Forced Oscillation of Nonlinear Impulsive Hyperbolic Partial Differential Equation with Several Delays

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1 Jourl of Applied Mhemics d Physics, 5, 3, Published Olie November 5 i SciRes hp://wwwscirporg/ourl/mp hp://dxdoiorg/436/mp5375 Forced Oscillio of Nolier Impulsive Hyperbolic Pril Differeil Equio wih Severl Delys Vdivel Sdhsivm, Jypl Kvih, Thgr R Pos Grdue d Reserch Deprme of Mhemics, Thiruvlluvr Goverme Ars College, Rsipurm, Idi Received 7 Ocober 5; cceped 4 November 5; published 7 November 5 Copyrigh 5 by uhors d Scieific Reserch Publishig Ic This wor is licesed uder he Creive Commos Aribuio Ieriol Licese (CC BY hp://creivecommosorg/liceses/by/4/ Absrc I his pper, we sudy oscillory properies of soluios for he olier impulsive hyperbolic equios wih severl delys We esblish sufficie codiios for oscillio of ll soluios Keywords Oscillio, Hyperbolic Equio, Impulsive, Delys Iroducio The heory of pril fuciol differeil equios c be pplied o my fields, such s biology, populio growh, egieerig, corol heory, physics d chemisry, see he moogrph [] for bsic heory d pplicios The oscillio of pril fuciol differeil equios hs bee sudied by my uhors see, for exmple []-[7], d he refereces cied herei The heory of impulsive pril differeil sysems mes is begiig wih he pper [8] i 99 I rece yers, he ivesigio of oscillios of impulsive pril differeil sysems hs rced more d more eio i he lierure see, for exmple [9]-[3] Recely, he ivesigio o he oscillios of impulsive pril differeil sysems wih delys c be foud i [4]-[9] To he bes of our owledge, here is lile wor repored o he oscillio of secod order impulsive pril fuciol differeil equio wih delys Moived by his observio, i his pper we sudy he oscillio of olier forced impulsive hyperbolic pril differeil equio wih severl delys of he form How o cie his pper: Sdhsivm, V, Kvih, J d R, T (5 Forced Oscillio of Nolier Impulsive Hyperbolic Pril Differeil Equio wih Severl Delys Jourl of Applied Mhemics d Physics, 3, hp://dxdoiorg/436/mp5375

2 V Sdhsivm e l wih he boudry codiios d he iiil codiio Here r ( u( x, = u( x, p( x, f( u( x, q( x, f ( u( x, σ F( x,,, ( x, = G (, α (,, (,, β ( u x = x u x u x, = x,, u x,, =, =,, u h( x u = g( x,, ( x, ( (,, (, u= ϕ x x (3 [ ] u x, =Φ x,, x, δ, (4 N is bouded domi wih boudry smooh eough d is he Lplci i he Euclide N-spce ( x C ([ δ ] Φ,,, N, γ is ui exerior orml vecor of, δ mx{ σ } =, I he sequl, we ssume h he followig codiios re fulfilled: r, PC,, σ is posiive cos, p( x,, q ( x, re clss of fucios which re (H piece wise coiuous i wih discoiuiies of firs id oly =, =,, d lef coiuous =, =,, f ( u f ( u (H f ( u, f ( u C(, ; C is posiive cos, C is posiive cos, for u u ; h x C, ; F x, PC, ;, ϕ, x PC, ; u ( < < < < <, lim = (H3 u( x, d heir derivives u(, oly =, =,,, d lef coiuous, gx d x re piecewise coiuous i wih discoiuiies of firs id = u( x, = u( x,, u( x u( x ( x,, u x,, β ( x,, u( x, PC(,, =,,, (H4 α d here exis posiive coss,, b, b d b such h for =,,, Le us cosruc he sequece { } { } { }, ( x ξ α,, ξ ( x ξ β,, b b ξ = σ where σ (, =,, =,, = σ d, < =,, u x for which he followig codiios re vlid: If δ, he u( x, = Φ ( x, If =, he u( x, coicides wih he soluio of he problem ( d ( ((3 wih iiil codiio By soluio of problem (, ( ((,(3 wih iiil codiio (4, we me h y fucio (, 3 If, { } \ { σ } <, he (, u x coicides wih he soluio of he problem ( d ( ((3 49

