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, fi (t) íåïðåðûâíû íà [−T , T ];f (t) Ny1 (t)y10 .. .. y (t) = . , y0 = . .yN (t)yN 0⇒ ∃M, N > 0, òàêèå ÷òî||A(t)|| ≤ M,||f (t)|| ≤ Níà [−T , T ].Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Ïóñòü (1) èìååò íåïðåðûâíîå è íåïðåðûâíî äèôôåðåíöèðóåìîå ðåøåíèå y = y (t). Ïîëàãàÿ 0 ≤ t ≤ T ,ïîëó÷èìdy (t), y (t) = By (t), y (t) + 2Re(f , y ),dtB = A(t) + A∗ (t) = B ∗ .Òàê êàê||B|| =pλmax (B ∗ B) = max{|λmin (B)|, |λmax (B)|}(ñìîòðè óïðàæíåíèå 1 8), ||B|| ≤ 2||A|| (ñìîòðè óïðàæíåíèå 29) è2Re(f , y ) ≤ 2|(f , y )| ≤ 2||f || · ||y || ≤ ||f ||2 + ||y ||2 ⇒d||y (t)||2 ≤ ||B|| · ||y (t)||2 + ||f ||2 + ||y ||2 ≤dt≤ (2M + 1)||y (t)||2 |||f (t)||2 ≤ C1 (M)||y (t)||2 + N 2 .Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Óìíîæàÿ îáå ÷àñòè íà e −C1 t è èíòåãðèðóÿ, ïîëó÷àåì:||y (t)||2 ≤ e C1 t ||y0 ||2 + N 2e C1 t − 1,C1èëè||y (t)||2 ≤ e C1 (M)T ||y0 ||2 + N 2e C1 (M)T − 1, 0 ≤ t ≤ T.C1 (M)(2)Ïðèìåíÿÿ çàìåíó τ = −t (ñì.
2), ïîëó÷èì àíàëîãè÷íóþîöåíêó è ïðè −T ≤ t ≤ 0.Èòàê, ïðè âñåõ t ∈ [−T , T ]||y (t)||2 ≤ e C1 t ||y0 ||2 + N 2≤ e C1 (M)T ||y0 ||2 + N 2e C1 t − 1≤C1e C1 (M)T − 1, 0 ≤ t ≤ T.C1 (M) ñëó÷àå çàäà÷è Êîøè 0y (t) = A(t)y (t) + f (t), t ∈ [−T , T ],y (t0 ) = y 0, t0 ∈ (−T , T )(3)0(1 )Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.îöåíêà (3) ïåðåïèøåòñÿ òàê:||y (t)||2 ≤ e 2C1 (M)T ||y0 ||2 + N 2e 2C1 (M)T − 1,C1 (M)0− T ≤ t ≤ T . (3 )Òàê êàê√0a2 + b 2 ≤ a + b , òî èç (3), (3 ) ⇒s1e C1 (M)·|t| − 1≤||y (t)|| ≤ e 2 C1 (M)·|t| ||y0 || + NC1 (M)≤1e 2 C1 T ||y0 ||s+Nèe C1 T − 1C1s||y (t)|| ≤ e C1 T ||y0 || + N(4)e 2C1 T − 1.C1Ñ ïîìîùüþ ýòèõ îöåíîê ïîëó÷èì òåîðåìó åäèíñòâåííîñòè:Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Ïóñòü åñòü äâà ðåøåíèÿ y I ,II (t) ⇒ 4(t) = y I (t) − y II (t) è04 (t) = A(t)4(t),4(0) = 0.t ∈ [−T , T ],(5) ñèëó (4) äëÿ (5) ïîëó÷àåì:||4(t)|| ≤1e 2 C1 Ts·0+0e C1 T − 1= 0.C1Ðàññìîòðèì òåïåðü çàäà÷ó Êîøè (1) ïðè f (t) = 0.0y (t) = A(t)y (t), t ∈ [−T , T ];y (0) = y 0 ∈ C N (èëè R N ).0(1 )Åñëè ïðåäïîëîæèòü, ÷òî çàäà÷à Êîøè ðàçðåøèìà, òî01) âñå ðåøåíèÿ çàäà÷è Êîøè (1 ) îáðàçóþò ëèíåéíîåïðîñòðàíñòâî;2) åãî ðàçìåðíîñòü ðàâíà N ;Ëèíåéíûé äèôô.
