Ứng dụng phương pháp newton-kantorovich giải phương trình tích phân phi tuyến

ữủ tọ ỏ t ỡ t tợ t P
t ữớ ữợ ú ù t t t
t ữỡ ự ồ
tú tr ổ ừ t ú t õ ỵ tự tr
qt t t ừ
t ỡ t ồ
t Pỏ ồ rữớ
ồ sữ ở tr tử tự t ồ
ú ù tr sốt q tr ồ t ự t
ỡ trữớ
P r Pú t ồ t ủ t t ồ
t t tốt
ỷ ớ ỡ ũ ỗ ú ù
t tr sốt õ ồ t s

ở t




▲❮■ ❈❆▼ ✣❖❆◆
❚æ✐ ①✐♥ ❝❛♠ ✤♦❛♥ ❧✉➟♥ ✈➠♥ ❧➔ ❝æ♥❣ tr➻♥❤ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛ r✐➯♥❣ tæ✐ ❞÷î✐
sü ❤÷î♥❣ ❞➝♥ ❝õ❛ P●❙✳ ❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤✳
❚r♦♥❣ ❦❤✐ ♥❣❤✐➯♥ ❝ù✉ ❧✉➟♥ ✈➠♥✱ tæ✐ ✤➣ ❦➳ t❤ø❛ t❤➔♥❤ q✉↔ ❦❤♦❛ ❤å❝
❝õ❛ ❝→❝ ♥❤➔ ❦❤♦❛ ❤å❝ ✈î✐ sü tr➙♥ trå♥❣ ✈➔ ❜✐➳t ì♥✳
▼ët sè ❦➳t q✉↔ ✤➣ ✤↕t ✤÷ñ❝ tr♦♥❣ ❧✉➟♥ ✈➠♥ ❧➔ ♠î✐ ✈➔ ❝❤÷❛ tø♥❣ ✤÷ñ❝
❝æ♥❣ ❜è tr♦♥❣ ❜➜t ❦ý ❝æ♥❣ tr➻♥❤ ❦❤♦❛ ❤å❝ ♥➔♦ ❝õ❛ ❛✐ ❦❤→❝✳

❍➔ ◆ë✐✱ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✶

❚→❝ ❣✐↔


▼ö❝ ❧ö❝

▼ð ✤➛✉

✶

❈❤÷ì♥❣ ✶✳ ❑✐➳♥ t❤ù❝ ❜ê trñ

✹

✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹

✶✳✷✳ ◆❣✉②➯♥ ❧þ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✸✳ ❚♦→♥ tû t✉②➳♥ t➼♥❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✹✳ ✣↕♦ ❤➔♠ ❋r➨❝❤❡t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✾

✶✳✺✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✷

✶✳✻✳ P❤÷ì♥❣ ♣❤→♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❧✐➯♥ q✉❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✻✳✶✳ P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✹

✶✳✻✳✷✳ P❤÷ì♥❣ ♣❤→♣ ♥❤➙♥ s✉② ❜✐➳♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✺

❈❤÷ì♥❣ ✷✳ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤

✶✽

✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❧➔♠ trë✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✽

✷✳✶✳✶✳ ❚♦→♥ tû ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✾

✷✳✶✳✷✳ ❚♦→♥ tû ❦❤æ♥❣ ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✸

✷✳✷✳ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻

✷✳✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤

✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✻


✐✐✐

✷✳✷✳✷✳ ▼ët sè ✤à♥❤ ❧þ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲
❑❛♥t♦r♦✈✐❝❤ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✷✼

❈❤÷ì♥❣ ✸✳ Ù♥❣ ❞ö♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✸✼
✸✳✶✳ Ù♥❣ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✲❑❛♥t♦r♦✈✐❝❤ ❣✐↔✐ ♣❤÷ì♥❣
tr➻♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼

✸✳✶✳✶✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ ❋r❡❞❤♦❧♠ ❞↕♥❣
❯r②s♦❤♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✸✼

✸✳✶✳✷✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ ❱♦❧t❡rr❛ ✳ ✳ ✳ ✳

✸✾

✸✳✶✳✸✳ ❚❤✉➟t t♦→♥ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥
t❤❡♦ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✲❑❛♥t♦r♦✈✐❝❤ ✳ ✳ ✳ ✳ ✳ ✳

