Existence of Nonoscillatory Solutions for a Class of N-order Neutral Differential Systems

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1 Vo 3 No Mod Appd Scc Exsc of Nooscaoy Souos fo a Cass of N-od Nua Dffa Sysms Zhb Ch & Apg Zhag Dpam of Ifomao Egg Hua Uvsy of Tchoogy Hua 4 Cha E-ma: chzhbb@63com Th sach s facd by Hua Povc aua sccs fud poc (NO 6D74 Absac Usg fxd po hom hough h mhod of sucua compsso mappg o s of boudd couous fuco hs pap quaav suds h xsc of ooscaoy souos fo a cass of -od ua dffa sysms ad obas som suffc codos of ooscao xsc of souo fo hs cass sysms Kywods: Nua Dffa sysm Nooscaoy souo Compsso mappg Fxd po hom Ioduco Cosd h xsc of ooscaoy souos fo -od ua dffa sysms [ x p x( ] Q f ( x( ( wh s a posv g ad h foowgs a aways sasfd: ( A p Q C [ R C [ R m ( ( m ( ; A f ( y C[ RR ( fo ough bg wh y yf ( y ( hods ( A3 Fuco sasfs oca Lpschz codo fo y xss cosa L ad whch ags ha f ( y f ( y L y y wh y y ( Th quao ( souo s oscaoy f has abay bg zo; Ohws s ooscaoy O h cuy 9's appad may paps (Ya Juag 99( Wag Zhb 995( Wag Law 995( L Guaghua Yu Yuahog & L ShZhog 997( Wag Guagp ( L Hogf & Wag Zhb abou oscaoy souos fo hgh od ua dffa quao Ths acs hav gv som souo oscaoy codos Howv h hav fw sachs fo ooscaoy souo Th mos as sach s dscussd by Zhag gg pap (Zhag gg 996 fo h xsc of ooscaoy souo x ( hx ( ( ( p( xg ( ( ( ( m Afwads Sh Jahu Yu Jash suds h mo ga quao ha ( pap (Sh Jahua & Yu 44

2 Mod Appd Scc Fbuay 9 Jash 996: ( x px ( f( x ( x ( (3 Ths pap sabshd h suffc codo fo boudd posv souo xsc of quao (3wh ua m p ( ad P ( p I c yas paps (Tag Xahua & Yu Jash ( Lu Kayu hav cosdd h spca suao of quao (3 ( x ( px ( ( Qx ( (4 whch sabshd h suffc codo of souo xsc of quao (4 wh ua m p ( has a chag amog Ths pap uss aach compsso fco pcp dscusss ooscao xsc of souo fo h ga quao ha quao (of paps (Zhag gg 996( Sh Jahua & Yu Jash 996( Tag Xahua & Yu Jash ( Lu Kayu ad obas h suffc codos of ooscao xsc of souo fo hs cass sysms whos cocusos a dff fom paps (Zhag gg 996( Sh Jahua & Yu Jash 996( Tag Xahua & Yu Jash ( Lu Kayu Cocusos ad Poof mma : ( aach compsso mappg pcp Supposs mappg A: s a compsso mappg h comp dsac spac ( h A mus xs oy a fxd po x sasfyg Ax x Thom : Suppos ( A 3 hod ad xs posv cosa c ad ogav umb whch sasfyg msup m ( s p s Q s ds ( h h quao xss vuay posv souo x ( ad x c Poof: Fom hom h a T ad sasfyg h foowg quao wh T ( ( ( ; ( s a boudd couous fuco s Do s C x(:[ R C h ( s p s Q s ds ( C s a aach spac Do x( C : x( h foowg poof s ha h s s a boudd cosd covx subs [ Rad df C I s obvous ha s bouday Fo abay xysc ax ( a y a [] s a covx s If x ad m x wh w hav x x x x x x x x x x x x Hc x Fom h abov s a boudd cosd covx subs Df mappg C sup x 45