óð - èÿ ñ ïåðåìåííûìè êîýôô.03) ðåøèâ N ðàç çàäà÷ó Êîøè (1 ) äëÿ N ëèíåéíî - íåçàâèñèìûõy0 , ìû ïîñòðîèì N ëèíåéíî - íåçàâèñèìûõ ðåøåíèé çàäà÷è0Êîøè (1 ).Ïóñòü y [k] (t), k = 1,...,N - ñèñòåìà ëèíåéíî - íåçàâèñèìûõ0ðåøåíèé çàäà÷è Êîøè (1 ), ïðè ýòîì:(0y [k] (t) = A(t)y [k] (t), t ∈ [−T , T ],[k]y [k] (0) = y0 ,ãäåy1k (t)y [k] (t) = ...
,y1k 0 (t)[k]y0 (t) = ... .yN k (t)yN k 0 (t)Ñîñòàâèì ìàòðèöó Y (t): Y (t) = yij (t) , i, j = 1,...N .Êàê è â ñëó÷àå ïîñòîÿííûõ êîýôôèöèåíòîâ, ìîæíî óòâåðæäàòü,÷òîdetY (0) = 0 ⇒ detY (t) ≡ 0, t ∈ [−T , T ], èdetY (0) 6= 0 ⇒ detY (t) 6= 0,t ∈ [−T , T ].Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Î÷åâèäíî, ÷òî Y (t) ìîæíî íàçâàòü ôóíäàìåíòàëüíûì0ðåøåíèåì çàäà÷è Êîøè (1 ).0Ëþáîå ðåøåíèå çàäà÷è Êîøè (1 ) âûðàæàåòñÿ ÷åðåç Y (t) òàê:y (t) = Y (t)C =NXCk y [k] (t),k=1 C1 .. ãäå C = .
, C = Y −1 (0)y0 , òî åñòücNy (t) = Y (t) · Y −1 (0)y0 .(6)0Ñàìà ôóíäàìåíòàëüíàÿ ìàòðèöà ðåøåíèé çàäà÷è Êîøè (1 )Y (t) - ðåøåíèå ñëåäóþùåé çàäà÷è Êîøè:0Y (t) = A(t)Y (t), t ∈ [−T , T ],Y (0) = Z , detZ 6= 0.(7)Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Êàê è äëÿ ïîñòîÿííûõ êîýôôèöèåíòîâ ñïðàâåäëèâà ôîðìóëà,ñâÿçûâàþùàÿ detY (t) è detY (0) (ñì. 3):NXdakk (t) · detY (t) ={detY (t)} =dtk=1= Tr A(t) · detY (t).R− 0t Tr A(s) dsÓìíîæàÿ ïîñëåäíåå íà e, ïîëó÷àåì:d − R0t Tr A(s) ds{edetY (t)} = 0, òî åñòüdtRtTr A(s) ds0detY (t) = eY (0),(8)ôîðìóëó Ëèóâèëëÿ.Âåðíåìñÿ ê ôîðìóëå (6).  ñèëó (4)1||y (t)|| ≤ e 2 C1 (M)|t| ||y0 ||,òî åñòü1||Y (t) · Y −1 (0)y 0|| ≤ e 2 C1 (M)|t| ||y0 ||Ëèíåéíûé äèôô.
