On some systems of convolution-type first-order integrodifferential equations on the semiaxis

We study a class of vector convolution-type integrodifferential equations on the semiaxis used for the description of various applied problems of mathematical physics. By using a special three-factor decomposition of the original mathematical integrodifferential operator, we prove the solvability of...

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Date:	2009
Main Authors:	Khachatryan, A. Kh., Khachatryan, Kh. A., Хачатрян, А. Х., Хачатрян, Х. А.
Format:	Article
Language:	Russian English
Published:	Institute of Mathematics, NAS of Ukraine 2009
Online Access:	https://umj.imath.kiev.ua/index.php/umj/article/view/3099
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Journal Title:	Ukrains’kyi Matematychnyi Zhurnal
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Ukrains’kyi Matematychnyi Zhurnal

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author	Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. Хачатрян, А. Х. Хачатрян, Х. А.
author_facet	Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. Хачатрян, А. Х. Хачатрян, Х. А.
author_sort	Khachatryan, A. Kh.
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datestamp_date	2020-03-18T19:45:12Z
description	We study a class of vector convolution-type integrodifferential equations on the semiaxis used for the description of various applied problems of mathematical physics. By using a special three-factor decomposition of the original mathematical integrodifferential operator, we prove the solvability of these equations in certain functional spaces.
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fulltext	UDK 517.968.78 A. X. Xaçatrqn, X. A. Xaçatrqn (Yn-t matematyky NAN Armenyy, Erevan) O NEKOTORÁX SYSTEMAX YNTEHRO- DYFFERENCYAL\|NÁX URAVNENYJ PERVOHO PORQDKA TYPA SVERTKY NA POLUOSY We investigate one class of vector integro-differential equations of a type of convolution on a semiaxis, by which a number of applied problems of the mathematical physics are described. By using the special three-factor decomposition of the initial matrix integro-differential operator, we prove the solvability of considered equations in certain functional spaces. DoslidΩeno odyn klas vektornyx intehro-dyferencial\nyx rivnqn\ typu zhortky na pivosi, qky- my opysu[t\sq nyzka prykladnyx zadaç matematyçno] fizyky. Z dopomohog special\noho try- faktornoho rozkladu vyxidnoho matematyçnoho intehro-dyferencial\noho operatora dovedeno rozv’qznist\ cyx rivnqn\ u konkretnyx funkcional\nyx prostorax. 1. Vvedenye. Vektorn¥m yntehro-dyfferencyal\n¥m uravnenyem vyda − + = + − ∞ d dx A x g x k x t t t dti i i i ij j j ϕ ϕ λ ϕ( ) ( ) ( ) ( ) ( ) 0 ∫∫∑ =j m 1 , (1.1) i = 1, 2, … , m, x ∈ +∞( , )0 , opys¥vagtsq rqd prykladn¥x zadaç estestvoznanyq [1 – 4]. V çastnosty, urav- nenyq vyda (1.1) voznykagt v πkonometryke, kynetyçeskoj teoryy metallov, v teoryy veroqtnostej y dr. [1, 2, 4]. Ysxodq yz prykladn¥x soobraΩenyj estest- venno predpoloΩyt\, çto yskomaq vektor-funkcyq ϕ ϕ ϕ ϕ= …( , , , )1 2 m T ∈ ∈ M, hde T — znak transponyrovanyq, M — klass absolgtno neprer¥vn¥x vektor-funkcyj na ( , )0 +∞ medlennoho rosta, t.8e. ϕ ε∈ = = … ∈ +∞ ∀ >{� df f f f f f ACm T j( , , , ) : ( , ),1 2 0 0 : e f x x j mx j − → → +∞ = … }ε ( ) , , , , ,0 1 2 . Zdes\ AC( , )0 +∞ — prostranstvo absolgtno neprer¥vn¥x na ( , )0 +∞ funk- cyj. Çysla Ai > 0 , i = 1, 2, … , m, qvlqgtsq parametramy uravnenyq (1.1). Funkcyy gi , i = 1, 2, … , m, neotrycatel\n¥ na ( , )0 +∞ y prynadleΩat pros- transtvu L( , )0 +∞ , funkcyy λ j : 0 1≤ ≤λ j , j = 1, 2, … , m, yzmerym¥ na ( , )0 +∞ . Qdern¥e matryc¥-funkcyy kij neotrycatel\n¥ na ( , )0 +∞ , pryçem k L R M Rij ∈ 1( ) ( )∩ , k x dx aij ij( ) = −∞ +∞ ∫ , i, j = 1, 2, … , m, (1.2) hde matryca ( ) ,aij i j m m = × 1 udovletvorqet uslovyg substoxastyçnosty aij i m j= ≤ = ∑ df 1 1µ , j = 1, 2, … , m, max 1 1 ≤ ≤ = j m jµ . (1.3) © A. X. XAÇATRQN, X. A. XAÇATRQN, 2009 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1277 1278 A. X. XAÇATRQN, X. A. XAÇATRQN Skalqrnoe uravnenye (1.1) rassmatryvalos\ v konkretn¥x sluçaqx, a ymenno, kohda λ = const, a qdro k predstavlqet soboj vpolne monotonnug funkcyg