1611689230-c82f95de5ae5b8e93bc9da7770e8f930 (826746), страница 10
Текст из файла (страница 10)
= A(t)4y , 4y = ... ,4fN4yN÷.ò.ä.Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - ÏèêàðàÑëåäñòâèå 1.Ïóñòü êðîìå óñëîâèé ëåììû âûïîëíåíî åùå íåðàâåíñòâîkz − w k ≤ Nk ·ÒîãäàtZkZ4f (τ )dτ k = k0≤Z0t|t|k.k!tA(τ )4y (τ )dτ k ≤0|t|k+1kA(τ )k · k4y (τ )kdτ ≤ L · Nk ·.(k + 1)!Ïðèñòóïèì ê äîêàçàòåëüñòâó òåîðåìû ñóùåñòâîâàíèÿ.Îïðåäåëèì ñëåäóþùóþ áåñêîíå÷íóþ ïîñëåäîâàòåëüíîñòüôóíêöèé, çàäàííûõ ïðè|t| ≤ T0 :y [0] (t) = 0,Rty [1] (t) = 0 f (τ , y [0] (τ ))dτ ,...Rty [k] (t) = 0 f (τ , y [k−1] (τ ))dτè ò.ä.Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - ÏèêàðàÎ÷åâèäíî, ÷òîky [0] (t)k ≤ Rïðè|t| ≤ T0 .Ïðåäïîëîæèì, ÷òîky [k−1] (t)k ≤ RÒîãäà,ky[k]ïðè|t| ≤ T0 .kf (t, y [k−1] (t))k ≤ F(t)k ≤ Z0tèkf (τ , y [k−1] (τ ))kdτ ≤≤ F |t| ≤ FT0 ≤ FR≤ R.FÈòàê, äîêàçàíî, ÷òîky [k] (t)k ≤ Rky [k] (t)k ≤ F |t|ïðèè|t| ≤ T0 .Ëåììà Àäàìàðà.
Òåîðåìà Ëèíäëåôà - ÏèêàðàÏîêàæåì, ÷òî{y [k] (t)}íåïðåðûâíîé ôóíêöèèïîñëåäîâàòåëüíîñòüy = y (t)íà⇒ê íåêîòîðîé|t| ≤ T0 .Ðàññìîòðèì ðÿäkXy [i] (t) − y [i−1] (t) = y [k] (t).i=1Èòàê, ýòîò ðÿä ðàâíîìåðíî ñõîäèòñÿ⇔ðàâíîìåðíî ñõîäèòñÿïîñëåäîâàòåëüíîñòü{y [k] (t)} ⇒ y (t).Òàê êàêy[i+1]Z[i](t) − y (t) =tf (τ , y [i] (τ )) − f (τ , y [i−1] (τ )) dτ =0Z=tA(τ ) y [i] (τ ) − y [i−1] (τ ) dτ ,0òî èìååìky [1] (t) − y [0] (t)k = ky [1] (t)k ≤ F |t|,Ëåììà Àäàìàðà.
