Orbital Euclidean stability of the solutions of impulsive equations on the impulsive moments

Transcription

1 Pure ad Appled Mahemacs Joural 25 4(: -8 Publshed ole Jauary (hp://wwwscecepublshggroupcom/j/pamj do: 648/jpamj254 ISSN: (Pr ISSN: (Ole Orbal ucldea sably of he soluos of mpulsve equaos o he mpulsve momes Kaya Georgeva Dshleva Faculy of Appled Mahemacs ad Iformacs echcal Uversy of Sofa Sofa Bulgara mal address: kgd@u-sofabg o ce hs arcle: Kaya Georgeva Dshleva Orbal ucldea Sably of he Soluos of Impulsve quaos o he Impulsve Momes Pure ad Appled Mahemacs Joural Vol 4 No 25 pp -8 do: 648/jpamj254 Absrac: Curves gve a paramerc form are suded hs paper Curves are couous o he lef he geeral case her correspodg parameers belog o he defoal ervals whch s possble o o cocde for he dffere curves Moreover he pos of dscouy (f hey exs are frs kd (jump dscouy ad hey are specfc for each curve Upper esmaes of he ucldea dsace bewee wo such curves are foud he resuls obaed are used sudes of he soluos of mpulsve dffereal equaos Suffce codos for he orbal ucldea sably of he soluos of such equaos respec o he mpulsve effecs o he al codo ad mpulve momes are foud hs ype of sably s roduced ad suded here for he frs me Keywords: ucldea Dsace Paramerc Curves Impulsve Dffereal quaos Orbal ucldea Sably Iroduco Fdg he upper esmaes of he ucldea dsace bewee paramerc curves (cludg he case where he curves are pecewse couous s a mpora ask For he coveece we wll cosder ha he parameer of hese curves represes he me Qualave research relaed o he orbal ucldea sably of he soluos of mpulsve dffereal equaos ca sar afer fdg he above-meoed esmaes ad more precse of he echology for her recevg I s kow ha he rajecores of hese equaos are curves whch are pecewse couous hey are couous o he lef had sde her correspodg erval of exsece hese pos of dscouy are frs kd (see [2] [3] [] [2] [4] [9] ad [24] I he case where he dffereal equaos have varable mpulsve momes her o cocdg soluos (rajecores possess dffere ses of breakpos (see [4] [5] [8] [9] ad [6] herefore he paper we explore ad assesses he ucldea dsace bewee paramerc curves whch are pecewse couous o her lef had sde ad whch possess specfc (ow momes of dscouy of he frs kd he obaed resuls are appled o he sudy of orbal ucldea sably whch s specfc for he equaos wh mpulsve effecs ad s roduced hs paper Should be oed ha he mpulsve dffereal equaos are covee mahemacal apparaus for a descrpo of dyamc pheomea subjeced o he dscree shor erm exeral flueces Due o s wde applcao he qualave properes of hese equaos are examed serously (see [] [7] [] [3] [5] - [8] [2] - [23] ad [25] 2 Prelmary Remarks ad Resuls We wll use he followg oaos Le he pos a( a a2 a ad b( b b2 b R he her do produc wll be deoed by: a b = ab a b 2 2 ab ucldea dsace bewee boh pos s: ( a b ( 2 ( 2 ( 2 = a b a b a b 2 2 ucldea orm a of po a s a = a a = a a a he ex equaly s vald ( a b a b =

