PFA lection 5 (А.С. Терсенов - Лекции (2021г))

Ïðèëîæåíèå òåîðåìû ØàóäåðàÍîâîñèáèðñêèé ãîñóäàðñòâåííûé óíèâåðñèòåòÌåõàíèêî-ìàòåìàòè÷åñêèé ôàêóëüòåòÊàôåäðà ïðèêëàäíîé ìàòåìàòèêèÊóðñ ëåêöèé ïî ïðèêëàäíîìóôóíêöèîíàëüíîìó àíàëèçó. ËÅÊÖÈß 5À.Ñ. Òåðñåíîââ ðàìêàõ ðåàëèçàöèè Ïðîãðàììû ðàçâèòèÿ ÍÈÓ-ÍÃÓ íà2009-2018 ã.2017Ïðèëîæåíèå òåîðåìû ØàóäåðàÐàññìîòðèì äèôôåðåíöèàëüíîå óðàâíåíèåx 00 (t) = f (t, x, x 0 ),(1)óäîâëåòâîðÿþùåå êðàåâûì óñëîâèÿx(0) = 0,x(p) = 0.(2)Ðàññìîòðèì òàêæå óðàâíåíèåx 00 (t) = g (t)(3)ñ òåìè æå êðàåâûìè óñëîâèÿìè (2).Oäíîðîäíàÿ êðàåâàÿ çàäà÷àx 00 = 0,x(0) = 0,x(p) = 0èìååò òîëüêî òðèâèàëüíîå ðåøåíèå.

Äåéñòâèòåëüíî, ðåøåíèåóðàâíåíèÿ èìååò âèä x = c1 t + c2 , ãäå c1 , c2 ïîñòîÿííûå.Ïîäñòàâëÿÿ ýòó ôóíêöèþ â êðàåâûå óñëîâèÿ, ìû ëåãêîïîëó÷àåì c1 = c2 = 0.Ïðèëîæåíèå òåîðåìû ØàóäåðàÎòñþäà íåìåäëåííî âûòåêàåò, ÷òî ðåøåíèå çàäà÷è (3), (2)x 00 (t) = g (t),x(0) = 0,x(p) = 0åäèíñòâåííî. Äëÿ äîêàçàòåëüñòâà ýòîãî ôàêòà ïðåäïîëîæèì,÷òî ñóùåñòâóåò äâà ðåøåíèÿ x1 (t) è x2 (t). Ðàññìîòðèì èõðàçíîñòü x(t) = x1 (t) − x2 (t).

Îíà óäîâëåòâîðÿåò îäíîðîäíîéêðàåâîé çàäà÷åx 00 = 0,x(0) = 0,x(p) = 0è, ñëåäîâàòåëüíî, x(t) ≡ 0. Îòêóäà x1 (t) ≡ x2 (t).Ïðèëîæåíèå òåîðåìû ØàóäåðàÔóíêöèÿ Ãðèíà ëèíåéíîãî äèôôåðåíöèàëüíîãî îïåðàòîðàÏóñòü L ëèíåéíûé äèôôåðåíöèàëüíûé îïåðàòîð âòîðîãîïîðÿäêà, ïîðîæäåííûé äèôôåðåíöèàëüíûì âûðàæåíèåìl(x) = c0 (t)x 00 (t) + c1 (t)x 0 (t) + c2 (t)x(t)(4)è êðàåâûìè óñëîâèÿìèα11 x(0) + α12 x 0 (0) + α21 x(p) + α22 x 0 (p) = 0.(5)β11 x(0) + β12 x 0 (0) + β21 x(p) + β22 x 0 (p) = 0.(6)Ïðèëîæåíèå òåîðåìû ØàóäåðàÔóíêöèÿ Ãðèíà ëèíåéíîãî äèôôåðåíöèàëüíîãî îïåðàòîðàÎïðåäåëåíèå 5.1.Ôóíêöèåé Ãðèíà îïåðàòîðà L íàçûâàåòñÿ ôóíêöèÿ G (t, s),óäîâëåòâîðÿþùàÿ ñëåäóþùèì óñëîâèÿìG (t, s) íåïðåðûâíà äëÿ âñåõ t , s èç [0, p].Ïðè ëþáîì ôèêñèðîâàííîì s ∈ (0, p) ôíóêöèÿ G (t, s)èìååò íåïðåðûâíûå ïðîèçâîäíûå ïåðâîãî è âòîðîãîïîðÿäêîâ ïî t â êàæäîì èç èíòåðâàëîâ [0, s), (s, p], ïðè÷åìïðîèçâîäíàÿ ïåðâîãî ïîðÿäêà èìååò ïðè t = s ñêà÷îêG 0 (s + 0, s) − G 0 (s − 0, s) =1.c0 (s) êàæäîì èç èíòåðâàëîâ [0, s), (s, p] ôóíêöèÿ G (t, s)ðàññìàòðèâàåìàÿ êàê ôóíêöèÿ îò t , óäîâëåòâîðÿåòóðàâíåíèþ (4) l(G ) = 0 è êðàåâûì óñëîâèÿì (5), (6).Ïðèëîæåíèå òåîðåìû ØàóäåðàÎáðàùåíèå äèôôåðåíöèàëüíîãî îïåðàòîðà ñ ïîìîùüþ ôóíêöèèÃðèíàÒåîðåìà 5.2.Åñëè çàäà÷à Lx = 0 èìååò òîëüêî òðèâèàëüíîå ðåøåíèå, òî äëÿëþáîé ôóíêöèè g (t), íåïðåðûâíîé íà îòðåçêå [0, p], ñóùåñòâóåòåäèíñòâåííîå ðåøåíèå çàäà÷è Lx = g .

