A soluio of equaio (.)[(.)] is called oscillaory if i has arbirarily large zeros oherwise, i is called ooscillaory. Fially, equaio (.)[(.)] is called oscillaory if all is soluios are oscillaory. I Secio we provide sufficie codiios for he oscillaio of all soluios of (.). Several paricular cases of (.) have bee discussed i he lieraure. o cie a few examples, he differeial equaio d dx dx d d d - - r() +q() x x=, has bee sudied by [5]-[]. A more geeral equaio ha (.3) - d dx dx - r() ψ (x) +q() x x=, d d d (.3) (.4) has bee cosidered by [] ad []. Our resuls iclude, as special cases, kow oscillaio heorems for (.3), (.4). I paricular, we exed ad improve he resuls obaied i [3], [7], [] ad [4]. I Secio 3 we will esablish oscillaio crieria for equaio (.). Several paricular cases of (.) have bee discussed i he lieraure. o cie a few examples, he differeial equaio - d dx dx r() ψ (x) +q()g(x)=, d d d esablished by [6] ad [9] cosidered a special case of his equaio as (.5) Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural
Differeial Equaios ad Corol Processes, 4, 8 - d dx dx r() d d d +q()g(x)=, Our resuls i his secio geeralize ad improve [7], [], [3] ad [8]. (.6). OSCILLAION RESULS FOR (.) I order o discuss our mai resuls, we eed he followig well-kow iequaliy which is due o Hardy e al. [4]. Lemma.. If X ad Y are oegaive, he λ λ λ X +( λ-)y -λxy, λ>, where equaliy hol if ad oly if X=Y. heorem.. Suppose, i addiio o codiios ϕ (,z) z for all z, (.) < ψ(x) γ for all x, (.) ha here exis differeiable fucios k, ρ :[, ) (, ) wih ρ() ad he coiuous fucio H : D {(,s): s } R ad h : D {(,s): >s } R, ad H has a coiuous ad oposiive parial derivaive o D wih respec o he secod variable such ha H(,)= for, H(,s)> for >s, ad - δ - (H(,s)k(s))=h(,s)(H(,s)k(s)) for all (,s) D. δs he equaio (.) is oscillaory if where γρ(s)r(s)r (, s) lim sup {H(,s) ρ(s)k(s)q(s)- }=, (.3) H(, ) / d ρ(s) R(,s)=h(,s)+(H(,s)k(s)) +p(s). ρ(s) Proof. O he corary we assume ha (.) has a ooscillaory soluio x(). We suppose wihou loss of geeraliy ha x()> for all [, ). We defie he fucio ω() as Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 3
Differeial Equaios ad Corol Processes, 4, 8 - dx dx r() ψ(x) d d ω()= ρ() for -. x x hus - d dx dx r() ψ(x) dx ()r() (x) () () d d d ρ ψ dω dρ d = ω ()+ ρ () ( ). - d ρ() d x x x his ad equaio (.) imply - dω() dρ() ω() - ω()- ρ()[q()+p() ϕ(, )] ( )[ γρ()r()] ω(). d ρ() d ρ() From (.) we obai - dω() dρ() ω - ()- ρ ()q()-p() ω () ( )[ γρ ()r()] ω (). d ρ() d Muliply he above iequaliy by H(,s)k(s) ad iegrae from o we obai Sice d ρ(s) H(,s)k(s) ρ(s)q(s) H(,s)k(s) ω(s) ρ(s) dω(s) H(,s)k(s) ω(s) -H(,s)k(s)p(s) ω(s)-h(,s)k(s) ( ). /( -) [ γρ(s)r(s)] dω(s) d H(,s)k(s) =H(,)k() ω()+ (H(,s)k(s)) (s) ω he previous iequaliy becomes =H(,)k() ω()- h(,s)(h(,s)k(s)). - ω(s). Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 4
