Existence of Nonoscillatory Solution of High Order Linear Neutral Delay Difference Equation

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1 oder Appied Sciece ovember, 008 Existece of oosciatory Soutio of High Order Liear eutra Deay Differece Equatio Shasha Zhag, Xiaozhu Zhog, Pig Yu, Wexia Zhag & ig Li Departmet of athematics Yasha Uiversity Qihuagdao 000, Chia Supported by the Foudatio for the atura sciece of Hebei provice of Chia (Z007) Abstract Cosider the high order eutra deay differece equatio with positive ad egative coefficiets + Δ [ x( ) + px ( τ)] + R( x ) ( δ ) R( x ) ( δ ) o Where p R; τ (); δ, δ ;{ R( )},{ R( )} are positive rea sequeces. A sufficiet coditio for the existece of the evetuay positive soutio of the above equatio is set forward i terms of Ri( ), i,, ( 0 ). This resut got rid of a quite strog tetative of existig iterature, which improve the reevat theorem. Keywords: Positive ad egative coefficiets, eutra differece equatio, Evetuay positive soutio. Itroductio owadays, as the rapid deveopmet of computer sciece, automatio techoogy, bioogy ad umerica research, eutra deay differetia equatios osciatio study attracts the attetio of may schoars, ad some research achievemets have bee tae i this area. At the same time, a differece equatio with positive ad egative coefficiets attractig more attetio becomes a ew study fied. However, oy a few research resuts have bee achieved i this fied, ad rare resuts o bouded positive soutios i eutra deay differece equatio with positive ad egative coefficiets. Referece studies the existece of positive soutios i secod-order eutra deay differece equatio with positive ad egative coefficiets Δ [ x( ) + px ( τ)] + R( x ) ( δ) R( x ) ( δ) o, ad gets a sufficiet coditio for the existece of positive soutios i this equatio. Aother referece promotes it to the higher order, that is, it focuses o foowig + order differece equatio with positive ad egative coefficiets + Δ [ x( ) + px ( τ)] + R( x ) ( δ ) R( x ) ( δ ) o () + z ; p R; τ {,, }; δ, δ {0,,, };{ R ( )},{ R ( )} are positive rea sequeces. Where For coveiece, ow the basic cocepts ad mars i this paper are isted as foows: Δ Said for the forward differece operator Δ y ( ) y ( + ) y ( ); Z Said for the set composig by a iteger; R {()} x Said for the rea umber set. Set a Z, ad a ( ) { aa, +, }, (0), is amed as the soutio of differece equatio () whe the {()} x satisfied the equatio (). Whe ( ), ad x ( ) > 0, {()} x is evetuay positive soutio which meas that positive itegra exists. Whe positive itegra existed, where ( ), ad x ( ) < 0, {()} x is evetuay egative soutio. Osciatio refers to that {()} x or evetuay egative, otherwise it caed as oosciatio. Lemma Assume (i) Ri( ), i,, ( 0 ) () are either evetuay positive 7

