On the Stability and Boundedness of Solutions of Certain Non-Autonomous Delay Differential Equation of Third Order

Transcription

1 Applie Mahemaic, 6, 7, Publihe Online March 6 in SciRe. hp:// hp://.i.rg/.436/am On he Sabili an Bunene f Sluin f Cerain Nn-Aunmu Dela Differenial Equain f Thir Orer Akinwale L. Oluim *, Daniel O. Aam Deparmen f Mahemaic, Lag Sae Univeri, Oj, Nigeria Deparmen f Mahemaic, Feeral Univeri f Agriculure, Abekua, Nigeria Receive 3 Januar 6; accepe March 6; publihe 4 March 6 Cprigh 6 b auhr an Scienific Reearch Publihing Inc. Thi wrk i licene uner he Creaive Cmmn Aribuin Inernainal Licene (CC BY. hp://creaivecmmn.rg/licene/b/4./ Abrac In hi paper, we u cerain nn-aunmu hir rer ela ifferenial equain wih cninuu eviaing argumen an eablihe ufficien cniin fr he abili an bunene f luin f he equain. The cniin ae cmplemen previul knwn reul. Eample i al given illurae he crrecne an ignificance f he reul baine. Kewr Ampic Sabili, Bunene, Lapunv Funcinal, Dela Differenial Equain, Thir-Orer Dela Differenial Equain. Inrucin Thi paper cnier he hir rer nn-aunmu nnlinear ela ifferenial + + ( + ( ( (, ( p,,, ( r, ( r, a h b g r c f r = (. r i equivalen em * Crrepning auhr. Hw cie hi paper: Oluim, A.L. an Aam, D.O. (6 On he Sabili an Bunene f Sluin f Cerain Nn-Aunmu Dela Differenial Equain f Thir Orer. Applie Mahemaic, 7, hp://.i.rg/.436/am.6.764

2 A. L. Oluim, D. O. Aam =, =, ( = a h, b g c f + b g r ( ( ( + c f + p,,, r, r,, r where r γ, r β, β ( ( < <, β an γ are me piive cnan, γ will be eermine laer, + a, b, c, h,, g, f, p (,,, r, r,,, + = [, are real value funcin cninuu in heir repecive argumen n ,,,,, an + repecivel an + = [,. Beie, i i uppe ha he erivaive f (, g ( are cninuu fr all,, wih f ( = g( =. In aiin, i i al aume ha he funcin h (,, f ( ( r, g( ( r an p(,,, ( r, ( r, aif a Lipchi cniin in,, ( r, ( r an ; hrughu he paper (, ( an ( are repecivel abbreviae a, an. Then he luin f (. are unique. In applie cience, me pracical prblem are aciae wih Equain (. uch a afer effec, nnlinear cillain, bilgical em an equain wih eviaing argumen (ee []-[3]. I i well knwn ha he abili f luin pla a ke rle in characeriing he behavir f nnlinear ela ifferenial equain. Sabili i much mre cmplicae fr ela equain. Thu, i i wrhwhile cninue inveigae he abili an bunene f luin f Equain (. an i variu frm. Equain f he frm (. in which a(, b( an c are cnan ha been uie b everal auhr Saek [4] [5], Zhu [6], Afuwape an Omeike [7], Aemla an Aramw [8], Ya an Meng [9], Tunc [3] an Aemla e al [] menin a few. The bain he abili, unifrm bunene an unifrm ulimae bunene f luin. In a equence f reul, Omeike [] cnier he fllwing nnlinear ela ifferenial equain f he hir rer, wih a cnan eviaing argumen r, + + ( + ( ( = a b g c h r p an eablihe cniin fr he abili an bunene f luin when p( = an Tunc [] cnier a imilar em wih a cnan eviaing argumen r f he frm + ψ ( + ( + ( ( p(,, ( r,, ( r, a b g c h r = (. p while an bain he cniin fr i bunene f luin. Reul baine are nw eene nn-aunmu ela ifferenial Equain (.. Reul baine in hi wrk are cmparable in generali he reul f Saek [7] n analgu hir rer ifferenial equain which ielf generalie an analgu hir-rer reul f Zhu [5], an al cmplemen eiing reul n hir rer ela ifferenial equain. We eablih ufficien cniin fr he abili (when p an bunene (when p f luin f Equain (. which een an imprve he reul f Omeike [] an Tunc []. An eample i given illurae he crrecne an ignificance f he reul baine. Nw, we will ae he abili crieria fr he general nn-aunmu ela ifferenial em. We cnier: ( θ n where f : I C H i a cninuu mapping, an fr H H, here ei L( H >, wih = f,, = + r θ,, (.3 { ( n φ } φ [ ] f, =, C : = C r,, : H H ( φ f L H when φ H. Definiin.. ([8] An elemen ψ C i in he ω -limi e f φ, a, Ω ( φ, if (,, φ i efine a n, wih ( φ ψ a n where n [, an here i a equence { } n, n n 458