✹✶

✸✳✷✳ ❱➼ ❞ö ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✹✷

✸✳✸✳ Ù♥❣ ❞ö♥❣ ❣✐↔✐ ❣➛♥ ✤ó♥❣ tr➯♥ ♠→② t➼♥❤ ✤✐➺♥ tû ❜➡♥❣ ♣❤➛♥
♠➲♠ ▼❛♣❧❡ ✶✸

✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺✼

❑➳t ❧✉➟♥

✻✶

❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

✻✷


ị

C

số ự

C[a;b]

tt số tỹ tử tr [a, b]

L2[a;b]

tt số tỹ ữỡ t tr [a, b]

Dk[a;b]

tt số õ tử
k tr [a, b]

l2

tt ỳ số tỹ ự x = (xn ) s


ộ

|xn |2 ở tử

n=1

L(X, Y ) tt t tỷ t t tử tứ X Y
N

số tỹ

N

số tỹ ổ

R

số tỹ

Rk

ổ tỹ k

ỉ

ủ rộ

.


t tú ự


é
ỵ ồ t
sữ tr t tữ t 1975 ợ
ổ tr ự

P ố tố ữ ỗ ỹ ổ ụ

ự t ồ ợ ỹ ỵ tt ự
ử ợ ữ t tr ổ
ữỡ rở ữỡ t
ữỡ tr t f (x) = 0 ữỡ tr

F (x) = 0



ữủ ỳ ổ F : X Y t ữỡ
rở õ ỹ ữủ ở tử tợ ừ ữỡ
tr

xn+1 = xn [F (xn )]1 F (xn ), n = 0, 1, 2, ...



ữỡ ữủ ồ ữỡ ttr
r ỵ t tr r ữủ
ừ tr x0 ở tử ừ ữỡ
tr
õ õ ừ tr sỷ ử ổ ử t
qt t ừ t số ụ ữ r ữủ
ổ tự tờ qt ừ t ợ ự ử ữỡ tr
t ữỡ tr t t


