ClassificationofNonOscillatorySolutionsofNonlinearNeutralDelayImpulsiveDifferentialEquations

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1 Global Joural of Scece Froer Research: F Maheacs ad Decso Sceces Volue 8 Issue Verso. Year 8 Type: Double Bld Peer Revewed Ieraoal Research Joural Publsher: Global Jourals Ole ISSN: & Pr ISSN: Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos By U. A. Abasewere, I. M. Esuabaa, I. O. Isaac & Z. Lpscey Uversy of Uyo Absrac- I hs paper, a geeral class of secod order olear eural delay pulsve dffereal equao of he for y( ) p ( ) y( τ ) + fj (,y( gjl ( ),,y( gjl) )) =, R +, = j = y( ) p y( ) τ + f (,y( g ( ),,y( g ( )))) =, R,= j jl jl + = j = s cosdered. We classfys o-oscllaory soluos o four ypes of soluo ses, aely (,,) ( b,a, ) (,,) (,,d) Λ, Λ, Λ ad Λ ad esablsh ecessary ad suffce codos for he exsece of hese o-oscllaory soluos by eas of Schauder-Tychooff fxed po heore ad Lebesgue s Moooe Covergece Theore. Soe exaples are gve o llusrae he obaed resuls. GJSFR-F Classfcao: MSC : 35R ClassfcaoofNoOscllaorySoluosofNolearNeuralDelayIpulsveDfferealEquaos Srcly as per he coplace ad regulaos of: 8. U. A. Abasewere, I. M. Esuabaa, I. O. Isaac & Z. Lpscey. Ths s a research/revew paper, dsrbued uder he ers of he Creave Coos Arbuo-Nocoercal 3. Upored Lcese hp://creavecoos.org/lceses/by-c/3./), perg all o coercal use, dsrbuo, ad reproduco ay edu, provded he orgal wor s properly ced.

2 Ref. B. G. Zhag, Zhu Shalag, Oscllao of secod-order olear delay dyac equaos o e scales, Copu. Mah. Appl., 49 (5), Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos U. A. Abasewere α, I. M. Esuabaa σ, I. O. Isaac ρ & Z. Lpscey Ѡ Absrac- I hs paper, a geeral class of secod order olear eural delay pulsve dffereal equao of he for y( ) p ( ) y( τ ) + fj (,y( gjl ( ),,y( gjl) )) =, R +, = j = y( ) p y( ) τ + f (,y( g ( ),,y( g ( )))) =, R,= j jl jl + = j = s cosdered. We classfys o-oscllaory soluos o four ypes of soluo ses, aely (,,) ( b,a, ) (,,) (,,d) Λ, Λ, Λ ad Λ ad esablsh ecessary ad suffce codos for he exsece of hese ooscllaory soluos by eas of Schauder-Tychooff fxed po heore ad Lebesgue s Moooe Covergece Theore. Soe exaples are gve o llusrae he obaed resuls. I. Iroduco A survey of rece sudes eural pulsve dffereal equaos reveal ha os of such wors revolve aroud he ques for oscllaory codos for pulsve dffereal equaos, wh or whou delay, lear or olear ([], [3], [5], [6], [7], [8], [], [3], [4] ). The develope of oscllaory ad o-oscllaory crera for olear pulsve dffereal equaos has so far araced very lle aeo. I fac, he cocep of o-oscllao for olear eural pulsve equaos presely suffers alos coplee eglec. I hs sudy, we aep o classfy he o-oscllaory soluos of a geeral class of secod order olear eural delay pulsve dffereal equaos o dffere soluo ses ad ae coscous effors o provde codos for he exsece of hese soluos. Auhor α: Depare of Maheacs ad Sascs, Uversy of Uyo P.M.B. 7, Uyo, Awa Ibo Sae, Ngera. e-al: ubeeservces@yahoo.co Auhor σ Ѡ: Depare of Maheacs, Uversy of Calabar, P.M.B. 5, Calabar, Cross Rver Sae, Ngera. e-als: esuabaaa@gal.co, zlpcsey@yahoo.co Auhor ρ: Depare of Maheacs/Sascs, Awa Ibo Sae Uversy, P.M.B. 67, Io Apade, Awa Ibo Sae, Ngera. e-al: doggrace@yahoo.co Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year Global Jourals