2 2 Kaya Georgeva Dshleva: Orbal ucldea Sably of he Soluos of Impulsve quaos o he Impulsve Momes Le he oempy ses A B R he ucldea dsace bewee boh ses s: ( { ( { a b a A b B A B = f f If a leas oe of he ses A ad B s empy he for he coveece we wll cosder ha ( A B = Furher by A ad A are deoed he coour ad closure of se A respecvely Remark he ex properes are vald for he ucldea dsace bewee he ses R Le he ses A B C D R ad cosa λ R he: A B < ( 2 If A B ( A B = 3 ( A B = ( { a A { b B : ( a b lm = 4 If ( A B = ad A B are bouded A B 5 If A B ( A B = 6 ( A B = ( A B 7 ( A B = ( B A 8 ( λ A λb = λ ( A B 9 If A B ad C D A C B D ( ( If se A s bouded ad ( A B < he se B s also bouded heorem Suppose ha he oempy ses A A2 A k B B B R 2 he ad s ( U A B p= k Uq= s p q { ( Ap Bq p q m = 2k = 2s Proof Sce ( p= k = 2 k A U Ap ( j = 2s Bj U Bq q= s he usg propery 9 of he prevous remark we oba ( U A B p= k Uq= s p q ( A B = 2 k j = 2 s j From he equales above he heorem s rue Le he fucos g g : R R ad he cosas R We roduce he paramerc curves: ad γ [ ] { ( g = > { g ( γ = > Smlarly we roduce he curves: ( ] [ ( γ γ γ ( ( γ γ γ defed he half ope ad ope ervals respecvely Remark 2 Le ad he followg defoal equaos relag o he ucldea dsace bewee he curves γ γ ad he ad [ ] uform dsace bewee he curves [ ] γ [ ] are vald respecvely: ( [ ] γ ad γ γ = f f ( g ( g ( { { R ( γ [ ] γ [ ] ( ( = sup { ( g g { g ( ( = sup g As see from he remark above he uform dsace s defed oly whe he curves have a commo defoal erval Smlar defoal equaos for he ucldea ad uform dsace bewee he curves are vald whe hey are defed he half-ope ad ope ervals I he ex wo heorems we wll use he oaos: = m { { m = = m { { m heorem 2 Suppose ha he fucos g g C R R = m 2 he equaly s sasfed he he followg esmae vald ( [ ] γ γ ( m m R γ γ Proof he saeme follows from Remark ad Remark 2 Acually we have

3 Pure ad Appled Mahemacs Joural 25 4(: -8 3 ( ( [ ] γ γ m m m = γ γ γ ( m m { ( g ( g ( m f ( g g { ( g ( g ( m ( m m R m m m γ γ γ γ γ { m = f f ( ( { m sup = γ γ he heorem s proved he ex heorem ehaces he prevous heorem 3 Suppose ha: he fucos g g : R R ad hey are couous o he lef had sde R m 2he equaly s sasfed he he followg esmae s vald: ( ( ( ] γ γ m m { R γ γ ( ( ( m ( ( ( g ( ( γ ( ( ( ( g γ ( ( g γ γ g ( Proof We wll prove he saeme of heorem uder he addoal assumpo ha he ex equales are vald: ad he oher hree cases are cosdered smlarly I s clear ha hs case he ervals ad herefore < = ad < = ( ( ( g ( ( γ = g = ( ( ( ( ( ( g γ = g = (2 akg o accou ha fuco g s couous o he lef had sde a a po we deduce ha γ Aalogously ( ] g ( γ ( ] = ( = ( γ ( γ g We apply propery 6 of Remark ad we derve ( ( ( ] ( ( ] γ γ γ γ ( = = ( g ( γ ( g ( γ ( ] = g ( γ ( γ ( g ( ( ( ( ( ( γ γ g g Usg heorem by he equaly above we receve: ( ( ( ] ( ( γ g γ γ { ( m ( ( ( g ( ( γ γ ( γ from where as we ge o accou ( ad (2 we fd: ( ( ( ] ( ( γ g γ γ { ( m ( ( ( ( g ( γ ( ( g ( ( ( ( g R γ γ γ ( γ he heorem 3 s proved Smlarly we prove he saeme: heorem 4 Suppose ha: he fucos g g : R R are couous o he rgh had sde R m 2 he equaly s sasfed he he ex esmae s vald: ( [ γ γ m m { R γ γ ( m ( ( g γ ( ( g γ ( g ( γ ( ( g γ As a cosequece of he heorem above we wll formulae he followg saeme relag o he ucldea dsace bewee couous curves heorem 5 Suppose ha: he fucos g g C R R m 2 he equaly s sasfed he he ex esmae s vald: ( [ ] = ( ( ] γ γ [ = γ γ γ γ ( = ( γ γ ( ( (