Ýòî ðåøåíèå äàåòñÿ ôîðìóëîéZp(7)x(t) = G (t, s)g (s) ds0Ïðèëîæåíèå òåîðåìû ØàóäåðàÔóíêöèÿ Ãðèíà ëèíåéíîé çàäà÷èÑ ïîìîùüþ ôóíêöèè Ãðèíà ëåãêî çàïèñàòü ðåøåíèå çàäà÷è (3),(2 )ZtZp1x(t) = − (p − t) s g (s)ds + t (p − s)g (s)ds  .(8)p0tÔóíêöèÿ Ãðèíà èìååò âèä(1p (p − t)s, 0 ≤ s ≤ t ≤ p,G (t, s) =1p (p − s)t, 0 ≤ t ≤ s ≤ p.(9)Òàêèì îáðàçîì, ìîæíî çàïèñàòüZpx(t) = −G (t, s)g (s)ds,0x 0 (t) = −ZpGt (t, s)g (s)ds.0(10)Ïðèëîæåíèå òåîðåìû ØàóäåðàÎöåíêè ðåøåíèÿ ÷åðåç ôóíêöèþ ÃðèíàÈç ïðåäñòàâëåíèé (9), (10) âûòåêàþò ñëåäóþùèå îöåíêèZpp0 ≤ G (t, s) ≤ ,4Zp01p2G (t, s)ds = t(p − t) ≤ ,28ZpZt0|Gt (t, s)|ds =|Gt (t, s)|ds +|Gt (t, s)|ds =0t1 2p[t + (p − t)2 ] ≤ .2p2Èç ïîëó÷åííûõ âûøå îöåíîê è èç (10) âûòåêàþò ñëåäóþùèåîöåíêè íà x è x 0|x(t)| ≤p2max |g (s)|,8 s∈[0,p]|x 0 (t)| ≤pmax |g (s)|.2 s∈[0,p](11)Ïðèëîæåíèå òåîðåìû ØàóäåðàÒåîðåìà ñóùåñòâîâàíèÿÒåîðåìà 5.3.Ïóñòü f (t, x, x 0 ) íåïðåðûâíà è îãðàíè÷åíà|f (t, x, x 0 )| ≤ mïðè 0 ≤ t ≤ p è âñåõ (x, x 0 ).