óð - èÿ ñ ïåðåìåííûìè êîýôô.è1||Y (t)Y −1 (0)y0 ||≤ e 2 C1 (M)|t| .||y0 ||||y0 ||6=0||Y (t) · Y −1 (0)|| = sup(9)Èñïîëüçóÿ Y (t), ïîëó÷èì ôîðìóëó äëÿ ðåøåíèÿ çàäà÷è Êîøè(1).0Óìíîæèì y = A(t)y + f (t) ñëåâà íà Y −1 (t).  èòîãå ïîëó÷èì0{Y −1 y } = Y −1 f , òî åñòüy (t) = Y (t) · Y−1(0)y0 +ZtY (t) · Y −1 (s)f (s)ds.0Ïîñêîëüêó(10)1||Y (t) · Y −1 (s)|| ≤ e 2 C1 (M)·|t−s|ñì. (9) , òî èç (10) ñëåäóåò îöåíêà||y (t)|| ≤1e 2 C1 (M)·|t| ||y0 ||tZ1e 2 C1 (M)·|t−s| ||f (s)||ds ≤+0Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.≤1e 2 C1 (M)·|t| ||y0 ||1e 2 C1 (M)·|t| − 1+ 2N,C1 (M)åùå îäèí âàðèàíò àïðèîðíîé îöåíêè çàäà÷è Êîøè (1).Ñëó÷àé îäíîãî ëèíåéíîãî óðàâíåíèÿ ïðîèçâîëüíîãî ïîðÿäêà ñêîýôôèöèåíòàìè, çàâèñÿùèìè îò t , t ∈ [−T , T ]:0 Lx = x (N) + a1 (t)x (N−1) + ... + aN−1 (t)x ++aN (t) = F (t), t ∈ [−T , T ],0x(0) = α1 , x (0) = α2 , ...
x (N−1) (0) = αN ,(11)ãäå ai (t), i = 1,...,N íåïðåðûâíû íà [−T , T ].Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Ñâåäåì (11) ê (1):x(t) x 0 (t) 0dy = dt = Ay (t) + f (t), t ∈ [−T , T ];...(N−1)00x(t)(1 ) α1 .. y (0) = y0 = . ;αN01 0 ...00 00 1 ...0 .. ..A=, f = . .. 0 00 0 ...1 F (t)−aN. . . −a1Èç òåîðåìû ñóùåñòâîâàíèÿ è åäèíñòâåííîñòè ðåøåíèÿ çàäà÷èÊîøè (1) ïîëó÷àåì ñóùåñòâîâàíèå è åäèíñòâåííîñòü ðåøåíèåçàäà÷è (11):Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Åñëè ai (t), i = 1,...,N , f (t) - íåïðåðûâíûå ôóíêöèè íà [−T , T ],òî äëÿ ëþáûõ êîíñòàíò αi , i = 1,...,N , âñåãäà ñóùåñòâóåò íà[−T , T ] íåïðåðûâíîå è N ðàç íåïðåðûâíî - äèôôèðåíöèðóåìîåðåøåíèå x = x(t), óäîâëåòâîðÿþùåå óðàâíåíèþ Lx = F (t) èíà÷àëüíûì óñëîâèÿì.Ðåøåíèå x = x(t) çàäà÷è Êîøè (11) îäíîçíà÷íî îïðåäåëÿåòñÿíà÷àëüíûìè óñëîâèÿìè.Ïðè F (t) ≡ 0 ôóíäàìåíòàëüíàÿ ìàòðèöà ðåøåíèé çàäà÷è Êîøè00(1 )Φ(t) = y [1] (t), ...
