Aled ad Comuaoal Maemacs 6; 5(): 5-55 ://wwwsceceulsggoucom/j/acm do: 648/jacm653 ISSN: 38-565 (P); ISSN: 38-563 (Ole) Te Saly of g Ode Ma-Tye Dffeece Equao a Ca-og * L Lue Ta Xue Scool of Maemacs ad Sascs Guag Nomal Uvesy Gul Ca Emal addess: cog_ma@63com (a Ca-og) * Coesodg auo To ce s acle: a Ca-og L Lue Ta Xue Te Saly of g Ode Ma-Tye Dffeece Equao Aled ad Comuaoal Maemacs Vol 5 No 6 5-55 do: 648/jacm653 Receved: Feuay 5 6; Acceed: Mac 8 6; Pulsed: Al 7 6 Asac: I s ae we vesgae e saly of followg ma-ye dffeece equao = + a ma + m wee < m < < m < < < w { m m m } { } = φ a > ( ) > ( j= ) ad ma{ } a e al j > values ae osve y cosucg a sysem of equaos ad ay fuco we sow e equao as a uque osve equlum soluo ad e osve equlum soluo s gloally asymocally sale Te cocluso of s ae eeds ad sulemes e esg esuls Keywods: Dffeece Equaos Posve Soluo Covegece Gloally Sale Ioduco I maemacs ecusve elao wc s dffeece equao s a kd of ecuso fomula o defe a sequece: e sequece of eac em s defed as a fuco Dffeece equao s e dscezao of dffeeal equaos Te dffeece sysem s desced e maemacal model of dscee sysem s a moa ac of dyamcal sysem e alcao of s eoy s adly oadeg o vaous felds suc as ecoomcs ecology yscs egeeg cool eoy comue scece ad so o (see [-4]) Te saly ad gloal eavo s oe of e o sos eseaces aou dffeece equao model e cocluso as a cea gudg ole o oduco acces I ece yeas moe ad moe eseaces o e dyamc eavos of ge ode olea dffeece equaos ave ee suded (see[5-9]) Oe of e classes of suc dffeece equaos ae ma-ye dffeece equaos (see [-9]) I [6] Amle suded e olea dffeece equao sowed a e uque osve equlum + = + soluo = + s gloally asymocally sale: I [7] Fa suded e ge ode dffeece equao = ad gave a suffce codo fo s f ( + k ) gloal asymocal saly ese esuls ae aled o e = a+ dffeece equao + k I [8] Su suded gloal eavo of e ma-ye dffeece equao = ma{/ A / } + m oved a f A ( ) ad sua < s a eodc sequece e evey osve soluo of s equao s eveually eodc w eod m I [9] Sevć suded eavo of osve soluos of e followg ma-ye sysem of dffeece equaos y ma{ c } y ma{ c } = = + + y oved a f c () e evey osve soluo of e
Aled ad Comuaoal Maemacs 6; 5(): 5-55 5 sysem coveges o () I s ae we vesgae e gloal saly of followg ma-ye dffeece equao = + ma + m () a wee = m < m < < m < < < w { m m m } { } = ϕ a > ( ) > ( j= ) ad P> ma{ j } a e al values ae osve y cosucg a sysem of equaos ad ay fuco we wll fomulae ad ove e equao as a uque osve equlum soluo ad e osve equlum soluo s gloally asymocally sale Te cocluso of s ae eeds ad sulemes e esg esuls s cocluso as a cea gudg ole o oduco acces as a maemacal model Fo coveece we deoe l = ma{ m } A = = ma{ } So > A Some Defos I s seco we wll oduce some defos (see[]) wc wll e eeded Defo A [] Le I e some eval of umes ad le f : I I I e a couously dffeece fuco A dffeece equao of ode (k+) s a equao of e fom + k = f ( ) = A o I s called equlum soluo of e dffeece equao f = f ( ) a s = fo all k Defo [] Te equlum s called locally sale