2 Vo., o. oder Appied Sciece (ii) There exists a arge eough positive iteger T, to every a > 0 ad () (iii) p ± () > hod, where ar ( ) R ( ) 0 hod, the Equatio () has a evetuay positive soutio. The proofig processes coud be foud i refereces.. ai resuts ad proof Amog the sufficiet coditios for the existece of oosciatory soutio i differetia equatio(), the coditio () may oo too restrictive; we wi deete the strog coditio (), permit p, to get the overa sufficiet coditio for existece of evetuay positive soutio o the p vaue i equatio (). Theorem Cosider the differece equatio with positive ad egative coefficiets + Δ [ ( ) + ( τ)] + ( ) ( δ) ( ) ( δ) x px R x R x o Where τ δ δ [ ) z ; p R; ();, ; R( ), R ( ) C(,, R ) If coditio () is hod, where p, the equatio () has a evetuay positive soutio. B be the Baach space which is composed of a bouded rea sequeces x x ( ) Proof: Let the sup orm x sup x ( ). The proof of Theorem wi be divided ito five caims. Caim. p From coditio (), choose a arge eough positive iteger > ), so that whe >, 0 R( ) (5) + R( ) () + i ( 0), defie hod. Defie a subset as A { x B : x( ) +, ( 0 )}, the it is easy to see A is a bouded, cosed, ad covex subset of B. Defie a mappigt : A B as foows + iτ ( + ) + ( s + ) i j + ( i ) τ j j j j j Tx( ) [ R()( sxsδ) R()( sxsδ) ] Tx( ) 0 < Ceary, Tx is cotiuous. ext we wi prove that T is a sef-mappig i A. Whe, x A, by usig (5)ad (7), we have iτ + ( i) τ + Tx( ) ( + ) + + ( s + ) R( s) x( s δ) i j + ( i ) τ j + ( i ) τ j j j j j + iτ ( + ) + ( + ) ( s + ) R ( s) i j + ( i ) τ j j j j j ( + ) + ( + ) ( s + ) R ( s) j j j j j j ( + ) + ( + ) s R ( s) ( + ) + ( + ) + + (7) 7

3 oder Appied Sciece ovember, 008 Furthermore, i view of () ad (7), we have: Tx( ) ( + ) ( + ) s R ( s) ( + ) ( + ) + Ceary, whe 0 <, we have x( ) +, thus Tx A which meas that T is the sef- mappig i A. ext we wi prove that T is a cotractio mappig i A. Whe >, for every x, x A, we have: + iτ Tx ( ) Tx ( ) ( s + ) i j + ( i ) τ j j j j j [ δ δ ] R () s x ( s ) x ( s ) j j j j j j s R() s x x x + x ( s + ) R ( s) x x Where q +, so that Tx ( ) Tx ( ) q x x. Ceary, whe 0, we have Tx( ) Tx( ) q x x. Thus T is a cotractio mappig i A. Above a, accordig to Baach cotractio mappig, T has a fixed poit o A such thattx x. Ad x {()} x satisfies equatio (7), so we have + iτ x ( ) + x ( τ ) (+ ) + ( s + ) [ R()( sxs δ ) R()( sxs δ )] j j j j j j i j + ( i ) τ j j j j j [ δ δ ] (+ ) + ( s + ) R ( s) x( s ) R ( s) x( s ) To get the + order differece equatio from the above equatio, + Δ [ ( ) + ( τ)] + ( ) ( δ) ( ) ( δ) x x R x R x o Thus, the fixed poit {()} x is a positive soutio of equatio (). This competes the proof of Caim. Caim. 0 p < From coditio (), choose a sufficiety arge positive iteger max { T, + δ}, where δ max { τ, δ, δ } that ( p) 0 R( ) (8) p p R( ) (9) [ R( ) + R( ) ] < p (0), so hod. Where ad are positive costat, p. + A { x B : x( ), ( )}, the it is easy to see A is a bouded, cosed, ad covex subset of B. Let 0 75

4 Vo., o. oder Appied Sciece Defie a mappigt : A B as foows p px( τ) + Cs+ [ R( s) x( s δ) Tx( ) R()( s x sδ ) ] () Tx( ) 0 < Ceary, Tx is cotiuous. ext we wi prove that T is a sef- mappig i A. Whe, ad x A, by usig (8) ad (), we have Tx( ) p + C R ( s) x( s δ ) s+ ( ( p)) p+ s R ( s) p + Furthermore, i view of (9) ad (), we have: Tx( ) p px( τ) C R ( s) x( s δ ) s+ p p s R ( s) ( p p ) p p Ceary, whe 0 <, we have x( ). Thus Tx A which meas that T is the sef- mappig i A. ext we wi prove that T is a cotractio mappig i A. Whe >, for every x, x A, we have: Tx ( ) Tx ( ) px ( τ) + px ( τ) + C R ( s) x ( sδ ) x ( s δ ) + s+ C R ( s) x ( sδ ) x ( sδ ) s+ px x + x x s( R( s) + R( s)) x x p+ s ( R( s) + R( s)) From coditio (0), where 0 < q <, so that Tx( ) Tx( ) q x x. Ceary, whe 0 <, we have Tx( ) Tx( ) q x x. Thus T is the cotractio mappig i A. Accordig to Baach cotractio mappig, T has a fixed poit o A such thattx x. Ad x {()} x we have p px( τ) + Cs+ [ R( s) x( s δ) Tx( ) R()( s x sδ ) ] Tx( ) < 0 So this fixed poit x {()} x is a positive sequece. 7

