ISSN 47-9 Oscllo Crer For Eve Order Noler Nerl Dfferel Eos Wh Med Argmes ABSTRACT E Thd SPdmvh S Pels Rmj Ise for Advced Sd Mhemcs Uvers of Mdrs Che 600 005 Id ehd@hooco Acdem Mlr Dermeo de Cêcs Ecs e Nrs Av Code Csro Gmrães 70- Amdor Porgl sdrels@gmlcom Ths er dels wh he oscllo crer for h order oler erl med e dfferel eos of he form d where s eve osve eger d re oegve coss d coss C[ 0 0 d d re ros of odd osve egers wh emles re rovded o llsre he m resls Kewords: Oscllo; Eve order; Med rgmes; Noler erl dfferel eos Mhemcs Sjec Clssfco 00: 4C5 re osve rel Some Cocl for Iovve Reserch Peer Revew Reserch Plshg Ssem Jorl: Jorl of Advces Mhemcs Vol 5 No edor@crworldcom wwwcrworldcom memercrworldcom 575 P g e D e c 6 0
ISSN 47-9 INTRODUCTION I hs er we sd he oscllor ehvor of ll solos of h order oler erl dfferel eos wh med rgmes of he form d where s eve osve eger d re oegve coss d coss C[ 0 0 d d re ros of odd osve egers wh re osve rel As s csomr solo s clled oscllor f hs rrrl lrge eros d ooscllor f s evell osve or evell egve Eos d re clled oscllor f ll s solos re oscllor Dfferel eos wh deled d dvced rgmes lso clled med dfferel eos or eos wh med rgmes occr m rolems of ecoom olog d hscs see for emle [4894] ecse dfferel eos wh med rgmes re mch more sle h del dfferel eos for dee reme of dmc heome The coce of del s reled o memor of ssem he s eves re morce for he crre ehvor d he coce of dvce s reled o oel fre eves whch c e kow he crre me whch cold e sefl for decso mkg The sd of vros rolems for dfferel eos wh med rgmes c e see [769] I s well kow h he solos of hese es of eos co e oed closed form I he sece of closed form solos rewrdg lerve s o resor o he lve sd of he solos of hese es of dfferel eos B s o e cler how o formle l vle rolem for sch eos d esece d eess of solos ecomes comlced sse To sd he oscllo of solos of dfferel eos we eed o ssme h here ess solo of sch eo o he hlf le The rolem of smoc d oscllor ehvor of solos of h order del d erl e dfferel eos hs receved gre eo rece ers see for emle [-7] d he refereces ced here However here re few resls regrdg he oscllor roeres of erl dfferel eos wh med rgmes I[0] he hors eslshed some oscllo crer for he followg med erl eos d c h c h g g 4 where s eve osve eger c h c h g g 5 c c h h d re rel mers d g d I[5] he hor eslshed some oscllo resls for he followg med erl eo g re osve coss 0 6 wh d re oegve rel vled fcos I[5] he hors eslshed some oscllo heorems for he followg secod order med erl dfferel eo where 0 d re ros of odd osve egers d C[ 0 [0 7 re osve coss d I[6] he hors eslshed some oscllo crer for he followg secod order med erl dfferel eo d 0 8 576 P g e D e c 6 0
ISSN 47-9 0 where d re ros of odd osve egers wh C[ [ 0 0 re osve coss d Clerl eos 4 d 5 wh d re secl cses of eos d d eo 6 wh d s secl cse of eo resecvel Moreover eos 7 8 d 9 wh re secl cses of of eos d resecvel Moved he ove oservo hs er we sd he oscllor ehvor of eos d for dffere vles of d Therefore or resls geerle d eed hose of [5056] I Seco we rese some sffce codos for he oscllo of ll solos of eos d Emles re rovded Seco o llsre he m resls SOME PRELIMINARY LEMMAS I hs seco we shll o some sffce codos for he oscllo of ll solos of he eos d Before rovg he m resls we se he followg lemms whch re