Hece we have H(,s)k(s) ρ(s)q(s) H(,)k() ω() Differeial Equaios ad Corol Processes, 4, 8 d ρ(s) + H(,s)k(s) ω(s) + h(,s)(h(,s)k(s)) ω(s) ρ(s) H(,s)k(s) ω(s) + H(,s)k(s)p(s) ω(s) ( ). /( -) [ γρ(s)r(s)] H(,s)k(s) ρ(s)q(s) H(,)k() ω() H(,s)k(s) ω(s) ω /( -) [ γρ(s)r(s)] + R(,s)((H(,s)k(s)) (s) ( ). (.4) Defie / X = [ γρ(s)r(s)] [ R(, s)], Y = γρ ω / /( ) H(,s)k(s)[ (s)r(s)] (s) -. Sice >, he by Lemma., ( )/ ( )H(,s)k(s) γρ(s)r(s) R(,s)(H(,s)k(s)) (s) (s) ω ω R(,s), /( ) [ γρ(s)r(s)] for all >s. Moreover, by (.4) we also have for every, or γρ(s)r(s)r (,s) H(,s)k(s) ρ(s)q(s) H(,)k() ω()+, γρ(s)r(s)r (,s) H(,s)k(s) ρ(s)q(s)- H(,)k() () ω H(, )k() ω(). (.5) We use he above iequaliy for = o obai Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 5
Differeial Equaios ad Corol Processes, 4, 8 γρ(s)r(s)r (,s) H(,s)k(s) ρ(s)q(s)- H(, )k( ) ω( ). herefore, γρ(s)r(s)r (,s) H(,s)k(s) ρ(s)q(s)- γρ(s)r(s)r (,s) = H(,s)k(s) ρ(s)q(s)- γρ(s)r(s)r (,s) + H(,s)k(s) ρ(s)q(s)-. H(, ) k(s) ρ(s) q(s) +k( ) ω( ), for all. his gives γρ(s)r(s)r (,s) lim sup H(,s)k(s) ρ(s)q(s)- H(, ) k(s) ρ(s)q(s)+k() ω() <, which coradics he assumpio (.3). his complees he proof. Corollary.. If he codiio (.3) is replaced by he codiios lim sup H(,s)k(s) ρ(s)q(s) =, H(, ) lim sup ρ(s)r(s)r (,s) <, H(, ) he he coclusio of heorem. remais valid. heorem.. Suppose ha (.) ad (.) are saisfied ad le he fucios H, h, ad k be he same as i heorem.. Moreover, assume ha ad H(,s) <if lim if, (.6) s H(, ) Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 6
Differeial Equaios ad Corol Processes, 4, 8 lim if ρ(s)r(s)r (,s)<, (.7) H(, ) hold. If here exiss a fucio Ω C([, ),R) such ha /( ) Ω+ ( s ) lim sup =, (.8) ad for every, ( k(s) ρ(s)r(s) ) /( ) γρ(s)r(s)r (,s) lim if { Hs (, ) ρ(s)k(s)q(s)- } (), (.9) H(,) Ω where Ω () = max{ Ω(),}, he equaio (.) is oscillaory. + Proof. O he corary we assume ha (.) has a ooscillaory soluio x(). We suppose wihou loss of geeraliy ha x()> for all [, ). Defiig ω () as i he proof of heorem., we obai (.4) he we ge γρ(s)r(s)r (,s) Hs (, ) ρ(s)k(s)q(s)- k() ω()-j(,) H(,) where γρ(s)r(s)r (,s) ( -)/ (, ) = { ( (, )k(s)) R(,s) (s) H(,) Hs ω ( ) Hs (, )k(s) + ω(s) ( -) }, /( ) J ( γρ( srs ) ( )) for all. hus, by (.9), we have ad Ω() k() ω() for all (.) limsup J(,)< for all. (.) Le o ( -)/ ( ) = R(,s)( (, )k(s)) ω(s), H(, ) F Hs ( ) ( ) () = (,)k(s)( γρ()()) ω(s) -, H(, ) G Hs srs for >. he by (.4) ad (.) we ge ha o Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 7
