Interval Oscillation of Nonlinear Differential Equation with Damped Term

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1 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 Inerv Oiion of Noniner Differeni Equion wih Dmped Term Yun-Hui Zeng Deprmen of Mhemi nd Compuion Siene, Hengyng Norm Univeriy,Hunn, 4,Chin Abr- In hi pper,we udy oiion of eond-order noniner differeni equion wih dmped erm nd oninu-ou diribued rgumenby uing verging funion nd Rii ehnique, ome new uffiien ondiion re ebih-ed, whih re differen from mo known one in he ene h hey re bed on he informion ony on equene of ubinerv of,, rher hn on he whoe hf-ine Keyword- Noniner; Inerv Oiion; Coninuou Diri-bued Argumen I INTRODUCTION Thi pper i onerned he oiory of eond-order noniner differeni equion wih dmped erm nd on- inuou diribued rgumen Where ( r () ( y()) k( y())) p() k( y()) + q (, ) f( y (), yg ( (, ))) d( ), () Throughou hi pper,we wy ume h ( H ) r (), p () CIR (, ), nd p ( ), r () I [, ), R(, ) for I, d, I [, ) r () ( H ) f ( xy, ) CR ( RR, ) nd f ( xy, ) h igned x nd y if hey hve me ign, nd f ( xy, ) f( xf ) ( x), f ( y) where he funion f ( x ) nd ( ) f( x) k, xr, y k, y, where k nd k re nonnegive onn H k ( ) for y ; nd ( y) y( g(, )) k( y( )) k,, [, ], for y R, g(, ) 4 ( H 4) q (, ) CI ( [, ], R ), g (, ) C([, ) [, ], R), g (, ) for [, ], g (, ) h oninuou nd poiive pri derivive on I [, ] wih repe o he fir vribe nd nondereing wih repe o he eond vribe, repeivey im inf g (, ) [, b] ( H 5) u( ) C([, ], R) i nondereing, nd he inegr of () i in he ene of Riemunn-ieje y ():, R, i ed ouion of () if (), In he eque i wi wy be umed h ouion of () exi for ny A ouion y() of () i ed oiory if i h rbirry rge zero, oherwie i i ed nonoiory We re h funion y ifie () for When ( y), k( y( )) y( ), q (, ) f( y (), yg ( (, ))) d( ) repe q ( ) f( y ( )) g( y ( )), Equion () redue o () ( r() y()) p() y() q() f( y()) g( y()), () - 7 -

2 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 nd when r() ndp(), Equion () redue o () y() q() f( y()) g( y()) () The oiion of () nd for i priur e uh he iner equion y() q() y()), (4) he noniner equion ( ry ( ) ( )) q ( ) f( y ( )), (5) nd he noniner equion wih dmping erm ( r() y()) p() y() q() f( y()), (6) h been diued by numerou uhor by ever differen mehod (ee Pper [-6] nd he referene herein) An imporn oo in he udy of oiory behvior of Equion () (6) i he verging ehnique whih goe bk fr he i reu of Winner nd Hrmn [9] providing uffiien ondiion for he oiion of (4) The reu of Winner w improved by Kmenev [], nd exenion of Kmenev rierion hve been obined for (4) by Phio [5], where for (5) by Li [] For (6) he reu of Phio hve been exended by Gre [8], nd Gre reu hve been refined reeny by Li [6] From he rum eprion heorem i i er h he oiion of (4) i ony n inerv propery, ie, if here exi equene of ubinerv [ i, bi] ofi, where i, uh h for eh I here exi ouion of (4) h h e wo zero in [, b] hen every ouion of (4) i oiory i i In 99 EL-Syed [5] ebihed n inereing inerv rierion for he oiion of fored eond-order equion, bu he reu i no very hrp, beue omprion wih equion of onn oeffiien i ued in he proof In997, Hung [7] preened inerv rieri for he oiion nd he nonoiion of he eond-order iner differeni Equion (5) nd Li nd Agrw [, 4] ebihed more gener reu for noniner fored Equion (5) We remrk h, Kong [], Li [7], Li nd Agrw [, 4] empoyed he ehnique in he work of Phio [5] nd obined ever inerv rieri oiion reu for he eond-order iner Equion (5) nd noniner Equion () (4), However, hey nno be ppied o he noniner differeni Equion () In hi pper by empoying generized Ri ehniqu-e [], we h preen ever new inerv rieri for he oiion of (), ie, rieri whih invove he behvior of () (or of r, p, q, f, g nd k) ony on equene of ubinerv of I our reu invove Kmenev ype ondiion nd improve nd exend he reu of Kmenev [], Li Agrw [, 4], nd Li [8] Oher reed oiion reu n refer o [,, 4, nd ] An imporn onribuion of our udy i h we h dipene wih hee ondiion We y h funion H H(, ) beong o of funion X, denoed by H X, if H C( D, R ), D (, ):, whih ifie where And h pri derivive H nd H on uh h where h, h L ( D, R) o H (,), H (, ), for, (7) H H h(, ) H (, ) h(, ) H (, ),, (8) For hor nonion we define he funion II MAIN RESULTS AND PROOF Q () q (, ) d ( ) We fir prove wo Lemm whih wi be uefu for ebihing oiion rieri for Eq() - 8 -

