Researh Inveny: Inernaional Journal Of Engineering And Siene Issn: 78-47, Vol, Issue 4 (Feruary 03), Pp 36-4 WwwResearhinvenyCom New Osillaion Crieria For Seond Order Nonlinear Differenial Equaions Xhevair Beqiri Sae Universiy of Teova Asra: Consider he seond order nonlinear differenial equaions wih damping erm and osillaion s naure of ( ) ))' p( ) ) q( ) f ( x( )) k( )) 0 0 () o used osillaory soluions of differenial equaions ( ( ) ))' ( ) f ( x( )) k( )) 0 () where () and () saisfy ondiions given in his work paper Our resuls exend and improve some previous osillaion rieria and over he ases whih are no overed y known resuls In his paper, y using he generalized Riai s ehnique and posiive funion (, of Philo we ge a new osillaion and nonosillaion rieria for () Key Wor: equaions, differenial, inerval, rieria, damping, seond order e I Inroduion In his paper we are eing onsidered he osillaion soluions in he seond order nonlinear funional differenial equaion: ( ) ))' p( ) ) q( ) f ( x( )) k( )) 0, 0 () where, p, q, C([ 0, ], ) and f, g(, ) In he following we shall assume he ondiions A) For all I, p ( ) 0, ( ) 0 r, for I [, ), and d r ( ) A) q () is a real value and loally inegraed over I A3) xf ( x) 0, and f '( x) k 0 A4) k( )) 0 By he soluion of equaion () or () we onsider a funion x ( ), [ x, ) [ 0, } whih is wie oninuously differeniale and saisfies equaion () or () on he given inerval The numer depen on ha pariular soluion x () under onsideraion We onsider only nonrivial soluions A soluion x() of () or () is said o e osillaory if here exiss a sequene } of poins in he inerval [ 0, }, so ha lim and ( ) 0 n n x, n N n { n n, oherwise i is said o e nonosillaory An equaion is said o e osillaory if all is soluions are osillaory, oherwise i is onsidered ha is nonosillaory equaion The Condiions for osillaory soluions of he seond order differenial equaions () are sudied y many auhors (see [5], [6], e) ere, we give some ondiions for oeffiiens where he equaion () has osillaory soluions and we also ake ino aoun he resul we have oained in he previous researhes, here we presen more generalized rieria ha define osillaion soluion of he equaion () o used osillaory soluions of ()In his paper are presened heorems, ha use generalized Riai ype ransformaions, and averaging ehnique, whih explain resuls for osillaory naure of differenial equaions Also, our resuls exend and improve a numer of exising resuls (see [5], [9], [0], e) 36
Wha follows is, E (, ( ) denoe and E( e q( dl p () l B( ) ( q( ) dl 4 k II New Osillaion Crieria For Seond Order Nonlinear Main Resul ha we will use in he following heorem Theorem : The equaion () is osillaory if for p ( ) 0,, and Proof: For and he equaion dl E ( B () (3) q( dl q( ) E( e, we have E' ( ) E( ), where E ( ) 0, ) ( E( ) ) ))' E( ) p( ) f ( x( )) k( )) 0 (4) may e redued in () We se ha () is osillaory if and only if he equaion (4) is osillaory Assume ha () is nonosillaory Then here exiss a nonosillaory soluion x () of () So we may assume ha x ( ) 0 on [, ), for some We show ha x '( ) 0, for From (4) we oain ha ( E( ) ) ))' E( ) p( ) x( ) k( )) 0 from where E ( ) ) ) is no inreasing for Assume ha E ( ) ) ) 0 for some Pu E( ) ) ) L, hen for, we have E( ) ) ) L Dividing oh sides y E ( ) ) and inegraing from o ( ), we oain Beause x( ) x( ) L dl E( L dl E( is ending o large, whih is a onradiion Therefore, x '( ) 0, for In ase ha ( ) 0 Considering he funion, where, we onlude ha x ( ) 0 x, pu y( ) x( ) So we have '( ) 0 x 37, for suffiienly ) x '( ) W () f ( x( )) we have p( ) W ( ) f '( x( )) W '( ) W ( ) q( ) k( )) ) ) and from A3), A4) we oain p( ) W ( ) k W '( ) W ( ) q( ) ) ), (5) ()