vyda k x e d sx s a b ( ) ( )= −∫ σ , 0 < a < b ≤ + ∞, hde σ — monotonno neub¥vagwaq funkcyq na a b,[ ) , udovletvorqgwaq ra- venstvu [2 – 8] 2 1 1 s d s a b σ ( ) =∫ . V nastoqwej rabote yssledugtsq system¥ (1.1) v predpoloΩenyy, çto ysko- maq vektor-funkcyq ϕ prynadleΩyt klassu M. V rqde sluçaev svojstva sootvetstvugweho yntehro-dyfferencyal\noho operatora uravnenyq (1.1) opy- s¥vagtsq svojstvamy eho symvola. Naybolee sloΩn¥m y ynteresn¥m qvlqetsq uravnenye v tex osob¥x (neπllyptyçeskyx) sluçaqx, kohda symvol uravnenyq v¥roΩdaetsq na vewestvennoj osy. V predlahaemoj rabote m¥ kak raz y ras- smatryvaem uravnenye (1.1) v neπllyptyçeskom sluçae. Osnovnoj podxod k re- ßenyg system¥ (1.1) — πto faktoryzacyq ysxodnoho yntehro-dyfferencyal\- noho operatora v vyde proyzvedenyq odnoho prostejßeho matryçnoho dyffe- rencyal\noho operatora y dvux matryçn¥x yntehral\n¥x operatorov typa svertky. Postroennaq faktoryzacyq svodyt reßenye zadaçy (1.1) – (1.3) k re- ßenyg trex bolee prost¥x system uravnenyj. Pervaq yz nyx — systema dyf- ferencyal\n¥x uravnenyj pervoho porqdka dyahonal\noho typa, reßenye koto- roj zapys¥vaetsq avtomatyçesky. Vtoraq — dyahonal\naq systema yntehral\- n¥x uravnenyj typa Vol\terra vtoroho roda. Sootvetstvugwyj matryçn¥j operator πtoj system¥ v rqde sluçaev qvlqetsq obratym¥m v konkretn¥x funkcyonal\n¥x prostranstvax. Norma obratnoho operatora kontrolyruetsq s pomow\g vvedennoho svobodnoho parametra α = diag ( , , , )α α α1 2 … m , α j > 0 , j = 1, 2, … , m . Y, nakonec, tret\e uravnenye predstavlqet soboj vektornoe yn- tehral\noe uravnenye poçty svertoçnoho typa. Zdes\, yspol\zuq metod¥ teoryy vektorn¥x yntehral\n¥x uravnenyj Vynera – Xopfa, teoryy funkcyj dejstvy- tel\noj peremennoj y rqd druhyx metodov, dokaz¥vaem razreßymost\ πtoj sys- tem¥ v klasse M. 2. Nekotor¥e klass¥ yntehral\n¥x operatorov. Pust\ E+ — odno yz sledugwyx banaxov¥x prostranstv: Lp ( , )0 +∞ , p ≥ 1, M ( , )0 +∞ , Cu ( , )0 +∞ , … (zdes\ Cu ( , )0 +∞ — prostranstvo neprer¥vn¥x funkcyj na ( , )0 +∞ , ymeg- wyx koneçn¥j predel v + ∞ ), a L L1 1≡ −∞ +∞( , ) . Oboznaçym çerez Ω klass matryçn¥x yntehral\n¥x operatorov Vynera – Xopfa [9]: K ∈ Ω , esly ( ) ( ) ( ) ( )K f x K x t f t dt= − +∞ ∫ 0 , (2.1) hde K x k xij i j m m ( ) ( ) , = ( ) = × 1 , k Lij ∈ 1 , i, j = 1, 2, … , m, (2.2) a ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 O NEKOTORÁX SYSTEMAX YNTEHRO-DYFFERENCYAL\|NÁX URAVNENYJ … 1279 f f f f E E E Em T m m = … ∈ ≡ × × … ×+ × + + +( , , , )1 2 � �� . (2.3) Netrudno ubedyt\sq, çto operator K perevodyt E m + × v sebq, pryçem v lg- bom yz prostranstv E m + × ymeet mesto neravenstvo K E j m ij i m m k x dx + × ≤ ≤ ≤ −∞ +∞ = ∫∑max ( ) 1 1 . (2.4) Dejstvytel\no, pust\ f f f fm T= …( , , , )1 2 ∈ E m + × — proyzvol\naq funkcyq. Ymeem K Kf f k x t f t dtE i E i m ij j j m m + × + ≡ = − = +∞ = ∑ ∫( ) ( ) ( ) 1 01 ∑∑∑ = + i m E 1 ≤ ≤ k x t f t dt K f xij j j m E i m ij j E ( ) ( ) ( ) ( )− = +∞ == ∫∑∑ + 011 ++== ∑∑ j m i m 11 ≤ ≤ K fij E j E j m i m + +== ∑∑ 11 . S druhoj storon¥, yzvestno, çto norma skalqrnoho yntehral\noho operatora Vynera – Xopfa v lgbom yz prostranstv E+ udovletvorqet neravenstvu [9 – 11] K k x dxij E ij + ≤ −∞ +∞ ∫ ( ) , i, j = 1, 2, … , m. (2.5) Sledovatel\no, s uçetom (2.5) ymeem K f K fE j m ij E j E j m i m j m m + × + + ≤ ≤ ≤ ≤ == ≤ ≤∑∑max max 1 11 1 kk x dx fij E i m m( ) + × −∞ +∞ = ∫∑ 1 , otkuda, v svog oçered\, sleduet neravenstvo (2.4). Vvedem sledugwye podklass¥ Ω Ω± ⊂ verxnyx y nyΩnyx matryçn¥x yn- tehral\n¥x operatorov typa Vol\terra: V± ±∈ Ω , esly ( ) ( ) ( ) ( )V+ += −∫f x V x t f t dt x 0 , ( ) ( ) ( ) ( )V− − ∞ = −∫f x V t x f t dt x , (2.6) hde V x v xij i j m m± ± = × = ( )( ) ( ) , 1 , v Lij ± ∈ +∞1 0( , ) , f E m∈ + × , i, j = 1, 2, … , m. (2.7) ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1280 A. X. XAÇATRQN, X. A. XAÇATRQN Netrudno ubedyt\sq, çto Ω Ω Ω= ⊕+ − . (2.8) Yzvestno takΩe [11, 12], çto esly K ∈ Ω , V± ±∈ Ω , to V K− ∈ Ω , KV+ ∈ ∈ Ω. Pust\ ΩΛ — sledugwyj rasßyrenn¥j klass matryçn¥x yntehral\n¥x ope- ratorov: KΛ ΛΩ∈ , esly ( ) ( ) ( ) ( ) ( )KΛ Λf x K x t t f t dt= − +∞ ∫ 0 , f E m∈ + × , (2.9) hde K zadaetsq posredstvom formul¥ (2.2), a matryca-funkcyq Λ ymeet vyd Λ = …( )diag λ λ λ1 