1611689219-34138cb19538412e83350f6586eb365d (826743), страница 5
Текст из файла (страница 5)
Ðàññìîòðèì ïîñëåäîâàòåëüíîñòü ôóíêöèé {y [k] (x)}, k = 0, 1, . . . ,çàäàâàåìóþ ðåêóððåíòíî:y [0] (x) = y0 ,y [1] (x) = y0 +y [2] (x) = y0 +Rxx0Rxf (ξ, y0 )dξ,f (ξ, y [1] (ξ))dξ,x0...y [k] (x) = y0 +Rxf (ξ, y [k−1] (ξ))dξ, k > 1.x0Ïîêàæåì, ÷òîÀ) Ïîñëåäîâàòåëüíîñòü {y [k] (x)} êîððåêòíî îïðåäåëåíà äëÿ x ∈ [x0 − h, x0 + h];Á) Ïîñëåäîâàòåëüíîñòü {y [k] (x)} ðàâíîìåðíî ñõîäèòñÿ ïðè k → ∞ äëÿ x ∈ [x0 −h, x0 +h];Â) Ïðåäåë ïîñëåäîâàòåëüíîñòè {y [k] (x)} åñòü ðåøåíèå ÇÊ(x0 , y0 ) äëÿ x ∈ [x0 −h, x0 +h].Èòàê, ïî ïîðÿäêó.À) Ïîñëåäîâàòåëüíîñòü {y [k] (x)}, k = 0, 1, . .
. , êîððåêòíî îïðåäåëåíà äëÿx ∈ [x0 − h, x0 + h].Èíà÷å ãîâîðÿ, èíòåãðàëû, ñòîÿùèå â ïðàâîé ÷àñòè êîððåêòíî îïðåäåëåíû, äëÿýòîãî äîñòàòî÷íî ïîêàçàòü, ÷òî ïîäûíòåãðàëüíàÿ ôóíêöèÿ îïðåäåëåíà â òî÷êàõ, ãäåïðîèñõîäèò èíòåãðèðîâàíèå, ò.å. (x, y [k] (x)) ∈ D ïðè x ∈ [x0 − h, x0 + h], k = 0, 1, . . .Ìû äîêàæåì åù¼ áîëåå ñòðîãèé ðåçóëüòàò: (x, y [k] (x)) ∈ Ï ïðè x ∈ [x0 − h, x0 + h],k = 0, 1, . . . , ò.å. ãðàôèêè ôóíêöèé {y [k] (x)}, k = 0, 1, . . . , öåëèêîì ëåæàò â Ï ïðèx ∈ [x0 − h, x0 + h].Ñíà÷àëà ïîêàæåì, ÷òî ãðàôèêè íå âûõîäÿò çà ïðåäåëû Ï ïî îñè OX , ò.å.
|x−x0 | 6a ïðè x ∈ [x0 − h, x0 + h] (çàìåòèì, ÷òî x ∈ [x0 − h, x0 + h] ýêâèâàëåíòíî|x − x0 | 6 h):b|x − x0 | 6 h = min a,6 a.FÒåïåðü ïîêàæåì, ÷òî ãðàôèêè íå âûõîäÿò çà ïðåäåëû Ï ïî îñè OY , ò.å. |y [k] − y0 | 6 bïðè x ∈ [x0 − h, x0 + h], k = 0, 1, . . . Äîêàæåì ýòî ïî èíäóêöèè.Ïðè k = 0: |y [0] − y0 | = |y0 − y0 | = 0 < b.Ïðè k = 1: x x Z ZZx [1]|y − y0 | = y0 + f (ξ, y0 )dξ − y0 = f (ξ, y0 )dξ 6 |f (ξ, y0 )|dξ 6 x0x026x0 Zxïîñêîëüêó (ξ, y0 ) ∈ Ï, òî |f (ξ, y0 )| 6 F 6 F dξ = Fx0bbF min a,6 F = b.FF x Z dξ = F |x − x0 | 6 F h =x0Ïóñòü ïðè k = n âåðíî |y [n] − y0 | 6 b, òîãäà ïðè k = n + 1: x x Z ZZx[n][n][n][n+1]|y− y0 | = y0 + f (ξ, y (ξ))dξ − y0 = f (ξ, y (ξ))dξ 6 |f (ξ, y (ξ))|dξ 6 x0x0x0[n][n]ïîñêîëüêó ïî èíäóêöèîííîìó ïðåäïîëîæåíèþ(ξ, y (ξ)) ∈ Ï, òî |f (ξ, y (ξ))| 6 F 6 x x ZZbb F dξ = F dξ = F |x − x0 | 6 F h = F min a,6 F = b.FFx0x0Èíäóêöèîííîå ïðåäïîëîæåíèå äîêàçàíî.