3 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 5 I wha follows, we recall soe of he basc oos ad defos ha wll be of porace as we advace hrough he arcle. Usually, he soluo y() for [,T) of a gve pulsve dffereal equao or s frs dervave y () s a pece-wse couous fuco wh pos of dscouy [,T),. Therefore, order o splfy he saees of he r asseros, we roduce he se of fucos PC ad PC whch are defed as follows: Le r N, D : = [T, ) R ad le S: = { } E, where E s our subscrp se whch ca be he se of aural ubers N or he se of egers Z, be fxed. Throughou hs dscusso, we wll assue ha he elees of he sequece S: = { } E are he oes of pulsve effecs ad sasfy he followg properes: C.: If { } s defed for all N, C.: If { } s defed for all l = ±. ± he < < < ad l = +. Z, he <, < + for Z, ad We deoe by PC(D,R) he se of all values ψ:d R whch s couous for all D, S. They are fucos fro he lef ad have dscouy of he frs d r a he pos for S. By PC (D,R), we deoe he se of fucos ψ:d Rhavg j dervave d ψ PC(D,R), j r j d ([], [4]). r To specfy he pos of dscouy of fucos belogg o PC ad PC, we r shall soees use he sybols PC(D,R;S) ad PC (D,R;S), r N. The soluo y() of a pulsve dffereal equao s sad o be.. 3. Fally posve (fally egave) f here exs T such ha y() s defed ad s srcly posve (egave) for T ([9]); Oscllaory, f s eher fally posve or fally egave; ad No-oscllaory, f s eher fally posve or fally egave ([], []). II. Saee of he Proble Here, we are cosderg he secod order olear eural pulsve dffereal equao of he for y( ) p ( ) y( τ ) + fj (,y( gjl ( ),,y( gjl) )) =, R +, S = j = y ( ) py( ) τ + fj (,y ( gjl ( ),, y ( gjl ( ) ))) =, R +, S. = j = (.) Ref. D. D. Baov ad P. S. Seoov, Oscllao Theory of Ipulsve Dffereal Equaos, Ieraoal Publcaos Orlado, Florda, 998. We roduce he followg codos: τ> ([ ) + ) C.:, p, p PC,,R, =,,, ad here exss δ (,] such ha 8 Global Jourals

4 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Ref. L. H. Erbe, Q. Kog ad B. G. Zhag,Oscllao Theory for Fucoal Dffereal Equaos, Deer, New Yor, 995. C.: js ([ ) ) js j = j = p + p δ,, R ; g C,, R, l g =, j =,,,, s =,,, ; C.3: [ ) wheever C.4: ( ) ( ) f PC, R,R, xf,x,,x > ; j j x f, x,, x > for x x >, =,,,, j =,,,. Moreover, j l Se ( ) ( ) fj,y,, y fj,x,, x fj,y,y fj,x,,x ( ) ( ) x y ad y x >, =,,,, j =,,, ; = ( τ ) = + x y p y. (.) Our a hs paper s o gve he classfcao of o-oscllaory soluos of equao (.). Bu frs, we defe soe coceps ad esablsh he followg leas whch wll be useful he dscusso of he a resuls. Theore.: (Schauder-Tychooff fxed po heore) Le X be a locally covex lear space, S a copac covex subse of X, ad le T:S Sbe a couous appg wh T(S) copac. The T has a fxed po S. Theore.: (Lebesgue s Moooe Covergece Theore) Le (A, µ, ) be a easure space ad f,f,f, 3 a powse o-decreasg sequece of [, valued ) easurable fucos. Le l f (): = f() for all A, he f s easurable ad l f dµ= f d µ. A A Lea. ad. are exesos of Lea 4.5. ad 4.5. o pages 4 ad 43 respecvely of he oograph by Erbe e al [] Lea.: Le y() be a fally posve (or egave) soluo of equao (.). If ly( ) =, he x() s fally egave (or posve) ad l x( ) =. Oherwse, x() s fally posve (or egave). Proof: Le y() be a fally posve soluo of equao (.). Fro he sae x, x orx, x x> orx< equao (.), ( ) > ( ) < fally. Also, fally. If l y( ) =, fro equao (.), follows ha l x( ) oooc, so =, l x ( ) = whch ples ha ( ) l x =. Sce x() s x >, x >. Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year Global Jourals