3 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 Lemm Suppoe h umpion 5 y () on, b [, ) For ny funion v () ( H ) ( H ) hod, nd h y() i ouion of () uh h C (, I R ), Dfine r ( ) ( y ( )) k( y( )) w () v () yg ( (, )), () on, b hen for ny H (, ) X, we hve H (,) bkkvqd () () Hbw (,) () Proof From () nd (), we hve for b b 4kk b kp () v() rv ()() h (,) b( ) Hb (,) d (),, () ()( () ( ()) ( ())) () v () v r y k y vr () () ( y ()) k( y()) y( g (, )) g(, ) w w v () yg ( (, )) y ( g(, )) v () ()( () ( ()) () v p k y w v () yg ( (, )) q (, ) f( y ( ), yg ( (, ))) d( )) yg ( (, )) v ()( r () ( y ()) k( y()) y( g (, )) g(, ) y ( g(, )) v() p() v() k( y()) w () v () yg ( (, )) v () q (, ) f( y (), yg ( (, ))) d( ) yg ( (, )) vr () () ( y ()) k( y()) y( g (, )) g(, ) y ( g(, )) v() p() v() k( y()) w () kkvq () () v () yg ( (, )) v() p() v() k( y()) w () kkvq () () v () yg ( (, )) () () ( ()) ( ()) ( (, )) (, ) v r y k y y g g y ( g(, )) v() r() ( y()) k( y()) y( g(, )) g(, ) vr () () ( y ()) k( y()) w () In view of v() p() [ ] w () kkvq () () v () r () ( y ()) ( H ),We obin by he bove equiy y( g(, )) g(, ) vr () () ( yk () ( y()) () w (4) yied Muipying (5) by (, ) v() p() w() [ k ] w() kkv () Q () kk w (), v () r () vr () () p () v() w() kkv() Q() kk4 w () [ k ] w( ) (5) vr () () r () v () H, inegring i wih repe o from o, for b,, nd uing (7) nd (8) - 9 -

4 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 H (, kkvqd ) ( ) ( ) kk Hwd (, ) ( ) H (, ) w( d ) vr ()() H k p () v() (, )( ) wd ( ) Hw (, ) ( ) [ h(, ) Hw (, ) ( ) kk vr ()() H (, ) w( )] d H k kkh (, ) H (, w ) ( ) [ w ( ) vr ()() vr ( ) ( ) kp () v( ) h k k r ( ) v ( ) ( (, ) ( ) (, ))] kp () v( ) 4 k k r ( ) v ( ) H (, w ) ( ) p () v() (, )( ) w ( ) H rv ()()[ h (,) ( ) H (,)] d 4kk Now, eing b, in he bove inequiy, we obin () kp () v() rv ()()[ h (,) ( ) H (,)] d Lemm Suppoe h umpion ( H ) ( H 5 ) hod And h y() be ouion of () uh h y () on, [, ), for ny funion v () C (, I R ),e w () be defined by () on, Then for ny H X, (, ) () () Hw (, ) () H kkvqd 4kk kp () v() rv ()() h (, ) ( ) H (, ) d (6) Proof Simir o he proof of Lemm, we hve (6) The foowing heorem i n immedie reu from Lem-m nd Lemm Theorem Aume h ( H) ( H5) hod nd h for ome (, b), nd for ome H X, H (, ) H (, kkvqd ) () () Hb (,) b H (,) bkkvqd ()() 4 k kh(, ) kp () v() rv ()() h (, ) ( ) H (, ) d () () ()() (,) ( ) (,) 4 kkh 4 ( b, ) b kp v rv h b Hb d (7) Then every ouion of () h e one zero inb, Proof Suppoe he onrry hen wihou o of generiy we my ume h here i ouion y of ()uh h y() for (, b) From Lemm nd, we find h boh () nd (6) hod Dividing () nd (6) by Hb (,) nd H (, ), repeivey, nd hen dding hem, we obin - -