( -) H(, s)k(s) lim sup G()-F() lim sup { ω(s) /( ) H(, ) ( γρ( s) r( s)) Now, we claim ha ( ) -R(,s) H(,s)k(s) Suppose o he corary ha Differeial Equaios ad Corol Processes, 4, 8 ( -)/ ω (s) } - lim supj(, )<. (.) /( -) ω(s) k(s) <. (.3) /( ) ( ρ( srs ) ( )) /( -) ω(s) k(s) =. (.4) /( ) ( ρ( s) r( s)) By (.6), here is a posiive cosa η saisfyig H(,s) if lim if > η. (.5) s H(, ) O he oher had, by (.4) for ay posiive umber µ here exiss a > such ha so for all From (.5) we have /( -) ω(s) γ µ k(s) for all, ρ srs η /( ) ( ( ) ( )) ( ) s /( -) ( ) γ ω(u) G () = H(,s)d ku ( ) du /( ) H(, ) ( ρ( uru ) ( )) s /( -) ( ) γ δh(,s) ω(u) = d k ( u ) du /( ) H(, ) δs ( ρ( u) r( u)) s ( ) γ δh(,s) ω(u) H(, ) d ( ) δs ku ( ρ ( uru ) ( )) /( -) /( ) ( ) H(,s) µ H (, ) =. (.6) γ µ γ δ ( ) η H(, ) δs ηh(, ) du Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 8
Differeial Equaios ad Corol Processes, 4, 8 H (, ) lim if > η>. H(, ) H (, ) So here exiss such ha H(, ) for all, ad sice µ is arbirary cosa, we coclude ha η for all. herefore by (.6) G() η lim G()=. (.7) Nex, cosider a sequece { } =i (, ) wih lim = ad such ha [ ] [ ] lim G( )-F( ) = limsup G()-F(). I view of (.), here exiss a cosa M such ha G( )-F( ) M for all sufficie large. (.8) I follows from (.7) ha his ad (.8) give lim G( )=. (.9) lim F( )=. (.) he, by (.8) ad (.9), F( ) -M - - > G( ) G( ) for large eough. hus, F( ) > G( ) for large eough. his ad (.) imply ha F ( ) lim =. - G ( ) (.) O he oher had, by he Holder's iequaliy, we have Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 9
Differeial Equaios ad Corol Processes, 4, 8 ( -)/ ( ) = R(,s)(H(,s)k(s)) ω(s) H(, ) F ω(s) H(,s)k(s) H(, ) ( ρ( srs ) ( )) ( )/ /( -) /( ) γ ρ()() srsr (,s) ( ) H(,) G ( ) ( ) γ H(, ) ( )/ / ( )/ / ρ()() srsr (,s), ad herefore, γ - ( ) F ( ) G ( ) ( ) H(, ) γ ( ) ( ) ηh(, ) for all large. I follows from (.) ha ρ( s) r( s) R (,s) ρ( s) r( s) R (,s) ha is, lim ρ( s) r( s) R (,s)=, (.) H(, ) lim ρ( s) r( s) R (,s)=, H(, ) which coradics (.7). Hece, (.3) hol. he, i follows from (.) ha /( ) /( ) Ω ( s ) ω( s ) + /( ) /( ) k( s) k( s) <, ( k(s) ρ(s)r(s) ) ( ρ(s)r(s) ) which coradics (.8). his complees he proof of heorem.. heorem.3. Suppose ha (.) ad (.) are saisfied ad le he fucios H, h, ρ ad k be he same as i heorem.. Moreover, assume ha Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural
Differeial Equaios ad Corol Processes, 4, 8 lim if Hsρ (, ) (s)k(s)q(s)<, (.3) H(, ) ad (.6) hold. If here exiss a fucio Ω C([, ),R) such ha (.8) ad (.9) hold, he equaio (.) is oscillaory. Proof. Wihou loss of geeraliy, we may assume ha here exiss a soluio x() of equaio (.) such ha x() o [, ) for some sufficiely large. Defie ω() as of heorem.. As i he proofs of heorem. ad., we ca obai (.4), (.5) ad (.). From (.3) i follow ha limsup G()-F() k( ) ω( ) -lim if H (, s) ρ(s)k(s)q(s)<, (.4) H(, ) where F() ad G() are defied as i he proof of heorem.. By (.9) we have Ω( ) limif H (, s) ρ(s)k(s)q(s) H(, ) his ad (.9) imply ha lim if H(, ) ρ(s)r(s)r (, s). lim if ρ(s)r(s)r (, s) <. H(, ) Cosiderig a sequece { } = i (, ) wih lim = ad such ha lim ρ(s)r(s)r (, s) H(, ) = limif ρ(s)r(s)r (, s)<. (.5) H(, ) Now, suppose ha (.4) hol. Wih he same argume as