4 4 Kaya Georgeva Dshleva: Orbal ucldea Sably of he Soluos of Impulsve quaos o he Impulsve Momes m m { R γ γ ( m ( ( ( g ( γ ( g ( ( ( g γ g γ γ he followg heorem summarzes heorem 3 ad heorem 4 heorem 6 Suppose ha: he fucos g g : R R 2 here exss a umber k N such ha he followg equales are sasfed: < < < < k < < < < k < < < < < where m m 2 2 { { m m k k = m = { { = m = m k k k k k k he: If he fucos g ad g are couous o he lef had sde R he he ex equaly s vald: ( ( ( ] γ γ k k m m { R γ ( γ ( ( m ( g ( γ ( ( ( ( ( γ ( ( g γ ( g ( ( 2 γ = k 2 If he fucos g ad g g are couous o he rgh had sde R he he ex equaly s vald: ( [ γ γ k k m m { R γ γ ( m ( g ( γ ( γ ( ( γ ( g ( g ( γ = 2 k Defo If he fucos g g g : R R 2 he fuco G ( ( γ γ [ ] where R We wll say ha he fucos g ad ucldea equvale f ( R G ( = g are: :lm = I hs case we deoe g g 2 Uformly ucldea equvale f ( ( R : lm G ( = he oao hs case s Remark 3 Le { g g = < he 2 G ( 2 = ( γ γ [ ] 2 2 ( [ ] [ ] [ ] 2 2 ( γ γ [ ] = G ( = γ γ γ γ e fuco G s mooocally decreasg for G Furhermore ( lm G ( always exss Cosequely he lm Remark 4 I s clear ha f wo fucos are uformly ucldea equvale he hey are ucldea equvale he oppose s o rue Ideed f here exss a cosa { > such ha: ( lm G = ad ( lm G = cos > he: From he frs equaly follows ha ( lm G = e he fucos are ucldea equvale 2 By he secod equaly follows ha ( ( ( lm G lm G = cos > e he fucos are o uformly ucldea equvale 3 Ma Resuls he followg al value problem of mpulsve dffereal equaos s a objec of vesgao he paper: = f ( x < (3 d ( ( ( ( x = x I x = 2 (4

5 Pure ad Appled Mahemacs Joural 25 4(: -8 5 ( x x = (5 where: Fuco f : R D R D s a oempy doma R he mpulsve fucos I : D R ( Id I : D D = 2 Id s a dey R x R D A al po ( he momes 2 < < 2 < are amed mpulsve he soluo x( x of problem (3 (4 (5 we defe as follows: For he soluo of he cosdered problem cocdes wh he soluo of problem whou mpulses f ( x x( x d = = We roduce he oao x x( x = 2 A he mome he mpulsve effecs sasfyg he equaly below akes place: ( = ( ( ( = x I ( x = x x x x x I x x 2 For < 2 he soluo of problem (3 (4 (5 cocdes wh he soluo of problem whou mpulses = f ( x x( = x d We deoe by x x( x = A he mome 2 he mpulsve effecs s performed: ( 2 = ( 2 2 ( ( 2 = x I ( x = x x x x x I x x ec I s clear ha he soluo s a pecewse couous fuco wh breakpos 2 whch he soluo s couous o he lef had sde Furher we wll use he oaos: ( x = x x ( ( ( ( ( x = x x I x x = Id I x = 2 Alog wh he ma problem we cosder also he perurbed problem: = f ( x < (6 d ( ( ( ( x = x I x = 2 (7 ( x x where he al po s ( x momes are 2 2 = (8 R D ad he mpulsve < < < he soluo of he perurbed problem s deoed by ( Le he cosas By ( ] 2 2 x x R ad < < 2 2 χ 2 s deoed he rajecory of problem (3 (4 (5 defed for < 2 Aalogously by χ ( 2 s deoed he rajecory of problem (6 (7 (8 locked < he followg equales are vald: 2 ad χ χ ( ( ] 2 2 We deoe { ( x x < 2 < 2 = 2 { x ( x < 2 < 2 = 2 = m { ad { m = = 2 Defo 2 We wll say ha he soluo of he ma problem (3 (4 (5 s (uformly orbal ucldea sable o he al po ( x ad he mpulsve momes 2 f: ( δ δ R δ δ < ( δx = cos R : ( ( x R D < δ x x < δ x ( 2 2 δ < = e he soluos ( ( x( x x x x x ad ( x x are (uformly ucldea equvales he ma purpose of hs sudy s o fd he suffce codos whch guaraee he propery uformly orbal ucldea sably of he soluos of he cosdered equaos Furher we wll refer o he problem ( ( f x x x d = = (9 χ he soluo of hs problem s deoed by ( x Defo 3 We wll say ha he soluos of sysem (9 are