Òîãäà çàäà÷à (1), (2) èìååò ïîêðàéíåé ìåðå îäíî ðåøåíèå, óäîâëåòâîðÿþùåå îöåíêàì|x(t)| ≤mp 2,8|x 0 (t)| ≤mp.2Çàìå÷àíèå. Äîñòàòî÷íî, ÷òîáû ôóíêöèÿ f (t, x, x 0 ) áûëà2mp0îïðåäåëåíà òîëüêî äëÿ |x(t)| ≤ mp8 , |x (t)| ≤ 2 .Ïðèëîæåíèå òåîðåìû ØàóäåðàÏëàí äîêàçàòåëüñòâà Òåîðåìû 5.3.Çàìåíà èñõîäíîé íåëèíåéíîé çàäà÷è ëèíåéíîé çàäà÷åé ñïðàâîé ÷àñòüþ, êîòîðàÿ ïîðîæäàåòñÿ ôóíêöèÿìè, ïðèíàäëåæàùèìè íåêîòîðîìó êëàññó ôóíêöèé, îáðàçóþùèõ âûïóêëîå ìíîæåñòâî.Ïîñòðîåíèå âçàèìíîîäíîçíà÷íîãî îòîáðàæåíèÿ T , ïåðåâîäÿùåãî ýòè ôóíêöèè â ðåøåíèÿ ëèíåéíîé çàäà÷è.Ëèíåéíàÿ çàäà÷à èçíà÷àëüíî êîíñòðóèðóåòñÿ òàê, ÷òîáûíåïîäâèæíàÿ òî÷êà îòîáðàæåíèÿ T ÿâëÿëàñü ðåøåíèåìèñõîäíîé íåëèíåéíîé çàäà÷è.Äîêàçûâàåì, ÷òî îòîáðàæåíèå T ÿâëÿåòñÿ âïîëíå íåïðåðûâíûì îòîáðàæåíèåì.Ïðèìåíÿåì âòîðóþ Òåîðåìó Øàóäåðà.Ïðèëîæåíèå òåîðåìû ØàóäåðàÄîêàçàòåëüñòâî Òåîðåìû 5.3.Ïóñòü D áàíàõîâî ïðîñòðàíñòâî íåïðåðûâíî äèôôåðåíöèðóåìûõ íà 0 ≤ t ≤ p ôóíêöèé h(t) ñ íîðìîép0||h|| = max max |h(t)|, max |h (t)| .0≤t≤p4 0≤t≤p2Ðàññìîòðèì ôóíêöèþ h(t) â øàðå ||h(t)|| ≤ mp8 .

Êàæäîéòàêîé ôóíêöèè ïîñòàâèì â ñîîòâåòñòâèå x(t), êîòîðûé ÿâëÿåòñÿ ðåøåíèåì çàäà÷èx 00 (t) = f (t, h, h0 ),x(0) = 0,x(p) = 0.Ðåøåíèå ýòîé çàäà÷è åäèíñòâåííî =⇒ ìû ïîñòðîèëèîòîáðàæåíèåT : h 7−→ x.(12)Ïðèëîæåíèå òåîðåìû ØàóäåðàÑâîéñòâà îòîáðàæåíèÿTÈññëåäóåì òåïåðü ñâîéñòâà ýòîãî îòîáðàæåíèÿ.Èç óñëîâèÿ |f (t, x, x 0 )| ≤ m ïðè 0 ≤ t ≤ p è âñåõ (x, x 0 ), è,â ñèëó îöåíêè (11), ìû ïîëó÷àåì, ÷òî ðåøåíèÿ çàäà÷è (12)óäîâëåòâîðÿþò íåðàâåíñòâàì|x(t)| ≤mp 2,8|x 0 (t)| ≤mp,2ñëåäîâàòåëüíî, èìååì, ÷òîT : ||h|| ≤mp 2mp 27−→ ||x|| ≤,882ò.å.

îòîáðàæåíèå T ïåðåâîäèò øàð ðàäèóñà mp8 ïðîñòðàíñòâà D â ñåáÿ (ò.å. âûïóêëîå ìíîæåñòâî â ñåáÿ) .Ïðèëîæåíèå òåîðåìû ØàóäåðàÑâîéñòâà îòîáðàæåíèÿTÎñòàëîñü ïîêàçàòü, ÷òî T íåïðåðûâíîå êîìïàêòíîå îòîáðàæåíèå è ïðèìåíèòü âòîðóþ òåîðåìó Øàóäåðà.Ðàññìîòðèì h1 (t) è h2 (t) òàêèå, ÷òî||h1 (t)|| ≤mp 2,8||h2 (t)|| ≤mp 2,8x1 = T (h1 ),x2 = T (h2 ).Òîãäà èç (10) è îöåíîê ôóíêöèè Ãðèíà ïîëó÷àåìZp|x1 − x2 | ≤|G (t, s)||f (s, h1 (s), h10 (s)) − f (s, h2 (s), h20 (s))|ds ≤0p4Zp0|f (s, h1 (s), h10 (s)) − f (s, h2 (s), h20 (s))|ds.(13)Ïðèëîæåíèå òåîðåìû ØàóäåðàÑâîéñòâà îòîáðàæåíèÿ00Zp|x1 (t)−x2 (t)| ≤T|Gt (t, s)||f (s, h1 (s), h10 (s))−f (s, h2 (s), h20 (s))|ds ≤000Zpmax |f (s, h1 (s), h1 (s)) − f (s, h2 (s), h2 (s))||Gt (t, s)|ds ≤ (14)0pmax |f (s, h1 (s), h10 (s)) − f (s, h2 (s), h20 (s))|.2Èç îöåíîê (13), (14) è íåïðåðûâíîñòè ôóíêöèè f âûòåêàåò, ÷òî||T (h1 ) − T (h2 )|| −→ 0 ïðè ||h1 (s) − h2 (s)|| −→ 0è, ñëåäîâàòåëüíî, îïåðàòîð T íåïðåðûâåí íà D .Ïðèëîæåíèå òåîðåìû ØàóäåðàÒåîðåìà Àðöåëà.Ïóñòü íà êîíå÷íîì ïðîìåæóòêå I äàíî ñåìåéñòâî {x(t)}, ñîñòîÿùåå èç ôóíêöèé x(t) ðàâíîìåðíî îãðàíè÷åííûõ è ðàâíîñòåïåííî íåïðåðûâíûõ.