3 A. L. Oluim, D. O. Aam φ = + θ,, φ fr r θ. n n φ, he luin f (., Definiin.. ([8] [3] A e Q CH i an invarian e if fr an Q,, φ,, an ( φ Q fr [,. Lemma ([8,3] An elemen φ CH i uch ha he luin ( φ f (.3 wih ( φ = φ i efine H H, Ω φ i a nn-emp, cmpac, invarian e an i efine n [ n [, an ( φ < fr [, hen i ( ( φ ( φ, Ω a. Lemma ([8] [3] Le V (, : I CH iin. V (, φ =, an uch ha: W φ( V (, φ W φ where W ( r, W ( r are wege; V (.3 (, φ fr φ C H. Then he er luin f (.3 i unifrml able. If we efine Z φ CH V(.3 ( φ luin f (.3 i ampicall able prvie ha he large invarian e in Z i { } φ be a cninuu funcinal aifing a lcal Lipchi cn- { :, } = =, hen he er Q =. The fllwing will be ur main abili reul (when p fr (... Saemen f Reul Therem In aiin he baic aumpin impe n he funcin a(, b(, c(, h (,, g(, f( an p, le u aume ha here ei piive cnan δ,,, a, abc,,, µ,, M an M uch ha he fllwing cniin are aifie: f ( f ( c; c >, f ( =, δ >, ; g( h (, + a; g ( =, b >, ; 3 < c b ; b c an < a a ; 4 h (, ; a < b µ c an f ( M, g ( M, fr all,. Then, he er luin f em (. i ampicall able, prvie ha an ( ( b > µ > (. c a b µ c β µ a + a β γ < min ;. M + M β + M µ + µ M + M β + M µ + (. Prf Our main l i he fllwing Lapunv funcinal V V (,, = efine a ( = + V,, c f ξ ξ µ b g η η ( η η η µ µ + a h, a c f ( r + r + + λ θ θ+ δ θ θ, where λ an δ are piive cnan which will be eermine laer. We al aume ha where < c b. lim c = c, lim b = b, (.3 459

4 A. L. Oluim, D. O. Aam B he aumpin a( > an h, + a, frm (.3 we have ( ( ξ ξ + µ η η + a + µ + + µ c f ( + r + + r + V,, c f b g The Lapunv funcinal (.4 can be arrange in he frm Frm Therem, λ θ θ δ θ θ. V a + + ( µ a 4a a ( η g b + µ bc ηη c η b µ { f ( b} c f f c + µ + + ξ ξ ξ b b r + r + + λ θ θ+ δ θ θ. µ a µ a > an Thu, here i a δ > uch ha µ > which make a a µ >. (.4 (.5 a + + ( µ a δ + δ. 4a a B ( an (3 f Therem, we have ha he hir erm n he righ in (.5 an ne w erm give Uing (.6, (.7 an (.8 in (.5, we have ( η (.6 g b µ b c ηη, c η b (.7 µ { f ( + b} + c f f c µ b ξ ξ ξ b µ c f ( ξ f ( ξ ξ. b µ V (,, c f ( ξ f ( ξ ξ δ δ + + b r + r + + λ θ θ+ δ θ θ. + δ + δ + λ, θ θ δ θ θ r + + r + where δ δ3 = (.8 (.9 46

5 A. L. Oluim, D. O. Aam an inegral an r + r + λ θ θ δ θ θ are nn-negaive. ha Thu, fr me piive cnan δ δ3 an D = δ δ3, δ, δ mall enugh uch δ, where ( V,, D + +. (. Fr he ime erivaive f he Lapunv funcinal (.3, alng a rajecr f he em (., we have V (,, = c f b g c f ( a h(, a ξ ξ + µ η η µ η η η + + b g( c f ( a h( a h ( ( µ µ, +, ηηη + b g + c f r r ( ( + µ b g + µ c f r r δ + λr + λ r δ δ + λr + r. r r r r Frm (4 f Therem, f ( M, g ( M an uing uv u + v, we have ha ( c f M r Mr + M. r r ( µ c f µ M r µ Mr + µ M ( r Similarl, we bain b g ( ( ( Mr + M r r µ b g ( ( ( µ Mr + µ M r r Thu, V (,, c f b g c f ( a h(, a ξ ξ + µ η η µ η η η + + r b g µ c f µ a h, + a h, ηηη + ( M + M + λ r + ( µ M + µ M + δ r + M ( µ + λ( r ( r + M ( µ + δ ( r (. r (. (. 46