Òàêèì îáðàçîì, ïîêàçàëè, ÷òî ãðàôèêè ôóíêöèé {y [k] (x)}, k = 0, 1, . . . , öåëèêîì ëåæàò â Ï ïðè x ∈ [x0 − h, x0 + h].Á) Ïîñëåäîâàòåëüíîñòü {y [k] (x)} ðàâíîìåðíî ñõîäèòñÿ ïðè k → ∞ äëÿx ∈ [x0 − h, x0 + h].Ñâåä¼ì èññëåäîâàíèå ðàâíîìåðíîé ñõîäèìîñòè ïîñëåäîâàòåëüíîñòè ê ðàâíîìåðíîé ñõîäèìîñòè ñóììû ðÿäà, äëÿ ýòîãî ïðåäñòàâèì y [k] (x) â âèäå ñóììû ðÿäà:y [k] (x) = y [0] (x) + y [1] (x) − y [0] (x) + y [2] (x) − y [1] (x) + · · · + y [k] (x) − y [k−1] (x) =y0 +kXy [i] (x) − y [i−1] (x)i=1Òåïåðü íàñ èíòåðåñóåò ðàâíîìåðíàÿ ñõîäèìîñòü ñóììûkPy [i] (x) − y [i−1] (x) äëÿ x ∈i=1[x0 −h, x0 +h] ïðè k → ∞.
Ïðèìåíèì çäåñü ïðèçíàê ñõîäèìîñòè Âåéåðøòðàññà, íàéä¼ì∞Pñõîäÿùèéñÿ ÷èñëîâîé ðÿäbi , ìàæîðèðóþùèé íàø: äëÿ âñåõ x ∈ [x0 − h, x0 + h]i=1 [i]y (x) − y [i−1] (x) 6 bi . Äëÿ ýòîãî äîêàæåì ïî èíäóêöèè, ÷òîi [i]y (x) − y [i−1] (x) 6 F Li−1 |x − x0 | äëÿ x ∈ [x0 − h, x0 + h], i > 1.i!Ïðè i = 1: x x Z ZZx [1] y (x) − y [0] (x) = y0 + f (ξ, y0 )dξ − y0 = f (ξ, y0 )dξ 6 |f (ξ, y0 )|dξ 6 x0x0x0 x x Z Z|f (ξ, y0 (ξ))| 6 F 6 F dξ = F dξ = F |x − x0 |.x027x0Ïðè i = 2:ZxZx [2] [1][1]y (x) − y (x) = y0 + f (ξ, y (ξ))dξ − y0 − f (ξ, y0 )dξ =x0x0 x ZxZZx [1][1] f (ξ, y (ξ))dξ − f (ξ, y0 )dξ = f (ξ, y (ξ)) − f (ξ, y0 ) dξ 6 x0x0x x0Z [1][1] f (ξ, y (ξ)) − f (ξ, y0 ) dξ 6 ïîñêîëüêó (ξ, y (ξ)), (ξ, y0 ) ∈ Ï, âîñïîëüçóåìñÿ (***) 6x0 x Z [1] [1] L y (ξ) − y0 dξ 6 èç äîêàçàííîãî âûøå |y (ξ) − y0 | 6 F |ξ − x0 | 6x0 x xZZ LF |ξ − x0 | dξ = LF |ξ − x0 | dξ =x0x0åñëè x > x0 , òîZxZx|ξ − x0 | dξ =x0Zxåñëè x < x0 , òî(x − x0 )22x0Zx|ξ − x0 | dξ = −x0(ξ − x0 )dξ =(x − x0 )2(ξ − x0 )dξ = −2= FL|x − x0 |2.2x0Ïóñòü ïðè i = n âåðíîn [n]y (x) − y [n−1] (x) 6 F Ln−1 |x − x0 | äëÿ x ∈ [x0 − h, x0 + h],n!òîãäà ïðè i = n + 1:ZxZx [n+1] [n][n+1][n]y(x) − y (x) = y0 + f (ξ, y(ξ))dξ − y0 − f (ξ, y (ξ))dξ =x0x0 xZ ZxZx [n+1][n][n+1][n] f (ξ, yf (ξ, y(ξ)) − f (ξ, y (ξ)) dξ 6(ξ))dξ − f (ξ, y (ξ))dξ = x0xx0 0xZ [n+1][n] f (ξ, y(ξ)) − f (ξ, y (ξ)) dξ 6x0[n+1][n]ïîñêîëüêó (ξ, y(ξ)), (ξ, y (ξ)) ∈ Ï, âîñïîëüçóåìñÿ (***) 628 x Z [n+1][n] L y(ξ) − y (ξ) dξ 6 èç ïðåäïîëîæåíèÿ èíäóêöèè 6x0 x xZZnn+1|ξ−x|0n LF Ln−1 = F Ln |ξ − x0 | dξ = F Ln |x − x0 | .dξn!(n + 1)!x0x0Èíäóêöèîííîå ïðåäïîëîæåíèå äîêàçàíî.