2( ), ( ), , ( )x x xm , (2.10) 0 1≤ ≤λ j x( ) , j = 1, 2, … , m, — yzmerym¥e funkcyy na ( , )0 +∞ . Operator¥ yz klassa ΩΛ y yx svojstva budut yspol\zovan¥ v dal\nejßyx rassuΩdenyqx. Ubedymsq, çto esly V− −∈ Ω , KΛ ΛΩ∈ , to V K− ∈Λ ΛΩ . Dejstvytel\no, pust\ f = ( , , , )f f fm T 1 2 … ∈ E m + × — proyzvol\naq funkcyq. Ymeem V K− − +∞+∞ ( ) = − −∫Λ Λf x V t x K t y y f y dy dt x ( ) ( ) ( ) ( ) ( ) 0 ∫∫ , , otkuda, menqq porqdok yntehryrovanyq, poluçaem V K− − +∞+∞ ( ) = − −      ∫∫Λ Λf x V t x K t y dt y x ( ) ( ) ( ) ( 0 )) ( )f y dy = = V z K x y z dz y f y dy T x− +∞+∞ − +       =∫∫ ( ) ( ) ( ) ( ) ( 00 Λ −− +∞ ∫ y y f y dy) ( ) ( )Λ 0 , hde T x V z K x z dz( ) ( ) ( )= +− +∞ ∫ 0 . (2.11) Yz teorem¥ Fubyny (sm. [13] ) sleduet, çto funkcyq T x( ) = T xij i j m m ( ) ,( ) = × 1 , T Lij ∈ 1 . Sledovatel\no, V K− ∈Λ ΛΩ . Oçevydno, çto vklgçenye V K− ∈ Ω qvlqetsq neposredstvenn¥m sledstvyem posledneho pry Λ = …diag ( , , , )1 1 1 . Çto kasaetsq K VΛ ΛΩ+ ∈ , to πto ne vsehda verno. 3. Vspomohatel\n¥e fakt¥. Dlq dal\nejßeho yzloΩenyq neobxodym¥ sledugwye fakt¥. ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 O NEKOTORÁX SYSTEMAX YNTEHRO-DYFFERENCYAL\|NÁX URAVNENYJ … 1281 Lemma 3.1. 1. Pust\ f Wp n∈ +∞−1 0( , ) , p ≥ 1, n , p ∈ N , — vewestvennaq funkcyq na ( , )0 +∞ . Tohda pry lgbom α > 0 funkcyq F x( ) = x t xe +∞ − −∫ α( ) × × f t dt( ) , x > 0, ymeet sledugwee svojstvo hladkosty: F ∈ Wp n ( , )0 +∞ . 2. Esly f Wp n∈ −∞ +∞−1( , ) , p ≥ 1, p, n ∈ N , to pry lgbom α > 0 Φ( )x = = e f t dtt x x − −+∞ ∫ α( ) ( ) ∈ Wp n ( , )−∞ +∞ . Dokazatel\stvo. Po yndukcyy moΩno lehko proveryt\ spravedlyvost\ formul¥ F x f x f x f xk k k k( ) ( ) ( )( ) ( ) ( ) ( )= − − − … −− − −1 2 1α α + + α αk t x x e f t dt− − +∞ ∫ ( ) ( ) , k = 0, 1, 2, … , n. (3.1) 1. Pust\ f Wp n∈ +∞−1 0( , ) , tohda dlq zaverßenyq dokazatel\stva p.81 dos- tatoçno pokazat\, çto F Lp∈ +∞( , )0 , p ≥ 1. Snaçala rassmotrym sluçaj p > 1. Dlq proyzvol\noho δ > 0 rassmotrym yntehral I e f t dt dxt x x p δ α δ = − − +∞ ∫∫ ( ) ( ) 0 . Prymenqq neravenstvo Hel\dera, ymeem I e f t dt dxt x x p δ α δ ≤      − − +∞ ∫∫ ( ) ( ) 0 = e f t e dt dxt x p t x q x p − − − − +∞ ∫∫      α α δ ( )/ ( )/( ) 0 ≤ ≤ e f t dt e dtt x p x p t x x − − +∞ − − +∞ ∫ ∫        α α( ) / ( )( ) 1             ∫ 1 0 /q p dx δ = = e f t dt dxt x p x p − − +∞ − ∫∫     α δ α ( ) ( ) 0 11 , p, q > 1, 1 1 1 p q + = . Menqq porqdok yntehryrovanyq v poslednem yntehrale, poluçaem I e f t dt dx e f t p t x p t x δ α α α ≤     + − − − − −1 1 ( ) ( )( ) ( )) p x dt dx δ δδδ ∞ ∫∫∫∫       00 = = 1 1 α α α α α    + − − − p p t x p t xf t e e dx dt f t e e dx d( ) ( ) tt t 000 δ δ δ ∫∫∫∫ ∞      ≤ ≤ 1 1 11 0α α α α αδ δ δ     + − − ∞ ∫∫ p p p tf t dt f t e e dt( ) ( )       ≤ ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1282 A. X. XAÇATRQN, X. A. XAÇATRQN ≤ 1 1 0α αδ δ     +       =    ∞ ∫∫ p p pf t dt f t dt( ) ( )  < +∞ ∞ ∫ p pf t dt( ) 0 , otkuda v sylu proyzvol\nosty çysla δ > 0 zaklgçaem, çto F prynadleΩyt Lp ( , )0 +∞ . Pust\ teper\ p = 1. Tohda, yspol\zuq ravenstvo (3.1) y teoremu Fu- byny, moΩno takΩe dokazat\, çto F prynadleΩyt Lp ( , )0 +∞ , k = 0, 1, 2, … … , n. Spravedlyvost\ p.82 dokaz¥vaetsq analohyçno. Lemma dokazana. 4. Zadaça faktoryzacyy. 4.1. Postanovka zadaçy. Pust\ vse πlement¥ matryc¥ A A A Am= …diag ( , , , )1 2 poloΩytel\n¥, t.8e. Ai > 0 , i = 1, 2, … , m . Uravnenye (1.1) zapyßem v operatornoj forme ( )D J K− + + =A gΛ ϕ 0 , (4.1) hde D — matryçn¥j dyfferencyal\n¥j operator vyda D = diag ( ,D D , … … , D) , ( ) ( )D x d dx ρ ρ = , J — edynyçn¥j matryçn¥j operator porqdka m × m , a KΛ zadaetsq posredstvom formul (2.1) – (2.3). Rassmotrym sledugwug zadaçu faktoryzacyy: dlq kaΩdoj dyahonal\noj matryc¥ α = diag ( , , , )α α α1 2 … m , α j > 0 , j = 1, 2, … , m, y zadann¥x opera- torov AJ y KΛ ΛΩ∈ najty takye operator¥ U ∈ −Ω , WΛ ΛΩ∈ , çtob¥ ymelo mesto razloΩenye D J K D J J U J W− + = − − −A Λ Λ( ) ( ) ( )α . (4.2) Faktoryzacyg (4.2) budem ponymat\ kak ravenstvo yntehral\n¥x operatorov, dejstvugwyx v W W W H m 1 1 1 1 1 1 1 10 0 0 0( , ) ( , ) ( , ) ( ,,+∞ × +∞ × … × +∞ ≡ +∞× )) . (4.3) 4.2. Osnovnaq faktoryzacyonnaq teorema. Osnovn¥m rezul\tatom na- stoqwej rabot¥ qvlqetsq sledugwaq faktoryzacyonnaq