i heorem., we coclude ha (.7) is saisfied. By (.4), here exiss a cosa M such ha (.8) is fulfilled. he, followig he procedure of he proof of heorem., we see ha (.) hol, which coradics (.5). his coradicio proves ha (.5) fails. he remaider of he proof is similar o ha of heorem., so we omi he deails. his complees he proof of heorem.3. heorem.4. Suppose ha (.) ad (.) are saisfied ad le he fucios H, h, ρ ad k be he same as i heorem.. Moreover, suppose ha Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural
Differeial Equaios ad Corol Processes, 4, 8 lim sup ρ(s)r(s)r (, s)<, (.6) H(, ) ad (.6) hold. If here exiss a fucio Ω C([, ),R) such ha (.8) hold for every ad γρ(s)r(s)r (,s) limsup H(, s) ρ(s)k(s)q(s)- (), (.7) H(,) Ω he equaio (.) is oscillaory. he proof of heorem.4 ca be carried ou as he proof of heorem. ad herefore i will be omied. Remark.. If = ad p(), r() ad ψ(x), he heorem.,. exed ad improve heorem i [7]. Remark.. If p(), r() ad ψ(x), he heorem.,.3 ad.4 exed ad improve heorem, 4 ad 3 of Li [3], respecively. Remark.3. If p(), he heorem.-.4 exed ad improve heorem, 4, 6 ad 5 i [], respecively. Example.. Cosider he differeial equaio We oe ha Le he d d - x ( ) dx -5/ ( + e ) + x()=, >. d = ad ψ (x)=+e - x(). ρ(s)=, k(s)=s² ad H(,s)=(-s)². γρ(s)r(s)r (,s) limsup H(,s)k(s) ρ(s)q(s)- H(, ) 3 9 6 = limsup s s s. (- ) + = s s Hece, his equaio is oscillaory by heorem.. while, Ayalar ad iryaki [], fails. Example.. Cosider he differeial equaio We o ha d d +cos² +3x² dx +3cos² +x² d dx + + x=, =. d +3x² ψ x 3 = γ, =. +x² < ( ) = Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural
Differeial Equaios ad Corol Processes, 4, 8 If we ake ρ()=, k()=, H(,s)=(-s)², he γρ(s)r(s)r (,s) lim sup H(,s)k(s) ρ(s)q(s)- H(, ) 3 +cos²s s s = lim sup (-s) s s (-) + 4 +3cos²s s s s s lim sup (-s) s s + (-) 4 s s =. Hece, his equaio is oscillaory. Oe such soluio of his equaio is x()=cos. 3. OSCILLAION RESULS FOR (.) heorem 3.. Suppose ha (.) ad g'(x) ( ψ (x) g(x) ) δ - /( -) > for x, (3.) hold, ad le he fucios H, h ad k be he same as i heorem.. Moreover, suppose ha here exis ρ C¹([, ),(, )). he equaio (.) is oscillaory if where β ρ(s)r(s)q (,s) lim sup H(,s)k(s) ρ(s)q(s)- =, H(, ) δ d ρ(s) β - ρ (s) / = ad Q(,s)= h(,s)- p(s) (H(,s)k(s)). Proof. Wihou loss of geeraliy, we may assume ha here exiss a soluio x() of equaio (.) such ha x() o [, ) for some sufficiely large. Defie ω() as hus, - dx dx r() ψ(x) d d ω()= ρ() for. g(x) Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 3
