Asymptotic Behavior of Solutions of Nonlinear Delay Differential Equations With Impulse

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1 P a g e Vol Issue7Ver,oveber Global Joural of Scece Froer Research Asypoc Behavor of Soluos of olear Delay Dffereal Equaos Wh Ipulse Zhag xog GJSFR Classfcao - F FOR 3 Absrac Ths paper sudes he asypoc behavor of soluos of he secod-order olear delay dffereal equaos wh pulses: =,, = r x p x q x f x x = a x x, x b x = Z, ad soe suffce codos are obaed Keywords-AsypocBehavor,Secod-order olear Delay Dffereal Equao, Ipulses I I ITRODUCTIO [], XLu suded he asypoc behavor of soluo of he forced olear eural dffereal equao wh pulses: [ x p x τ ] Σ q f x = h,, = x x = b x, Z I [],he auhors researched he effecve suffce codos for he asypoc sably of he rval soluo of pulsve delay dffereal equao: x Σ p x τ =,, = x x = b x, =,, I hs paper,we dscuss he asypoc behavor of a class of secod-order olear delay dffereal equao wh pulses The equao s: =,, = r x p x q x f x x = a x x, x b x = Z,, where < < <, l =,ad a, b, =,, are cosa x h x x h x x = l, x = l, =,, h h h h r, p, q, h C [,, R,, =,, ; < < < Le PC deoes he se of fucoφ :[, ] R, whch s couous he se [, ] \{ : =,, } ad ay have dscoues of he frs d ad s couous fro lef a he pos suaed he erval, ] For ay, φ PC, a fuco x s sad o be a soluo of ad ad sasfyg he al value codo: x = φ x, = x x, φ= x, x = [, ] 3,, Depof Maheacs,shaax Isue of Educao, Shaax,x'a 76

2 Global Joural of Scece Froer Research Vol Issue7Ver,oveber P a g e 3 he erval [,, f x :[, R sasfes 3 ad for,,,, =,,,, =,,, x, x s couously dffereal ad sasfes ; for [,, x, x, x ad x exs, x = x, x = x ad sasfes Because ca be rasfored o oe-order dffereal equaos wh pulses, so he exsece ad sole of soluos of ca be deduced by [3] A soluo of ad s called eveually posve egave f s posve egave for all suffcely large,ad s called oscllaory f s eher eveually posve or eveually egave Oherwse, s called ooscllaory II MAI LEMMAS Throughou hs paper, we assue ha he followg codos hold: H r r, p d p, q q, =,,, r, p, q R H for all [,, he ergrao H = f s ds coverges; Σ b < where b = ax{ b } ;, p s H a b ds du = = = = u 3 l l exp[ ] l r u r p s H 4 b exp[ ] ds > = r Lea Suppose ha x s a soluo of equaos ad, ad here exss T such ha x >, T, If H3 hold,he x >, x >, where, ], =,, Proof Frs, we prove x >, for all T Oherwse,here exss soe such ha T, x <, he x = b x fro,we ge p s p s p s [ r x exp[ ds ]] q x exp[ ds ] f exp[ ds] r s r s = = r x exp[ ds] s decreasg o, ] ad, ], p s p s r x ex p [ d s ] r x r b x r p s x b x exp[ d ] s r o r p s r x b x exp[ d ] s r r p s r r b b x exp[ ds] r p s r = b b x exp[ ds] By duco,we have,for all = r p s x b x exp[ ] d s = r p s = [ q x f ]exp[ ds ] < = Hece,

3 Page 4 Vol Issue7Ver,oveber Global Joural of Scece Froer Research p s Because r x exp[ ds] s decreasg o, ], so, r s r ps x b x exp[ d ] s ] r,, rs Iegrag he above equaly fro s o, we have u ps x x s b r x exp[ ds] du s s ru, < <, Le s,, we ge u ps x x b r x exp[ ds] du ru u ps a x b r x ex p [ ds] du ru ru ex x a a x a b r x u ps b b r x exp[ ds] du ru By duco, we ge, for all u ps p[ ds ] du ps x a x r x a b ds du u l exp[ ] = = = l= ru because of x >, x < T, s coraco o he codo H 3 Hece, x > for all r x exp[ ds] s decreasg o, ], hus, p s r s ps ps r x exp[ ds] r x exp[ ds] herefore, x,, ] The proof s coplee Theore Le H H3 holdsuppose ha q, q s ds =, 4 = = ad here exss cosa λ > such ha for suffcely large r q s ds λ < r p 5 = T ad where r [, ], q = ax{} q,, q = ax{ q}, Theevery ooscllaory soluo of ad eds o zero as Proof: Choose a posve eger such ha 5holds for ad b r p λ = < le x be a ooscllaory soluo of ad We wll assue ha x s eveually posve, he case where x s eveually egave s slar adoedle x > for, By Lea, we ow ha x >, for r = < 6 y = rx psx sds q s xsds H bx The for,, =,,, ; =,, Defe