teorema. Teorema 4.1. Pust\ KΛ ΛΩ∈ , A � 0 — dyahonal\naq matryca s polo- Ωytel\n¥my πlementamy. Tohda dlq kaΩdoho α = diag ( , , , )α α α1 2 … m , α j > > 0, j = 1, 2, … , m, ymeet mesto faktoryzacyq (4.2), hde U ∈ −Ω , WΛ ∈ ∈ ΩΛ — operator¥, qdra kotor¥x zadagtsq sootvetstvenno po formulam U x u x u x u xm( ) ( ), ( ), , ( )= …( )diag 1 2 , u x A e xj j j A xj( ) ( ) ( )= − −α θ , (4.4) θ( ) , , , , x x x = ≥ <     1 0 0 0 esly esly W x t W x t t W x w xij i j m mΛ Λ( , ) ( ) ( ), ( ) ( ) , = − = ( ) = × 1 , (4.5) w x e k x t dt Wij A t ij i( ) ( ) ( , )= + ∈ −∞ +∞− +∞ ∫ 1 1 0 , i, j = 1, 2, … , m, ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 O NEKOTORÁX SYSTEMAX YNTEHRO-DYFFERENCYAL\|NÁX URAVNENYJ … 1283 a Λ( )t — dyahonal\naq matryca, zadavaemaq posredstvom (2.10). Krome to- ho, ymegt mesto sledugwye apryorn¥e ocenky: w x dx A ij i m i j i m ( ) max≤    ≤ ≤ −∞ +∞ = ∫∑ 11 1 µ , (4.6) dw dx dx ij i m j −∞ +∞ = ∫∑ ≤ 1 2µ , j = 1, 2, … , m. (4.7) Dokazatel\stvo. Pust\ Yα — obratn¥j operator matryçnoho dyffe- rencyal\noho operatora αJ D− v prostranstve H m 1 1 0, ( , )× +∞ . Zametym, çto Yα ∈ −Ω y ymeet vyd Yα α α α= …diag ( , , , )Y Y Y m1 2 , ( ) ( ) ( )( )Y x e t dtj j t x x α αρ ρ= − − +∞ ∫ , (4.8) α j > 0 , ρ ∈ +E , j = 1, 2, … , m. Oboznaçym çerez PΛ proyzvedenye operatorov Yα y K P Y KΛ Λ Λ: = α . Yz rezul\tatov p.82 sleduet, çto PΛ ΛΩ∈ , qdro kotoroho ymeet vyd P x t P x t tΛ Λ( , ) ( ) ( )= − , P x p xij i j m m ( ) ( ) , = ( ) = × 1 , (4.9) p x e k t x dtij t ij i( ) ( )= +− +∞ ∫ α 0 , i, j = 1, 2, … , m. PokaΩem, çto funkcyy pij udovletvorqgt dvum ocenkam p x dxij i m i j i m ( ) max≤    ≤ ≤ −∞ +∞ = ∫∑ 11 1 α µ , (4.10) dp dx dx ij j i m ≤ −∞ +∞ = ∫∑ 2 1 µ , j = 1, 2, … , m. (4.11) Ymeem p x dx e k d dxij x ij x i( ) ( )( )= − − +∞ −∞ +∞ −∞ +∞ ∫∫∫ α τ τ τ , hde, yzmenqq porqdok yntehryrovanyq, s yspol\zovanyem teorem¥ Fubyny y s uçetom (1.2) poluçaem p x dx aij i ij( ) = −∞ +∞ ∫ 1 α , otkuda y sleduet neravenstvo (4.10). ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1284 A. X. XAÇATRQN, X. A. XAÇATRQN DokaΩem neravenstvo (4.11). Yz (4.9) vydno, çto dp dx k x e k t dt ij ij i t x ij x i= − + − − +∞ ∫( ) ( )( )α α , otkuda ymeem dp dx dx a ij ij −∞ +∞ ∫ ≤ 2 , i, j = 1, 2, … , m. (4.12) Oçevydno, çto yz (4.12) sleduet neravenstvo (4.11). Operator D J K− +A Λ predstavym v vyde D J K D J J J K− + = − + − +A AΛ Λα α . S uçetom (4.8), (4.9) yz posledneho ravenstva naxodym D J K D J J P Y− + = − − − −( )A AΛ Λ( ) ( )α α α . (4.13) Oboznaçym çerez J R+ α rezol\ventu matryçnoho operatora J Y− −( )α αA v prostranstve H m 1 1 0, ( , )× +∞ . Tohda netrudno ubedyt\sq, çto Rα ∈ −Ω y Rα = diag ( , , , )R R R mα α α1 2 … , hde ( ) ( ) ( ) ( )( )R x A e t dti i i i A t x x α ρ α ρ= − − − +∞ ∫ , ρ ∈ +E , i = 1, 2, … , m. (4.14) Yspol\zuq (2.11), zaklgçaem, çto R Pα Λ ΛΩ∈ . Sledovatel\no, W P R PΛ Λ Λ ΛΩ≡ + ∈α . (4.15) Prost¥my v¥çyslenyqmy moΩno ubedyt\sq, çto qdro operatora WΛ zadaetsq posredstvom (4.5). Takym obrazom, yz (4.13) s uçetom (4.14), (4.15) poluçaem faktoryzacyg (4.2). Netrudno ubedyt\sq, çto matryçn¥j operator WΛ dej- stvuet v prostranstve H m 1 1 0, ( , )× +∞ . Ocenky (4.6) y (4.7) dokaz¥vagtsq, kak y (4.10), (4.11). Teorema dokazana. 4.3. Svqz\ meΩdu perv¥my momentamy qder K y W . Dlq dal\nejßeho yzloΩenyq nam suwestvenno ponadobqtsq suwestvovanye y znak pervoho momen- ta matryçnoho qdra W (v komponentnom sm¥sle). S πtoj cel\g v sledugwej lemme m¥ ustanovym svqz\ meΩdu perv¥my momentamy matryçn¥x qder K y W. Lemma 4.1. Esly A � 0 y suwestvuet ν( ) ( )K tK t dt≡ −∞ +∞ ∫ < + ∞, to su- westvuet y ν( )W , pryçem dlq yx πlementov ymegt mesto sootnoßenyq ν ν ( ) ( ) w k A a A ij ij i ij i = − 2 , i, j = 1, 2, … , m. ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 O NEKOTORÁX SYSTEMAX YNTEHRO-DYFFERENCYAL\|NÁX URAVNENYJ … 1285 Dokazatel\stvo. Yz (4.5) na osnovanyy teorem¥ Fubyny ymeem ν( ) ( ) ( )w xw x dx x e k x t dt dxij ij A t ij i= = +− +∞ −∞ +∞ ∫ 0 ∫∫∫ −∞ +∞ = = e xk x t dx dtA t ij i− −∞ +∞+∞ +∫∫ ( ) 0 , otkuda, oboznaçaq τ = x + t, okonçatel\no poluçaem ν τ τ τ ν ( ) ( ) ( ) ( ) w e t k d dt k A kij A t ij ij i i i= − = −− +∞ ∫ 0 jj A td te dti( )τ τ −∞ +∞ −∞ +∞ − +∞ ∫∫ ∫ 0 = = ν( )k A a A ij i ij i − 2 , i, j = 1, 2, … , m. Lemma dokazana. 