f fo evey ε > ee ess δ> suc a f { } s a soluo of dffeece equao w al = k values sasfed + + + < δ k k e < ε fo all k Defo C [] Te equlum s called a gloal aaco f fo evey soluo { } of dffeece equao we ave lm = = k Defo D [] Te equlum of dffeece equao s called gloally asymocally sale f s a locally sale ad s also a gloal aaco of e dffeece equao 3 Ma Resuls I s seco we fomulae ad ove some lemmas ad ma eoems s ae oa a evey osve soluo of () as o e e ulmae fom of gloally asymocally sale Teoem Equao () as a uque osve equlum soluo = + A Poof Sce we ave = + A = + A > so equao () as a uque osve equlum = + A # Equao a s + a ma = + A = e oly fed o fo e soluo of s equao s = deoe y D a s A D= ecause A so D > Lemma Fo ay eal ume D f al values [ ] e [ ] ( ) l Poo f Sce = + a ma D = = A = + A D= so A fo ay [ ] we ave l = + a ma m A + A + = A A = + a ma m + = + ( A) A = > A A
53 a Ca-og e al: Te Saly of g Ode Ma-Tye Dffeece Equao suose fo evey k ee s [ ] e = + a ma k+ k m k k A + A + = y duco fo evey we ave [ ] # Le = = D fo ay defe e sysem of equaos as follows: = + a ma + () = + a ma + (3) a s = + A = + A + + Lemma Fo evey ad lm ee s < < + + = lm = Poof Oaed y aove defo (-3) we ave = + A = + A < + A= = + A = + A > + A= so = < < = ecause = + A = + A = + A = + A so < < < + A = + A= > + A = + A= y duco ee s < < fo + + evey Ta s = < < < = Accodg o e moooe ouded eoem we kow e lms of { } { } ae esece Le = lm = lm Take lms o o sdes of (-3) e a s = + ma a = + ma a = + A = + A eefoe = P+ A= P+ A so ( ) = Sce lm = lm = # Teoem Te uque equlum equao () s locally sale Poof Se so = = + A a s = D= ad as defed A Lemma Fo evey ε> w < ε< m{ D } accodg o Lemma ad local oudedess ee ess suc a ε< < < < + ε Take < δ= m{ } a s ( δ + δ) [ ] Te fo evey ( δ + δ) l we ave + a ma = + + a ma = + a s [ ] ( ε + ε) Smlaly y duco ee s [ ] of ( ε + ε) fo evey l Accodg o eoem e equlum > A = + = + a m ma = + A s locally sale # A ( A) Teoem 3 Te uque equlum soluo = + A
Aled ad Comuaoal Maemacs 6; 5(): 5-55 54 of equao () s gloally asymocally sale Poof I Teoem we ave oved = + A s locally sale e we wll ove = + A s gloal aaco Se = ma{ D} ad as defed l+ Lemma Followg Lemma fo evey ee s [ ] = [ ] So = + A ( l+ ) + = + a ma l+ m l+ l+ + A = = + A ( l+ ) + = + a ma l+ m l+ l+ lm = y Defo C we kow = + A s gloal aaco Accodg o Defo D s ovously a e equlum = + A of equao () s gloally asymocally sale # 4 Eamle Cosde oe of eamle of dffeeal equao (): 5 3 5 = + ( + 5 + 35 ) ma + 4 5 = 3 6 9 (4) wee e al values ( + ) Ovously 9 sasfes e codos of Teoem 3 so e uque equlum = 38 of equao (4) s gloally asymocally sale y gvg e al value assgme e followg fgues - sow e gloal asymoc saly If al values = = = = equlum = 38 9 s gloally asymocally sale (see Fgue ) If alvalues = = = 5 = = = = = 3 6 9 7 8 = = equlum = 38 s gloally asymocally (see Fgue ) 4 5 + A = y duco ee s [ ] fo evey ( l + ) + Smlaly we ave [ ] fo evey ( l+ ) + y duco [ ] fo evey kl+ ( ) + wee k= Followg Lemma we kow lm k k = lm = so Fgues Te soluo of equao (4) we al values = = = = 9 Fgue Te soluo of equao (4) we al values = = = 5 3 6 9 = = = = = = = 4 5 7 8