, y [N] (t) =x1...xN0 x0...xN 1=....x1(N−1) . . .xN(N−1)Ïðè ýòîì äëÿ ëþáîãî ðåøåíèÿ çàäà÷è Êîøè (11)x(t) =NXk=1ck xk (t).Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Çàìåòèì, ÷òî îïðåäåëèòåëü ÂðîíñêîãîdetΦ(t) 6= 0, ïðè÷åìdetΦ(t) = detΦ(0) · exp{−Z0ta1 (s)ds}.(12)Ðåøåíèå çàäà÷è Êîøè (11) ïðè F (t) 6= 0 ìîæíî ïîëó÷àòü ñïîìîùüþ ôîðìóëû (10).Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Ñóùåñòâîâàíèå ðåøåíèÿ çàäà÷è Êîøè (1).Ïóñòü ìíîæåñòâî D = (−T , T ) × {||y − y0 || < K } - îáëàñòü âR × R N , K - ïîñòîÿííàÿ.Ïðåäïîëîæèì, ÷òî ïðàâàÿ ÷àñòü ñèñòåìû (1) (âåêòîðA(t)y (t) + f (t)) îïðåäåëåíà â D .Ñõåìà ðàññóæäåíèé.1. Ïîñòðîåíèå ε - ïðèáëèæåííûõ ðåøåíèé çàäà÷è Êîøè (1).2.
Ñóùåñòâîâàíèå ïîñëåäîâàòåëüíîñòè ïðèáëèæåííûõ ðåøåíèé,ñõîäÿùåéñÿ ê ðåøåíèþ çàäà÷è (1).Èòàê, ïåðâûé ýòàï.Îïðåäåëåíèå 1.Íåïðåðûâíàÿ âåêòîð - ôóíêöèÿ ϕ(t), îïðåäåëåííàÿ íàèíòåðâàëå I = (-T,T), íàçûâàåòñÿ ε - ïðèáëèæåííûì ðåøåíèåìçàäà÷è Êîøè (1), åñëè1) t, ϕ(t) ∈ D , åñëè t ∈ I;2) ϕ ∈ C 1 − T , T , çà èñêëþ÷åíèåì, âîçìîæíî, êîíå÷íîãî÷èñëà òî÷åê S íà I , â êîòîðûõ ϕ(t) èìååò ðàçðûâû ïåðâîãîðîäà.03) ||ϕ (t) − A(t)ϕ(t) − f (t)|| < ε, åñëè t ∈ I \S .Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Åñëè âûïîëíåíî óñëîâèå 2), òî ãîâîðÿò, ÷òî âåêòîð - ôóíêöèÿϕ(t) èìååò êóñî÷íî - íåïðåðûâíóþ ïðîèçâîäíóþ íà èíòåðâàëåI , îáîçíà÷àÿ ýòîò ôàêò ñëåäóþùèì îáðàçîì: ϕ ∈ Cp1 (I ).Îáîçíà÷èì:M = max kA(t)y + f (t)k,|t|≤Tα = min(T ,ky − y0 k ≤ K ,(13)K).MÏðåäëîæåíèå 1.Äëÿ ëþáîãî ε > 0 íà èíòåðâàëå |t| ≤ α ñóùåñòâóåò ε ïðèáëèæåííîå ðåøåíèå çàäà÷è Êîøè (1) òàêîå, ÷òî φ(0) = y0 .ÄîêàçàòåëüñòâîÏóñòü ε > 0.
Ïîñòðîèì íà [0, α] - ε - ïðèáëèæåííîå ðåøåíèå(äëÿ îïðåäëåííîñòè ðàññìàòðèâàåòñÿ ïðàâàÿ ïîëîâèíà îòðåçêà[−α, α]).Òàê êàê âåêòîð A(t)y + f (t) íåïðåðûâåí íà D , òî îíðàâíîìåðíî íåïðåðûâåí íà D (òåîðåìà Êàíòîðà),Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.ñëåäîâàòåëüíî, ∀ε > 0 ∃δε > 0 òàêîå, ÷òîkA(t)y + f (t) − A(t̃)ỹ − f (t̃)k ≤ ε, åñëè |t − t̃| ≤ δε ,(14)ky − ỹ k ≤ δε , (t, y ) ∈ D, (t̃, ỹ ) ∈ D.Ðàçäåëèì îòðåçîê [o, α] íà n ÷àñòåé òî÷êàìè:0 < t1 < ... < tn = α,δε).MÈç òî÷êè (0, y0 ) ïðîâåäåì íàïðàâî ïðÿìóþ ñ óãëîâûìèêîýôôèöåíòàìè{A(0)y0 + f (0)}max |tk − tk−1 | ≤ min(δε ,(15)äî ïåðåñå÷åíèÿ ñ ïëîñêîñòüþ t = t1 â òî÷êå (t1 , y 1 ).Ïðîäîëæàÿ ïðîöåññ, ïîñòðîèì ëîìàíóþ ëèíèþ, êîòîðàÿïåðåñåêàåòñÿ ñ ïëîñêîñòüþ t = α, íå âûõîäÿ çà ãðàíèöó îáëàñòèD , ñì.
