Novel Bounds for Solutions of Nonlinear Differential Equations

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1 Appe Mahemacs, 5, 6, 8-94 Pubshe Oe Jauary 5 ScRes. hp:// hp://.o.org/.436/am.5.68 Nove Bous for Souos of Noear Dfferea Equaos A. A. Maryyu S.P. Tmosheo Isue of Mechacs of NAS of Urae, Kyv, Urae Ema: ceer@mech.ev.ua Receve 4 November 4; accepe December 4; pubshe 9 Jauary 5 Copyrgh 5 by auhor a Scefc Research Pubshg Ic. Ths wor s cese uer he Creave Commos Arbuo Ieraoa Lcese (CC BY). hp://creavecommos.org/ceses/by/4./ Absrac I hs paper he esmaes for orms of souos o oear sysems are obae va a egra equay. As a appcao we cosere affe coro sysems a sysems of equaos for sychrozao of moos. Keywors Noear Sysems, Nove Bous for Souos, Saby, Sychrozao. Irouco The probem of esmag he orms of souos o oear sysems of orary fferea equaos remas urge ue o exesve appcao of he aer he escrpo of rea processes may mechaca, physca a oher aure sysems. Usuay, o oba he esmaes of orms of souos o ear a weay oear equaos, he Growa-Bema emma s appe (see, for exampe, []-[3] a bbography here). The eveopme of he heory of oear equaes has subsaay wee he possbes for obag he esmaes of orms of souos o oear sysems a has gve a mpeus o her appcao he quaave heory of equaos (see, for exampe, [4]-[6]). Boh ear a oear egra equaes are effcey use for he eveopme of he rec Lyapuov meho, parcuar, for he vesgao of moo boueess a saby of oear weay coece sysems [7]. The prese paper s ame a obag ew esmaes of orms of souos for some casses of oear equaos of perurbe moo. The paper s arrage as foows. I Seco he saeme of he probem s gve vew of some resus of papers [] [3]. Seco 3 preses ma resus o obag he esmaes of orms of souos for some casses of oear sysems of fferea equaos. I hs regar, severa resus from [8] are ae o accou. How o ce hs paper: Maryyu, A.A. (5) Nove Bous for Souos of Noear Dfferea Equaos. Appe Mahemacs, 6, hp://.o.org/.436/am.5.68

2 A. A. Maryyu I Seco 4 wo appcao probems are cosere: a probem o sabzao of souos o affe sysem (cf. [8]) a a probem o esmao of vergece of souos a sychrozao (cf. [9]). I Seco 5 he possbes of appcao of hs approach for souo of moer probems of oear yamcs a sysems heory are scusse.. Saeme of he Probem Coser a oear sysem of orary fferea equaos of perurbe moo x ; ( f C, ) A s a -marx wh he eemes couous o ay fe - = of probem () exss a s uque for a < a where, erva. I s assume ha souo x x (,, x) (, ). x A x f x,, x x, = = () Equaos of ype () are fou may probems of mechacs (see, for exampe, [] [] a bbography here). Moreover, hese equaos may be reae as he oes escrbg he perurbao of he sysem of ear equaos A x, x x, = = () I orer o esabsh boueess a saby coos for souos of sysem () s ecessary o esmae he orms of souos uer varous ypes of resrcos o sysem () a vecor-fuco of oeares sysem (). The purpose of hs paper s o oba esmaes of orms of souos o some casses of oear orary fferea Equaos () erms of oear a pseuo-ear egra equaes. 3. Ma Resus Frs,we sha eerme he esmae of he orm of souos x of sysem () uer he foowg assumpos: A. For a here exss a oegave egrabe fuco b such ha A. For a a such ha (cf. []) for a (, ). A b for a ; u here exss a couous oegave egrabe fuco wu (, ), (, ) (, ) f x w x w, =, x Here a esewhere a Euca orm of he vecor x a a specra orm of he marx cosse wh are use. Theorem. For sysem () e coos of assumpos A a A be sasfe, he for ay souo x = x,, x wh he a vaues x : x c, c < < he equay hos for a. If here exs: x c b s x s w s, x s s (3) (a) a couous a oegave fuco v for a (b) a couous, oegave a oecreasg fuco a g u for u such ha w, zexp b( s) s exp b( s) s v g( z),, z, 83