Òåîðåìà Ëèíäëåôà - Ïèêàðàky [2] (t) − y [1] (t)k ≤ FL|t|22,...ky [i+1] (t) − y [i] (t)k ≤ FLi|t|i+1.(i + 1)!|t| ≤ T0 :Ñëåäîâàòåëüíî, ïðèky [i+1] (t) − y [i] (t)k ≤F Li |t|i F (LT0 )i≤ ·.L i!Li!Òàê êàê ðÿä∞XFk=1ñõîäèòñÿ, òî èL·F LT0(LT0 )k=(e− 1)!k!L{y [i] (t)} ⇒ y (t).Ïåðåõîäÿ ê ïðåäåëó â ñîîòíîøåíèè[i]Zy (t) =0tf (τ , y [i−1] (τ ))dτ ,Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - Ïèêàðàïîëó÷èìZty (t) =(3)f (τ , y (τ ))dτ .0ky [i] (t)k ≤ F |t| ≤ R , òî(3) ïî t , ïîëó÷àåì: 0y (t) = f (t, y )y (0) = 0Ïðè ýòîì, òàê êàêÄèôôåðåíöèðóÿèky (t)k ≤ R .Åäèíñòâåííîñòü.Äîêàçàòåëüñòâî âåäåì îò ïðîòèâíîãî: ïóñòü åñòü äâà ðåøåíèÿz(t)èw (t),óäîâëåòâîðÿþùèå óñëîâèÿì:kz(t)k, kw (t)k ≤ R,0z (t) = f (t, z(t)),Òîãäà, äëÿz(0) = w (0) = 0,0w (t) = f (t, w (t)).4y = z(t) − w (t)0(4y ) = A(t)4y (t)4y (0) = 0Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - ÏèêàðàèkA(t)k ≤ L.Òåïåðü åäèíñòâåííîñòü ðåøåíèÿ ñëåäóåò èç òåîðåìûåäèíñòâåííîñòè äëÿ ëèíåéíûõ ñèñòåì.Òàêèì îáðàçîì, äîêàçàíàÒåîðåìà Ëèíäëåôà - ÏèêàðàÅñëè ïðàâàÿ ÷àñòü ñèñòåìû óðàâíåíèé (1) óäîâëåòâîðÿåòóñëîâèÿì 1), 2), òî ðåøåíèå çàäà÷è Êîøè (1) äëÿ íååñóùåñòâóåò è åäèíñòâåííî.Çàìå÷àíèå 3.Ìåòîä ïîñëåäîâàòåëüíûõ ïðèáëèæåíèé ïîçâîëÿåò îïðåäåëèòüðåøåíèå çàäà÷è Êîøè ñ ëþáîé çàäàííîé òî÷íîñòüþ:y (t) = y [k] (t) + y [k+1] (t) − y [k] (t)++ y [k+2] (t) − y [k+1] (t) + ...
⇒ky (t) − y [k] (t)k ≤ ky [k+1] (t) − y [k] (t)k++ ky [k+2] (t) − y [k+1] (t)k + ... ≤Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - Ïèêàðà≤F·Lk T0k+1Lk+1 T0k+2++ ... .(k + 1)!(k + 2)!(4)Çàìå÷àíèå 5.Äëÿ ñïðàâåäëèâîñòè òåîðåìû ñóùåñòâîâàíèÿ äîñòàòî÷íî ëèøüóñëîâèÿ 1). Îäíàêî äëÿ åäèíñòâåííîñòè ðåøåíèÿ 1)íåäîñòàòî÷íî: òðåáóþòñÿ äîïîëíèòåëüíûå óñëîâèÿ íàf (t, y )(íàïðèìåð óñëîâèÿ 2)).Âîîáùå ãîâîðÿ, óñëîâèå 2) ìîæíî íåñêîëüêî îñëàáèòü, íîâîîáùå áåç äîïîëíèòåüíûõ óñëîâèé îáîéòèñü íåëüçÿ.Ïðèìåð 1.N = 1,y ≥ t 2,ïðè |y | < t 2 ,f (t, y ) =−2t ïðè y ≤ −t 2 . 2tïðèy2ty (0) = 0Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - Ïèêàðà2y=tf(t,y) = 2tyf(t,y) = _tyf(t,y) = _t0f(t,y) = -2t2y=-tÐèñ.
3Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - ÏèêàðàËåãêî âèäåòü, ÷òîf (t, y ) íåïðåðûâíà âñþäó, â òîì ÷èñëåt = 0, y = 0.0Ðåøåíèå óðàâíåíèÿ y (t) = f (t, y ) äàåòñÿ ôîðìóëîé: 200 t + C (C ≥ 0) ïðè y ≥ t 2 ,0000y (t) =C t 2 (|C | ≤ 1) ïðè |y | < t 2 ,000000−t 2 + C (C ≤ 0) ïðè y ≤ −t 2 .è âòî÷êåÒåïåðü î÷åâèäíî, ÷òî ó çàäà÷è Êîøè (1) áåñêîíå÷íî ìíîãî ýòîì0000y (t) = C t 2 (|C | ≤ 1).ñëó÷àå fy ñóùåñòâóåò ëèøü ïðè |y | =6 t 2! 0 ïðè y ≥ t 2 ,2ïðè |y | < t 2 ,fy = t0 ïðè y ≤ −t 2 .ðåøåíèé:11.