3 4 If, { σ } u x coicides wih he soluio of he problem ( ((3 d he followig equios or <, he (, V Sdhsivm e l (, (, (, ( (, r u( x = u x p x f u x ( ( σ q x, f u x, F x,,, x, = G ( = ( = { } { } u x, u x,, u x, u x,, for \ σ, α ( β { σ } { } u x, = x,, u x,, u x, = x,, u x,, for i i Here he umber i is deermied by he equliy = i We iroduce he oios: The soluio u C ( Γ C ( Γ {( x ( x } = Γ =, :,, ; Γ = Γ, {( x ( x } = Γ =, :,, ; Γ = Γ, = = p mi p x, d q mi q x, x of problem (, ( ((,(3 is clled ooscillory i he domi G if i is eiher eveully posiive or eveully egive Oherwise, i is clled oscillory This pper is orgized s follows: Secio, dels wih he oscillory properies of soluios for he problem ( d ( I Secio 3, we discuss he oscillory properies of soluios for he problem ( d (3 Secio 4 preses some exmples o illusre he mi resuls Oscillio Properies of he Problem ( d ( To prove he mi resul, we eed he followig lemms Lemm Suppose h λ is he miimum posiive eigevlue of he problem d ( x x ( x λη ( x, x, η = η h x x = x η,, η is he correspodig eigefucio of λ The λ > d η ( x >, x Proof The proof of he lemm c be foud i [] Lemm Le u( x, C ( Γ C ( Γ be posiive soluio of he problem (, ( i G The he fucios re sisfies he impulsive differeil iequliy η η v = u x, x dx d F x, x dx r v λ v Cp v Cq v( σ R, (5 v v (6 ( b b, =,, (7 493

4 V Sdhsivm e l where η R = x g x, d S hs eveully posiive soluio Proof Le u( x, be posiive soluio of he problem (, ( i G Wihou loss of geerliy, we my ssume h here exiss T, T u x, >, u x, σ >,,,,, for > > such h ( ( x, [, = muliplyig Equio ( wih ( x For,,,,, d he iegrig ( wih respec o x over yields η, which is he sme s h i Lemm d d r u( x, η( x d x u( x, η( x d x p( x, f( u( x, η( x dx d d = By Gree s formul, d he boudry codiio we hve ( ( σ η η q x, f u x, x d x F x, x d x u η u x, x d x u ds u d x g hu u h ds u dx η = η η = ( η ( ( η ( λη = η x g x, d S λ u x, η x d, x where ds is he surfce eleme o Also from codiio (H, d Jeso s iequliy we c esily obi ( η η p x, f u x, x d x Cp u x, x dx ( ( σ η ( σ η q x, f u x, x d x Cq u x, x dx Thus, v( > Hece we obi he followig differeil iequliy where d d r u( x, η( x d x λ u( x, η( x d x Cp u( x, η( x dx d d ( σ η η Cq u x, x d x g x, x d S, r v λ v Cp v C q v( σ R, η R = x g x, d S For, =, =,,, from ( d codiio (H4, we obi Accordig o η v = u x, x d, x we obi (, u x u x (, ( x, u b b u ( x, 494