5. Razreßymost\ uravnenyj (1.1) – (1.3). V πtom punkte budem rassmatry- vat\ vopros¥ razreßymosty system¥ (1.1). Pry πtom vospol\zuemsq faktoryza- cyej (4.2). Vvedem summ¥ ρ j ij ii m a A = = ∑ 1 , j = 1, 2, … , m, (5.1) y oboznaçym ρ ρ= ≤ ≤ max 1 j m j . (5.2) Esly ρ < 1, to zadaçu (1.1) – (1.3) uslovno nazovem dyssypatyvnoj, esly ρ = 1 — konservatyvnoj, a esly ρ > 1 — krytyçeskoj. Rassmotrym πty sluçay ot- del\no. 5.1. Dyssypatyvnaq zadaça (1.1) – (1.3). Pust\ çyslo ρ, zadavaemoe po- sredstvom (5.1), (5.2), men\ße edynyc¥. Tohda, yspol\zuq faktoryzacyg (4.2), uravnenye (1.1) zapys¥vaem v vyde ( ) ( ) ( )D J J U J W− − − + =α ϕΛ g 0 . (5.3) Reßenye system¥ (5.3) svodytsq k posledovatel\nomu reßenyg sledugwyx svq- zann¥x system uravnenyj: ( )D J− = −α ψ g , (5.4) ( )J U− =χ ψ , (5.5) ( )J W− =Λ ϕ χ . (5.6) Snaçala rassmotrym uravnenye (5.4). Zapyßem eho v vyde d dx g x xi i i i ψ α ψ+ =( ) ( ) , i = 1, 2, … , m. (5.7) ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1286 A. X. XAÇATRQN, X. A. XAÇATRQN S uçetom lemm¥ 3.1 netrudno ubedyt\sq, çto dyahonal\naq systema dyfferen- cyal\n¥x uravnenyj (5.7) v prostranstve H m 1 1 0, ( , )× +∞ ymeet edynstvennoe re- ßenye vyda ψ ψ ψ ψ( ) ( ), ( ), , ( )x x x xm T= …( )1 2 , ψ α j t x j x e g t dtj= − − +∞ ∫ ( ) ( ) , (5.8) j = 1, 2, … , m. Rassmotrym teper\ systemu uravnenyj (5.5): χ ψ α χα i i i i t x i x x x A e t dti( ) ( ) ( ) ( )( )= + − − − +∞ ∫ , i = 1, 2, … , m. (5.9) Yspol\zuq lemmu 3.1, prost¥my v¥çyslenyqmy moΩno ubedyt\sq, çto uravnenye (5.9) v H m 1 1 0, ( , )× +∞ ymeet edynstvennoe reßenye, zadavaemoe posredstvom for- mul¥ 0 ≤ = − − +∞ ∫χi A t x i x x e g t dti( ) ( )( ) , i = 1, 2, … , m. (5.10) Perejdem k rassmotrenyg system¥ (5.6): ϕ χ ω λ ϕi i ij j j j m x x x t t t dt( ) ( ) ( ) ( ) ( )= + − +∞ = ∫∑ 01 , i = 1, 2, … , m, (5.11) yly v matryçnoj forme ϕ χ ϕ( ) ( ) ( ) ( ) ( )x x W x t t t dt= + − +∞ ∫ Λ 0 , (5.12) hde ϕ( )x = ϕ ϕ ϕ1 2( ), ( ), , ( )x x xm T…( ) , χ( )x = χ χ χ1 2( ), ( ), , ( )x x xm T…( ) , a mat- ryc¥-funkcyy W y Λ zadagtsq posredstvom formul (4.5) y (2.10) sootvet- stvenno. Dlq πtoj system¥ rassmotrym sledugwye yteracyy: ϕ χ ϕ( ) ( )( ) ( ) ( ) ( ) ( )n nx x W x t t t dt+ +∞ = + −∫1 0 Λ , (5.13) hde ϕ( )k = ϕ ϕ ϕ1 2 ( ) ( ) ( )( ), ( ), , ( )k k m k T x x x…( ) , k = 0, 1, 2, … , ϕ( )0 = ( , , , )0 0 0… T . Zametym, çto posledovatel\nost\ vektor-funkcyj ϕ( )n = ϕ1 ( )( )n x( , ϕ2 ( )( )n x , … , ϕm n T x( )( )) monotonno vozrastaet po n n: ( )ϕ +1 � ϕ( )n , t.88e. ϕ j n x( )( )+1 ≥ ϕ j n x( )( ) , j = 1, 2, … , m, n = 0, 1, 2, … . ∏tot prostoj fakt doka- Ωem8po yndukcyy. Pust\ n = 0, tohda yz (5.10) vydno, çto ϕ( )( )1 x = χ( )x � ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 O NEKOTORÁX SYSTEMAX YNTEHRO-DYFFERENCYAL\|NÁX URAVNENYJ … 1287 � ϕ( )( )0 x . PredpoloΩym teper\, çto pry n = k utverΩdenye verno, y dokaΩem eho pry n = k + 1. Ymeem ϕ χ ϕ( ) ( )( ) ( ) ( ) ( ) ( )k kx x W x t t t dt+ +∞ += + −∫2 0 1Λ . (5.14) Poskol\ku qdro W x tΛ ( , ) predstavlqet soboj neotrycatel\nug matrycu- funkcyg, yz (5.14) okonçatel\no poluçaem ϕ( ) ( )k x+2 � χ ϕ ϕ( ) ( ) ( ) ( ) ( )( ) ( )x W x t t t dt xk k+ − = +∞ +∫ Λ 0 1 . (5.15) Yspol\zuq (1.2), (5.13), analohyçn¥m obrazom moΩno dokazat\ sledugwee: χ ϕ( )n , n = 1, 2, … ; ϕ( ) , ( , )n mH∈ +∞× 1 1 0 , n = 0, 1, 2, … . (5.16) V sylu (5.16) s uçetom monotonnosty ϕ( )n = ϕ1 ( )( )n x( , ϕ2 ( )( )n x ,…, ϕm n T x( )( )) yz (5.13) poluçym ϕ χ λ ϕ( ) ( , ) ( , ) ( ) ( )n L L ij jm m w x t t+ +∞ +∞× ×= + −1 0 0 1 1 jj n j m i m t dt dx( )( ) 0101 +∞ = +∞ = ∫∑∫∑ ≤ ≤ χ ρ ϕL n L m m 1 1 0 1 0 × ×+∞ + +∞ +( , ) ( ) ( , ) , (5.17) hde çerez f L m 1 0× +∞( , ) oboznaçena norma f fL i L i m m 1 10 0 1 × +∞ +∞ = = ∑( , ) ( , ) , f = ( , , , )f f fm T 1 2 … . Sledovatel\no, yz (5.17) ymeem ϕ( ) ( , ) n L m + +∞× 1 01 ≤ χ ρ L m 1 0 1 × +∞ − ( , ) . Takym obra- zom, yspol\zuq teoremu B. Levy o monotonnoj sxodymosty (sm. [14]), zaklgçaem, çto posledovatel\nost\ vektor-funkcyj ϕ( )n = ϕ1 ( )n( , ϕ2 ( )n ,…, ϕm n T( )) , n = 0, 1, 2, … , poçty vsgdu v ( , )0 +∞ ymeet predel ϕ( )x = ϕ ϕ1 2( ), ( )x x( , … … , ϕm Tx( )) ∈ L m 1 0× +∞( , ) . DokaΩem, çto predel\naq vektor-funkcyq qvlqetsq reßenyem