Îáñóæäåíèå óòâåðæäåíèé ëîêàëüíîé òåîðåìûñóùåñòâîâàíèÿ. Äîñòàòî÷íûå óñëîâèÿ ñóùåñòâîâàíèÿðåøåíèÿ â öåëîì. Ïðîäîëæåíèå ðåøåíèé. Òåîðåìà îïîêèäàíèè êîìïàêòàÄîêàçàëè òåîðåìó îá îäíîçíà÷íîé ðàçðåøèìîñòè çàäà÷è Êîøè0y = f (t, y ),y (0) = 0,(1)ïðè íåêîòîðûõ ïðåäïîëîæåíèÿõ îòíîñèòåëüíî ïðàâîé ÷àñòèf (t, y ).Íàïîìíèì, ÷òî:1) f (t, y ) îïðåäåëåíàè íåïðåðûâíà âΩ̄ = {(t, y ) |t| ≤ T , ky k ≤ R}, ïðè ýòîì kf (t, y )k ≤ F â Ω̄.2) Êîýôôèöèåíòû ìàòðèöû fy (t, y ) îïðåäåëåíû è íåïðåðûâíûâ Ω̄ è kfy (t, y )k ≤ L.Òåîðåìà îá îäíîçíà÷íîé ðàçðåøèìîñòè çàäà÷è Êîøè (1).Ïóñòü T0 = min{T , RF }. Òîãäà ïðè |t| ≤ T0 ñóùåñòâóåòíåïðåðûâíî - äèôôåðåíöèðóåìàÿ y (t) òàêàÿ, ÷òî âûïîëíåíî (1)Ïðîäîëæåíèå ðåøåíèé.
Òåîðåìà î ïîêèäàíèè êîìïàêòàèky (t)k ≤ F |t| ≤ FT0 ≤ R.Âåêòîð - ôóíêöèÿ y (t) îïðåäåëåíà óñëîâèÿìè 1), 2) îäíîçíà÷íî.Ïðèìåð 1 (N = 1).0y = 1 + t 2 + y 2 , y (0) = 0.Äëÿ ïðèìåíåíèÿ òåîðåìû íóæíî çíàòü T , R . Ïóñòü T = 2,R = 5. Òîãäà |f (t, y )| ≤ 30, òî åñòü F = 30. Ïðè ýòîìT0 = min{2, 16 } = 16 .Èòàê, â òåîðåìå óòâåðæäàåòñÿ ñóùåñòâîâàíèå ðåøåíèÿ ïðè|t| ≤ 16 .Ïðåäïîëîæåíèå 2) âûïîëíåíî:|fy | ≤ 10 (= L).Ìîæåò ïîêàçàòüñÿ, ÷òî ñèëüíîå îãðàíè÷åíèå íà äëèíó îòðåçêàìû ïîëó÷èëè ïîòîìó, ÷òî â ñàìîì íà÷àëå âûáðàëè íå î÷åíü6,áîëüøèå T , R . Ïóñòü T = 7, R = 10.
Òîãäà F = 150 è T0 = 25òî åñòü äëèíà îòðåçêà ïî t , íà êîòîðîì îïðåäåëåíî ðåøåíèåy (t) åùå ìåíüøå, ÷åì â ïðåäûäóùåì ñëó÷àå.Ïðîäîëæåíèå ðåøåíèé. Òåîðåìà î ïîêèäàíèè êîìïàêòàÍà ñàìîì äåëå ðåøåíèå çàäà÷è Êîøè è íå ìîæåò áûòüîïðåäåëåíî ïðè âñåõ |t| < ∞.Äåéñòâèòåëüíî, ïóñòü t ≥ 0. Òîãäà000y = 1 + t 2 + y 2 , òî åñòü y ≥ 1 + y 2 , èëè {arctg y } ≥ 1.Îòñþäàarctg y (t) ≥ t, òî åñòü y (t) ≥ tg t.Ñëåäîâàòåëüíî, èíòåðâàë ñóùåñòâîâàíèÿ ðåøåíèÿ íå ìîæåòñîäåðæàòü îòðåçîê [0, π2 ], òàê êàê y (t) íà íåì íå ìîæåò áûòüîãðàíè÷åííûì, à ñëåäîâàòåëüíî, è íåïðåðûâíûì.