5 v v ( b b, =,, Hece, we obi h v( > is posiive soluio of impulsive differeil iequliies (5-(7 V Sdhsivm e l This complees he proof Lemm 3 Le u( x, C ( Γ C ( Γ be posiive soluio of he problem (, ( i G If we furher f u f u u, d he impulsive differeil iequliy (5, d ssume h ( =, r v λ v Cp v Cq v( σ R, (8 v (9 v ( b b, =,, ( hve o eveully posiive soluio, he ech ozero soluio of he problem (-( is oscillory i he domi G Proof Le u( x, be posiive soluio of he problem (, ( i G Wihou loss of geerliy, we my ssume h here exiss T >, > T such h u( x, >, u( x, σ >,,,,, for ( x, [, From Lemm, i follows h he fucio v( is eveully posiive soluio of he iequliy (5 which is cordicios, x,,, he he fucio If u( x < for [ ( x = u( x ( x [,,,,,, is posiive soluio of he followig impulsive hyperbolic equio d sisfies r u x = u x p x f u x ( (, (, (, ( (, (, ( (, σ (,,, (, q x f u x F x x = G ( = α ( β ( u x, x,, u x,, u x, = x,, u x,, =, =,, u h( x u = g( x,, ( x, d d r u( x, η( x d x λ u( x, η( x d x Cp u( x, η( x dx d d ( σ η η Cq u x, x d x g x, x d S, r v λ v Cp v C q v( σ R, 495

6 V Sdhsivm e l where η R = x g x, d S For, =, =,,, from ( d codiio (H4, we obi Accordig o η (, x x b v = x, x d, x we obi (, ( x, ( x, v b v ( b b, =,, Thus, i follows h he fucio η v = x, x dx is posiive soluio of he iequliy (8-( for > T which is lso cordicio This complees he proof Now, if we se g i he proof of Lemm 3, he we c obi he followig lemm Lemm 4 Le u( x, C ( Γ C ( Γ be posiive soluio of he problem (, ( i G If we furher f u f u u, d he impulsive differeil iequliy (5, d ssume h ( =, r v v Cp v λ ( σ Cq v, v ( v ( ( b b, =,, (3 hs o eveully posiive soluio, he ech ozero soluio of he problem (, sisfyig he boudry codiio u h( x u =, ( x,, is oscillory i he domi G u x, be posiive soluio of he problem (, ( i G Wihou loss of geerliy, we my Proof Le ssume h here exiss T, T > > such h ( σ u x, >, u x, >,,,,, for ( x [,, From Lemm, i follows h he fucio v( is eveully posiive soluio of he iequliy (5 which is cordicio If u( x, < for ( x, [,, soluio of he followig impulsive hyperbolic equio,,,,,, is posiive he he fucio u( x = u( x ( x [ 496

7 V Sdhsivm e l d sisfies r u x = u x p x f u x ( (, (, (, ( (, (, ( (, σ (,,, (, q x f u x f x x = G ( = α ( β ( u x, x,, u x,, u x, = x,, u x,, =, =,, u h( x u =, ( x,, d d r u( x, η( x d x λ u( x, η( x d x Cp u( x, η( x dx d d ( σ η Cq u x, x dx, λ ( σ r v Cp v Cq v, For, =, =,,, from ( d codiio (H4, we obi Accordig o η (, x x ( x, ( x, v = x, x d, x we obi (, b b, =,, v v ( b Thus i follows h he fucio η b > T which is lso cordicio This complees he proof v = x, x d, x is posiive soluio of he iequliy (-(3 for Lemm 5 Assume h (A he sequece { } sisfies < < <, lim = ; (A m PC, is lef coiuous for =,, ; (A3 for =,, d, where p, q C(, fucio from The m p m q,, (, m dm e, d d e re coss PC deoe he clss of piecewise coiuous o, wih discoiuiies of he firs id oly =, =,, 497

8 V Sdhsivm e l ( ( s ( m m d exp p s ds d exp p r dr q s ds < < s< < < < < < d exp p s d s e Proof The proof of he lemm c be foud i [] Lemm 6 Le v( be eveully posiive (egive soluio of he differeil iequliy (-(3 Assume h here exiss T such h v( > v( < for T If < < s b lim ds = (4 ( = l hold, he ( for [ T, ] (, ], where l = { T} Proof The proof of he lemm c be foud i [] We begi wih he followig heorem Theorem If codiio (4, d he followig codiio hold, where F here exiss T >, T, such h From Lemm 4, we ow h h v r F s s < < sb mi : lim d =, (5 exp( δ ( λ Cp w C q = r he every soluio of he problem (, ( oscilles i G Proof Le u( x, be ooscillory soluio of (, ( Wihou loss of geerliy, we c ssume h u x, >, u( x, σ >, =,,, for ( x, [, v is posiive soluio of (-(3 Thus from Lemm 6, we c fid for For,, =,,, defie w >, The we hve, h for Subsiue (6-(8 io ( d he we obi, Hece we hve w = r, v r v w v = We my ssume h exp ( d exp ( d (, v =, hus we hve v = w s s (6 = w ws s (7 = exp d exp d (8 v w w s s w w s s ( ( σ ( ( ( r w exp w( s ds r w exp w s ds w exp w s ds λ exp w s ds Cp exp w s ds C q exp w s ds 498

