New Results on Oscillation of even Order Neutral Differential Equations with Deviating Arguments

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Advace i Pue Maheaic 9-53 doi: 36/ap3 Pubihed Oie May (hp://wwwscirpog/oua/ap) New Reu o Ociaio of eve Ode Neua Diffeeia Equaio wih Deviaig Ague Abac Liahog Li Fawei Meg Schoo of Maheaica Sye Sciece aiha Coege ai a Chia Schoo of Maheaica Sciece Qufu Noa Uiveiy Qufu Chia E-ai: 397@63co fweg@qfueduc Received Jauay 9 ; evied Mach 5 ; acceped Mach I hi pape we poi ou oe a iae i [6] evie he we obai oe ew ociaio eu fo ceai eve ode eua diffeeia equaio wih deviaig ague Ou eu exed ipove ay ow ociaio cieia becaue he aice u geeaie Meg Xu eu Keywo: Ociaio Neua Diffeeia Equaio Deviaig Ague Ioducio Ociaio of oe eve ode diffeeia equaio have bee udied by ay auho Fo iace ee [-7] he efeece heei We dea wih he ociaoy behavio of he eve ode eua diffeeia equaio wih deviaig ague of he fo x pi x i i q f x () whee i eve houghou hi pape i i aued ha: (A ) pi q C R f CR R uf u fo u f u i o-deceaig o R i ; (A ) i C R i i i i ; C R i (A 3 ) ; (A ) hee exi a coa M uch ha f xg x M x fo x ; (A 5 ) pi p p hee exi a fuc- i io q C R uch ha q q i : By a ouio of Equaio () we ea a fucio aifie Equa- We eic ou aeio o hoe x of Equaio () which exi o oe haf-ie x wih fo ay x A oivia ouio of Equaio () i caed ociaoy if i ha abiaiy age eo ohewie i i aid o be oociaoy Equaio () i aid o be ociaoy if a of i oivia ouio ae ociaoy Recey Meg Xu [6] udied Equaio () obaied oe ufficie codiio fo ociaio of he Equaio () we i he ai eu of [6] a foow Foowig Phio [5] we ay ha a fucio x which ha he popey ha x pi x i i C x R fo oe x io () o x ouio up x : H H beog o a fucio ca W deoe by H W if H CD R whee D : which aifie: (H ) H H fo ; (H ) H ha a coiuou o-poiive H paia deivaive aifyig he codiio: S H hh S fo oe h Loc D R C i a o-deceaig fucio heoe A ([6 heoe ]) Aue ha (A ) - (A 5 ) hod e he fucio H haify (H ) (H ) uppoe iup CF C G () Copyigh SciRe

5 L Z LI E AL ho fo evey C C whee F H q d H h G H H p he evey ouio of Equaio () i ociaoy heoe B ([6 heoe ]) Aue ha (A )-(A 5 ) hod H h ae he ae a i heoe A uppoe ha H if i if H i up G If hee exi a fucio (3) () C R uch ha fo a i if CF G (5) C i up d (6) whee ax he evey ouio of Equaio () i ociaoy I heoe A B fucio G houd be G o each of he codiio () () (5) (6) ha a ay a codiio Meawhie he Riccai fucio i o we-defied hee exi oe a eo i he poof of he heoe he pupoe of hi pape i fuhe o eghe ociaio eu obaied fo Equaio () by Meg Xu [6] I ou pape we edefie he fucio F G povide oe ew ociaio cieia fo ociaio of Equaio () Mai Reu I he eque we eed he foowig ea: Lea ([]) Le x be a ie diffeeiabe fucio o of oe ig x o which ( ) aifie x x he: (I ) hee exi a uch ha i x i ae of oe ig o ; (I ) hee exi a ube h 3 5 whe h 6 whe i odd i eve o uch ha i i i x x xx fo i h ; fo i h h Lea ([]) If x i a i Lea x x fo he fo evey hee exi a coa N uch ha x N X fo a age Lea 3([7]) Suppoe ha x i a eveuay poiive ouio of Equaio () e x pi x i i he hee exi a ube uch ha heoe Aue ha (A ) - (A 5 ) hod e he fucio H h aify (H ) (H ) uppoe iup MF G N (7) ho fo evey fo oe whee F H q H h G H H p he evey ouio of Equaio () i ociaoy Poof Suppoe o he coay ha x i a oociaoy ouio of Equaio () ha x i eve- uay poiive (whe x i eveuay egaive he poof i iia) Le be defied a i Lea 3 he foowig he poof of heoe i [6] wihou o of geeaiy aue hee exi a uch ha x ( ) N (by ea ) M q fo a Le he we have (o a [6]) Copyigh SciRe