Ýòîò ïðèìåðïîêàçûâàåò, ÷òî äàæå åñëè ïðàâàÿ ÷àñòü "õîðîøàÿ ôóíêöèÿ",íå âñåãäà ìîæíî óòâåðæäàòü, ÷òî ðåøåíèå çàäà÷è Êîøèîïðåäåëåíî ïðè ëþáûõ çíà÷åíèÿõ t .Òåì íå ìåíåå, ìîæíî óñòàíîâèòü íåêîòîðûå äîñòàòî÷íûåóñëîâèÿ íà f (t, y ) è íà íà÷àëüíûå äàííûå äîïîëíèòåëüíûå ê 1),2), êîòîðûå áû îáåñïå÷èâàëè ñóùåñòâîâàíèå ðåøåíèÿ y = y (t)çàäà÷è Êîøè (1) ïðè ëþáûõ t ≥ 0. äàëüíåéøåì ðàññìàòðèâàþòñÿ ëèøü ÷àñòíûå ñëó÷àè çàäà÷èÏðîäîëæåíèå ðåøåíèé. Òåîðåìà î ïîêèäàíèè êîìïàêòàÊîøè (1), à èìåííî:0t > 0,y = f (y ),y (0) = y0 .0(1 )≡ Àâòîíîìíûå ñèñòåìû.Êðèòåðèé ïðèíàäëåæèò À.Ì. Ëÿïóíîâó.Èñïîëüçóþòñÿ âñïîìîãàòåëüíûå ôóíêöèèH(y ) = H(y1 , ...
, yN ):Ë.1) Ôóíêöèÿ H(y ) îïðåäåëåíà ïðè ky k ≤ R è ÿâëÿåòñÿ âíóòðèè íà ãðàíèöå ýòîãî øàðà íåïðåðûâíîé è èìåþùåé íåïðåðûâíûå÷àñòíûå ïðîèçâîäíûå∂H, j = 1,...,N .∂yjË.2) H(y ) ≥ 0 ïðè ky k ≤ R , ïðè÷åì H(0) = 0 è H(y ) > 0 ïðè0 < ky k < R .Ë.3) Íåïðåðûâíàÿ ôóíêöèÿ J(y ):J(y ) = −NXi=1fi (y )∂H∂iiÏðîäîëæåíèå ðåøåíèé.
Òåîðåìà î ïîêèäàíèè êîìïàêòàóäîâëåòâîðÿåò óñëîâèþ J(y ) ≥ 0 ïðè ky k ≤ R .0Èíîãäà âìåñòî Ë.3 ðàññìàòðèâàþò åãî óñèëåíèå (óñëîâèå Ë.3 ):0Ë.3 ) Íåïðåðûâíàÿ ôóíêöèÿ J(y ), îïðåäåëåííàÿ â óñëîâèè Ë.3)â äîïîëíåíèå ê íåðàâåíñòâóJ(y ) ≥ 0 ïðè ky k ≤ Róäîâëåòâîðÿåò åùå óñëîâèþ J(y ) > 0 ïðè 0 < ky k ≤ R .Áåç äîêàçàòåëüñòâà òåîðåìà ñóùåñòâîâàíèÿ ðåøåíèÿ çàäà÷è0Êîøè (1 ) â öåëîì, òî åñòü ïðè âñåõ T ≥ 0 (ñì.