3 A. A. Maryyu he for a [ β ) hos rue, where he equay, exp, (4) x G G c v s s b s s G s a fuco coverse wh respec o he fuco G( u ) : u s G u G u =, < u cu, a he vaue β s eerme by he correao (c) If, aoay, here exss a cosa he equay (4) s sasfe for a u g s β = sup : G( c) v( s) s om G. a > such ha v s, g s a,.e. β = for he vaues c (, a ) Proof. Le he rgh-ha par of equay (3) be equa. p exp b s s. Usg equay (3) a co- o (b) of Theorem we ge p b p bs s = b x w x exp, Sce he fuco g s oecreasg a we ge he equay b p vg x exp bs s exp bs s. x p exp bs s, p v g( p ), p( ) c. = Hece, by he Bhar emma (see [], p. ) we have p G G( c) v( s) s, for a (, β ). Ths mpes esmae (4). To prove he seco assero of Theorem we oe ha he couaby coo for fuco he equay or G c v s s u s g s p s 84

4 A. A. Maryyu Ths equay s sasfe for ay c (, a ) we have c s s s. v( s) s g( s) = g( s) g( s) u u c for whch coo (c) of Theorem hos rue. Sce c < a, Hece foows ha for c (, a ) s s. v( s) s < g( s) g( s) a c he vaue β =. Ths proves Theorem. Furher we sha coser sysem () uer he foowg assumpo. c for a a a cosa > such ha A 3. There exs a oegave egrabe fuco for a (, ). (, ) f x c x x Theorem. For he sysem of Equaos () e coos of Assumpos A a A 3 be sasfe. The x = x,, x he esmae for he orm of souos x hos rue for a wheever x exp b s s ( ) x c( s) exp ( ) b( τ) τ s s (5) ( ) x c( s) exp ( ) b( τ) τ s <. (6) Proof. Le x be he souo of sysem of Equaos () wh he a coos x A Equao () yes he esmae of he orm of souo Uer coos A a 3 s = x,. x he form. (7) x x b s x s s c s x s s We rasform equay (7) o he pseuo-ear form x x b s c s x s x s, s (8) a appyg he Growa-Bema emma [] arrve a he esmae for a. Furher, for esmao of he expresso exp x x b s c s x s s (9) exp c( s) x( s) s he foowg approach s appe (cf. [8]). Desgae x = ψ for a a from equay (9) oba 85

5 A. A. Maryyu ψ x exp ( ) ( b( s) c( s) ψ ( s) ) s. () Mupyg boh pars of equay () by he expresso we ge ( ) c exp ( ) cs ψ ( s) s, ( ) c ψ exp ( ) cs ψ ( s) s ( ) x c exp ( ) bs s. Ths mpes ha ( ) x c exp ( ) b( s) s exp ( ) c( s) ψ ( s) s. Iegrag he obae equay bewee he ms a we arrve a s ( ) x c( s) exp ( ) b( τ) τ s exp ( ) c( s) ψ ( s) s. Uer coo (6) hs esmae mpes exp ( ) c( s) ψ ( s) s. ( ) x c( s) exp ( ) b( τ) τ s s Moreover, equay () becomes ψ x exp ( ) b( s) s. ( ) x c( s) exp ( ) b( τ) τ s s Ths equay yes esmae (5) for a for whch coo (6) s sasfe. Ths compees he proof of Theorem. Iequay (7) s a para case of equay (3) a s represeao pseuo-ear form (8) aows us o smpfy he proceure of obag he esmae of orm of souos o sysem (). Theorem has a seres of coroares as appe o some casses of sysems of orary fferea equaos. Coroary. Coser sysem () for A for a f x,, x x. = = () Ths s a esseay oear sysem,.e. a sysem whou ear approxmao. Such sysems are fou he coserao of sysems wh ry frco, eecroacousc wavegues a oher probems. Sysems wh secor oeary (see []) are cose o hs ype of sysems. If coo A 3 s fufe wh he fuco c such ha c s s >, 86