9 or ( σ λ r w r w Cp C q exp w s d s,, λ ( σ r w Cp C q exp w s d s, From bove iequliy d codiio b, δ Thus w w for which implies h d From (-(3, we obi Le i is esy o see h he fucio r w Cp exp w C q, λ ( δ ( The ccordig o Lemm 5, we hve ( b b w r r r w = =, v v λ ( δ ( r w Cp exp w C q, b w ( r w, =,, exp( δ ( λ Cp w C q F = r b b w w( r r F ( s ds < < s< < b = w( r r F( s ds < < < < < s b Sice w, he ls iequliy cordics codiio (5 This complees he proof 3 Oscillio Properies of he Problem ( d (3 Nex we cosider he problem ( d (3 To prove our mi resul we eed he followig lemms Lemm 3 Suppose h λ is he smlles posiive eige vlue of he problem λ ( x Ψ x Ψ x =, x, Ψ =, x, d Ψ ( x is he correspodig eige fucio of λ The λ > d Ψ ( x >, x Proof The proof of he lemm c be foud i [] u x, C Γ C Γ be posiive soluio of he problem (, (3 i G The he fucio V Sdhsivm e l w is oicresig for Lemm 3 Le re sisfies he impulsive differeil iequliy Ψ F x, x dx 499

10 V Sdhsivm e l where hs he eveully posiive soluio Proof Le (, r v v Cp v λ ( σ Cq v Q, v (9 v ( ( b b, =,, ( Ψ Q = ϕ ( x, d S N = Ψ v u x, x d x u x be posiive soluio of he problem (, (3 i G Wihou loss of geerliy, we my ssume h here exiss T, T u x, >, u x, σ >,,,,, for > > such h ( ( x, [, For,,,,, Ψ x, which is he sme s h i Lemm 3 d he iegrig ( wih respec o x over yields = muliplyig equio ( wih d d r u( x, ( x d x u( x, ( x d x p( x, f( u( x, ( x dx d d Ψ = Ψ Ψ By Gree s formul, d he boudry codiio we hve where ds is he surfce eleme o From codiio (H, we c esily obi ( ( σ q x, f u x, Ψ x d x F x, Ψ x d x u Ψ u x, Ψ x d x= Ψ u ds u Ψdx Ψ = ϕ( x, ds u( λψ dx Ψ = ϕ( x, d S u( x,( λψ( x d, x ( Ψ Ψ p x, f u x, x d x Cp u x, x dx ( ( σ Ψ ( σ Ψ q x, f u x, x d x Cq u x, x d x The proof is similr o h of Lemm d herefore he deils re omied Lemm 33 Le u( x, C ( Γ C ( Γ be posiive soluio of he problem (, (3 i G If we furher f u f u u, d he impulsive differeil iequliy (9, d ssume h ( =, r v λ v Cp v Cq v( σ Q, ( 5