5 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 5 Therefore, x() < fally. If o, he x() < { }such ha oba ly, he lsup y >. We show ha x() > fally. If l, Thus, l x exss a sequece fally. If y() = y = axy ad l y( ) < s ubouded, he here exss a sequece = ( τ) =. Fro equao (.), we x y p y y p. (.3) = = =, whch s a coradco. If y() s bouded, he here { } such ha l ad l y( ) lsup y( ) sequeces { p ( ) } ad { y τ ( ) } Whou loss of geeraly, we ay assue ha exs. Hece = =. Sce he are bouded, here exss coverge subsequeces. l y τ ad l p, =,,, l x = l y p y τ = = ( ) lsupy p >, whch, aga, s a coradco. Therefore, x() > fally. A slar proof ca be repeaed f y() < fally. Lea.: Assue ha l p ( ) = P (,] = egave) soluo of equao (.). If l x( ) =( or ), he = ( or ). Proof: Le y() ly, ad y() s a fally posve (or l x = a R, he ly( ) be a fally posve soluo of equao (.), he y() x() If =, he =. Now we cosder he case ha l x ly x() s bouded whch ples, by equao (.3), ha y() = a. If p fally. l x = a R. Thus, s bouded. Therefore, here exss a sequece { } such ha l = l y lsup y whou loss of geeraly, we ay assue ha ( ) ( ) exs. Hece ha s, a ad =. As before, l p ad l y τ, =,,, = l x( ) = l y( ) l p ( ) l y( τ ) lsupy( )( p) = a l supy() p,. (.4) Noes 8 Global Jourals

6 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos O he oher had, here exss { } loss of geeraly, we assue ha Hece such ha l y( ) = l f y(). Whou l p ( ) ad l y( τ ), =,,, exs. Ref. L. H. Erbe, Q. Kog ad B. G. Zhag,Oscllao Theory for Fucoal Dffereal Equaos, Deer, New Yor, 995. or a = l x( ) = l y( ) lp ( ) l y( τ) ( ) = a p lf y lf y p Cobg equales (.4) ad (.5), we oba ly( ) argue ca be repeaed f y() <. We are ow ready o prove he followg resuls. III. Ma Resuls. (.5) a =. A slar p Here, Theore 3., 3., 3.3, 3.4, 3.5 are exesos of Theore 4.5., 4.5., 4.5.3, 4.5.4, foud o pages 44, 45, 49, 5, 5, respecvely, beg her eural delay versos as defed he wor by Erbe e al ([]). Theore 3.: Assue ha l p ( ) p [,) = =. Le y() be a o-oscllaory soluo of equao (.). Le Λ deoe he se of all o-oscllaory soluos of equao (.), ad defe The Proof: (,,) { y : l y( ), l x( ), l ( x ( ), x( )), } Λ = Λ = = =, ( b,a,) { y : l y( ) b: a, l x( ) a, l ( x ( ), x( )) p, } Λ = Λ = = = = (,,) { ( ( )), } Λ = y Λ : ly =, l x =, l x, x =, (,,d) { y : ly( ), l x( ), l ( x ( ), x( )) d, } Λ = Λ = = =. (,,) ( b,a, ) (,,) (,,d) Λ=Λ Λ Λ Λ. Whou loss of geeraly, le y() be a fally posve soluo of equao (.). If ly( ) =, he by Lea., l x( ) = ad ( ) l x, x =, ha s,, (,, ) y. If ly, he by Lea., x() > fally ad herefore ples ha Λ, Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year Global Jourals