5 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 H (, ) (, ) () () H kkvqd Hb (,) b H (,) bkkvqd () () 4 uk k H (, ) kp () v() rv ()() h (, ) ( ) H (, ) d 4 ukk4h ( b, ) b kp () v() rv ()() h (,) b ( ) Hb (,) d, whih onrdi umpion (7) Theorem Aume h ( H) ( H5) hod If nd imup H ( kkvq, ) ( ) ( ) imup H ( kkvq, ) ( ) ( ) kp rv ()() () v() [ h (, ) ( ) H(, ) d, (8) 4k k kp rv ()() () v() [ h (, ) ( ) H(, ) d (9) 4k k For ome H X, v C (, I R ), nd for eh, hen every ouion of () i oiory Proof For ny T, e T, nd in (8) we hooe, Then exi n, uh h H (, kkvq ) () () 4kk4 rv ()()( h(, ) In (9) we hooe, Then exi b, uh h b H (,) bkkvq () () 4kk4 kp () v() ( ) H(, )) d () kp rv ()()( h(, ) () v() ( ) H (, )) d () Combining () nd () we obin (7) The onuion hu ome from Theorem For he e where h( ) h ( ) The ub of X onining uh H : H( ) X, we hve h, nd denoe hem h ( ) H ( ) i denoed by X, Appying Theorem o X, we obin Theorem Aume h ( H) ( H5) hod If for eh T, here exi vc (,, R ), H X, nd, Ruh h T, nd H ( ) kk[ v( ) Q( ) v( ) Q( )] d [()() rv r( v )( )] h(, d ) 4k k 4 [ r( v ) ( ) p( v ) ( ) k k rv () () ] pvh ()()( ) H ( d ) 4 kp () v( ) kp ( ) v( ) [()()( rv ) 4 k k r( ) v( ) ( ) ] H( ) d, () 4 r ( ) v ( ) r( ) v( ) hen every ouion of () i oiory b Proof Le b Then Hb ( ) H ( ) H ( ) nd for ny w L[, b], we hve b wd () w( d ) - -

6 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 Hene b Hb ( wd ) ( ) H ( w ) ( d ) Thu () impie h (7) hod for H X, vc (,, R ), nd herefore every ouion of () i oiory by Theorem The bove oiion rieri, we n obin differen uffiien ondiion for oiion of ouion of () by hoie of H (, ) Le H (, ) ( ),, where i onn ed o he foowing Corory Corory Aume h ( H) ( H5) hod Then every ouion of () i oiory provided for eh nd (,, R ), uh h he foowing wo inequiie hod: ome, here exi funion nd Define nd e where i onn v C imup ( ) () () kp () v( ) ()()( )] d, () v Q rv 4 kk r ( ) v ( ) imup ( ) () () kp () v( ) ()()( )] d (4) k k v Q rv 4 kk r ( ) v ( ) Proof The proof i imir o h of Theorem Theorem 4 Aume h 5 for eh R() d, r (), H (, ) [ R () R ( )],, ( H ) ( H ) hod, nd im R ( ) Then, every ouion of () i oiory provided nd ome he foowing wo inequiie hod: kk imup ( R( ) R( )) kkq( ) R () p () ] d 4 kkr ( ) 4( ) (5) nd Proof Le v () Then, we hve kk imup ( R( ) R( )) kkq( ) R () p () ] d 4 kkr ( ) 4( ) (6) h(, ) [ R() R( )] r () nd h (, ) [ R() R( )] r () - -