essel he roofs of or oscllo heorems Lemm Le 0 B 0 If A B he A d The roof m e fod [] The A B A B Lemm [] Le C [ 0 R If A B A B s evell of oe sg for ll lrge 9 he here ess for some 0 d eger l 0 l wh l eve for 0 or l odd for 0 sch h l 0 mles h k 0 for k 0 l d l mles h lk k 0 for k l l Lemm [] Le e s Lemm Assme h s o decll ero o ervl [ 0 here ess 0 sch h 0 here ess T sch h for ll T Lemm 4 [5]Sose : 0 rel mer The he followg hold I If! for ll If lm 0 d he for ever 0 [ R s coos d evell oegve fco d s osve lms hold for some 0 he he el s s!! s ds hs o evell osve solo whch ssfes j 0 evell j 0 577 P g e D e c 6 0
ISSN 47-9 II If lms hold for some 0 he he el s s!! s ds hs o evell osve solo whch ssfes j j 0 evell j 0 Lemm 5 [4] Assme h for lrge s 0 for ll s[ ] where ssfes The [ ] 0 0 hs evell osve solo f d ol f he corresodg el hs evell osve solo [ ] 0 0 I [687] he hors vesged he oscllor ehvor of he followg eo where C[ 0 R [ ] 0 0 0 C[ 0 R lm d 0 osve egers Le The eo redces o he ler del dfferel eo s ro of odd 0 0 d s show h ever solo of eo osclles f lmf s ds e Oscllo Resls Frs we sd he oscllo of ll solos of eo Theorem Le for 0 d d ocresg fcos for 0 Assme h he dfferel eles re d c c 0 0 0 4 5 578 P g e D e c 6 0
ISSN 47-9 579 P g e D e c 6 0 where } m{ c hve o evell osve solo o evell osve decresg solo d o evell osve cresg solo resecvel The ever solo of eo s oscllor Proof: Le e ooscllor solo of eo Who loss of geerl we m ssme h s evell osvee here ess 0 sch h 0 for Se The 0 0 ll for 6 Ths 0 re of oe sg o ; [ As resl we hve wo cses: 0 for 0 for Cse: 0 for I hs cse we le 0 The Usg ove el eo we hve 0 or 0 c c hs osve solo whch s cordco Cse: 0 for Now we se 7 The Usg he moooc of d d Lemm he ove el we ge
ISSN 47-9 580 P g e D e c 6 0 Now sg 0 for he ove el we o 0 8 whch mles h he fco 0 re of oe sg We shll rove h 0 evell If o he 0 v Hece v Usg he ls el 6 we o 0 v v v or 0 v c v c v Usg he rocedre of cse v s osve solo of cordco Ths we cosder wo ossle cses:: 0 evell 0 evell Cse: Assme h 0 for ll The from 4 we hve 0 for 4 d 0 We clm h 0 for 4 To rove ssme 0 for 4 The dfferee 5 we hve 0 or 0 w w w where 0 w o [ 4 Sce he fco w s decresg o [ 4 we hve 0 0 4 for w cordco Ths 0 for 4 d from 6 we hve 0 0 4 d for 9 Now sg he moooc of we o 4 The from he ove el d 6 we hve
ISSN 47-9 0 4 0 hs osve decresg solo cordco Cse: Assme h 0 Scse: Assme h 0 o for ll Now we cosder he followg wo scses: for ll Proceedg s Cse d sg he moooc of Usg he ls el 6 d he moooc of we o Ths oce g s osve decresg solo of he eo whch s cordco Scse: Assme h 0 d hs wh 6 mles for ll The we hve hs osve cresg solo whch ssfes cordco Ths comlees he roof 0 we 0 for d Corollr Le for 0 d d d ocresg fcos for 0 If lmf re! s s s ds 0 e s s lms s ds 0!! 4 d 58 P g e D e c 6 0
ISSN 47-9 s s lms s ds!! where 0 he ever solo of eo s oscllor 5 Proof: Le e osve solo of for 0 The we hve 0 for ll Frher 0 for ll oherwse s Hece we hve 0 0 d 0 for The Lemm d Lemm we o From d he moooc of we hve 0! Comg he ls wo eles we o Le 0 0! The we see h! s osve solo of eo 0 B ccordg o he Lemm 5 d he codo codo grees h el 4 hs o osve solo whch s cordco Hece hs o evell osve solo Moreover vew of Lemm 4 II d he codo el 8 hs o evell osve solo whch ssfes 7 whch s cordco Hece hs o evell osve decresg solo Also vew of Lemm 4 I d he codo el 9 hs o evell osve decresg solo whch ssfes 0 whch s cordco Hece hs o evell osve cresg solo Ne we cosder he eo d rese sffce codos for he oscllo of ll solos Theorem Le for 0 odecresg fcos for 0 Assme h he dfferel eles 6 d d re c c 0 7 d 0 0 8 9 where c m{ } hve o evell osve solo o evell osve decresg solo d o evell osve cresg solo resecvel The ever solo of eo s oscllor 58 P g e D e c 6 0