6 A. A. Maryyu for ay (, ), <, =,,,, he x x c s x s. s Appyg o hs equay he same proceure as he proof of Theorem s easy o show ha f for a, he x ( ) x c s s > x ( ) x c( s) s for a. Comme. Esmae () s obae as we by a mmeae appcao of he Bhar emma (see []) o he equay wh he fuco Φ ( u) = x, >,. Coroary. I sysem () e f( x, ) (, ) uous wh respec o (, ). x x c s x s s Bxx, where B : s a -marx co- x Coser a sysem of o-auoomous ear equaos wh pseuo-ear perurbao ( ) A Bx, x, x x. () = = (3) Assume ha coo A s sasfe a here exss a oegave egrabe fuco h such ha for a (, ). x Equao (3) mpes ha Bx, h x, (4) x x b s x s h( s x s. s (5) Appyg o equay (5) he same proceure as he proof of Theorem we ge he esmae x whch hos rue for he vaues of [, ) x exp b s s x h s exp b τs for whch ( τ) s (6) s x h s exp b s >. ( τ) τ (7) Comme. If equay (5) fucos b h esmae (see [4]) = = for a, he Theorem yes he 87

7 A. A. Maryyu x exp( ) ( ) x x exp for a [ τ ),, where τ s eerme by he formua x τ =. x co,,, Coroary 3. I sysem () e f(, x) = A x Ax x = x x x for a =,3,,. Furher we sha coser he sysem of oear equaos where A C(, ), where = A ( x ), x = x, (8) = are ( ) A. Assume ha here exs oegave egrabe o [, ) fucos A I vew of (9) we ge from (8) he equay -marces wh he eemes couous o ay fe erva a,,,, b, =,,,, such ha A b = (9). () x x A s x s s x b s x s s Iequay () s presee pseuo-ear form = = Hece x x b( s) b ( s) x( s) x( s). s = x x exp b( s) b ( s) xs s. () = exp b s x s s. = We sha f he esmae of he expresso Iequay () mpes ha he esmae r x x exp ( ) b( s) br ( s) xs s r = r x exp ( ) b( s) ( ) br ( s) x( s) s. r = s rue. Mupyg boh pars of hs equay by he egave expresso we ge r ( ) b exp ( ) br ( s) x( s) s r = b x exp b s x s s b x exp b s s. r r r = Summg up boh pars of hs equay from = o we f 88

8 A. A. Maryyu r b x exp br ( s) x( s) s ( ) b x exp ( ) b( s) s. = r= = Iegrao of hs equay bewee a resus he foowg equay exp b s x s s b s x exp b τ s ( τ) = = From hs equay we f ha exp b ( s) x( s) s ( ) b ( s) x exp ( ) b( τ) τ s = Hece foows he esmae = x whch s va for a [, ) x exp b ( s) s ( ) b ( s) x exp ( ) b( τ) τ s = such ha () of (5). The oaos x ( 5) s ( ) x b ( s) exp ( ) b( τ) τ s > = Esmae (5) aows boueess a saby coos for souo of sysem () o be esabshe he foowg form. Theorem 3. If coos A a A 3 of Theorem are sasfe for a ( x, ) a here exss a cosa β > such ha x ( 5 < β for a ), where β may epe o each souo, he he souo x (,, x ) of sysem () s boue. Theorem 4. If coos A a A 3 of Theorem are sasfe for a ( x, ) a f( x, ) = for x =, a for ay ε > a? here exss a δ (, ε ) > such ha f x < δ (, ε ), he he esmae x ( 5 < ε s sasfe for a ), he he zero souo of sysem () s sabe. The proofs of Theorems 3 a 4 foow mmeaey from he esmae of orm of souos x he form β x < ε mea ha he rgh ha par of equay (5) mus sa- < a ( 5) sfy hese equaes uer approprae a coos. Smar asseros are va for he sysems of Equaos (), (3) a (8) erms of esmaes (), (6) a (). 4. Appcaos 4.. Sabzao of Moos of Affe Sysem Coser a affe sysem wh may corog boes = Ax G( x, ) u Bu, (3) Cx, y = (4) x x, = (5) 89