7 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 Sine nd rh () (,) d rh () (,) d In view of im R ( ) I foow h r () r () [ R () R ()] d r () r () [ R () R ()] d Now, from (5) nd (7), we hve imup ( R( ) R( )) k k Q( ) R () rh d [ ( ) ( )] R R im ( ) (, ) 4 kkr ( ) 4 kk( ) [ ( ) ( )] R R (7) im ( ) (, ) 4 kkr ( ) 4 kk( ) rh d (8) r () p () [ (, ) (, )] 4 k kr ( ) r ( ), h H d r ( ) imup ( R( ) R( )) k kq( ) d R () imup [ h (, ) R () 4 k k r() r () h (, ) p( ) 4 k kr ( ) r ( ) H(, )] r ( ) p( ) d imup [ (, )] H d R () 4 k k r() r () imup [( R ( ) R ( )) R () p () p () [ kkq ( ) ] d ( ( ) ( )) R R ] d 4 k kr ( ) kk r ( ) imup h (, ) d () 4 () R kk4r imup ( R( ) R( )) ( kkq( ) R () p () ) 4 k kr ( ) p () ( ( ) ( )) kkr ( ) R R d 4 kk( ) ie, (8) hod Simiry (6) nd (8) impy h (9) hod Thu by Theorem every ouion of () i oiory ACKNOWLRDGEMENT Thi rerh w uppored by eduion foundion of Hunn Provine(C54), Cuivion of young eher of Hengyng Norm Univeriy of Chin nd Conru progrm of he key diipine in Hunn Provine[()76] REFERENCES [] J W Bker, Oiion heorem for eond order dmped noniner differeni equion, SIAM J Mh An 97,5, 7-4 [] G J Buer, The oiory behvior of eond order noniner differeni equion wih dmping, J Mh An App 977,57, 7-89 [] T A Buron nd R C Grimmer, Sbiiy of ( ru ) f ( u) g( u), Monh Mh 97,74, - [4] R Byer, B J Hrri nd M K Kwong, Weighed men nd Oiion ondiion for eond order mrix differeni equion, J - -

8 Communiion in Informion Siene nd Mngemen Engineering Mr, Vo I, PP 7-4 Differeni Equion 986,6, [5] M A E-Syed, An oiion rierion for fored eond order iner differeni equion Pro Amer Mh So 99,8, 8-87 [6] S R Gre, Oiion heorem for noniner differ-eni equion of eond order, JMh An App 99,7, -4 [7] S R Gre nd B S Li, Inegr verging ehni-que for he oiion of eond order noniner differeni equion, J Mh An App 99,49, 77- [8] S R Gre nd B S Lind C C Yeh, Oiion heorem for noniner eond order differeni equion wih noniner dmping erm, SLAM J Mh An984,5, 8-9 [9] P Hrmn, On nonoiory iner differeni equion of eond order, Amer J Mh 95,74,89-4 [] I V Kmenev, Oiion rieri reed o verging of ouion of eond order differeni equion,(n Ruin), Differeni nye Urvnenyi 974,,46-5 [] Q Kong, Inerv rieri for oiion of eond order iner ordinry differeni equion, J MhAn App 999,9, 58-7 [] W T Li, Oiion of erin eond-order noniner differeni equion, J Mh An App 998,7, -4 [] W T Li nd R P Agrw, Inerv oiion rieri for eond order noniner differeni equion wih dmping, Compuer Mh Appi,4(/), 7- [4] W T Li nd R P Agrw, Inerv oiion rieri reed o inergr verging ehnique for erin noniner differeni equion, J Mh An App,45, 7-88 [5] Ch G Phio, Oiion heorem for iner differen-i equion of eond order, Arh Mh (Be)989,5, [6] W T Li nd M Y Zhng nd X L Fei, Oiion rieri for eond order noniner differeni equion wih dmping erm, Indin J Pure App Mh999, (), 7-9 [7] C C Hung, Oiion nd nonoiion for eond order iner differeni equion, J Mh An App,997,, 7-7 [8] W T Li, Inerv oiion rieri for eond order noniner differeni equion wih dmping, Tiwnee Joun of Mhmi,,Vo 7\, [9] W T Li nd R P Agrw, Inerv rieri for eond-order noniner perurbed differeni equion, Compuer Mh Appi, 4,47, Yun Hui Zeng(978), me, nive of Hengyng, Hunn, P R Chin Curreny,he i n oie profeor wih Hengyng Norm Univeriy Hi min reerh inere inude biiy nd oiion heory of differeni equion - 4 -