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íèæå òåîðåìó Ïåàíî - Êîøè).Çàìå÷àíèå 3.Ëþáàÿ ïîñëåäîâàòåëüíîñòü ϕn ⇒ ðåøåíèþ (åñëè åñòüåäèíñòâåííîñòü).Ïðèìåð 1.Ðàññìîòðèì ñëåäóþùóþ çàäà÷ó Êîøè:10x = x3,x(0) = 0.(21)Òîãäà ìíîæåñòâî ðåøåíèé çàäà÷è (21) ìîæíî çàäàòü òàê:ϕc (t) = 0,ϕc (t) =(0 ≤ t ≤ c);2(t−c) 23,3(c < t ≤ 1).(22)Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.Òàêèì îáðàçîì, â îáùåì ñëó÷àå òðåáîâàíèå íåïðåðûâíîñòèïðàâîé ÷àñòè óðàâíåíèÿ (èëè ñèñòåìû) íå äîñòàòî÷íî äëÿîáîñíîâàíèÿ åäèíñòâåííîñòè ðåøåíèÿ çàäà÷è Êîøè.Íåîáõîäèìû äîïîëíèòåëüíûå óñëîâèÿ.Ñôîðìóëèðóåì íàèáîëåå îáùóþ òåîðåìó î ñóùåñòâîâàíèèðåøåíèÿ çàäà÷è Êîøè äëÿ äèôôåðåíöèàëüíîãî óðàâíåíèÿïåðâîãî ïîðÿäêà.Òåîðåìà 3 (Êîøè - Ïåàíî).Åñëè f (t, y ) ∈ C â ïðÿìîóãîëüíèêåP = {|t − t0 | ≤ a, |y − y0 | ≤ b}, (a, b > 0),òî äëÿ|t − t0 | ≤ α = min(a,b), (M = max |f (t, y )|)PMñóùåñòâóåò ðåøåíèå y (t) ∈ C 1 çàäà÷è Êîøè 0y = f (t, y ),y (t0 ) = y0 .(23)Ëèíåéíûé äèôô.
óð - èÿ ñ ïåðåìåííûìè êîýôô.Äîêàçàòåëüñòâî ïðîâîäèòñÿ àíàëîãè÷íî äîêàçàòåëüñòâóòåîðåìû 2: ïðè ïîìîùè ìåòîäà ëîìàíûõ Ýéëåðà.Çàìå÷àíèå 4.Òåîðåìà 3 ãàðàíòèðóåò ñóùåñòâîâàíèå ðåøåíèÿ çàäà÷è Êîøè âìàëîì, òîëüêî íà îòðåçêå Ïåàíî, |t − t0 | ≤ α.Çàìå÷àíèå 5.Ñëîæíûé ïðèìåð íååäèíñòâåííîñòè ðåøåíèÿ çàäà÷è Êîøèâïåðâûå, â 1925 ãîäó, áûë ïîñòðîåí Ì.À. Ëàâðåíòåâûì (ñì.òàêæå Ô. Õàðòìàí, Îáûêíîâåííûå äèôôåðåíöèàëüíûåóðàâíåíèÿ, Ìîñêâà, Ìèð, 1970, ñ. 31-35). ýòîì ïðèìåðå íà êàæäîì îòðåçêå [t0 , t0 + ε] è [t0 − ε, t0 ] äëÿëþáîãî ïîëîæèòåëüíîãî ε çàäà÷à Êîøè (23) èìååò áîëååîäíîãî ðåøåíèÿ.Îêàçûâàåòñÿ, ÷òî íå ñóùåñòâóåò ïðåäåëüíîãî (ìèíèìàëüíîãî)òðåáîâàíèÿ íà ïðàâóþ ÷àñòü óðàâíåíèÿ, ãàðàíòèðóþùåãîåäèíñòâåííîñòü ðåøåíèÿ çàäà÷è Êîøè.Ïðèâåäåì òåïåðü îäèí ðåçóëüòàò Îñòãóäà î åäèíñòâåííîñòè.Ëèíåéíûé äèôô.