5 oder Appied Sciece ovember, 008 To get the + order differece equatio from the above equatio, + Δ [ ( ) + ( τ)] + ( ) ( δ) ( ) ( δ) x px R x R x o Thus, the fixed poit {()} x is a positive soutio of equatio (). This competes the proof of Caim. Caim. p > From coditio (), choose a arge eough positive iteger > > 0 p ( ) + R( ) () p( ) ( + ) R( ) () [ R( ) + R( ) ] < p () hod. Where ad are positive costats, ( p + ad p( <, where τ max { δ, δ } + +, so that 0 Let A { x B : x( ), ( 0)}, the it is easy to see A is a bouded, cosed, ad covex subset of B. Defie a mappigt : A B as foows x ( τ) + Cs+ τ [ R( sxs ) ( δ) p p p + τ Tx( ) R()( s x sδ ) ] Tx( ) 0 < Caim. < p < From coditio (), choose a arge eough positive iteger max { T, + δ}, where δ max { τ, δ, δ } ( + p) ( + p) (5) 0 R( ) () ( + p) ( + p) 5 R( ) (7) [ R( ) + R( ) ] p (8), so that hod. Where, 5 are positive costats, ad 0< 5 <. Let A { x B : 5 x( ), ( 0)}, the it is easy to see A is a bouded, cosed, ad covex subset of B. Defie a mappigt : A B as foows + p px( τ) + Cs+ [ R( s) x( s δ) Tx( ) R()( s x sδ) ] (9) Tx( ) < 0 77

6 Vo., o. oder Appied Sciece Caim 5. p < From coditio (), choose a arge eough positive iteger > > 0 ( 7 )( p+ ) (0) R( ) 8 ( 8)( + p) R( ) () 8 [ R( ) + R( ) ] < p (), where τ max { δ, δ } + +, so that 0 hod. Where, 7 are positive costats, ad 8 0< 7 < <. 8 Let A { x B : 7 x( ) 8, ( 0)}, the it is easy to see A is a bouded, cosed, ad covex subset of B. Defie a mappigt : A B as foows + x ( τ) + Cs+ τ [ R( sxs ) ( δ) p p p + τ Tx( ) R()( s x sδ ) ] Tx( ) 0 < () Due to the proofs of Caim,, 5 are simiar as Caim, these proofs are eft out. Refereces Cheg,Jifa & Z.Aie.Existece of oosciatory Soutio to Secod Order Liear eutra Deay Equatio.System Sciece ad athematics,00,() Liu,Yue-hua & Che,Ya-bo.Existece of oosciatory Soutio of Secod Order Liear eutra Differece Equatio..Joura of Hua Agricutura Uiversity(atura Scieces),00,7():7-79. Li,Wexia.eutra Partia Fuctioa Differetia Equatio. Joura of Ahui Uiversity (atura Sciece Editio), 00,0():-. Liag,Jigcui, Zhog,Xiaozhu,Wag,Doghua & Ge,Lixia.Existece of oosciatory Soutio of Third Order Liear eutra Deay Differece Equatio. Joura of Yasha Uiversity, 00, 0():0-0. Su,Shu-rog & Asymptptic.Behavior ad Existece of Positive Soutio for the Higher Order eutra Differece Equatios.Joura of Egieerig athematics,00,9():5-0. Xig,Haiog & Zhog,Xiaozhu. Existece of oosciatory Soutio of High Order Liear eutra Deay Differece Equatio with Positive ad egative Coefficiets. Joura of Yasha Uiversity,007,():-0. 78