syste- m¥8(5.11). Yz (5.13) sleduet, çto ϕ χ λ ϕi n i ij j j j x x w x t t t dt( ) ( ) ( ) ( ) ( ) ( )+ +∞ = ≤ + −∫1 011 m ∑ , i = 1, 2, … , m, otkuda, perexodq k predelu pry n → ∞, poluçaem ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1288 A. X. XAÇATRQN, X. A. XAÇATRQN ϕ χ λ ϕi i ij j j j m x x w x t t t dt( ) ( ) ( ) ( ) ( )≤ + − +∞ = ∫∑ 01 , i = 1, 2, … , m. (5.18) S druhoj storon¥, χ λ ϕ ϕi ij j j n j m ix w x t t t dt x( ) ( ) ( ) ( ) (( )+ − ≤ +∞ = ∫∑ 01 )) , i = 1, 2, … , m, otkuda v sylu teorem¥ Lebeha (sm. [14]) ymeem χ λ ϕ ϕi ij j j j m ix w x t t t dt x( ) ( ) ( ) ( ) ( )+ − ≤ +∞ = ∫∑ 01 , i = 1, 2, … , m. (5.19) Takym obrazom, yz neravenstv (5.18) y (5.19) sleduet, çto ϕ( )x = ϕ1( )x( , ϕ2( )x , … , ϕm Tx( )) qvlqetsq reßenyem system¥ (5.11). Teper\ ubedymsq, çto ϕ ∈ +∞×H m 1 1 0, ( , ) . Poskol\ku funkcyy ki j prynadleΩat L R M R1( ) ( )∩ , i, j = = 1, … , m, yspol\zuq ravenstva dw dx k x A e k t dt ij ij i A t x ij x i= − + − − +∞ ∫( ) ( )( ) , i, j = 1, 2, … , m, s uçetom yzvestnoj teorem¥ y dyfferencyruemosty pod znakom yntehrala (sm. [15]) poluçaem d dx d dx w x t x t t di i ij j m j j ϕ χ λ ϕ= + ∂ − ∂ +∞ = ∫∑ ( ) ( ) ( ) 01 tt , i = 1, 2, … , m. (5.20) Dalee, tak kak χi W∈ +∞1 1 0( , ) , i = 1, 2, … , m, s uçetom (4.5) yz (5.20) poluçaem ϕ ∈ +∞×H m 1 1 0, ( , ) . Tem sam¥m dokazana sledugwaq teorema. Teorema 5.1. PredpoloΩym, çto 0 g ∈ L m 1 0× +∞( , ) , 0 ≤ λ j x( ) ≤ 1, j = 1, 2, … , m, — yzmerym¥e funkcyy. Tohda esly ρ < 1, to zadaça (1.1) – (1.3) v prostranstve H n m 1 0, ( , )× +∞ ymeet poloΩytel\noe reßenye. Yspol\zuq lemmu 3.1, poluçaem takoe sledstvye. Sledstvye. PredpoloΩym, çto 0 g ∈ L p m× +∞( , )0 , 0 ≤ λ j x( ) ≤ 1, j = 1, 2, … , m, — yzmerym¥e funkcyy, k W Rij p n∈ −1( ) ∩ W Rn ∞ −1( ) , i, j = 1, 2, … , m, y udovletvorqgt uslovyg (1.2). Tohda esly ρ < 1, to zadaça (1.1) – (1.3) v prostranstve H p n m , ( , )× +∞0 ymeet poloΩytel\noe reßenye, hde H p n m , ( , )× +∞0 ≡ ≡ Wp n ( , )0 +∞ × Wp n ( , )0 +∞ × … × Wp n ( , )0 +∞ . 5.2. Konservatyvnaq zadaça (1.1) – (1.3). Ymeet mesto sledugwaq te- orema. Teorema 5.2. PredpoloΩym, çto 0 g ∈ L m 1 0× +∞( , ) , 0 ≤ λ j x( ) ≤ 1, j = 1, 2, … , m, — yzmerym¥e funkcyy na ( , )0 +∞ . Tohda v sluçae ρ = 1, esly ν( )ki j ≤ ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 O NEKOTORÁX SYSTEMAX YNTEHRO-DYFFERENCYAL\|NÁX URAVNENYJ … 1289 ≤ a A i j i , i, j = 1, 2, … , m, zadaça (1.1) – (1.3) v klasse M ymeet pokomponentno poloΩytel\noe reßenye s asymptotykoj ϕ( ) ( )t dt o S t dt xx =      ∫∫ 00 , x → + ∞, hde S x( ) — poloΩytel\naq pokomponentno vozrastagwaq vektor-funkcyq, dlq kotoroj S( )0 = ( , , , )1 1 1… T , pryçem esly ν( )ki j = a A i j i , i, j = 1, 2, … , m, to S x( ) = O x( ) , x → + ∞, a esly ν( )ki j < a A i j i , i, j = 1, 2, … , m, to S x( ) = = O( )1 , x → + ∞. Dokazatel\stvo. Yspol\zuq faktoryzacyg (4.2) y posledovatel\no reßaq uravnenyq (5.4) y (5.5) (kak v dokazatel\stve teorem¥ 5.1), poluçaem systemu uravnenyj (5.11). Zametym, çto v πtom sluçae ρ = max ( ) 1 1≤ ≤ −∞ +∞ = ∫∑ j m iji m w x dx = 1. Narqdu s urav- nenyem (5.11) rassmotrym vspomohatel\noe matryçnoe uravnenye F x x w x t F t dti i ij j j m ( ) ( ) ( ) ( )= + − +∞ = ∫∑χ 01 , i = 1, 2, … , m, yly v matryçnoj forme F x x W x t F t dt( ) ( ) ( ) ( )= + − +∞ ∫χ 0 . (5.21) V rabotax [10, 16] dokazano, çto esly suwestvuet ν( )W 0, to uravnenye8(5.21) ymeet poloΩytel\noe lokal\no yntehryruemoe reßenye, y yteracyy F x x W x t F t dtn n( ) ( )( ) ( ) ( ) ( )+ +∞ = + −∫1 0 χ , (5.22) F T( ) ( , , , )0 0 0 0= … , n = 0, 1, 2, … , poçty vsgdu v ( , )0 +∞ sxodqtsq k mynymal\nomu (pokomponentno) neotryca- tel\nomu reßenyg uravnenyq (5.21), pryçem F t dt o S t dt xx ( ) ( )=      ∫∫ 00 , x → + ∞ (o svojstvax funkcyy S sm. v formulyrovke teorem¥ 5.2). Sravnym yteracyy (5.13) s yteracyqmy (5.22). S uçetom χ � 0 poluçaem χ ϕ( )n ↑ po n y ϕ( )n F n( ) F poçty vsgdu v ( , )0 +∞ . Sledovatel\no, suwestvuet poçty vsgdu v ( , )0 +∞ predel ϕ( )x = lim ( )( ) n n x →∞ ϕ , pryçem 0 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1290 A. X. XAÇATRQN, X. A. XAÇATRQN ϕ( )x F x( ) . Netrudno ubedyt\sq, çto predel\naq vektor-funkcyq ϕ qv- lqetsq reßenyem uravnenyq (5.11), pryçem ϕ( ) ( )t dt o S t dt xx =      ∫∫ 00 , x → + ∞. (5.23) Analohyçn¥m obrazom proverqetsq, çto ϕ — absolgtno neprer¥vnaq funkcyq. Yz yntehral\noj asymptotyky (5.23) sleduet, çto ϕ = ( , , , )ϕ ϕ ϕ1 2 … m T ∈ M. Teorema dokazana. 