6 6 Kaya Georgeva Dshleva: Orbal ucldea Sably of he Soluos of Impulsve quaos o he Impulsve Momes uformly Lpschz sable wh a posve Lpschz cosa C f L ( δ L = cos > : ( ( ' '' ' '' R x x D x x < δl ( χ ( ' '' ' '' L χ x x < C x x he uform Lpschz sably was roduced 986 by F Daa ad S layd [6] We wll use he followg codos: x R D he soluo of HFor every po ( problem (9 exss ad s uque for he soluo s Lpschz sable wh Lpschz cosa C L H2here exss a posve cosa C such ha ( ( ( x R D f x Cf H3here exss a posve cosa C such ha C = 2 H4he mpulsve fucos I I 2 are equal Lpschz e here exss a posve cosa C I such ha ( ( ( x ' x'' D I x' I x '' C x ' x'' = 2 I heorem 7 Le he codos H-H4 be vald If CLC I < he he soluo of problem (3 (4 (5 s uformly orbal ucldea sable Proof From codo H3 follows ha lm = s fulflled We assume ha: f < δ ( where < δ < C = 2 δ δ < ad x x < δ x ( where δ x > Usg codo H3 ad equaly ( follows m 2 < = For coveece we dvde he proof o several pars Par We wll evaluae he orm of dfferece x x For hs purpose we wll x ( x ( suppose ha = he oher case s cosdered smlarly Gve all of ( ( ad codo H2 we ge ( x ( x x x ( x ( x x x ( x ( x x x ( ( x x f τ x τ x dτ δ δ δ x Cf x Cf Par 2 We wll esmae he orm of dfferece ( m m x x x( x Whou lmao we ca assume ha δx Cf δ < δl where he cosa δ L sasfes Defo 3 he from Par we ge ( ( x x x x δ < L Fally from codo H we have x ( m ( m x x x ( ( x( x( x ( ( L x x CLδx CLCf δ x x x m m C x x Par 3We wll evaluae he dfferece x ( x ( x x s vald he Assume ha he equaly = m = x ( x x ( x ( ( ( ( x x x x x x x x ( C δ C C δ L x f L Par 4We wll esmae he orm of dfferece ( = x( x x x x x Le as he prevous par of he proof we assume ha = he cosderao he oher case are smlar We cossely fd ( ( x x = x x x x ( I ( x ( x = x x ( I ( x( x x x x ( x x( x where ( ( I ( x( x I x x ( C C C C = qδ q δ f L I Lδx CICL δ CL q = CICL ad ( C C f L = CL x