7 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 54 ( ) > ad x (, ) x ( ) < fally. ( ( )) =. By Lea., we have ly( ) a x, x, l x, x If l x =, he by Lea., >, we oba x, x y Λ (,,) or (,,d) y Λ. If l x = a > exss, he = = b, ha s, p l y() =. Sce x, x < ( ) l x, x = d, where d= or, ( b,a,) y Λ. ad d>. The eher Ths coplees he proof of Theore 3.. I wha follows, we shall show soe exsece resuls for each d of ooscllaory soluo of equao (.). Theore 3.: Assue ha here exs wo cosas h > h > such ha ad p p h,p p h, =,,,, = = p exp h τ + exp h p exp h τ > p exp h exp h p exp h = = τ + τ (3.) ( p ( ) exp( hτ ) + exp( h ) p exp( h ( τ) ) ) exp( h ) = = x ( ( )) u fj u,exp hgj u,...,exp hgjl u du + j= + f,exp h g,, exp h g (3.) j j jl < j = (,,) fally. The equao (.) has a soluo y Λ. Proof: ([ )) PC, Se Le us deoe by B p ad defe he sup or B p he space of all bouded pece-wse couous fucos as follows: y : = sup y. p ( ) ( ) y B :exp h y exp h Ω= y y L,y y L, for,, :, ad for L h. The Ω s a oepy, closed covex bouded se B. p Noes 8 Global Jourals

8 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos For he sae of coveece, deoe ( ( )) ( ( )) f u,y g u = fj u,y gj u,, y gjl u j = f (,y( g( ))) = fj (,y( gj ( )),,y( gjl ( ))), j = (3.3) Noes ( ( )) ( ( )) f u,exp hg u = fj u,exp h j u,, exp hgjl u j= f (,exp( hg( ))) = fj (,exp( hgj ( )),, exp( hgjl ( ))). j= Defe a appg J o Ω as follows: where ( ) p ( y ) ( τ ) + py ( τ) ( u fu,ygu ) du = = (Jy)() = ( ) + f(,y( g( ) )),, T < exp( K(y)) + exp( K(y) ),,< T, ( T) l Jy Ky =, T T s suffcely large such ha s =,,,, for, T. Now, we see ha codo (3.) ples ha T τ ; js ( ( )) ( ) T < (3.4) (3.5) τ ; g ; =,,, ; j =,,, ; ( ) f u,exp h g u du + f,exp h g <, whle fro codo C., follows ha for a gve α ( δ ) ( α p ( ) ) L α ( δ ) L > = ( α p ) α ( δ) L. = Therefore, T ca be chose so large ha for ad, T, = ( ) ( ( )) α,, f u,exp h g u du p L T f ( ) exp h g( ) α p L, T < = (3.6) (3.7) Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year Global Jourals

9 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos exp( h( ) ) = ( exp( h( ) )) α+ τ α+ τ =. Hece fro equales (3.) ad (3.), follows ha ( Jy)( ) p( y ) ( τ ) + py ( τ ) = = Noes Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 56 ad Tha s, ( ) p exp h τ + p exp h τ = = exp h p exp h τ + exp h p exp h τ = = exp h for, T, ( ) Jy p exp h τ + p exp h τ = = ( ( )) ( ) ( ( ) ) < u f u,exp h g u du f,exp h g ( ) = exp h + exp h p exp h τ + exp h p exp h τ = = ( ( )) ( ) ( ( ) ) < u f u exp h g u du f,exp h g exp h for, T. exp h Jy exp h, T, ( ) By he defo of K(y) ad he saee I s clear ha Nex, we show ha exp h Jy exp h, T. h Ky h. Hece exp h T Jy T exp h T, exp h Jy exp h,, < T. 8 Global Jourals