11 v V Sdhsivm e l v (3 ( b b, =,, (4 hve o eveully posiive soluio, he ech ozero soluio of he problem (, (3 is oscillory i he domi G Proof The proof is similr o Lemm 3, d hece he deils re omied Fuhermore, if we se ϕ, he we hve he followig lemm Lemm 34 Le u( x, C ( Γ C ( Γ be posiive soluio of he problem (, (3 i G If we furher f u f u u, d he impulsive differeil iequliy (9, d ssume h ( =, λ ( σ r v Cp v Cq v, (5 v v (6 ( b b, =,, (7 hs o eveully posiive soluio, he ech ozero soluio of he problem (, sisfyig he boudry codiio u=, x,, is oscillory i he domi G Proof The proof is similr o Lemm 4, d hece he deils re omied Usig he bove lemms, we prove he followig oscillio resul Theorem 3 If codiio (4 d he followig codiio hold, where exiss T >, T, such h From Lemm 34, we ow h h v r F s s < < sb lim d =, (8 F λ = Cp r he every soluio of he problem (, (3 oscilles i G Proof Le u( x, be ooscillory soluio of (, (3 Wihou loss of geerliy, we c ssume h here u x, >, u( x, σ >, =,,, for ( x, [, v is posiive soluio of (5-(7 Thus from Lemm 6, we c fid for For,, =,,, defie w >, The we hve, h for w = r, v r v w v = We my ssume h, ( v =, hus we hve v = exp w s d s, (9 5

12 V Sdhsivm e l ( ( d ( = w exp ws d s, (3 exp = w w s s w exp w s d s We subsiue (9-(3 io (5 d c obi he followig iequliy, he we hve From (6-(7, we c obi I follows h Le The ccordig o Lemm 5, we hve ( ( ( r w exp w( s ds w exp w s ds λ exp w s d s Cp exp w s ds, λ r w Cp, ( b w r r = v v b = r w, =,, λ r w Cp, b w r w, = λ F = < < s< < Cp r b b w w ( r r F ( s ds b = w( r r F( s ds < < < < < s b Sice w, he ls iequliy cordics (8 This complees he proof Theorem 3 If codiio (4 d he followig codiio r C q s < < sb (3 lim d =, (3 hold for some q, he every soluio of he problem (, (3 oscilles i G Proof The proof is obvious d hece he deils re omied 4 Exmples I his secio, we prese some exmples o illusre he mi resuls Exmple 4 Cosider he impulsive differeil equio 5

13 V Sdhsivm e l ( π ( u( x, = ( π si u( x, ( π cos u( x, π ( π si u x, ( π si ( cos si ( x, >,, =,, 3, u x = (, u( x, ( u x, = u x,, =,, d he boudry codiio (33 u, = u π, =,,, =,, (34 Here =, π,,,,,, ( π, ( π si = = b = b = = r = =, π p = ( π cos, q ( π si, σ, f ( u u, f( u u, Moreover = = = = d ig { } = { } b lim ds ds = < < s < < s 3 4 = ds ds ds ds 3 < < s < < s < < s < < s 3 3 = = = = = = = = = π =, he π so (4 holds We e λ, C C, δ mx { σ }, w hus F ( π ( π cos ( π = = < < s < < s ( s cos, lim r( F( s ds lim ( π ( cos ( s ds = b lim cos d s= Hece (8 holds Therefore ll codiios of Theorem 3 re sisfied Hece every soluio of he problem (33, (34 oscilles i (, π [, I fc u( x, = si xcos is oe such soluio of he problem (33 d (34 Exmple 4 Cosider he impulsive differeil equio d he boudry codiio ( π ( u( x, = ( π si u( x, ( π cos u( x, 5π 3 ( π si u x, ( π cos cos ( x, >,, =,, 3, u x = (, u( x, ( u x, = u x,, =,, (35 53