7 Pure ad Appled Mahemacs Joural 25 4(: -8 7 Par 5Le δx = qδx q δ By repeag he reasog of he prevous four pars of he proof we reach he esmae ( x x qδ q δ = q δ q δ qδ x x Smlarly for each umber k we oba k k k k k δx ( δ δ δ( k x x q q q q k = 2 Par 6For each ( lm G R we have ( = lm χ χ [ ] lm x k x k k k k k xlm q lm ( q q q ( k k k δ δ δ δ ( ( k k q δ s q δ q δ k s = lm k s s2 ( q δ ( q δ ( 2 qδ k s k s ( k ( k s ( ( k s s lm q q q δ δ δ k s s 2 ( q q q { δ ( k s ( k s ( k = 2 ( δ δ δ( k s k s q lm q lm { δ = ( k s ( k s 2 ( k q k s δ q sup = k s k s 2 q q { δ ( ( (2 Le ε be a arbrary posve cosa Sce he cosa q sasfes he equales < q < he s clear ha s δ ε s = s N s s q < (3 q 2 ( ( ε : ( We fx s = s Sce he seres δ δ2 s coverge he s fulflled ( k N k > s ( : ε > k s k < q 2 ε sup { δ = ( k s ( k s 2 < (4 q 2 By (2 (3 ad (4 follows ha for every s sasfed ( ε G ( lm G < lm = R I meas ha he soluo of problem (3 (4 (5 s uformly orbal ucldea sable he heorem 7 s proved Refereces [] Akhme M Bekloglu M rgec kacheko V A mpulsve rao-depede predaor prey sysem wh dffuso Nolear Aalyss: Real World Applcaos Vol 7 5 ( [2] Alzabu J Abdeljawad xpoeal boudedess for soluos of lear mpulsve dffereal equaos wh dsrbued delay Ieraoal J of Pure ad Appled Mahemacs Vol 34 2 ( [3] Alzabu J xsece of perodc soluos of a ype of olear mpulsve delay dffereal equaos wh a small parameer J of Nolear Mahemacal Physcs Vol 5 ( [4] Baov D Dshlev A Populao dyamcs corol regard o mmzg he me ecessary for he regeerao of a bomass ake away from he populao Comes Redus de l Academe Bulgare Sceces Vol 42 2 ( [5] Baov D Dshlev A he pheomeo beag of he soluos of mpulsve fucoal dffereal equaos Commucaos Appled Aalyss Vol 4 ( [6] Daa F layd S Lpchz sably of olear sysems of dffereal equaos J of Mahemacal Aalyss ad Applcaos Vol 3 ( [7] Darlg R W R Norrs J R Dffereal equao approxmaos for Markov chas Probably Surveys Vol 5 ( [8] Dshlev A Baov D Couous depedece of he soluo of a sysem of dffereal equaos wh mpulses o he mpulse hypersurfaces J of Mah Aalyss ad Applcaos Vol 35 2 ( [9] Dshleva K Couous depedece of he soluos of mpulsve dffereal equaos o he al codos ad barrer curves Aca Mahemaca Scea Vol 32 3 ( [] Dshleva K Dffereably of soluos of mpulsve dffereal equaos wh respec o he mpulsve perurbaos Nolear Aalyss Seres B: Real World Applcaos Vol 2 6 ( [] Dog L Che L Sh P Perodc soluos for a wo-speces oauoomous compeo sysem wh dffuso ad mpulses Chaos Solos & Fracals Vol 32 5 ( [2] Hederso J hompso H Smoohess of soluos for boudary value problems wh mpulse effecs II Mah ad Compuer Modelg Vol 223 ( [3] Lu L Ye Y xsece ad uqueess of perodc soluos for a dscree-me SIP epdemc model wh me delays ad mpulses Ieraoal J of Compuaoal ad Mah Sceces Vol 5 4 (

8 8 Kaya Georgeva Dshleva: Orbal ucldea Sably of he Soluos of Impulsve quaos o he Impulsve Momes [4] Mohamad S Gopalsamy K Akça H xpoeal sably of arfcal eural eworks wh dsrbued delays ad large mpulses Nolear Aalyss: Real World Applcaos Vol 9 3 ( [5] Özbekler A Zafer A Pcoe ype formula for lear o-selfadjo mpulsve dffereal equaos wh dscouous lecroc J of Qualave heory of Dffereal quaos Vol 35 (2-2 [6] Shag Y he lm behavor of a sochasc logsc model wh dvdual me-depede raes Joural of Mahemacs Vol 23 ( [7] Samov G Almos perodc soluos of mpulsve dffereal equaos Sprger Hedelberg New York Dordrech Lodo (22 [8] Samova I Samov G Lyapuov-Razumkh mehod for asympoc sably of ses for mpulsve fucoal dffereal equaos lecroc J of Dffereal quaos Vol 48 (28 - [9] arboo J Nouyas S haprayoo C Asympoc behavor of soluos of mxed ype mpulsve eural dffereal equaos Advaces Dfferece quaos Vol 24:327 (24 [2] Wag F Pag G She L Qualave aalyss ad applcaos of a kd of sae-depede mpulsve dffereal equaos J of Compuaoal ad Appled Mah Vol 26 ( [2] Wag H Feg Xu Z Opmaly codo of he olear mpulsve sysem fed-bah fermeao Nolear Aalyss: heory Mehods & Applcaos Vol 68 ( [22] Wag W She J Luo Z Paral survval ad exco wo compeg speces wh mpulses Nolear Aalyss: Real World Applcaos Vol 3 ( [23] Wag X Sog Q Sog X Aalyss of a sage srucured predaor-prey Gomperz model wh dsurbg pulse ad delay Appled Mah Modelg Vol 33 ( [24] Wu S-J Meg X Boudedess of olear dffereal sysems wh mpulsve effec o radom momes Aca Mahemacae Applcaae Sca Vol 2 ( [25] Xa Y Posve perodc soluos for a eural mpulsve delayed Loka-Volerra compeo sysem wh he effec of oxc subsace Nolear Aalyss: Real World Applcaos Vol 8 (