Îòñþäà ñðàçó ñëåäóåò îöåíêà|y [i] (x) − y [i−1] (x)| 6 F Li−1Ðÿä∞Pbi ñ bi = F Li−1i=1∞Xi=1hiäëÿ x ∈ [x0 − h, x0 + h], i > 1.i!hiñõîäèòñÿ:i!bi =∞XFL∞F X Li hiF Lh==e −1 .i!L i=1 i!Lii−1 hi=1Çíà÷èò, ïîñëåäîâàòåëüíîñòü {y [k] (x)} ðàâíîìåðíî ñõîäèòñÿ ê íåêîòîðîé íåïðåðûâíîéôóíêöèè y(x) ïðè k → ∞ äëÿ x ∈ [x0 − h, x0 + h].Â) Ïðåäåë ïîñëåäîâàòåëüíîñòè {y [k] (x)} åñòü ðåøåíèå ÇÊ(x0 , y0 ) äëÿ x ∈[x0 − h, x0 + h].Ïî îïðåäåëåíèþ ïîñëåäîâàòåëüíîñòè {y [k] (x)}Zx[k]y (x) = y0 +f (ξ, y [k−1] (ξ))dξ.x0Ïðè k → ∞ ëåâàÿ ÷àñòü ðàâåíñòâà ðàâíîìåðíî ñõîäèòñÿ ê y(x) :y [k] (x) ⇒ y(x), äëÿ x ∈ [x0 − h, x0 + h].Ïðàâàÿ ÷àñòü ïðè ýòîì òàêæå ðàâíîìåðíî ñõîäèòñÿ:ZxZxf (ξ, y [k−1] (ξ))dξ ⇒ f (ξ, y(ξ))dξ, äëÿ x ∈ [x0 − h, x0 + h].x0x0Äîêàçûâàåòñÿ ýòî ïî îïðåäåëåíèþ ðàâíîìåðíîé ñõîäèìîñòè. Ïîñëåäîâàòåëüíîñòü {y [k] (x)}ðàâíîìåðíî ñõîäèòñÿ ê ôóíêöèè y(x)äëÿ x ∈ [x0 − h, x0 + h], ò.å.
äëÿ ëþáîãî x ∈e (ee (e[x0 −h, x0 +h] äëÿ ïðîèçâîëüíîãî εe íàéä¼òñÿ íîìåð Nε) òàêîé, ÷òî äëÿ âñåõ k > Nε)k|y (x) − y(x)| 6 εe. Ïîêàæåì ðàâíîìåðíóþ ñõîäèìîñòü èíòåãðàëîâ. Ðàññìîòðèì x xZ ZZx [k−1] f (ξ, y [k−1] (ξ))dξ − f (ξ, y(ξ))dξ = f (ξ, y(ξ) − f (ξ, y(ξ)) dξ 6 x0x0x0 xZ f (ξ, y [k−1] (ξ) − f (ξ, y(ξ)) dξ 6x029 Zxïîñêîëüêó (ξ, y [k−1] (ξ)), (ξ, y(ξ)) ∈ Ï, âîñïîëüçóåìñÿ (***) 6 L y [k−1] (ξ) − y(ξ) dξ .x0Åñëè |y k (x) − y(x)| 6 εe, òî x x xZZZZx = Le f (ξ, y [k−1] (ξ))dξ − f (ξ, y(ξ))dξ 6 Ledξε|x − x0 |.εdξ=Leεx0x0x0x0Òåïåðü äëÿ ëþáîãî x ∈ [x0 −h, x0 +h] äëÿ ïðîèçâîëüíîãî ε âîçüì¼ì íîìåð N = N (ε) =e ε , òîãäà äëÿ âñåõ k > N (ε)NLh x x ZZZx [k−1] [k−1] f (ξ, y(ξ) − y(ξ) dξ 6(ξ))dξ − f (ξ, y(ξ))dξ = L y x0x0x0 xZε L dξ = ε |x − x0 | 6 ε h = ε.Lh hhx0Ò.å. èíòåãðàëû ðàâíîìåðíî ñõîäÿòñÿ. ðåçóëüòàòå, ïîëó÷àåì ðàâåíñòâîZxy(x) = y0 +f (ξ, y(ξ))dξ ïðè x ∈ [x0 − h, x0 + h],x0çíà÷èò, y(x) ÿâëÿåòñÿ ðåøåíèåì èíòåãðàëüíîãî óðàâíåíèÿ, à ïî ëåììå 1 è ðåøåíèåìçàäà÷è Êîøè.Ëåììà 2 äîêàçàíà.III.