55 a Ca-og e al: Te Saly of g Ode Ma-Tye Dffeece Equao 5 Cocluso I s ae we vesgae e caaces of osve soluo of e ma-ye dffeece equao () Fs we sowed equao () as uque osve equlum = + A Te we oved wo useful lemmas y cg lemmas we sowed e ma eoems s ae a s e equlum soluo = + A of equao () s gloally asymocally sale A las we gve a eamle of dffeece equao () daw e ajecoy of e soluo y gvg wo dffee al values us uvely eflec e gloal asymoc saly Ackowledgemes Taks fo edos ad evewes' valuale commes ad suggesos fo movg s ae Ts eseac was suoed y NNSF of Ca (467) Scefc ad ecologcal eseac ojec of Guag colleges ad uveses fuded y Guag Deame of Educao (LX448 LX455) ad You Foudao of Guag Nomal Uvesy Refeeces [] El-Mewally Gloal eavo of a ecoomc model Caos Solos & Facals 33(3) 7 994-5 [] El-Mewally El-Aff M M O e eavo of some eeso foms of some oulao Models Caos Solos & Facals 36() 8 4-4 [3] Zou L ogua U Lag C e al Reseac o dffeece equao model affc flow calculao Joual of Cag cu Uvesy of Scece & Tecology 4 7-3 [4] uag C M Wag W P Alcaos of dffeece equao oulao foecasg model Advaced Maeals Reseac 4 79-8 [5] eeau K Foley J S Sevć S Te gloal aacvy of e aoal dffeece equao y = + y / y k m Poceedgs of e Ameca Maemacal Socey 35 7 33-4 [6] eeau K S Sevć S Te eavo of e osve soluos of e dffeece equao = A + ( / ) J Joual of Dffeece Equaos ad Alcaos (9) 6 99-98 [7] eg L Sevć S Peodcy of some classes of olomoc dffeece equaos Joual of Dffeece Equaos ad Alcaos (8) 6 87-835 [8] Iča Sevć S Some sysems of olea dffeece equaos of ge ode w eodc soluos Dyamcs of Couous Dscee ad Imulsve Sysems Sees 3A (3-4) 6 499 57 [9] Iča Sevć S Eveually cosa soluos of a aoa ldffeece equao Aled Maemacs ad Comuao 5 9 854-856 [] Elaasy E M El-Mewally A Elsayed E M Gloal eavo of e soluos of some dffeece equaos Advaces Dffeece Equaos 8() 683-689 [] Elsayed E M Iča Sevć S O e ma-ye equao = ma{ A / } As Comaoa 95 + 87-9 [] Sevć S Gloal saly of a ma-ye dffeece equao Aled Maemacs & Comuao 6() 354 356 [3] Su T X X J a C Dyamcs of e ma-ye dffeece equao = ma{/ A / } + m Joual of Aled Maemacs ad Comug (-) 73-8 [4] Sevć S O a symmec sysem of ma-ye dffeece Equaos Aled Maemacs ad Comuao 9(5) 3 847-84 [5] Sevć S O some eodc sysems of ma-ye dffeece equaos Aled Maemacs ad Comuao 8 483 487 [6] Amle A M Geogou D A Gove E A Ladas G O e ecusve sequece = α+ Joual of / + Maemaal Aalyss ad alcaos 33() 999 79-798 [7] Fa Y Wag L L W Gloal eavo of a ge ode olea dfeece equao Joual of Maemaal Aalyss ad alcaos 99() 4 3-6 [8] Su T X e Q L Wua X X J Gloal eavo of e ma-ye dffeece equao = ma{/ A / } Aled + m Maemacs ad Comuao 48 4 687-69 [9] Lu W P Sevć S Gloal aacvy of a famly of o-auoomous ma-ye dffeece equaos Aled Maemacs ad Comuao 8() 697-933 [] Gove E A Ladas G Peodces Nolea Dffeece Equaos Vol 4 New Yok: Cama& all/crc Pess 5