9 A. A. Maryyu where A s a G x, s a m- m m marx, he coro vecors u for a =,,,, B s a m-marx a he coro u, C s a cosa -marx, x s a vecor of he a saes of sysem (3). Wh regar o sysem (3) he foowg assumpos are mae: A 4. Fucos G (,) =, =,,,, for a. A 5. There exss a cosa m-marx K such ha for he sysem x, he fuamea marx Φ sasfes he esmae -marx wh couous eemes o ay fe erva, y ( A BKC) y = ( s) Φ Φ s Me, for s, where M a are some posve cosas. A 6. There exs cosas γ > a q > such ha G x, γ x, for a =,,,. The foowg assero aes pace. Theorem 5. Le coos of assumpos A4 - A 6 be sasfe a, moreover, qm q ( K C ) x q qs γ e s, I > q where γ = γ. The he coros =, =,,,, = u K y u K y sabze he moo of sysem (3) o he expoeay sabe oe. Proof. Le he coros u = Ky a u = Ky be use o sabze he moos of sysem (3). Beses, we have a = ( A BKCx ) G( x )( KCx ),. ( )( ) x =Φ Φ x Φ Φ s G s, x s K Cx s. s (6) I vew of coos of Theorem 5 we ge from (6) he esmae of orm of souo of sysem (3) he form We rasform equay (7) o he form Appyg Coroary 3 o equay (8) we ge ( s) γ x x Me Me KC x s. s (7) γ qs s x e x M Me KC x s e. s (8) q q 9

10 A. A. Maryyu for a. If coo x e M x q q qs γ qm ( KC ) x e I s = M x q q γ M ( KC ) x I q ( e ) M x. q q q γ M ( KC ) x. q q qm q ( K C ) x q qs γ e s, I > of Theorem 5 s sasfe, he q q γ M ( KC ) x > a for he orm of souo x we have he esmae for a, where M Ths compees he proof of Theorem Sycrozao of Moos = e x M x q q q γ M ( KC ) x The heory of moo sychrozaos sues he sysems of fferea equaos of he form (see [9] a bbography here) where (,, ) : [,] M. = µ f( x,, µ ), x = x, (9) f xµ wh respec o wh he pero T, a µ s a sma parameer. Aogse sysem (9) we sha coser a ajo sysem of equaos, f s a fuco couous wh respec o, x, µ a peroc 9

11 A. A. Maryyu where = µ g x, x = x, (3) T g( x) = f( sx,,) s. T Assume ha he eghborhoo of po vecor-fuco f a s para ervaves are couous. Desgae x for suffcey sma vaue of µ for ay [, T] f f M= max f( x,, µ ),,. [, T], x x, µ µ µ v j he I s cear ha he souos of Equaos (9) a (3) rema he eghborhoo x x for µ < M. Wh aowace for a we compe he correao x (, µ ) = x µ f( sxs, (, µ ), µ ) s x(, µ ) = x µ g( x( s, µ )), s ( µ ) ( µ ) = µ ( ( µ ) µ ) ( ( µ ) ) µ ( ( µ ) ) ( ( µ ) ) ( ) ( ) x x, x, f sxs,,, f sxs,,, s f sxs,,, f sx, s,, s (3) µ f s, x s, µ, g x s, µ. s As s show moograph [9] for he frs a hr summas correao (3) he foowg esmaes ho rue ( ) ( ) f sxs,, µ, µ f sxs,, µ, s Mµ, (3) ( ) ( ) f s, x s, µ, g x s, µ s MT 4 M Tµ. (33) To esmae he seco summa we assume ha here exs a egrabe fuco : ha for ay, [, T] ( < ) a > such ha he oma of vaues [, T] a x, x D. I vew of esmaes (3)-(34) we f from (3) N s s > (,,) (,,) N such f x f x N x x (34) 9