Ëîêàëüíàÿ åäèíñòâåííîñòü ðåøåíèÿ çàäà÷è ÊîøèÏóñòü∈ C(D), D íåïóñòîå îòêðûòîå ìíîæåñòâî, òî÷êà (x0 , y0 ) ∈ D,f ∈ C(D),ôóíêöèÿ ϕ(x) ðåøåíèå çàäà÷è Êîøè(x0, y0), îïðåäåë¼ííîå íà hc, di, ãðàôèê êîòîðîãî öåëèêîì â ïðÿìîóãîëüíèêå Ï,ôóíêöèÿ ψ(x) ðåøåíèå çàäà÷è Êîøè(x0, y0), îïðåäåë¼ííîå íà hec, die , ãðàôèê êîòîðîãî öåëèêîì â ïðÿìîóãîëüíèêå Ïòîãäàe.ϕ(x) = ψ(x) ïðè x ∈ hc, di ∩ hec, diËåììà 3.∂f∂yÄîêàçàòåëüñòâî.e.Ðàññìîòðèì ðàçíîñòü ϕ(x)−ψ(x) è ïîêàæåì, ÷òî îíà ðàâíà 0 ïðè x ∈ hc, di∩hec, diÏîñêîëüêó ϕ(x) ðåøåíèå çàäà÷è Êîøè(x0 , y0 ), òî îíà óäîâëåòâîðÿåò èíòåãðàëüíîìóóðàâíåíèþZxϕ(x) = y0 + f (ξ, ϕ(ξ))dξ ïðè x ∈ hc, di,x030ïîñêîëüêó ψ(x) ðåøåíèå çàäà÷è Êîøè(x0 , y0 ), òî îíà óäîâëåòâîðÿåò èíòåãðàëüíîìóóðàâíåíèþZxψ(x) = y0 + f (ξ, ψ(ξ))dξ ïðè x ∈ hc, di,x0òîãäà èìååì îöåíêó x x ZZZx |ϕ(x) − ψ(x)| = f (ξ, ϕ(ξ))dξ − f (ξ, ψ(ξ))dξ = (f (ξ, ϕ(ξ)) − f (ξ, ψ(ξ))) dξ 6 x0x0x0 xZ |f (ξ, ϕ(ξ)) − f (ξ, ψ(ξ))| dξ 6x0eãðàôèêè ôóíêöèé ϕ(x) è ψ(x) ëåæàò â Ï ïðè x ∈ hc, di∩hec, di , òî âîñïîëüçóåìñÿ (***) xZ6 L |ϕ(ξ) − ψ(ξ)| dξ .x0Ðàññìîòðèì ñëó÷àé x0 6 x.