4 Global Joural of Scece Froer Research Vol Issue7Ver,oveber Page 5 y = q r x r 7 = ad y y = b b x, =,, Thus, y s ocreasg o [, Se L= l y, we cla ha L R Oherwse, L =, he x > such ha y H < ad x = ax{ x : } Thus, we have: > y H us be ubouded by vrue of H ad 4Hece, s possble o choose r = < r x psx sds q s xsds bx x r p λ b >, = whch s a coradco ad so L R By egrag boh sdes of 7 fro o, we have: = q s r x s r ds = y s ds = y [ y y ] y < y L whch, ogeher wh 4 ples ha < x L[ R,, ad so l x = The proof s he coplee Lea Le x be a oscllaory soluo of equao ad, suppose ha here exss soe T,f H 4 hold, he x x, x x, where, ], =,, Proof:Fro he resul of Lea,we ow ha,f x > he, x >, x >, where,, ] we wll assue ha whe x > we have x x, x x,, ], he case x s egave s slar adoedfro Lea,we have ] x >, x >,,, he he x s creased We also obaed ps ps [ r x exp[ ds]] < [ r x exp[ ds]] < p s Hece, r x exp[ ds] s decreasg o, ] ad r s r ps r x b x exp[ ds], for all,we oba r ps x b x exp[ ds] = r By he codo H 4, we ge x < x,whch s a coracothe proof s coplee Theore Le H, H ad H 4 holdssuppose ha b <, 8 =

5 Page 6 Vol Issue7Ver,oveber Global Joural of Scece Froer Research ad here exss posve cosa λ ad r, ] such ha where lsup = lsup Q Q λ < r p,9 q, for large, Q = q s ds, = r = Q = sg r q s ds, The every oscllaory soluo ad eds o zero as Proof:Le x be a oscllaory soluo of ad We frs show ha x ad x are bouded Oherwse, x s ubouded whch ples ha here exss posve eger such ha l sup s x s = ad ad Se sup x s = sup x s,, s s r p λ b < 3 = r = < y = rx psx sds q s xsds H bx, whereb = ax{ b, } The 7 holds For, usglea we have whch ples r = y r x p x q s x s d s H b x = s r p x Q b sup x s H, s s = s s sup ys r p sup Q b sup x s sup Hs 4 Hece, lsup y = Fro 7we oce ha y s oscllaory, we see ha here s a such ha y = sup ys ad y = Fro 7 ad, we ge x r = by Lea We ow ha x s s oscllaory, hece, here s a > r such ha x r = Iegrag boh sdes of 7 fro r o, we oba

6 Global Joural of Scece Froer Research Vol Issue7Ver,oveber Page 7 y = y r q s r x s r d s r = r r p s x s ds q r s x s ds H r bx = r = whch ples ha q s r x s r d s r = = p s x s ds H r q s x s r r ds b, r x = < r y p Q r b su xp s H r 5 Fro 4 ad 5,we have = s sup sup sup s = s s r p Q r Q s b H s H r x s Le ad og ha l sup x s =, we have s r p λ b, = by 9, whch coradcs 3 ad so x s boudedby Lea, we ow ha x s bouded ex we wll prove ha µ = lsup x = To hs ed, we defe r = 6, z = rx psx sds q s xsds H bx he z s bouded ad for suffcely large, z r x p x Q sup x s H b x, hus, by H ad 8 s< lsup lsup β = z r p µ µ Q lsup = µ [ r p Q ] 7 o he oher had, we have by 6 for,, =,, =,,, = z = q r x r 8 Fro hs we see ha z s oscllaory Hece here exss a sequece { } such ha z x r =, =,, slar o 5 we ca oba by l =, l z = β, = ad 6ad 8,here s a >, such ha z p Q r su xp s H r bx, s r whch ples by 8 ad H ha

7 Page 8 Vol Issue7Ver,oveber Global Joural of Scece Froer Research lsup β µ [ p Q ] Ths,ogeher wh 7,yelds µ [ r p l sup Q l sup Q] Therefore,by 9we have µ r p λ, µ = by 9ad so, l x = Hece we ca oba ha l x = Thus, he proof s whch ples copleed III REFERECES [] XLu,JShe,asypoc behavor of soluos of pulsve eural dffereal equaos,j,applmahle [] AZhao,JYa,asypoc behavor of soluos of pulsve delay dffereal equaos,jmahaalappl,996, [3] LWe,YChe,Razuh ype heores for fucoal dffereal equaos wh pulsve,dyacs of couous ad pulsve syses,6999,389-4