ðèñ. 1 (íà ðèñóíêå y0 = 0, îñü Ot íàïðàâëåíà íå âïðàâî,à ââåðõ).Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.ty1t1t00y2Ðèñ. 1Ïîñòðîåííàÿ âåêòîð - ôóíêöèÿ ϕ(t) ÿâëÿåòñÿ ε ïðèáëèæåííûì ðåøåíèåì çàäà÷è Êîøè (1).Ïðîâåðèì, ÷òî ýòî äåéñòâèòåëüíî òàê:1) ϕ(0) = y0 .Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.2) ϕ(t) = ϕ(tk−1 ) + A(tk−1 )ϕ(tk−1 ) + f (tk−1 ) ··(t − tk−1 ), åñëè tk−1 < t < tk , k = 2, ... , n.Òîãäà î÷åâèäíî, ÷òî ϕ(t) ∈ Cp1 [0, α] èkϕ(t) − ϕ(t̃)k ≤ M|t − t̃|, t, t̃ ∈ [0, α].(16)Ñëåäîâàòåëüíî, èç (15) è (16) ñëåäóåò, ÷òîkϕ(t) − ϕ(tk−1 )k ≤ δε ,êîãäà tk−1 < t < tk , k = 2, ...
, n.3)0kϕ (t) − A(t)ϕ(t) + f (t) k = kA(tk−1 )ϕ(tk−1 )++f (tk−1 ) − A(t)ϕ(t) + f (t) k ≤ ε.(17)Òàêèì îáðàçîì, ïðåäëîæåíèå äîêàçàíî.Çàìå÷àíèå 1.Ðåêêóðåíòíîå ñîîòíîøåíèåϕk = ϕk−1 + (tk − tk−1 ) A(tk−1 )ϕ(tk−1 ) + f (tk−1 )Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô. îñíîâà ðÿäà âû÷èñëèòåëüíûõ àëãîðèòìîâ äëÿ íàõîæäåíèÿïðèáëèæåííûõ ðåøåíèé çàäà÷è (1).Ïåðåéäåì ê äîêàçàòåëüñòâó ñóùåñòâîâàíèÿ ðåøåíèÿ çàäà÷è (1).Äëÿ ýòîãî èñïîëüçóåì ïðèíöèï êîìïàêòíîñòè Àñêîëè - Àðöåëà.Îïðåäåëåíèå 2.Ìíîæåñòâî ôóíêöèé F = {f }, îïðåäåëåííûõ íà èíòåðâàëå I ,íàçûâàåòñÿ ðàâíîñòåïåííî íåïðåðûâíûì íà I , åñëè ∀ε > 0∃δε > 0 (íå çàâèñÿùåå îò f ) òàêîå, ÷òî kf (t) − f (t̃)k ≤ ε, êîãäà|t − t̃| ≤ δε , t, t̃ ∈ I .Òåîðåìà 1 (Àñêîëè - Àðöåëà).Ïóñòü âûïîëíåíû äâà ñâîéñòâà:1) Ñåìåéñòâî ôóíêöèé F = {f } ðàâíîìåðíî îãðàíè÷åíî;2) F = {f } ðàâíîñòåïåííî íåïðåðûâíî íà êîíå÷íîì èíòåðâàëå I .Òîãäà ñóùåñòâóåò ïîñëåäîâàòåëüíîñòü ôóíêöèé {fn }n∈N ∈ F ,ðàâíîìåðíî ñõîäÿùàÿñÿ íà I .Óêàçàíèå.Äëÿ äîêàçàòåëüñòâà òåîðåìû ìîæíî èñïîëüçîâàòü ïðèíöèïÁîëüöàíî - Âåéåðøòðàññà êîìïàêòíîñòè îãðàíè÷åííîãî îòðåçêà.Òåîðåìà 2.Ëèíåéíûé äèôô.