Îáîçíà÷èìZx|ϕ(ξ) − ψ(ξ)| dξ,u(x) =x0òîãäà u(x0 ) = 0, u(x) > 0 è u0 (x) = |ϕ(x) − ψ(x)|, à ïîäñòàâëÿÿ íîâîå îáîçíà÷åíèå âïîñëåäíåå íåðàâåíñòâî, ïîëó÷èìeu0 (x) 6 Lu(x) ïðè x ∈ hc, di ∩ hec, di,eu0 (x) − Lu(x) 6 0 ïðè x ∈ hc, di ∩ hec, di,eu0 (x)e−hx − Lu(x)e−hx 6 0 · e−hx ïðè x ∈ hc, di ∩ hec, di,0eu(x)e−hx 6 0 ïðè x ∈ hc, di ∩ hec, di.e , òîãäà äëÿ x0 6 x èìååìÇíà÷èò, ôóíêöèÿ u(x)e−hx óáûâàåò ïðè x ∈ hc, di ∩ hec, dieu(x0 )e−hx0 > u(x)e−hx ïðè x ∈ hc, di ∩ hec, di,e0 > u(x)e−hx ïðè x ∈ hc, di ∩ hec, di,e0 > u(x) ïðè x ∈ hc, di ∩ hec, di.Âîçâðàùàÿñü ê ïåðâîíà÷àëüíûì îáîçíà÷åíèÿìZx0>e|ϕ(ξ) − ψ(ξ)| dξ ïðè x ∈ hc, di ∩ hec, di.x031Ïîñêîëüêó x0 6 x è |ϕ(ξ) − ψ(ξ)| > 0, òî èç ïîëó÷åííîãî íåðàâåíñòâà ïîëó÷àåòñÿåäèíñòâåííûé âàðèàíòe|ϕ(ξ) − ψ(ξ)| = 0 ïðè x ∈ hc, di ∩ hec, di.Ëåììà 3 äîêàçàíà.IV.
Íåïðîäîëæàåìîå ðåøåíèå çàäà÷è Êîøè îïðåäåëåíî íà îòêðûòîìèíòåðâàëå.Ïóñòüf ∈ C(D),∈ C(D), D íåïóñòîå îòêðûòîå ìíîæåñòâî, òî÷êà (x0 , y0 ) ∈ D,ôóíêöèÿ y(x) íåïðîäîëæàåìîå ðåøåíèå çàäà÷è Êîøè(x0, y0), îïðåäåë¼ííîå íà hα, ωi,òîãäàhα, ωi íå ñîäåðæèò êîíöû: hα, ωi = (α, ω).Ëåììà 4.∂f∂yÄîêàçàòåëüñòâî.Ïðåäïîëîæèì ïðîòèâíîå, ïóñòü ω ∈ hα, ωi, hα, ωi = hα, ω]. Ïî îïðåäåëåíèþ, ãðàôèê ôóíêöèè öåëèêîì ëåæèò â D, çíà÷èò, (ω, y(ω)) ∈ D (îáîçíà÷èì y(ω) = yω ). Òîãäàïî ëåììå 2 ÷åðåç (ω, yω ) îáÿçàòåëüíî ïðîõîäèò ñâî¼ ðåøåíèå óðàâíåíèÿ y 0 = f (x, y),îáîçíà÷èì åãî ye(x), îïðåäåëåíî îíî íà íåêîòîðîì èíòåðâàëü÷èêå [ω − eh, ω + eh], ïðè÷¼ìye(ω) = yω .
Ðàññìîòðèì ôóíêöèþy(x) ïðè x ∈ hα, ω],z(x) =ye(x) ïðè x ∈ (ω, ω + eh].Ýòà ôóíêöèÿ, âî-ïåðâûõ, ÿâëÿåòñÿ ãëàäêîé íà hα, ω + eh]. Âîïðîñ ãëàäêîñòè âîçíèêàåò0òîëüêî ïðè x = ω , íî â íåé ïðåäåëû z ñëåâà è ñïðàâà ñîâïàäàþò:z 0 (x) = y 0 (x) = f (x, y(x)) = f (ω, y(ω)),x→ω−00z (x) x→ω+0x→ω−00= ye (x) x→ω+0x→ω−0= f (x, ye(x)) = f (ω, y(ω)).x→ω+0Âî-âòîðûõ, z(x) ÿâëÿåòñÿ ðåøåíèåì óðàâíåíèÿ y 0 = f (x, y) ïðè hα, ω + eh]:0z (x) ≡y(x) ïðè x ∈ hα, ω],=ye(x) ïðè x ∈ (ω, ω + eh]f (x, y(x)) ïðè x ∈ hα, ω],= f (x, z(x)) ïðè x ∈ hα, ω + eh].f (x, ye(x)) ïðè x ∈ (ω, ω + eh]Â-òðåòüèõ, z(x0 ) = y(x0 ) = y0 .
![](https://s.studizba.com/z.php?f=/templates/si/i/noavatar.png&w=100&h=100&t=1"){kind=avatar}