Ãîäóíîâ ñòð.145-150).Òåîðåìà 1.Ïðåäïîëîæèì, ÷òî ñóùåñòâóåò H(y ) òàêàÿ, ÷òî âûïîëíåíûË.1), Ë.2), Ë.3).  ýòîì ñëó÷àå ìîæíî âûáðàòü òàêîå ρ0(0 < ρ0 < R ), ÷òî åñëè ky0 k ≤ ρ0 , òî ñóùåñòâóåò íåïðåðûâíî äèôôåðåíöèðóåìàÿ ôóíêöèÿ y = y (t), îïðåäåëåííàÿ ïðèëþáîì t ≥ 0, óäîâëåòâîðÿþùàÿ íåðàâåíñòâó ky (t)k ≤ R è0òàêàÿ, ÷òî y (0) = y0 , y = f (y )(ìû, åñòåñòâåííî, ïðåäïîëàãàåì, ÷òî ïðàâàÿ ÷àñòü F (y )óäîâëåòâîðÿåò óñëîâèÿì 1), 2), îáåñïå÷èâàþùèõÏðîäîëæåíèå ðåøåíèé.
Òåîðåìà î ïîêèäàíèè êîìïàêòàñïðàâåäëèâîñòü òåîðåìû ñóùåñòâîâàíèÿ).0Åñëè æå ïðåäïîëîæèòü, ÷òî ñïðàâäåëèâî åùå è Ë.3 ), òîy (t) → 0 ïðè t → ∞.Ðàçúÿñíèì, êàêîâ ñìûñë ôóíêöèè J(y ).Ïóñòü íà íåêîòîðîì èíòåðâàëå ïî t îïðåäåëåíî ðåøåíèå y (t)0óðàâíåíèÿ y = f (y ).Ðàññìîòðèì ôóíêöèþ ϕ(t) = H y (t) è âû÷èñëèì åå0ïðîèçâîäíóþ ϕ (t).N X ∂H dyidy (t)=ϕ (t) = H y (t) =dt∂yidt0i=1=NX∂Hi=1òî åñòü− J y (t)∂yifi = −J y (t) , ïðîèçâîäíàÿ ïî t îò H(y ) âäîëü0ðåøåíèÿ y (t) çàäà÷è Êîøè (1 ).Òîãäà, åñëè J(y ) ≥ 0, òî ñ ðîñòîì t ϕ(t) íå âîçðàñòàåò.Ïðîäîëæåíèå ðåøåíèé. Òåîðåìà î ïîêèäàíèè êîìïàêòàÌîæíî ïîêàçàòü, ÷òî ôóíêöèÿ H(y ) ìîæåò ðàññìàòðèâàòüñÿäëÿ îöåíêè íîðìû âåêòîðà y è ïîýòîìó íåâîçðàñòàíèå H(y )ñâèäåòåëüñòâóåò î òîì, ÷òî ñ òå÷åíèåì âðåìåíè ðåøåíèå y (t)îãðàíè÷åíî.Íàïîìíèì, ÷òî â ïðèâåäåííîì âûøå ïðèìåðå 0y = 1 + t 2 + y 2,y (0) = y0èìåííî íåîãðàíè÷åííîå âîçðàñòàíèå y = y (t) íà êîíå÷íîìèíòåðâàëå áûëî ïðè÷èíîé òîãî, ÷òî y (t) íåëüçÿ áûëîîïðåäåëèòü ïðè âñåõ t .Ïðèìåð 2.0y1 = −y2 − y13 ,0y2 = y1 − y23 ,0(Óäîâëåòâîðÿåò Ë.1), Ë.2), Ë.3), Ë.3 !)H(y ) = y12 + y22 = ky k2 ,Ïðîäîëæåíèå ðåøåíèé.
Òåîðåìà î ïîêèäàíèè êîìïàêòàJ(y ) = 2(y14 + y24 ).Î÷åâèäíî, ÷òî óñëîâèÿ ëîêàëüíîé òåîðåìû ñóùåñòâîâàíèÿ(óñëîâèÿ 1), 2) ) âûïîëíåíû ïðè ëþáîì R > 0.Ðåøåíèå çàäà÷è Êîøè ñóùåñòâóåò ïðè âñåõ t ≥ 0 è ëþáûõíà÷àëüíûõ äàííûõ.Ïðèìåð 3.Äëÿ ñèñòåìû0y1 = −y2 ,0y2 = y1H(y ) = y12 + y22 . Òîãäà J(y ) = 0. Óñëîâèÿ 1), 2) âûïîëíåíû ïðè0ëþáîì R > 0, óñëîâèÿ Ë.1), Ë.2), Ë.3) âûïîëíåíû (íî íå Ë.3 ).Òåîðåìà ñóùåñòâîâàíèÿ â öåëîì ïî t ñïðàâåäëèâà, íîóòâåðæäàòü, ÷òî ïðè t → ∞ âñå ðåøåíèÿ → 0 ìû íå ìîæåì.Îïðåäåëåíèå 1.H(y ) - ôóíêöèÿ Ëÿïóíîâà.Ïðîäîëæåíèå ðåøåíèé. Òåîðåìà î ïîêèäàíèè êîìïàêòà.Ðàññìîòðèì çàäà÷ó Êîøè0y = f (x, y ) â G - îáëàñòè â R N ,(2)Ïðîäîëæåíèå ðåøåíèé.