3 Vo 3 No Mod Appd Scc ( p( x ( ( ( s Q( s c ( s ( Ax f ( s c x( ( s ds T; ( Ax( T T T T wh c sasfy quay c Th foowg w pov ha mappg A s sf ad compssv (Fo abay x ( ad T fom (ad (3 w hav Wh ( ( Ax( p( x( ( ( s c! (3 ( s Q ( s f ( s c x( ( s ds ( s p s Q s ds ( ( ( T w hav ( Ax ( Ax( T T T (5 Th abov dcas A A s a sf mappg (Whou oss gay suppos x( x( ad wh T fom quao(3 w hav Wh ( ( s Q ( s c! ( s ( s ( ( ( ( ( ( s ( p ( s Q( s ds x ( Ax ( ( Ax ( p ( x ( ( x ( ( T w hav f s c x s f s c x s ds x x x ( Ax ( Ax ( Ax( T ( Ax( T T x x Th abov dcas ha ( Ax s a compssv mappg Thfo xss fxd po x ( sasfyg ( Ax x amy (4 (6 (7 Tag x( cx ( ( p( x( ( ( s Q( s c ( s x ( f ( s c x ( ( s ds T; ( Ax ( T T T T fom (8 w hav (8 46

4 Mod Appd Scc Fbuay 9 ( ( ( ( ( ( s (! p( x( ( (! x p c x ( s Q ( s f ( s c x ( ( s ds ( s Q ( s f ( s x( ( s ds( T Dva quao (9 ms quao ( s go So x( cx ( s h vuay posv souo of quao(ad x c Th poof s compd Cooay : If codos ( A 3 a sasfd ad quas (9 p ( cc ( s Q ( s ds ( hodh h quao(xss vuay posv souo Poof: Fom(( w hav m sup p c ( m ( s Q ( s ds Th wh K= w hav ( s msup p( ( s Q( s ds L msup p ( s Q( s ds (3 Fom hom s obad ha h quao(xss vuay posv souo Cooay Wh codos ( A 3 hod f posv cosas p sasfy p ( p Q ( Q ( ( ( ; ad h quay ( p L Q hodh quao(xss vuay posv souo Poof Fom codo (4 Cooay (4 p s Q s ds ( s msup ( ( s msup p ( s Q ds L p Q Th quao ( xss vuay posv souo by hom I quao(wh f ( x( ( x( ( quao [ x p( x ] Q( x ( (5 47

5 Vo 3 No Mod Appd Scc s go Cooay 3 Wh codo ( A s sasfd ad f h xss cosa h quay ( s msup p ( s Q( s ds hodsh quao(5xss vuay posv souo Examp: Suppos ad d s x ( x ax ( ( bx ( Wh d x ( s a ( b ( a ( b ( whh h quao dos hav h vuay posv souo? s Souo: Sc s p fom hom w hav ( s ( s msup p ( s Q( s ds s ( s msup p( ( s [ as bs ] ds (! s ( s msup ( s ds s (s! s msup s s Whch dcas hs quao has vuay posv souos Rfcs L Guaghua Yu Yuahog & L ShZhog (997 Oscao of souos fo hgh od oa fucoa dffa quao Mahmacs pacc ad hoy 7( L Hogf & Wag Zhb ( Oscao fo hgh od oa m ag dffa quaos of vaab coffcs Joua of gg mahmacs 7(3-4 Lu Kayu ( Exsc of bouday posv souos fo hgh od ua dffa quao Joua of Hua Uvsy 7( -6 Sh Jahua & Yu Jash (996 Exsc of posv souos fo hgh od ua dffa quao Aca Mahmacs Sca 39( Tag Xahua & Yu Jash ( Exsc of posv souos fo hgh od ua dffa quao Joua of Hua Uvsy 7(4 7-9 Wag Guagp ( Oscao hom of a d of hgh od oa ua quao Mahmacs pacc ad hoy 3( Wag Law (995 Oscao codo fo hgh od oa ua quao Joua of Shadog Uvsy 3( 6-33 Wag Zhb (995 Oscao ad gadao fo hgh od ua quao of vaab coffcs Aca Mahmacs Appcaa Sca 8( Ya Juag (99 Oscao ad gadao of souos fo -od oa m ag dffa quaos souos Aca Mahmacs Sca 33( Zhag gg (996 Posv souos fo a d of ua quaos Aca Mahmacs Appcaa Sca 9( -3 48

k of the incident wave) will be greater t is too small to satisfy the required kinematics boundary condition, (19)

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