10 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Noes for, [, ) ad:, [, ) Jy Jy L, (3.8). Whou loss of geeraly, we assue ha ad :. Ideed, for T ad : T, usg codo (3.7) ad equaly (3.8), we have ha ( Jy)( ) ( Jy)( ) = ( Jy)( ) + ( Jy)( ) ( Jy)( ) ( Jy)( ) = p y τ + p y τ p y τ p y τ + ( ) ( ) ( ( )) + u fuygu du+ f,yg u fu,ygu du < < ( ) ( ) f,y g( ) p y p y p y p y = = τ τ + τ τ + ( ) ( ) + u f u y g u du u f u y g u du + ( ) ( ) + f,y g f,y g < < p y y p p y = = τ τ + τ + ( ) ( ) + ( u ) f u y g( u) du + f u,y g u du + p y y p p y = = + τ τ + τ + ( ) ( ) + f,y g + f,y g < ( p( ) + exp( h( τ ) )) L + f ( u,exp( hg( u) )) du + = Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 57 ( ( )) ( ) + p + exp h τ L + f,exp h g = < ( ) { ( )} = = = p + exp h τ + α p L + 8 Global Jourals

11 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos { ( )} = = = + p + exp h τ + α p L ( ) exp h L exp h L = = = τ +α + τ +α L L + Noes Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 58 L. For T ad : T, we have ( Jy)( ) ( Jy)( ) = ( Jy)( ) + ( Jy)( ) ( Jy)( ) ( Jy)( ) ( ) = exp K y + exp K y exp K y exp K y ( ) exp K y exp K y + exp K y exp K y L L + = L. For < T ad : < T, we oba ( Jy)( ) ( Jy)( ) ( Jy)( ) ( Jy)( ) + ( Jy)( ) ( Jy)( ) ( Jy)( ) ( Jy )(T) ( Jy)( T) ( Jy)( ) ( Jy)( ) ( Jy)( T) ( Jy)( T) ( Jy)( ) L L L L T + T + T + T L L = + = L. We have proved ha equaly (3.8) holds for all ad :. Therefore, JΩ Ω. Hece, J s pece-wse couous. Sce J Ω Ω,JΩ s uforly bouded. Se y Ω. I edaely ples ha where b > ad ( Jy)( ) b, for Jy Jy L ad :. Whou loss of geeraly, we se 8 Global Jourals

12 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos b = exp h,,. Noes Hece, for ay arbrarly pre-assged sall posve uber ε, here exss a suffcely large T > such ha wheever exp( h ) < ε, ( Jy)( ) ( Jy)( ) exp( h ) exp( h ) + ε (3.9) for, T, T ad : T. O he oher had, f we se λ= ε ad assue ha L <λ, he for all T ad : T, becoes clear ha ( Jy)( ) ( Jy)( ) ε (3.) Thus, fro equales (3.9) ad (3.), we ca affr ha JΩ s quasequcouous. Therefore, JΩ s relavely copac. By vrue of Schauder-Tychooff fxed po heore, he appg J has a fxed po y J such ha y = Jy. The y (,,) s a posve soluo of equao (.) ad y Λ. Ths coplees he proof of Theore 3.. Theore 3.3: Assue ha l p ( ) + l p = p [,) o-oscllaory soluo for b. Proof. The equao (.) has a = = ( b,a, ) y Λ b,a f ad oly f j j j = < j = u f u,b,..., b du+ f, b,..., b < (3.) y Λ be a fally posve soluo of equao (.). Fro Theore 3., we ow ha b> ad a>. Usg oaos equaos (3.3) ad (3.4), we oba fro equaos (.) ad (.), ) Necessy: Whou loss of geeraly, le ( ) = x f,y g x f,y g. = ( b,a,) Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 59 Iegrag fro s o for s, we have s ( ) ( ( ) ) s < x ( s) = f u,y g( u) du + f,y g. (3.) Aga, egrag equao (3.) fro T o, where T s suffcely large, we oba 8 Global Jourals