his ad equaio (.) imply From (.) ad (3.) we have Differeial Equaios ad Corol Processes, 4, 8 - d dx dx r() ψ(x) dω() dρ() d d d dx = ω ()+ ρ () d ρ() d g(x) d g'(x) ω() ( ψ(x) g(x) ) [ ρ()r()] - /( -) /( -) dω() dρ() ω() ω()- ρ()[q()+p() ϕ(, )] d ρ() d ρ(). g'(x) ω() ( ψ(x) g(x) ) [ ρ()r()] - /( -) /( -). dω() dρ() ω() ω()- ρ()q()-p() ω() δ. /( -) d ρ() d [ ρ()r()] Muliply he above iequaliy by H(,s)k(s) ad iegrae from o we obai d ρ(s) H(,s)k(s) ρ(s)q(s) H(,s)k(s) p(s) ω(s) ρ(s) Sice - dω(s) dω(s) - -H(,s)k(s) -δ H(,s)k(s)[ γρ(s)r(s)] ω(s). dω(s) δ -H(,s)k(s) =H(,)k() ω()+ (H(,s)k(s)) ω(s) δs he previous iequaliy becomes Defie ( -) =H(,)k() ω()- hs (, )(H(,s)k(s)) ω(s). ( -)/ ( -) H(,s)k(s) ρ(s)q(s) H(,)k() ω()+ Q(,s)(H(,s)k(s)) ω(s) βh(,s)k(s) ω(s) ( ). (3.) /( -) [ ρ(s)r(s)] Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 4
X Y = β = ( )/ / Differeial Equaios ad Corol Processes, 4, 8 [ ρ(s)r(s)] [ Q(, s)], ( )/ ( [ ] ) ( )/ / β ρ ω /( ) H(,s)k(s) [ (s)r(s)] (s). he use he lemma., we have /( ) ( ) ( )/ βh(,s)k(s) ω(s) β ρ(s)r(s)q (,s) Q(,s)(H(,s)k(s)) ω(s) ( ). /( ) [ ρ(s)r(s)] From (3.) we have ( ) β ρ(s)r(s)q (,s) H(,s)k(s) ρ(s)q(s)- H(,)k() ω(). he remaider of he proof procee as i he proof of heorem.. he proof is complee. Followig he procedure of he proof of heorem. ad.3, we ca also prove he followig heorems. heorem 3.. Suppose ha (.) ad (3.) hold, ad le he fucios H, h ad k be he same as i heorem.. If here exis wo fucios ρ C¹([, ),(, )) ad Ω C([, ),R) such ha lim if ρ(s)r(s)q (,s)<, (3.3) H(, ) ad ha for every, ( ) β ρ(s)r(s)q (,s) lim if { Hs (, ) ρ(s)k(s)q(s)- } (), (3.4) H(,) Ω ad (.8) hold, he every soluio of (.) is oscillaory. heorem 3.3. Suppose ha (.) ad (3.) hold, ad le he fucios H, h ad k be he same as i heorem.. If here exis wo fucios ρ C¹([, ),(, )) ad Ω C([, ),R) such ha (.8), (3.4) ad lim if Hsρ (, ) (s)k(s)q(s)<, (3.5) H(,) hold, he every soluio of (.) is oscillaory. heorem 3.4. Suppose ha (.) ad (3.) are saisfied. Le he fucios H, h ad k be he same as i heorem.. If here exis wo fucios ρ C¹([, ),(, )) ad Ω C([, ),R) such ha Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 5
Differeial Equaios ad Corol Processes, 4, 8 lim sup ρ(s)r(s)q (, s)<, (3.6) H(, ) ad for every, β ρ(s)r(s)q (,s) lim sup Hs (, ) ρ(s)k(s)q(s)- (), (3.7) H(,) Ω ad (.8) hold, he every soluio of (.) is oscillaory. Remark 3.. If p() ad =, he heorem 3. ad 3.4 exed ad improve heorem 4 ad 3 of Grace [3]. Remark 3.. If p() ad H(,s)=(-s)ⁿ from heorem 3., we obai heorem of Agarwal ad Grace []. Example 3.. Cosider he differeial equaio We oe ha Le he d d dx d - - 3 x ( ) + x ()=, >. =, r()= -, q()= - ad g '( x) = 3. ψ ( x ) ρ(s)=, k(s)=s² ad H(,s)=(-s)². β ρ(s)r(s)q (,s) lim sup H(,s)k(s) ρ(s)q(s)- H(, ) 3 3 = lim sup s s s. (- ) + + = 4s s 3 Hece, his equaio is oscillaory by heorem 3.. Example 3.. Cosider he differeial equaio +cos² +3x² dx +3cos² +x² d If we ake ρ()=, k()=, H(,s)=(-s)², he dx + d (+3cos²) 3 + (x+x )=, =. g '( x) = + = δ = ψ ( x ) x,. We o ha Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 6