óð - èÿ ñ ïåðåìåííûìè êîýôô.Íà îòðåçêå |t| ≤ α ñóùåñòâóåò ðåøåíèå ϕ(t) ∈ C 1 çàäà÷è (1).Äîêàçàòåëüñòâî.Ïóñòü εn - ìîíîòîííî óáûâàþùàÿ ïîñëåäîâàòåëüíîñòüïîëîæèòåëüíûõ ÷èñåë, εn → 0 ïðè n → ∞.Ïîñòðîèì ïîñëåäîâàòåëüíîñòü ϕn (t) ïðèáëèæåííûõ εn ðåøåíèé çàäà÷è (1).Òîãäà âûïîëíåíû ñâîéñòâà:1) ϕn (0) = y0 ;2) kϕn (t) − ϕn (t̃)k ≤ M|t − t̃| ⇒ kϕn (t)k ≤ ky0 k + K ,(18)òî åñòü ñåìåéñòâî ôóíêöèé ðàâíîìåðíî îãðàíè÷åíî èðàâíîñòåïåííî íåïðåðûâíî.Ñëåäîâàòåëüíî, â ñèëó ïðèíöèïà êîìïàêòíîñòè Àñêîëè Àðöåëà ñóùåñòâóåò ïîäïîñëåäîâàòåëüíîñòü ϕnk ⇒ ϕ íà [−α, α].Ïðåäåëüíàÿ ôóíêöèÿ ϕ(t) è åñòü ðåøåíèå.Äîêàæåì ýòî.Z tϕn (t) = y0 +A(s)ϕn (s) + f (s) + 4n (s) ds,(19)0Ëèíåéíûé äèôô.
óð - èÿ ñ ïåðåìåííûìè êîýôô.ãäå 1 ϕn − A(s)ϕn (s) + f (s) , â òî÷êàõ, ãäå∃ϕn ;4n (s) =0,èíà÷å. ñèëó íåðàâåíñòâà (17) è îïðåäåëåíèÿ ε - ïðèáëèæåííîãîðåøåíèÿ èìååì:k4n (t)k ≤ εn , àA(t)ϕnk (t) + f (t) ⇒ A(t)ϕ(t) + f (t) íà [−α, α].Ïåðåõîäÿ òåïåðü ê ïðåäåëó ïðè n, k → ∞ â ñîîòíîøåíèè (19)ïîëó÷àåì:Z tϕ(t) = y0 +A(t)ϕ(t) + f (t) dt.(20)0Äèôôåðåíöèðóÿ ðàâåíñòâî (20), ïðèõîäèì ê ñîîòíîøåíèÿì: 0ϕ = A(t)ϕ + f (t), t > 0,ϕ(0) = y0 .Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Òåîðåìà ïîëíîñòüþ äîêàçàíà.Çàìå÷àíèå 2.Ìåòîä, èñïîëüçîâàííûé ïðè äîêàçàòåëüñòâå òåîðåìû 2,íàçûâàåòñÿ ìåòîäîì ëîìàíûõ Ýéëåðà. Îí ïðèìåíÿåòñÿ íåòîëüêî â ñëó÷àå ëèíåéíûõ çàäà÷ (êàê ñèñòåìà óðàâíåíèé (1)) íîè â áîëåå îáùèõ ñèòóàöèÿõ (ñì.
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