L Z LI E AL 5 N M q Muipyig he above equaio wih epaced by H iegaig i fo o fo a by h (o a [6]) fo oe we obai M H q d H h N H NH H NH NH h H NH Hece we have MF G N fo a hi give i up MF N G which coadic (7) hi copee he poof of he heoe he aupio (7) i heoe ca fai coequey heoe doe o appy he foowig eu povide oe eeiay ew ociaio cieia fo Equaio () heoe Aue ha (A )-(A 5 ) hod he fucio H hf G be he ae a i heoe uppoe ha H if i if H If hee exi a fucio fo a fo oe NH (8) C R uch ha i up MF N G (9) h () i up whee ax he evey ouio of Equaio () i ociaoy Poof Aue o he coay ha () i o-ociaoy Foowig he poof of heoe wihou o of geeaiy aue fo a fo oe we obai M H q d H h NH NH So we ge MF G NB N whee B H H d Copyigh SciRe

5 L Z LI E AL he iup MF N G Niif B Fo a fo ay by (9) we have So epeciay N i if B () i if B ( ) N () Now we cai ha i up (3) Suppoe o he coay ha i up () By (8) hee i a poiive coa aifyig H if iif H (5) Le be ay abiay poiive ube fo () hee exi a uch ha fo a he fo we have B v v H d vdv H v v v H vdv H v H H By (5) hee exi a uch ha fo a H which ipie B fo a H Sice i abiay we have B i if B i which coadic () hu (3) ho he by () (3) we ge i up i up d which coadic () hi copee he poof Rea Le i heoe heoe educe o heoe A [6]; we obai he ae eu i heoe i which we oi he aupio () i heoe B [6] heefoe heoe ae geeaiaio ipovee of he eu obaied i [6] Rea Wih a appopiae choice of he fucio H h oe ca deive a ube of ociaio cieia fo Equaio () fo ou heoe Le () i a coa H h we have H i i H fo ay Coequey e uig heoe we have: Cooay Aue ha (A )-(A 5 ) (8) hod uppoe ha hee exi a fucio C R uch ha fo oe i up M q N (6) () (wih ) hod he evey ouio of Equaio () i ociaoy Exape Le coide he foowig ecod ode eua diffeeia equaio x p x q x (7) Copyigh SciRe

L Z LI E AL 53 p q ax i f x x i i hi cae M Le whee N by diec cacuaio we ge i up M q N i up i i co i co I i eay o veify ha () ho heefoe Equaio (7) i ociaoy by Cooay Howeve we ca eaiy fid ha i upg i up i o codiio () i heoe B i o aified hee how ha heoe B cao be appied o Equaio (7) Obviouy ou eu ae upeio o he eu obaied befoe 3 Acowedege he auho ae vey gaefu o he efeee fo hi/he vauabe uggeio Refeece [] R P Agawa S R Gace D ORega Ociaio heoy fo Diffeeia Equaio Kuwe Acadeic Dodech [] R P Agawa S R Gace he Ociaio of Highe Ode Diffeeia Equaio wih Deviaig Ague Copue & Maheaic wih Appicaio Vo 38 No 3-999 pp 85-99 doi:6/s898-(99)93-5 [3] Y Boa O Ai Ociaoy Behavio of Highe Ode Neua ype Noiea Foced Diffeeia Equaio wih Ociaig Coefficie Joua of Maheaica Aayi Appicaio Vo 9 No pp 3-39 doi:6/aa396 [] W N Li Ociaio of Highe Ode Deay Diffeeia Equaio of Neua ype he Geogia Maheaica Joua Vo 7 No pp 37-353 [5] Ch G Phio Ociaio heoe fo Liea Diffeeia Equaio of Secod Ode Achiv de Maheai Vo 53 No 5 989 p 83 doi:7/bf373 [6] F Meg R Xu Kaeev-ype Ociaio Cieia fo Eve Ode Neua Diffeeia Equaio wih Deviaig Ague Appied Maheaic Copuaio Vo 9 No 7 pp -8 doi:6/ac77 [7] Yu V Rogovcheo F ucay Ociaio Cieia Fo Secod-Ode Noiea Diffeeia Equaio wih Dapig Noiea Aayi: heoy Meho & Appicaio Vo 69 No 8 pp 8- doi:6/a75 Copyigh SciRe

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