k p( ) p ( ) W '( ) ( W ( ) ) q( ) ) k 4 k ) New Osillaion Crieria For Seond Order Nonlinear k p( ) p ( ) W '( ) ( W ( ) ) ( q( ) ) ) k 4 k ) Inegraing (6) from o ( ), we ge p ( k p( W ( ) W ( ) ( q( ) dl ( W ( ) dl 4 k k By means of (4) here exiss a 3, suh ha for 3, we gain k q() l W ( ) ( W ( ) dl k whih is impossile eause W ( ) 0, for To use he heorem aove we an prove ha for funion W () is ruh he following lemma Lemma : Assume ha for, p ( ) 0, q ( ) 0 and (3) are valid If he differenial equaions () have a posiive soluion, we have ) ) lim 0 f ( x( )) Proof: Le ( ) 0 (3) ha here exiss a suh ha x '( ) 0, for Pu ) ) W ( ) 0 f ( x( )) for, and onsider Riai inequaion p( ) W ( ) k W '( ) W ( ) q( ) ) ) i is ovious ha dividing wih W >0, we ge x, e a soluion of () From Theorem i follows ha from p ( ) 0, for W '( ) k W ( ) ) Inegraing he aove inequaion over [, ) and onsidering he ondiion r ( ) 0, and d, we have r ( ) W'( W ( W ( ) W ( ) W () k W ( ) from where for, we oain lim W ( ) 0, and 38
New Osillaion Crieria For Seond Order Nonlinear ere, we presen some suffiien ondiions for () o e osillaory o used he lass X of funions (, whih omes from Philos, where (, 0, for s 0, (, ) 0, are oninuous and have parial derivaives (, (, and d saisfying : (, h (, (, s (, and h (, (, s Theorem Le assumpions A) A4) hold and X If here exiss ( a, ) [ 0, ), ( a, ), suh ha h ( (, [ q( ( p( ( ) ] 4 ( ) kr s s h ( s q s p s 4 ( ) (6) [ ( ) ( ( ) ( ) ] 0 kr s s hen every soluion of eq () is osillaory Proof: Supposed o he onrary, ha x () e a non-osillaory soluion of (), say x ( ) 0 on [ 0, ) If we muliply equaion (5) y ( s, ) and inegrae i from o where (, ), s(, ) hen we have p( W ( (, q( (, W '( (, W ( k (, (, q( (, W ( / W( (, h (, (, p( W ( W ( k (, (, (, q( (, W ( / from ha k p() s ( ) h (, (, [ W ( ( W ( ) s k k (, h ( ( ) ( ) / ( ( ) ( ) ) 4 k (, s q s s W s s p s (7) Le in (7) and dividing i y hen wha we ge is s q s W ( ) ( ) 39
New Osillaion Crieria For Seond Order Nonlinear + (, ( p( ( ) h ( (8) 4 k (, By muliplying (5) y ( s, ) and inegrae i over (, ) where ( a, ), s(, ) we have p( W ( ( s, ) q( ( s, ) W '( ( s, ) from ha Le W ( k ( s, ) ( s, ) q( ( s, ) W ( / in (9) and dividing i y + h ( ( s, ) ( p( ( ) 4 k ( s, ) we ge s q s W ( ) ( ) + h ( (, ( p( ( ) (0) 4 k (, Adding (8) and (0) we have he following inequaliy h ( (, [ q( ( p( ( ) ] 4 ( ) kr s s (9) h ( s q s p s 4 ( ) () [ ( ) ( ( ) ( ) ] 0 kr s s whih onradis he ondiion (6), herefore, every soluion of equaion () is osillaory s, we have he following orollary If for (, (, 0 Corollary Assuming ha ondiion A) A4) hold, hen every soluion of equaion () is osillaory if for 0, he following inequaliies hold: and Proof: For rs ( ) limsup ( s [ q( ( p( ( ) ] 0 4 k ( s () rs ( ) limsup ( l [ q( ( p( ( ) ] 0 4 k ( l (3) (, (, we have h(, h(, ( from where similarly o use (6) we ge () and (3) The proof is omplee Example Consider he nonlinear differenial equaion of seond order 4 ( ))' ( os ) ) x( )( x ( ))( ( )) ) 0 40
New Osillaion Crieria For Seond Order Nonlinear where 4 4 We an see ha f ( x) x( x ), f '( x) 5x, g x ( ') x, for all x Equaion is osillaory eause he oeffiiens of equaion saisfy he ondiion given in he heorem Referenes [] X Wang, G Song, Osillaion rieria for seond order nonlinear damped differenial equaions, In Jour of inform and sys s, vol Nr 0,pg 73-8 [] Yuri V Rogovhenko;Inerval osillaion of a seond order nonlinear differenial equaion wih a damping erm, Disree and oninuous dynamial sysems supplemen, pp 883 89, 007 [3] MMA El Sheikh, RASallam, DI Elimy; osillaion rieria for nonlinear seond order damped differenial equaions, Inernaional journal of nonlinear siene, vol0, nr 3, 97 307, 00 [4] E Tun, Avi, Osillaion rieria for a lass of seond order nonlinear differenial equaions wih damped, Bull of Mah analysis and appli, vol4 iss (0) pg 40 50 [5] Xh Beqiri, E Koi, osillaion rieria for seond order nonlinear differenial equaions, Briish Journal of Siene, pg 73 80,0, Vol 6 () [6] R Kim, Osillaion and nonosillaion rieria for differenial equaions of seond order, Korean J Mah 00, N04, pp 39 40 [7] A Lomoaidze, J Sremr, On asillaion of seond order linear ordinary differenial equaions, Mem on differ equa and mah physis, vol 54, 0, pg 69 8 [8] MJSaad, N Kumaresan, K Ranavelu, Osillaion heorems for seond order nonlinear differenial equaions wih damping, Ins of mah sienes, Univer of Malaya, 0, pg - [9] A Tiryaki, A Zafer, Osillaion rieria for seond order nonlinear differenial equaions wih deviaing, Turk J Mah 4, 000, pg 85 96 4