5.3. Krytyçeskaq zadaça (1.1) – (1.3). Ymeet mesto sledugwaq teorema. Teorema 5.3. PredpoloΩym, çto v¥polnen¥ sledugwye uslovyq: 1) 0 g ∈ L m 1 0× +∞( , ) , 2) 0 ≤ λ j x( ) ≤ 1, λ j W∈ +∞∞ 1 0( , ) , pryçem c = max 1 1≤ ≤ =∑ j m iji m c < 1, hde c t w dij t i ij t = + ∈ +∞ − +∞ ∫vrai max ( ) ( ) ( , )0 λ τ τ τ , i, j = 1, 2, … , m. Tohda zadaça (1.1) – (1.3) v sluçae ρ > 1 v prostranstve Soboleva H m 1 1 0, , ( , )Λ × +∞ s vesom Λ( )x ymeet poloΩytel\noe reßenye. Zdes\ H m 1 1 0, , ( , )Λ × +∞ = W1 1 1 0, ( , )λ +∞ × … × W m1 1 0, ( , )λ +∞ , hde W j1 1 0, ( , )λ +∞ — pros - transtvo Soboleva s vesom λ j , j = 1, 2, … , m. Dokazatel\stvo. Zdes\ takΩe snaçala reßaem uravnenyq (5.4) y (5.5), zatem perexodym k rassmotrenyg uravnenyq (5.11). UmnoΩym sleva obe ças- ty8πtoj system¥ na matrycu-funkcyg Λ( )x . Tohda, oboznaçaq ξi x( ) = = λ ϕi ix x( ) ( ) , q xi ( ) = λ χi ix x( ) ( ) , i = 1, 2, … , m, poluçaem sledugwug systemu yntehral\n¥x uravnenyj otnosytel\no yskomoj vektor-funkcyy ξ = (ξ1 , ξ2 , … , ξm T) : ξ λ ξi i i ij j j m x q x x w x t t dt( ) ( ) ( ) ( ) ( )= + − +∞ = ∫∑ 01 , i = 1, 2, … , m. yly v matryçnoj forme ξ ξ( ) ( ) ( ) ( ) ( )x q x x W x t t dt= + − +∞ ∫Λ 0 . (5.24) Rassmotrym yteracyy ξ ξ( ) ( )( ) ( ) ( ) ( ) ( )n nx q x x W x t t dt+ +∞ = + −∫1 0 Λ , ξ( ) ( , , , )0 0 0 0= … T , n = 0, 1, 2, … . Lehko proveryt\, çto q ξ( )n ↑ po n (pokomponentno) y ξ( )n ∈ L m 1 0× +∞( , ) , n = 0, 1, 2, … . Sledovatel\no, s uçetom yzloΩennoho ymeem ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 O NEKOTORÁX SYSTEMAX YNTEHRO-DYFFERENCYAL\|NÁX URAVNENYJ … 1291 ξ( ) ( , ) ( , ) n L Lm mq+ +∞ +∞× ×≤1 0 0 1 1 8+ + λ ξi ij j n j m i x w x t t dt dx( ) ( ) ( )( )− + +∞ = +∞ = ∫∑∫ 1 0101 mm ∑ = = q x w x t t dt dxL i ij j n m 1 0 1 0 × +∞ + +∞ + −∫( , ) ( )( ) ( ) ( )λ ξ 0011 +∞ == ∫∑∑ j m i m , otkuda, yzmenqq porqdok yntehryrovanyq, s uçetom teorem¥ Fubyny poluçaem ξ( ) ( , ) ( , ) n L Lm mq+ +∞ +∞× ×≤1 0 0 1 1 8+ + ξ ξ( ) ( , ) ( , ) (n L ij L i m n m mc q c+ +∞ +∞ = + × ×≤ +∑1 0 0 1 1 1 1 )) ( , )L m 1 0× +∞ . Tohda ξ( ) ( , ) ( , ) n L Lm m c q+ +∞ +∞× ×≤ − 1 0 0 1 1 1 1 . Takym obrazom, yz poluçennoho s uçetom teorem¥ B. Levy sleduet, çto po- sledovatel\nost\ vektor-funkcyj ξ( )n = ξ1 ( )n( , ξ2 ( )n ,…, ξm n T( )) , n = 0, 1, 2, … , poçty vsgdu v ( , )0 +∞ sxodytsq k summyruemoj funkcyy ξ = (ξ1 , ξ2 , … … , ξm T) . Kak v dokazatel\stve teorem¥ 5.1, zdes\ proverqetsq, çto ξ qvlqet- sq reßenyem uravnenyq (5.24) y ξ ∈ +∞×H m 1 1 0, ( , ) . Poskol\ku 0 ≤ λ j x( ) ≤ 1, λ j x( ) ∈ W∞ +∞1 0( , ) , j = 1, 2, … , m , netrudno ubedyt\sq, çto ϕ ∈ ∈ H m 1 1 0, , ( , )Λ × +∞ . Teorema dokazana. 1. Lyfßyc E. M., Pytaevskyj L. M. Fyzyçeskaq kynetyka. – M.: Nauka, 1979. – T. 10. 2. Sargan J. D. The distribution of wealth // Econometrica. – 1957. – # 25. – P. 568 – 590. 3. Xaçatrqn X. A. O nekotor¥x yntehro-dyfferencyal\n¥x uravnenyqx, voznykagwyx v fy- zyçeskoj kynetyke // Yzv. NAN Respublyky Armenyq. Matematyka. – 2004. – 39, # 3. – S. 72 – 80. 4. Lat¥ßev A. V., Gßkanov A. A. ∏lektronnaq plazma v polubeskoneçnom metalle pry nalyçyy peremennoho πlektryçeskoho polq // Ûurn. v¥çyslyt. matematyky y mat. fyzyky. – 2001. – 41, # 8. – S. 1229 – 1241. 5. Xaçatrqn A. X., Xaçatrqn X. A. O razreßymosty odnoj kraevoj zadaçy fyzyçeskoj kyne- tyky // Yzv. NAN Respublyky Armenyq. Matematyka. – 2004. – 41, # 6. – S. 65 – 74. 6. Khachatryan A. Kh., Khachatryan Kh. A. On solvability of some integral-differential equation with sum-difference kernels // Int. J. Pure and Appl. Math. (India). – 2005. – 2, # 1. – P. 1 – 13. 7. Khachatryan A. Kh., Khachatryan Kh. A. On structure of solution of one integral-differential equation with completely monotonic kernel // Int. Conf. ”Harmonic Analysis and Approximations.” – 2005. – P. 42, 43. 8. Xaçatrqn A. X., Xaçatrqn X. A. K voprosu razreßymosty odnoho yntehro-dyfferencyal\- noho uravnenyq s poçty summarno-raznostn¥m qdrom // Matematyka v v¥sßej ßkole. – 2006. – 2, # 4. – S. 26 – 31. ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9 1292 A. X. XAÇATRQN, X. A. XAÇATRQN 9. Wiener N., Hopf N. Über eine Klasse singularer Integral eichungen Sitzing. – Berlin, 1931. – S. 696 – 706. 10. ArabadΩqn L. H., Enhybarqn N. B. Uravnenyq v svertkax y nelynejn¥e funkcyonal\n¥e uravnenyq // Ytohy nauky y texnyky. Mat. analyz. – 1984. – 22. – S. 175 – 242. 11. Enhybarqn N. B., Arutgnqn A. A. Yntehral\n¥e uravnenyq na poluprqmoj s raznostn¥my qdramy y nelynejn¥e funkcyonal\n¥e uravnenyq // Mat. sb. – 1975. – 97. – S. 35 – 58. 12. Enhybarqn N. B., ArabadΩqn L. H. O nekotor¥x zadaçax faktoryzacyy dlq yntehral\n¥x operatorov typa svertky // Dyfferenc. uravnenyq. – 1990. – 26, # 8. – S. 1442 – 1452. 13. Kolmohorov A. N., Fomyn V. S. ∏lement¥ teoryy funkcyj y funkcyonal\noho analyza. – M.: Nauka, 1981. 14. Natanson Y. P. Teoryq funkcyj vewestvennoj peremennoj. – M.: Nauka, 1974. – 480 s. 15. Fyxtenhol\c H. M. Kurs dyfferencyal\noho y yntehral\noho ysçyslenyq: V Z t. – M.: Nauka, 1966. – T. 2. 16. ArabadΩqn L. H. Ob odnom yntehral\nom uravnenyy teoryy perenosa v neodnorodnoj srede // Dyfferenc. uravnenyq. – 1987. – 23, # 9. – S. 1618 – 1622. Poluçeno 11.03.08 ISSN 1027-3190. Ukr. mat. Ωurn., 2009, t. 61, # 9
id	umjimathkievua-article-3099
institution	Ukrains’kyi Matematychnyi Zhurnal
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language	rus English
last_indexed	2026-03-24T02:36:13Z
publishDate	2009
publisher	Institute of Mathematics, NAS of Ukraine
record_format	ojs
resource_txt_mv	umjimathkievua/c9/597c13ee9a5dce186fcbbd5a56139ac9.pdf
spelling	umjimathkievua-article-30992020-03-18T19:45:12Z On some systems of convolution-type first-order integrodifferential equations on the semiaxis О некоторых системах интегро-дифференциальных уравнений первого порядка типа свертки на полуоси Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. Хачатрян, А. Х. Хачатрян, Х. А. We study a class of vector convolution-type integrodifferential equations on the semiaxis used for the description of various applied problems of mathematical physics. By using a special three-factor decomposition of the original mathematical integrodifferential operator, we prove the solvability of these equations in certain functional spaces. Досліджено один клас векторних інтегро-диференціальних рівнянь типу згортки на півосі, якими описується низка прикладних задач математичної фізики. З допомогою спеціального три-факторного розкладу вихідного математичного інтегро-диференціального оператора доведено розв'язність цих рівнянь у конкретних функціональних просторах. Institute of Mathematics, NAS of Ukraine 2009-09-25 Article Article application/pdf https://umj.imath.kiev.ua/index.php/umj/article/view/3099 Ukrains’kyi Matematychnyi Zhurnal; Vol. 61 No. 9 (2009); 1277-1292 Український математичний журнал; Том 61 № 9 (2009); 1277-1292 1027-3190 rus en https://umj.imath.kiev.ua/index.php/umj/article/view/3099/2946 https://umj.imath.kiev.ua/index.php/umj/article/view/3099/2947 Copyright (c) 2009 Khachatryan A. Kh.; Khachatryan Kh. A.
spellingShingle	Khachatryan, A. Kh. Khachatryan, Kh. A. Хачатрян, А. Х. Хачатрян, Х. А. Хачатрян, А. Х. Хачатрян, Х. А. On some systems of convolution-type first-order integrodifferential equations on the semiaxis
title	On some systems of convolution-type first-order integrodifferential equations on the semiaxis
title_alt	О некоторых системах интегро-дифференциальных уравнений первого порядка типа свертки на полуоси
title_full	On some systems of convolution-type first-order integrodifferential equations on the semiaxis
title_fullStr	On some systems of convolution-type first-order integrodifferential equations on the semiaxis
title_full_unstemmed	On some systems of convolution-type first-order integrodifferential equations on the semiaxis
title_short	On some systems of convolution-type first-order integrodifferential equations on the semiaxis
title_sort	on some systems of convolution-type first-order integrodifferential equations on the semiaxis
url	https://umj.imath.kiev.ua/index.php/umj/article/view/3099
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On some systems of convolution-type first-order integrodifferential equations on the semiaxis

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