ISSN 47-9 58 P g e D e c 6 0 Proof: Le e ooscllor solo of eo Who loss of geerl we m ssme h s evell osvee here ess 0 sch h 0 for Se d roceedg s he roof of Theorem we see h he fco 0 re of oe sg o ; [ As resl we hve wo cses: 0 for 0 for Cse: 0 for I hs cse we le 0 The Usg ove el eo we hve 0 or 0 c c hs osve solo whch s cordco Cse: 0 for Now we se 0 The Usg he moooc of d d Lemm he ove el we ge Now sg 0 for he ove el we o 0
ISSN 47-9 As he roof of Theorem cse we c esl see h 0 cses:: ' 0 evell ' 0 evell Cse: Assme h ' 0 for Ne we cosder wo ossle for ll The we hve 0 4 0 We clm h ' 0 for To rove ssme 0 for The dfferee 8 we hve or where ' 0 w o [ 4 4 0 ' ' ' ' 0 ' w w w Sce he fco cordco Ths ' 0 for 4 Now sg he moooc of we o w s decresg o [ 4 for ' 4 we hve 0 ' w 0 for 4 d from 9 we hve 4 d 0 for d 0 4 The from he ove el d 9 we hve hs osve decresg solo cordco Cse: Assme h ' 0 Scse: Assme h ' 0 o for ll 4 Now we cosder he followg wo scses: for ll Proceedg s Cse d sg he moooc of Usg he ls el 9 d he moooc of we o we 584 P g e D e c 6 0
ISSN 47-9 Ths oce g s osve decresg solo of he el 6 whch s cordco Scse: Assme h ' 0 for ll The we hve d hs wh 9 mles 0 hs osve cresg solo whch ssfes cordco Ths comlees he roof 0 for d 4 Corollr Le for 0 ocresg fcos for 0 If d d d re! lmf s s s ds 0 e 5 s s lms s ds 0!! 6 d s s lms s ds 0!! 7 he ever solo of eo s oscllor Proof: The roof s smlr o h of Corollr d hece he dels re omed Ne we cosder he eo d rese sffce codos for he oscllo of ll solos Theorem Le for d d Q m{ } P m{ } re osve fcos for 0 Assme h he dfferel eles Q 0 4 8 d 585 P g e D e c 6 0
ISSN 47-9 P 0 4 9 hve o evell osve decresg solo d o evell osve cresg solo The ever solo of eo s oscllor Proof: Le Se e evell osve solo of eo he here ess 0 d roceedg s he roof of Theorem we see h he fco for some Now we defe The 0 for d he 0 Usg he moooc of d Q 4 Now sg 0 for whch mles h he fco sch h 0 for s of oe sg o [ d Lemm he ove el we ge P 4 he ove el we o Q P 0 4 4 0 ' 0 evell ' 0 evell Cse: Assme h ' 0 0 d 4 we hve re of oe sg Ne we cosder wo ossle cses:: for ll Usg he fc h he fco The here ess 40 4 so h 0 s decresg o [ 4 d he we hve Usg he ls el 9 we o Q 0 4 hs osve decresg solo cordco for he eo 8 4 586 P g e D e c 6 0
ISSN 47-9 Cse: Assme h ' 0 s cresg o [ for ll so h ' 0 for ll Usg he fc h he fco 9 we hve P 0 4 hs osve cresg solo whch ssfes cordco The roof s ow comlee Corollr Le 0 for d 44 for d s s Q s ds 4 45!! lms where 0 d s s P s ds 4 46!! lms where 0 he ever solo of eo s oscllor Proof: The roof s smlr o h of Corollr d hece he dels re omed 4 Emles I hs seco we rese some emles o llsre he m resls Emle 4 Cosder he dfferel eo Here v 0 47 The oe c see h ll codos of Corollr re ssfed Therefore ll he solos of eo 4 re oscllor I fc s oe sch oscllor solo of eo 4 Emle 4 Cosder he dfferel eo where 0 Here e 9 e 9 e e v 7 e 79 If 744 79e 7 e 79 4 s 744 79e The oe c see h ll codos of Corollr re ssfed Therefore ll he solos of eo 4 re oscllor I fc e s oe sch oscllor solo of eo 4 s Emle 4 Cosder he dfferel eo 48 v 6 49 587 P g e D e c 6 0