12 A. A. Maryyu for a s. s ( ( ) ) x s, µ x s, µ µ MT 4 M T M µ µ N τ x τ, µ x τ, µ τ (35) Le here exs [,] x µ uer he same a co- for a µ µ os s esmae as foows for a [, T] µ such ha T ( ) µ ( µ ) µ MT 4M T M N s s > (36) <. The he orm of vergece of souos xµ (, ) a (, ) (, µ ) x (, µ ) x µ MT 4M T M µ T ( ) µ ( MT ( 4M T M ) µ ) µ N ( s) s a for µ < µ. Esmae (37) s obae from equay (35) by he appcao of Coroary. Comme 3. If esmae (34) = a N = M, he he appcao of he Growa-Bema emma o equay (35) yes he esmae of vergece bewee souos he form [9] for a [, T]. 5. Cocug Remars (, µ ) (, µ ) µ ( 4 ) µ exp( µ ) x x MT M T M MT I hs paper he esmaes of orms of souos o fferea equaos of form (), () a (3) are obae erms of oear a pseuo-ear egra equaes. Ths approach facaes esabshg he esmaes of orms of souos for some casses of sysems of equaos of perurbe moo fou varous appe probems (see [] [3]). Effcecy of he obae resus s usrae by wo probems of oear yamcs. I s of eres o eveop he obae resus he vesgao of souos o yamc equaos o me scae (see [4] [5]). I moograph [6] he egra equaes o me scae form a bass of oe of he mehos of aayss of souos o yamc equaos. Refereces [] Bema, R. (953) Saby Theory of Dfferea Equaos. Dover Pubcaos, New Yor, 66 p. [] Maryyu, A.A., Lashmaham, V. a Leea, S. (979) Saby of Moo: Meho of Iegra Iequaes. Nauova Duma, Kev. (I Russa) [3] Rao, M.R.M. (98) Orary Dfferea Equaos. Affae Eas-Wes Press Pv L., New Deh- Maras, 66 p. [4] Guows, R. a Razszews, B. (97) Asympoc Behavour a Properes of Souos of a Sysem of No Lear Seco Orer Orary Dfferea Equaos Descrbg Moo of Mechaca Sysems. Archwum Mecha Sosowaej, 6, [5] Maryyu, A.A. a Guows, R. (979) Iegra Iequaes a Saby of Moo. Nauova Duma, Kev. (I Russa) [6] Pachpae, B.G. (998) Iequaes for Dfferea a Iegra Equaos. Acaemc Press, Sa Dego. [7] Maryyu, A., Cheresaya, L. a Maryyu, V. (3) Weay Coece Noear Sysems: Boueess a Saby of Moo. CRC Press, Boca Rao. [8] Louarass, Y., Mazou, E.H.E. a Eaam, N. () A New Geerazao of Lemma Growa-Bema. Appe Mahemaca Sceces, 6, [9] Rozo, M. (97) Noear Oscaos a Saby Theory. Naua, Moscow. (I Russa) [] Demovch, B.P. (967) Lecures o Mahemaca Saby Theory. Naua, Moscow. (I Russa) (37) 93

13 A. A. Maryyu [] Brauer, F. (963) Bous for Souos of Orary Dfferea Equaos. Proceegs of he Amerca Mahemaca Socey, 4, hp://.o.org/.9/s [] Aesarov, A.Yu., Aesarova, E.B. a Zhabo, A.P. (3) Saby Aayss of a Cass of Noear Nosaoary Sysems va Averagg. Noear Dyamcs a Sysems Theory, 3, [3] N Doye, I., Zasazs, M., Darouach, M., Rahy, N.-E. a Bouazz, A. () Expoea Sabzao of a Cass of Noear Sysems: A Geeraze Growa-Bema Lemma Approach. Noear Aayss, 74, hp://.o.org/.6/j.a..7.5 [4] Babeo, S.V. a Maryyu, A.A. (3) Noear Dyamc Iequaes a Saby of Quas-Lear Sysems o Tme Scaes. Noear Dyamcs a Sysems Theory, 3, 3-4. [5] Boher, M. a Maryyu, A.A. (7) Eemes of Saby Theory of A. M. Lyapuov for Dyamc Equaos o Tme Scaes. Noear Dyamcs a Sysems Theory, 7, 5-5. [6] Maryyu, A.A. () Saby Theory of Souos of Dyamc Equaos o Tme Scaes. Phoex, Kev. (I Russa) 94