13 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos T ( ) ( ) x = x T + u T f u,y g u du+ T f u,y g u du + Sce u ( ) ( ) + T f,y g + T f,y g. (3.3) T < ( ) jh jh yg b ( jh u) for u T l y g u = b> ad l y g = b>, j=,,,, h=,,,, exss a T fro equao (3.3) we have such ha ( jh ) ad yg( ) here b for : T. Hece Noes Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 6 b b b b j j T j = T j = ( u T) f ( u,,, ) du+ ( T) f (,,, ) < x( ) x( T) whch ples ha codo (3.) holds. ) Suffcecy: Se b > ad Α> so ha Α< ( pb ). Fro codo (3.) here exss a suffcely large T so ha for, T we have τ, τ, =,,,, ad gjh ( ) o, gjh ( ), j =,,,, h =,,, ad ( ) j ( ) j ( ) Α b = b T j = b T < j = Le Ω p p u f u,b,..., b du f,b,...,b. (3.4) be he se of all pece-wse couous fucos y [, ) y b,,. Defe a appg J Ω Se ( Jy)( ) as follows: ( ( )) ( ) Α+ p ( y ) ( τ ) + py ( τ ) + ufu,ygu du + = = T =, ( ) ( ) + f u,y g u du+ f,y g + f,y g, T ( Jy)( T ),, < T. T < y () =, ; such ha (3.5) I edaely follows ha y = Ty,, =,,. (3.6) y < y =Α b,. By duco, we oba Α y y b,, =,,. + 8 Global Jourals

14 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Thus, l y ( ) y( ) exss ad y b, [, ) Α. By Lebesgue s oooe covergece heore, we oba fro equao (3.6) he resul Noes ( ( )) Α+ ( τ ) + ( τ ) + + = = T ( ) p y py ufu,ygu du fu,ygu du y( ) = + f,y g + f,y g u,, T T < y( T ),,< T. Theore 3., <Α y < b, fro ( b,a,) y Λ. Ths coplees he proof of Theore 3.3. Hece, y() s a posve soluo of equao (.). Sce Usg reasog aalogous o ha gve he proof of Theore 3.3 above, we ca verfy he followg resuls. Theore 3.4: Assue ha l p ( ) + l p = p [ o,) o-oscllaory soluo (,,d). The equao (.) has a = = y Λ, d f ad oly f ( ) fj u,dg j u,..., dgj u du fj,dgj,..., dgj j = < j = + <, (3.7) for soe d. Theore 3.5: Assue ha l p ( ) + l p = p [,). Furher assue ha = = ( ) fj u,dg j u,..., dgj u du fj,dgj,..., dgj j = < j = + < (3.8) for soe d ad j j j = j = u f u,b,..., b du+ f,b,..., b = (3.9) for soe d, where bd >. The equao (.) has a o-oscllaory soluo (,,) y Λ. We exae he followg o help llusrae he obaed resuls. Exaple 3.: Cosder y y ( ) + y 6 ( ) = ( ) 3 3 ( ) 3 y ( ) y ( ) + y 6 ( ) =, ( ) (3.) Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year Global Jourals