óð - èÿ ñ ïåðåìåííûìè êîýôô.Òåîðåìà 4.Åñëè ôóíêöèÿ f (x, y ) äëÿ ëþáîé ïàðû òî÷åê (x, y1 ) è (x, y2 )îáëàñòè G óäîâëåòâîðÿåò íåðàâåíñòâó|f (x, y2 − f (x, y1 )| ≤ Φ(|y1 − y2 |),ãäå Φ(u) > 0, 0 < u ≤ ε, Φ(u) - íåïðåðûâíà èZ cdu→ ∞, ε → 0,ε Φ(u)(24)òî ÷åðåç êàæäóþ òî÷êó (x0 , y0 ) ∈ G ïðîõîäèò ãðàôèê íå áîëåå0÷åì îäíîãî ðåøåíèÿ óðàâíåíèÿ y = f (x, y ).Çàìå÷àíèå 6. êà÷åñòâå Φ(u) ìîæíî âçÿòü: Ku , Ku · | ln(u)|,Ku| ln(u)| · ln(| ln(u)|), ... (K > 0 - ïîñòîÿííàÿ) (òî åñòü, åñëèâçÿòü Ku , òî ôóíêöèÿ f (x, y ) óäîâëåòâîðÿåò óñëîâèþ Ëèïøèöàïî àðãóìåíòó y ); åñëè îáëàñòü G âûïóêëà ïî y , òî ïîäõîäèòñëó÷àé, êîãäà f (x, y ) èìååò îãðàíè÷åííóþ ÷àñòíóþïðîèçâîäíóþ ïî y .Ëèíåéíûé äèôô.
óð - èÿ ñ ïåðåìåííûìè êîýôô.Äîêàçàòåëüñòâî òåîðåìû 4.Ïóñòü çàäà÷à Êîøè èìååò äâà ðåøåíèÿ: y1 (x) è y2 (x), ïðè÷åìy1 (x0 ) = y2 (x0 ) = y0 è ïóñòü (áåç îãðàíè÷åíèÿ îáùíîñòè) x0 = 0.Ðàññìîòðèì ôóíêöèþ z(x) = y2 (x) − y1 (x). Òîãäà ñóùåñòâóåòòî÷êà x1 > 0 òàêàÿ, ÷òî z(x1 ) 6= 0 (äëÿ îïðåäåëåííîñòèz(x1 ) > 0).Òîãäàd(y2 − y1 )dz=≤ Φ(|y2 − y1 |) < 2Φ(|y2 − y1 |),dxdxåñëè |y2 − y1 | > 0.Ïîñòðîèì ðåøåíèå y (x) óðàâíåíèÿdydx= 2Φ(y ), ïðè÷åìy (x1 ) = z(x1 ) = z1 .(25) ñèëó óñëîâèÿ (24) ðåøåíèå çàäà÷è Êîøè (25) ñóùåñòâóåò,åäèíñòâåííî è àñèìïòîòè÷åñêè ñòðåìèòñÿ ê âåùåñòâåííîéïðÿìîé (îñè àáñöèññ) ïðè x → −∞ (ñì.