14 V Sdhsivm e l = = =, b Here (, π, p u, = u π, =,,, =,, (36 x x = b =,,, r = π si, = = ( π, = ( = ( π cos, q ( σ = hu =, { } { } 5π, π si, f u = u, f u = u, d ig = I is esy o chec h he codiios of Theorem re sisfied Therefore, every soluio of he problem (35, (36 oscilles i (, π [, I fc problem (35 d (36 Acowledgemes u x, = si cos x is oe such soluio of he The uhors h Prof E Thdpi for his suppor o complee he pper Also he uhors express heir sicere hs o he referee for vluble suggesios Refereces [] Wu, JH (996 Theory of Pril Fuciol Differeil Equios d Applicios New Yor, Spriger hp://dxdoiorg/7/ [] Liu, AP (996 Oscillios of Ceri Hyperbolic Dely Differeil Equios wih Dmpig Term Mhemic Applice, 9, 3-34 [3] Liu, AP, Xio, L d Liu, T ( Oscillios of he Soluios of Hyperbolic Pril Fuciol Differeil Equios of Neurl Type Ac Alysis Fuciolis Applice, 4, [4] He, MX d Liu, AP (3 Oscillio of Hyperbolic Pril Differeil Equios Applied Mhemics d Compuio, 4, 5-4 hp://dxdoiorg/6/s96-33(95-3 [5] Shouu, Y ( Oscillio of Soluios for Forced Nolier Neurl Hyperbolic Equios wih Fuciol Argumes Elecroic Jourl of Differeil Equios,, -6 [6] Shouu, Y d Yoshid, N ( Oscillio of Nolier Hyperbolic Equios wih Fuciol Argumes vi Ricci Mehod Applied Mhemics d Compuio, 7, 43-5 hp://dxdoiorg/6/mc53 [7] Yoshid, N (8 Oscillio Theory of Pril Differeil Equios World Scieific, Sigpore hp://dxdoiorg/4/746 [8] Erbe, L, Freedm, H, Liu, XZ d Wu, JH (99 Compriso Priciples for Impulsive Prbolic Equios wih Applicio o Models of Sigle Species Growh The Jourl of he Ausrli Mhemicl Sociey Series B Applied Mhemics, 3, 38-4 hp://dxdoiorg/7/s334785x [9] Biov, DD d Simeoov, PS (989 Sysems wih Impulse Effec: Sbiliy Theory d Applicios Ellis Horwood, Chicheser [] Biov, DD d Simeoov, PS (993 Impulsive Differeil Equios: Periodic Soluios d Applicios Logm, Hrlow [] Ch, C d Ke, L (994 Remrs o Impulsive Quechig Problems Proceedigs of Dymics Sysems d Applicios,, 59-6 [] Zhg, LQ ( Oscillio Crieri for Hyperbolic Pril Differeil Equios wih Fixed Momes of Impulse Effecs Ac Mhemic Siic, 43, 7-6 [3] Mil m, VD d Myshis, AD (96 O he Sbiliy of Moio i he Presece of Impulses Siberi Mhemicl Jourl,, [4] Biov, DD, Kmo, Z d Michev, E (996 Moooe Ierive Mehods for Impulsive Hyperbolic Differeil Fuciol Equios Jourl of Compuiol d Applied Mhemics, 7, hp://dxdoiorg/6/377-47(959-x [5] Cui, BT, H, MA d Yg, HZ (5 Some Sufficie Codiios for Oscillio of Impulsive Dely Hyperbolic Sysems wih Robi Boudry Codiios Jourl of Compuiol d Applied Mhemics, 8, hp://dxdoiorg/6/cm46 [6] Cui, BT, Liu, YQ d Deg, FQ (3 Some Oscillio Problems for Impulsive Hyperbolic Differeil Sysems wih Severl Delys Applied Mhemics d Compuio, 46,

15 V Sdhsivm e l hp://dxdoiorg/6/s96-33(6- [7] Deg, LH d Ge, WG ( Oscillio Crieri of Soluios for Impulsive Dely Prbolic Equios Ac Mhemic Siic, 44, 5-56 [8] Yg, JC, Liu, AP d Liu, GJ (3 Oscillio of Soluios o Neurl Nolier Impulsive Hyperbolic Equios wih Severl Delys Elecroic Jourl of Differeil Equios, 3, - [9] Fu, XL d Shiu, LJ (3 Oscillio Crieri for Impulsive Prbolic Boudry Vlue Problem wih Dely Applied Mhemics d Compuio, 53, [] Ye, QX d Li, ZY (99 Iroducio o Recio Diffusio Equios Sciece Press, Beiig [] Lshmihm, V, Biov, DD d Simeoov, PS (989 Theory of Impulsive Differeil Equios World Scieific, Sigpore hp://dxdoiorg/4/96 [] Luo, JW ( Oscillio of Hyperbolic Pril Differeil Equios wih Impulses Applied Mhemics d Compuio, 33, hp://dxdoiorg/6/s96-33(7-x 55