Differeial Equaios ad Corol Processes, 4, 8 β ρ(s)r(s)q (,s) lim sup H(,s)k(s) ρ(s)q(s)- H(, ) (-s) s +cos²s s s = lim sup 4 s (-) + +3cos²s +3cos²s s s s s s s lim sup (-s) 4 s + =. (-) 4 4 s s s Hece, his equaio is oscillaory. Oe such soluio of his equaio is x()=cos. REFERENCES [] Agarwal, R.P., Grace, S.R. O he oscillaio of cerai order differeial equaios, Georgia Mah. J.7 (),- 3. [] Ayalar, B. ad iryaki, A. Oscillaio heorems for oliear secod-order differeial equaios, Compuers ad Mahemaics wih Applicaios 44 (), 59-538. [3] Grace, S.R. Oscillaio heorems for oliear differeial equaios of secod order, J. Mah. Aal. Appl. 7 (99), -4. [4] Hardy, G.H.; Lilewood, J.E. ad Polya, G. Iequaliies, d ediio, Cambridge Uiversiy Press, Cambridge, (988). [5] Hsu, H.B. ad Yeh, C. C. Oscillaio heorems for secod order half-liear differeial equaios, Appl. Mah. Le. 9 (996), 7-77. [6] Kusao,. ad Naio, Y. Oscillaio ad ooscillaio crieria for secod order quasiliear differeial equaios, Aca Mah. Hugar. 76(-) (997), 8-99. [7] Kusao,. ad Yoshida, N. Nooscillaio heorems for a class of quasiliear differeial equaios of secod order, J. Mah. Aal. ad Appl. 89(995), 5-7. [8] Li, H.J. ad Yeh, C. C. A iegral crierio for oscillaio of oliear differeial equaios, Mah. Japoica 4 (995), 85-88. [9] Li, H.J. ad Yeh, C.C. Nooscillaio crieria for secod order half-liear differeial equaios, Appl. Mah. Le. 8 (995), 63-7. [] Li, H.J. ad Yeh, C. C. Nooscillaio heorems for secod order quasiliear differeial equaios, Publ. Mah. Debrece 47/3-4(995), 7-79. [] Li, H.J. ad Yeh, C. C. Oscillaio crieria for oliear differeial equaios, Houso Jour. Mah. (995), 8-8. [] Li, H.J. ad Yeh, C. C. Oscillaio of half-liear secod order differeial equaios, Hiroshima Mah. Jour. 5(995), 585-594. [3] Li, H.J. Oscillaio crieria for half-liear secod order differeial equaios, Hiroshima Mah. J.5 (995), 57-583. [4] Li, W.. Zhog, C.K. ad Fa, X.L. Oscillaio crieria for secod-order halfliear ordiary differeial equaios wih dampig, Joural of mahemaics, volume 33, umber 3 fall (3). Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 7
Differeial Equaios ad Corol Processes, 4, 8 [5] Lia, W.C., Yeh, C.C. ad Li, H.J. he disace bewee zeros of a oscillaory soluio o a half-liear differeial equaios, Compuers Mah. Applic. 9 (995), 39-43. [6] Maojlovic, J. V. Oscillaio heorems for oliear differeial equaios of secod order, E. J. Qualiaive heory of Diff.Equ. (), -. [7] Philos, Ch.G. Oscillaio heorems for liear differeial equaios of secod order, Arch. Mah. 53 (989), 483-49. [8] Wag, J. O secod order quasiliear oscillaio, Fukcial. Ekvac.4 (998), 5-54. [9] Wog, P. J. Y. ad Agarwal, R. P. Oscillaory behavior of soluios of cerai secod order oliear differeial equaios, J. Mah. Aal. ad Appl. 98 (996),337-354. [] Wu, H.W. Wag, Q.R. ad Xu, Y.. Oscillaio ad Asympoics for oliear secod-order differeial equaios, compuers ad Mahemaics wih Applicaios 48 (4), 6-7. Elecroic Joural. hp://www.eva.ru/joural, hp://www.mah.spbu.ru/user/diffjoural 8