ðèñóíîê 2).Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.yy(x)z(x)z10x1xÐèñ. 2 òî÷êå (x1 , z1 ) ãðàôèêè ôóíêöèé y (x) è z(x) ïåðåñåêóòñÿ,ïðè÷åì000z (x1 ) < 2Φ(z1 ) = 2Φ y (x1 ) = y (x1 ).Òîãäà ñóùåñòâóåò ε > 0 òàêîå, ÷òî íà èíòåðâàëå (x1 − ε, x1 )Ëèíåéíûé äèôô. óð - èÿ ñ ïåðåìåííûìè êîýôô.z(x) > y (x). Íî ýòî æå íåðàâåíñòâî ñïðàâåäëèâî è ïðè âñåõε > 0, 0 < ε ≤ x1 .Äîêàæåì ýòî ìåòîäîì îò ïðîòèâíîãî.Âîçüìåì íàèáîëüøåå âîçìîæíîå çíà÷åíèå ε, ïðè êîòîðîìçíà÷åíèÿ z(x2 ) = y (x2 ), x2 = x1 − ε.Ñëåäîâàòåëüíî,00z (x2 ) ≥ y (x2 ) = 2Φ y (x2 ) = 2Φ z(x2 ) ,òàê êàê ïðàâåå x2 z(x) > y (x).0Ñ äðóãîé ñòîðîíû z (x2 ) < 2Φ z(x2 ) - ïðîòèâîðå÷èå.Çíà÷èò, ïðè âñÿêîì x , åñëè 0 ≤ x ≤ x1 ,z(x) ≥ y (x) > 0,ñëåäîâàòåëüíî, z(0) > 0 - ïðîòèâîðå÷èå.Òåîðåìà 4 (Îñòâóäà) äîêàçàíà.Ãëàâà II. Ñóùåñòâîâàíèå è åäèíñòâåííîñòü ðåøåíèéíåëèíåéíûõ ñèñòåì. Çàäà÷à Êîøè. Ïðîäîëæåíèåðåøåíèé10.
Ñóùåñòâîâàíèå è åäèíñòâåííîñòü ðåøåíèéíåëèíåéíûõ ñèñòåì äèôôåðåíöèàëüíûõ óðàâíåíèé ñäîñòàòî÷íî ãëàäêèìè ïðàâûìè ÷àñòÿìè. ËåììàÀäàìàðà. Òåîðåìà Ëèíäåëåôà - ÏèêàðàÏåðåõîäèì ê èçó÷åíèþ ñèñòåì äèôôåðåíöèàëüíûõ óðàâíåíèéäîñòàòî÷íî îáùåãî âèäà. À èìåííî ìû ðàññìîòðèì âîïðîñ îáîäíîçíà÷íîé ðàçðåøèìîñòè çàäà÷è Êîøè ñëåäóþùåãî âèäà:Çäåñüy (t) = y1 (t)...ddt y (t)= f (t, y ),y (t0 ) = y0 .(1)yN (t)- âåêòîð èñêîìûõ ôóíêöèé;Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - Ïèêàðày10 (t)y0 (t) = ...
- âåêòîð íà÷àëüíûõ äàííûõ;y (t)N 0f1 (t, y )f (t, y ) = ... - âåêòîð ïðàâûõ ÷àñòåé.fN (t, y )Äàëåå ïðåäïîëàãàåì, ÷òî:1)f (t, y )- îïðåäåëåíà è íåïðåðûâíà â îáëàñòèΩ̄ = {(t, y ) |t − t0 | ≤ T , ky − y0 k ≤ R,0< T < ∞,0< R < ∞}.Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - Ïèêàðàyy0 +Ry0t0 - Tt0 + Ty0- R0t0tÐèñ. 1Ëåììà Àäàìàðà.
Òåîðåìà Ëèíäëåôà - ÏèêàðàÑëåäîâàòåëüíî, âãäå 0<F <∞Ω̄ kf (t, y )k ≤ F ,- íåêîòîðàÿ ïîñòîÿííàÿ.Çàìå÷àíèå 1.Äåëàÿ â ñèñòåìå (1) çàìåíóτ = t − t0 ,z = y − y0 ,z = z(τ ),ïðåîáðàçóåì åå ê ñëåäóþùåìó âèäó:0z = f (t0 + τ , y0 + z) = f (τ , z),z(0) = 0.0(1 )Ñëåäîâàòåëüíî, â çàäà÷å Êîøè (1) ñðàçó ìîæíî ïîëàãàòü, ÷òît0 = 0, y0 = 0.À òîãäà óñëîâèå 1) ýêâèâàëåíòíî òàêîìó óñëîâèþ:f (t, y ) - îïðåäåëåíàè íåïðåðûâíàΩ̄ = {(t, y ) |t| ≤ T , ky k ≤ R}.Çíà÷èò kf (t, y )k ≤ F â Ω̄.âËåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - ÏèêàðàyR-TT0t-RÐèñ.
2Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - ÏèêàðàÇàìå÷àíèå 2Ïðåäïîëîæèì, ÷òî ñóùåñòâóåò ðåøåíèåy = y (t) çàäà÷è Êîøè (1) òàêîå, ÷òî ïðè0 ≤ t ≤ t1 èëè t1 ≤ t ≤ 0 çíà÷åíèÿ y (t) íåky k ≤ R (òî÷íåå öèëèíäðà Ω̄).âûõîäÿò èç øàðàÒîãäà èíòåãðèðóÿ óðàâíåíèå0y = f (t, y )ïîtîò 0 äî t1 , ïîëó÷èìZy (t1 ) =ky (t1 k ≤ t1f (t, y (t))dt,òî åñòü0Zt10f (t, y (t))dt ≤ F |t1 |.Èòàê, åñëè ìû õîòèì, ÷òîáû ïðè íåêîòîðîìêðèâàÿy = y (t)íå âûõîäèëà èç öèëèíäðàt = t1 èíòåãðàëüíàÿΩ̄, òî íàìäîñòàòî÷íî ïîòðåáîâàòü, ÷òîáûF |t1 | ≤ R,R.FËåììà Àäàìàðà.
Òåîðåìà Ëèíäëåôà - Ïèêàðàòî åñòü|t1 | ≤Ñëåäîâàòåëüíî, â äàëüíåéøåì ìîæíî ïîëàãàòü, ÷òîR|t| ≤ T0 ≤ min{ , T }.FÍèæå ìû äîêàæåì, ÷òî ïðè|t| ≤ T0 ∃!ðåøåíèå çàäà÷è Êîøè(1) òàêîå, ÷òîky (t)k ≤ R.Ïðè ýòîì ñïðàâåäëèâî íåðàâåíñòâîky (t)k ≤ F |t|.Åùå îäíî óñëîâèå íà ïðàâóþ ÷àñòü:2) â îáëàñòèΩ̄ñóùåñòâóåò íåïðåðûâíàÿ è, ñëåäîâàòåëüíî,îãðàíè÷åííàÿ ìàòðèöàfy (t, y ) =∂fi(t, y ) , i, j = 1,...,N ,∂yjïðè÷åìkfy k ≤ L,0< L < ∞.Ëåììà Àäàìàðà. Òåîðåìà Ëèíäëåôà - ÏèêàðàËåììà Àäàìàðà.Åñëèfóäîâëåòâîðÿåò 1),z(t), w (t)íåïðåðûâíûå âåêòîð -B ⊂ [−T , T ] è òàêèå,4f = f (t, z) − f (t, w ) ìîæíîôóíêöèè, îïðåäåëåííûå íà îòðåçêåkz(t)k, kw (t)k ≤ R ,òî÷òîïðåäñòàâèòü â âèäå4f = A(t) · 4y , 4y = z − w ,A = aij (t) , i, j = 1,...,N ;Z 1∂fi(t, λz(t) + (1 − λ)w (t))dλ.aij (t) =0 ∂yj(2)Çàìå÷àíèå ê ôîðìóëèðîâêå ëåììûÈç ñîîòíîøåíèé (2) ñëåäóåò, ÷òî êîýôôèöèåíòûíåïðåðûâíûå ôóíêöèè îòtè â ñèëó óñëîâèÿ 2)A(t) kA(t)k ≤ L. ñàìîì äåëå, òàê êàêZA(t) = k01∂fi(t, λz(t) + (1 − λ)w (t))dλk è∂yjkλz + (1 − λ)w k ≤ RíàB,òîËåììà Àäàìàðà.
Òåîðåìà Ëèíäëåôà - Ïèêàðà1ZkA(t)k ≤kfy (t, λz + (1 − λ)w kdλ ≤ L.0Äîêàçàòåëüñòâî ëåììû Àäàìàðà.Ïîñêîëüêó4fi (t, z) = fi (t, z) − fi (t, w ) =Z=10=ZNXj=101dfi (t, λz + (1 − λ)w )dλ =dλ∂fi(t, λz + (1 − λ)w )dλ (zj − wj ) =∂yj=NXaij (t) · 4yj ,òî åñòüj=14f14y14f = ... = 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.
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