6 A. L. Oluim, D. O. Aam =, hen If b g µ c f =. If, we can rewrie he erm a g b µ c f ( bb µ cc [ b µ c], (.3 where b (3 f Therem, b c > a An b ( an ( f Therem,. g a h(, η a ηη b µ c f ( µ g a b µ c f ( b c. Accring ( f Therem, h (, + a an b (3, µ µ > an cerainl a ( a hu, a a ( ( + (.4 µ + µ ah, µ a a, (.5 an b (3 an (4 f Therem, we have ha a h, ηηη ( ξ ξ µ ( η η µ c f + b g + c f fr all, an. Thu, frm (., (., (.3, (.4 an (.5, we have an If we che an uing r(, we bain V (,, ( b c a( a µ µ + + ( M + M + λ r + ( µ M + µ M + δ r + M ( µ + λ( β ( r + M ( µ + δ ( β (. r ( + ( β M µ λ = > ( + ( β M µ δ = > ( + ( + ( + ( β γ M M β M µ V (,, ( b µ c ( + ( + ( + ( β γ µ M M β M µ + µ a ( a. 46

7 A. L. Oluim, D. O. Aam Ching we have fr me δ 4 >. ( ( b µ c β µ a + a β γ < min ;, M + M β + M µ + µ M + M β + M µ + Finall, i fllw ha V ( ( V U φ. V (,, δ4 ( +, (.6,, = if an nl if = =, V φ < fr φ an fr Thu, (. an (.6 an he la aemen agree wih Lemma. Thi hw ha he rivial luin f (. i ampicall able. Hence, he prf f he Therem i nw cmplee. Remark. If r( i a cnan an (. i he cnan c-efficien ela ifferenial equain + a + b ( r + c ( r =, hen cniin (-(4 reuce he Ruh-Hurwi cniin a >, c > an ab > c. T hw hi we e a b c an h(, = a, g( ( r = b ( r an f ( ( r = c( r. Remark. If h(, = a an a = b = c = in (., he nn-aunmu Equain (. reuce he aunmu equain cniere in Saek [4]. 3. The Bunene f Sluin Therem We aume ha all he aumpin f Therem an hl, where P i a piive cnan. ( ( p(,,, r, r, q, q P < Then, here ei a finie piive cnan K uch ha he luin f Equain (. efine b he iniial funcin aifie he inequaliie fr all, where φ Prf f Therem = φ, = φ, = φ,, K K K r,, prvie ha ( ( b µ c β µ a + a β γ < min ;. M + M β + M µ + µ M + M β + M µ + A in Therem, he prf f Therem epen n he calar iffereniable Lapunv funcin V (,, efine in (.3. Since p, in (.. In view f (.6, V (,, V (. (,, + ( + µ p,,, ( r, ( r, 463

8 A. L. Oluim, D. O. Aam Since V (. fr all,,, hu V,, δ µ q Hence, i fllw ha V 4 5 q 5 q fr a cnan 5 5 =,. Making ue f he inequaliie < + an r (,, δ ( + + δ ( + δ ( + δ >, where δ { µ } B (., we have ( D V( Hence, where δ { 6 ma δ5, δ5d } r =. < +, i i clear ha V (,, δ5 ( + + q. + +,,, V (,, δ5 + D V (,, q V 6q 6V q (,, δ + δ (,, ( 6 ( δ6 δ6 ( δ6 ( δ Mulipling each ie f hi inequali b he inegraing facr ep δ q ( 6 6, we ge V (,, ep q( V,, q ep q δ q ep q. Inegraing each ie f hi inequali frm, we ge, where V ( ( (, (, ( V ep ( δ6 q( V ( ep δ6 q( ( 6 ( δ6 δ6 =, ( V,, V ep q + ep q. Since ep δ q ( an uing he fac ha ( δ ( δ q P fr all, hi implie V,, V ep 6P + ep 6P fr. Nw, ince he righ-han ie i a cnan, an ince (,, ha here ei a K > uch ha Frm he Equain (. hi implie V a K, K, K fr. K, K, K fr. The prf f Therem i nw cmplee. Remark 3. If r( i a cnan, g ( r g reul baine reuce Omeike [6] an a reul f Tunc []. 4. Cncluin =, h( + +, i fllw, = an p in (., he The luin f he hir-rer nn-aunmu ela em are ampicall able an bune accring 464