Òîãäà èç ýòîãî ñåì-âà ìîæíî âûáðàòü ðàâíîìåðíî ñõîäÿùóþñÿ íà I ïîñëåäîâàòåëüíîñòü ôóíêöèé.Îïðåäåëåíèå 5.4.Ñåìåéñòâî ôóíêöèé {x(t)} íàçûâàåòñÿ ðàâíîìåðíîîãðàíè÷åííûì, åñëè ñóùåñòâóåò òàêàÿ ïîñòîÿííàÿ M , ÷òî|x(t)| ≤ M ∀ x(t) ∈ {x(t)}.Îïðåäåëåíèå 5.5.Ôóíêöèè ñåìåéñòâà {x(t)} íàçûâàþòñÿ ðàâíîñòåïåííîíåïðåðûâíûìè, åñëè ∀ ε > 0 ∃ δ > 0, ÷òî ∀ x ∈ {x(t)} è ∀t1 , t2 ∈ I èìååò ìåñòî|t1 − t2 | ≤ δ ⇒ |x(t1 ) − x(t2 )| ≤ ε.Ïðèëîæåíèå òåîðåìû ØàóäåðàÑâîéñòâà îòîáðàæåíèÿTÄàëåå, äëÿ ëþáîãî x(t) èç îáëàñòè çíà÷åíèé îïåðàòîðà T , ò.å.2äëÿ x = T (h) ñ íåêîòîðûì h èç D0 = {h ∈ D, ||h|| ≤ mp8 },èìååì|x 00 (t)| ≤ m,|x 0 (t)| ≤Zp|Gt (t, s)||f (t, h, h0 )|ds ≤0mp.2Èç ýòèõ äâóõ íåðàâåíñòâ ñðàçó âûòåêàåò, ÷òî 0 x(t1 ) − x(t2 ) mp x (t1 ) − x 0 (t2 ) ≤ ≤ m,, t1 − t2 2t1 − t2îòêóäà íåìåäëåííî ïîëó÷àåì|x(t1 ) − x(t2 )| ≤mp|t1 − t2 |,2|x 0 (t1 ) − x 0 (t2 )| ≤ m|t1 − t2 |.Ïðèëîæåíèå òåîðåìû ØàóäåðàÑâîéñòâà îòîáðàæåíèÿTÒàêèì îáðàçîì,ñåìåéñòâà ôóíêöèé {x(t)} è {x 0 (t)} ðàâíîìåðíî îãðàíè÷åíû è ðàâíîñòåïåííî íåïðåðûâíû,ñëåäîâàòåëüíî, ïî òåîðåìå Àðöåëà ñåìåéñòâî {x(t)} ïðåä2êîìïàêòíî â D0 = {h ∈ D, ||h|| ≤ mp8 }.â èòîãå ìû ïîñòðîèëè îïåðàòîðT : D0 7−→ D0 ,êîòîðûé ÿâëÿåòñÿ âïîëíå íåïðåðûâíûì íà çàìêíóòîì âûïóêëîì ìíîæåñòâå =⇒ ïî âòîðîé òåîðåìå Øàóäåðà èìååòíåïîäâèæíóþ òî÷êó, êîòîðàÿ è ÿâëÿåòñÿ ðåøåíèåì çàäà÷è(1), (2)..