15 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 6 where q ( ) ( ) = ad q ( ) ( ) 3 3 = 6 I s obvous ha equaly (3.) holds. Therefore, equao (3.) has a ooscllaory soluo y Λ b,a,,b,a. I fac, y = s such a soluo, where a= ad b=. Exaple 3.: Cosder where y y ( ) + qy 3 = y 3 y + qy =, ( ) 3 3 q( ) ( ), q = =.. (3.) 5 For large ad 3 q ~M 5 ad q 3 ~M. I s obvous ha equales (3.8) (,,) ad (3.9) are sasfed. Fro Theore 3.5, equao (3.) has a soluo y Λ. = s such a soluo of equao (3.). I fac, y( ), Rear 3.: The above argues ca be appled o he equao y( ) p ( ) y( τ ) = fj (,y( gj ( )),,y( gj ( ))),, S = j = y ( ) py ( ) τ = fj (,yg ( j ( ) ),,yg ( j ( ) )),, S. = j = For sace, uder he assupos of Theore 3., we have (,,) ( b,a, ) ( α,, ) (,, ) Λ=Λ Λ Λ Λ. (3.) Therefore, Theores 3.3 ad 3.4 hold for equao (3.). Furherore, equao (3.) has a o-oscllaory soluo y (,, ) Λ f ( ) fj,dgj,..., dgj d fj,dgj,..., dgj j = < j = + < (3.3) for soe d. Noes 8 Global Jourals

16 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Refereces Référeces Referecas Noes. D. D. Baov ad P. S. Seoov, Oscllao Theory of Ipulsve Dffereal Equaos, Ieraoal Publcaos Orlado, Florda, B. G. Zhag, Zhu Shalag, Oscllao of secod-order olear delay dyac equaos o e scales, Copu. Mah. Appl., 49 (5), Q. Yag, L. Yag, S. Zhu, Ierval crera for oscllao of secod-order olear eural dffereal equaos, Copu. Mah. Appl., 46 (3), V. Lashaha, D. D. Baov ad P. S. Seoov, Theory of Ipulsve Dffereal Equaos, World Scefc Publshg Co. Pe. Ld. Sgapore, J. S. W. Wog, Necessary ad suffce codos for oscllao for secod order eural dffereal equaos, J. Mah. Aal. Appl. 5 (), R. Xu, F. Meg, Oscllao crera for secod order quas-lear eural delay dffereal equaos, Appl. Mah. Copu., 9 (7), Y. G. Su, S. H. Saer, Oscllao for secod-order olear eural delay dfferece equaos. Appl. Mah. Copu., 63 (5), F. Meg, J. Wag, Oscllao crera for secod order quas-lear eural delay dffereal equaos, J. Idoes. Mah. Soc (MIHMI), (4), I. O. Isaac, Z. Lpcsey & U. J. Ibo. Nooscllaory ad Oscllaory Crera for Frs Order Nolear Neural Ipulsve Dffereal Equaos, Joural of Maheacs Research, Vol. 3 Issue, (), I. O. Isaac ad Z. Lpcsey. Oscllaos of Scalar Neural Ipulsve Dffereal Equaos of he Frs Order wh varable Coeffces, Dyac Syses ad Applcaos,9, (), L. H. Erbe, Q. Kog ad B. G. Zhag,Oscllao Theory for Fucoal Dffereal Equaos, Deer, New Yor, U. A. Abasewere, E. F. Nse, I. U. Moffa, Oscllao Codos for a Type of Secod Order Neural Dffereal Equaos wh Ipulses, Aerca Joural of Appled Maheacs, Vol. 5, No. 4, 7, pp do:.648/ j.aja U. A. Abasewere, I. U. Moffa, Oscllao Theores for Lear Neural Ipulsve Dffereal Equaos of he Secod Order wh Varable Coeffces ad Cosa Rearded Argues, Appled Maheacs, Vol. 7 No. 3, 7, pp do:.593/j.a U. A. Abasewere, I. U. Moffa, Crera for Bouded (Ubouded) Oscllaos of eural Ipulsve Dffereal Equaos of he Secod Order wh Varable Coeffces, Ieraoal Joural of Maheacs Treds ad Techology (IJMTT), do:.4445/35373/ijmtt-v48p56, V48():8-3. Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year Global Jourals

17 Classfcao of No-Oscllaory Soluos of Nolear Neural Delay Ipulsve Dffereal Equaos Noes Global Joural of Scece Froer Research ( F ) Volue XVIII Issue I V erso I Year 8 64 Ths page s eoally lef bla 8 Global Jourals