9 A. L. Oluim, D. O. Aam he Lapunv her if he inequaliie (. an (. are aifie. Eample 3. We cnier nn-aunmu hir-rer ela ifferenial equain wih equivalen em f (3. a: = = in ( + ( ( r ( r + in + in + ( r ( r in + + in + 3 = ( ( r r = in in + in + in + in in + r in ( + in + r in ( r + + r + ( ( cmparing (. wih (3., i i ea ee ha + = a a = 3 in = b b = in = c c = in + The funcin The funcin 5 f = + in + 3, i i clear frm he equain ha g f ( = δ >, = + in +, i i clear frm he equain ha g 5 = b >, (3. (3. f ( = c, c > 465

10 A. L. Oluim, D. O. Aam (, = 8+ + h al, = + 7, a = > µ >, we che, µ = 3, we che, = 5 an = 8, we che, = 4 Since we have f = = M 5 g = = M < β <, we che, β =, γ < min, = I fllw ha r, if he ela i increae ben hi range a limi ccle appear, fllwe even- 375 uall b a peri-ubling cacae leaing cha. Finall, an (,,, (, (, p r r = r + + r + + ( ( q = = <. + Π Thu, all aumpin f Therem an Therem are hel. Tha i, er luin f Equain (. i ampicall able an all he luin f he ame equain are bune. Reference [] Afuwape, A.U., Omari, P. an Zanalin, F. (989 Nnlinear Perubain f Differenial Operar wih Nnrivial Kernel an Applicain Thir-Orer Periic Bunar Prblem. Jurnal f Mahemaical Anali an Applicain, 43, hp://.i.rg/.6/-47x( [] Crnin, J. (997 Sme Mahemaic f Bilgical Ocillain. SIAM Review, 9, -37. hp://.i.rg/.37/97 [3] Rauch, L.L. (95 Ocillain f a Thir-Orer Nnlinear Aunmu Sem in Cnribuin he Ther f Nnlinear Ocillain. Annal f Mahemaic Suie,, [4] Saek, A.I. (3 On he Sabili an Bunene f a Kin f Thir Orer Dela Differenial Sem. Applie Mahemaic Leer, 6, hp://.i.rg/.6/s (363-6 [5] Saek, A.I. (5 On he Sabili f Sluin f Sme Nn-Aunmu Dela Differnial Equain f Thir Orer. Ampic Anali, 43, -7. [6] Zhu, Y.F. (99 On Sabili, Bunene an Eience f Periic Sluin f a Kin f Thir Orer Nnlinear Dela Differenial Sem. Annal f Differenial Equain, 8, [7] Afuwape, A.U. an Omeike, M.O. (8 On he Sabili an Bunene f Sluin f a Kin f Thir Orer De- 466

11 A. L. Oluim, D. O. Aam la Differenial Equain. Applie Mahemaic an Cmpuain,, hp://.i.rg/.6/j.amc [8] Aemla, A.T. an Aramw, A.T. (3 Unifrm Sabili an Bunene f Sluin f Nn-Linear Dela Differenial Equain f Thir Orer. Mahemaical Jurnal f Okaama Univeri, 55, [9] Ya, H. an Meng, W. (8 On he Sabili f Sluin f Cerain Nn-Linear Thir Orer Dela Differenial Equain. Inernainal Jurnal f Nn-Linear Science, 6, [] Aemla, A.T., Ogunare, B.S., Oguniran, M.O. an Aeina, O.A. (5 Sabili, Bunene an Eience f Periic Sluin Cerain Thir-Orer Dela Diffrenial Equain wih Muliple Deviaing Argumen. Inernainal Jurnal f Differenial Equain, 5, Aricle ID: [] Omeike, M.O. (9 Sabili an Bunene f Sluin f Sme Nn-Aunmu Dela Differenial Equain f he Thir Orer. Analele Siinifice Ale Univeriaii Aleanru Ian Cua Din Iai Maemaica, 55, [] Tunc, C. (9 Bunene in Thir Orer Nnlinear Differenial Equain wih Bune Dela. Blein e Mahemaica, 6, -. [3] Tunc, C. (6 New Reul abu Sabili an Bunene f Sluin f Cerain Nn-Linear Thir Orer Dela Differenial Equain. The Arabian Jurnal fr Science an Engineering, 3,