ISSN 47-9 where Here 6 we c see h ll he codos of Corollr re ssfed Therefore ll he solos of eo 4 re oscllor I fc s s oe sch oscllor solo of eo 4 Emle 44 Cosder he dfferel eo Here e e v 87 6 e 87 e 4 87 87 e 4 e The oe c 6 e e see h ll codos of Corollr re ssfed ece codo Therefore ll he solos of eo 44 o ecessrl oscllor I fc e s oe sch ooscllor solo sce ssfes eo 44 We coclde hs er wh he followg remrk Remrk I wold e eresg o o oscllo resls for he eos o 4 whe 0 or 0 d or d 0 Refereces [] RPAgrwl SRGrce d DO Reg 000 Oscllo Theor for Dfferece d Fcol Dfferel Eos Klwer Acdemc Dordrech [] TAsel d HYoshd 00 Sl sl d comle ehvor mcrodmc models wh olc lg Dscree Dmcs Nre d Soce 5 8-95 [] LBeresk d EBrverm 998 Some oscllo rolems for secod order ler del dfferel eos J Mh Al Al 0 79-740 [4] DMDos 004 Eeso of he Kldor-Kleck models of sess ccle wh comol ced cl sock Jorl of Orgsol Trsformo d Socl Chge 6-80 [5] JDr d SKlsr 00Oscllo crer for secod order erl fcol dfferel eosplmhderece59 5- [6] LHEre Qgk Kog d BGZhg 995 Oscllo Theor for Fcol Dfferel Eos Mrcel Dekker New York [7] JMFerrer d SPels 006 Oscllor med dfferece ssems Hdw lshg cororo Advced Dfferece Eos ID -8 [8] RFrsh d HHolme 95 The Chrcersc solos of med dfferece d dfferel eo occrg ecoomc dmcs Ecoomerc 9-5 [9] GGdolfo 996 Ecoomc dmcs Thrd Edo Berl Srger-verlg [0] SRGrce 994Oscllo crer for -h order erl fcol dfferel eosjmhalal84 44-55 [] SRGrce d BSLll 984 Oscllo heorems for -h order oler dfferel eos wh devg rgmes Proc Amer Mh Soc90 65-70 [] IGör d GLds 99 Oscllo Theor of Del Dfferel Eos Clredo Press New York [] VIkovelev d CJVegs 005 O he oscllo of dfferel eos wh deled d dvced rgemes Elecroc Jorl of Dfferel Eo 57-6 [4] RWJmes d MHBel 98 The sgfcce of he chrcersc solos of med dfferece d dfferel eos Ecoomerc 6 6-4 [5] YKmr 985 Oscllo of fcol dfferel eos wh geerl devg rgmes Hroshm MhJ 5 445-49 [6] TKrs 000 No oscllo for fcol dfferel eos of med e JMhAlAl 45 6-45 [7] TKso 98 O eve order fcol dfferel eos wh dvced d rerded rgmes JDfferel 50 d 588 P g e D e c 6 0
ISSN 47-9 Eos45 75-84 [8] GSLddeVLkshmkhm d BGZhg 987 Oscllor Theor of Dfferel Eos wh Devo Argmes Mrcel DekkerIc NewYork [9] GLds d IPSvrolks 98 Oscllo csed severl rerded d dvced rgmes Jorl of Dfferel Eo44 4-5 [0] TL d EThd 0 Oscllo of solos o odd order oler erl fcol dfferel eos ElecJ Dff Es - [] ChGPhlos 98 A ew crer for he oscllor d smoc ehvor of del dfferel eos Bll Polsh Acd Sc Mh96-64 [] YVRogovcheko 999 Oscllo crer for cer oler dfferel eos JMhAlAl 9 99-46 [] STg CGo EThd d TL 0Oscllo heorem for secod order erl dfferel eos of med e Fr Es JMhSc6 75-85 [4] XHTg 00 Oscllo for frs order serler del dfferel eos JLodo MhSoc655- [5] EThd d ReRem 0 Oscllo heorems for secod order oler erl dfferel eos of med e Fcol Dfferel Eoso er [6] EThd d ReRem O Oscllo crer for secod order oler erl dfferel eos of med eelecjqlve Theor of Dff Es75-6 [7] QXZhg JRY LGo 00 Oscllo ehvor of eve order oler erl del dfferel eos wh vrle coeffces Com Mh Al59 46-40 589 P g e D e c 6 0