Òåîðåìà î ïîêèäàíèè êîìïàêòày (x0 ) = y0 ,(x0 , y0 ) ∈ G .(3)Ïðåäïîëîæèì, ÷òî ôóíêöèÿ f (x, y ) íåïðåðûâíà â îáëàñòè G .Îïðåäåëåíèå 2.G - îáëàñòü åäèíñòâåííîñòè äëÿ óðàâíåíèÿ (2), åñëè äâà ëþáûõðåøåíèÿ óðàâíåíèÿ (2) (èõ ãðàôèêè ïðèíàäëåæàò îáëàñòè G !),îïðåäåëåííûå íà ïðîìåæóòêå < a, b > è ñîâïàäàþùèå ïðèíåêîòîðîì x0 ∈< a, b >, ñîâïàäàþò íà âñåì ïðîìåæóòêå< a, b >.Äàæå ïðè ýòîì íåëüçÿ óòâåðæäàòü, ÷òî ðåøåíèå çàäà÷è Êîøè(2), (3) åäèíñòâåííî, òàê êàê ðåøåíèÿ ìîãóò îòëè÷àòüñÿîáëàñòÿìè îïðåäåëåíèÿ.Ïðèìåð 4.0y =yy (0) = 1.(4)Òîãäà e x : (−1, 1) → R + è e x : (−2, 2) → R + - ðåøåíèÿ çàäà÷èÊîøè (4).Ïîëíîå æå ðåøåíèå y (x) = e x îïðåäåëåíî ïðè âñåõ x ∈ R .Ïðîäîëæåíèå ðåøåíèé. Òåîðåìà î ïîêèäàíèè êîìïàêòàÎïðåäåëåíèå 3.Ðåøåíèå ϕ : < a, b >→ R n ïðîäîëæèìî âïðàâî çà òî÷êó b , åñëèñóùåñòâóåò ðåøåíèåψ : < a1 , b1 >→ R n , b1 > b , ñóæåíèå êîòîðîãî íà < a, b >ñîâïàäàåò ñ ϕ (ψ - ïðîäîëæåíèå ðåøåíèÿ ϕ âïðàâî).Àíàëîãè÷íî îïðåäåëÿåòñÿ ïðîäîëæåíèå ðåøåíèÿ âëåâî.ÑïðàâåäëèâàÒåîðåìà 2.Äëÿ òîãî, ÷òîáû ðåøåíèå ϕ : [a, b) → R n ïðîäîëæàëîñü âïðàâî,íåîáõîäèìî è äîñòàòî÷íî, ÷òîáû ñóùåñòâîâàë êîíå÷íûé ïðåäåë:∃ lim ϕ(x) = η,x→b−0(b, η) ∈ G .Äîêàçàòåëüñòâî.1.Íåîáõîäèìîñòü ñëåäóåò èç îïðåäåëåíèÿ 2.2.Äîñòàòî÷íîñòü.Ïóñòü ϕ1 : (α, β) → R n , b ∈ (α, β) - ðåøåíèå óðàâíåíèÿ (2) ñíà÷àëüíûìè äàííûìè (b, η), òî åñòü ϕ1 (b) = η . ñèëó òåîðåìû Ïåàíî ñóùåñòâóåò ðåøåíèå ñôîðìóëèðîâàííîéçàäà÷è Êîøè.Ïðîäîëæåíèå ðåøåíèé.
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