✷

❙❛✉ ❦❤✐ ❋r❡❞❤♦❧♠ ✤÷❛ r❛ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ tr♦♥❣
♠ët ❜➔✐ ❜→♦ ❝õ❛ ♠➻♥❤ ♥➠♠ 1903✱ ❧þ t❤✉②➳t t♦→♥ tû t➼❝❤ ♣❤➙♥ ✤➣ ♣❤→t
tr✐➸♥ r➜t ♠↕♥❤ ♠➩ ✈➔ ❝â ù♥❣ ❞ö♥❣ tr♦♥❣ ♥❤✐➲✉ ❧➽♥❤ ✈ü❝✱ tr♦♥❣ ✤â ❝â ❝↔
❧þ t❤✉②➳t ❝❤✉é✐ ❋♦✉r✐❡r ✈➔ t➼❝❤ ♣❤➙♥ ❋♦✉r✐❡r✳ ❚✉② ✈➟②✱ t♦→♥ tû t➼❝❤ ♣❤➙♥
t✉②➳♥ t➼♥❤ ✈➝♥ ❝❤÷❛ ✤→♣ ù♥❣ ✤÷ñ❝ ♠ët sè ❜➔✐ t♦→♥ tr♦♥❣ t❤ü❝ t➳ ❝ô♥❣
♥❤÷ tr♦♥❣ ❝→❝ ♥❣➔♥❤ ❦❤♦❛ ❤å❝ ❦❤→❝✳ ▼ët t❤í✐ ❣✐❛♥ s❛✉ ✤â ❧þ t❤✉②➳t t♦→♥
tû t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✤÷ñ❝ ✤➲ ❝➟♣ ✈➔ ✤➣ ✤→♣ ù♥❣ ✤÷ñ❝ ②➯✉ ❝➛✉ ❝õ❛
❚♦→♥ ❤å❝ ✈➔ t❤ü❝ t✐➵♥✳
❱✐➺❝ ❣✐↔✐ ①➜♣ ①➾ ❝→❝ ❜➔✐ t♦→♥ ❝â þ ♥❣❤➽❛ t❤ü❝ t➳ q✉❛♥ trå♥❣✱ ✤➦❝ ❜✐➺t
tr♦♥❣ ❣✐❛✐ ✤♦↕♥ ❤✐➺♥ ♥❛② ✈î✐ sü ❤é trñ ❝õ❛ ♠→② t➼♥❤ ✤✐➺♥ tû ✈✐➺❝ ♥➔②
❝➔♥❣ trð ❧➯♥ ❝â ❤✐➺✉ ❧ü❝✳ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✲❑❛♥t♦r♦✈✐❝❤ ①➙② ❞ü♥❣
✤÷ñ❝ ❞➣② ①➜♣ ①➾ ❤ë✐ tö ✤➳♥ ♥❣❤✐➺♠ ✈î✐ tè❝ ✤ë ❤ë✐ tö tèt✱ ❝â t❤✉➟t t♦→♥
rã r➔♥❣✱ ❝â t❤➸ ❝➔✐ ✤➦t ✤÷ñ❝ ❝→❝ ❝❤÷ì♥❣ tr➻♥❤ ❝❤♦ ♠→② t➼♥❤ ✤✐➺♥ tû t❤ü❝
❤✐➺♥✳
❱î✐ ❝→❝ ❧þ ❞♦ ♥❤÷ tr➯♥ ❝❤ó♥❣ tæ✐ ♠♦♥❣ ♠✉è♥ ✤÷ñ❝ t➻♠ ❤✐➸✉✱ ♥❣❤✐➯♥
❝ù✉ s➙✉ ❤ì♥ ✈➲ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✲❑❛♥t♦r♦✈✐❝❤ ✈➔ ù♥❣ ❞ö♥❣ ✈➔♦ ❣✐↔✐
♠ët ❧î♣ ❜➔✐ t♦→♥ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❦❤♦❛ ❤å❝ tü ♥❤✐➯♥✱ ❦✐♥❤ t➳✱ ❦ÿ
t❤✉➟t ✲ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥✳ ❉÷î✐ sü ✤à♥❤ ❤÷î♥❣ ✈➔ ❤÷î♥❣
❞➝♥ ❝õ❛ P●❙✳ ❚❙✳ ❑❤✉➜t ❱➠♥ ◆✐♥❤✱ ❝❤ó♥❣ tæ✐ q✉②➳t ✤à♥❤ ❝❤å♥ ✤➲ t➔✐
✧Ù♥❣ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✲❑❛♥t♦r♦✈✐❝❤
❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ ✧

❧➔♠ ✤➲ t➔✐ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣ t❤↕❝ s➽ ♥❣➔♥❤ ❚♦→♥ ❣✐↔✐ t➼❝❤✳


✸

✷✳ ▼ö❝ ✤➼❝❤ ♥❣❤✐➯♥ ❝ù✉
❚r➻♥❤ ❜➔② ❧þ t❤✉②➳t ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥✲❑❛♥t♦r♦✈✐❝❤ ✈➔ ù♥❣

❞ö♥❣ ✤➸ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥✱ ✤ç♥❣ t❤í✐ ♥❣❤✐➯♥ ❝ù✉
❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ tr➯♥ ♠→② t➼♥❤ ✤✐➺♥ tû✳

✸✳ ◆❤✐➺♠ ✈ö ♥❣❤✐➯♥ ❝ù✉
✲ P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✕ ❑❛♥t♦r♦✈✐❝❤✳
✲ Ù♥❣ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✕ ❑❛♥t♦r♦✈✐❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥✳

✹✳ ✣è✐ t÷ñ♥❣ ✈➔ ♣❤↕♠ ✈✐ ♥❣❤✐➯♥ ❝ù✉
Ù♥❣ ❞ö♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤
♣❤➙♥ ♣❤✐ t✉②➳♥✳

✺✳ P❤÷ì♥❣ ♣❤→♣ ♥❣❤✐➯♥ ❝ù✉
✲ ◆❣❤✐➯♥ ❝ù✉ ❧þ ❧✉➟♥✱ t➔✐ ❧✐➺✉ ❝❤✉②➯♥ ❦❤↔♦✳
✲ P❤➙♥ t➼❝❤✱ tê♥❣ ❤ñ♣ ❦✐➳♥ t❤ù❝✳

✻✳ ✣â♥❣ ❣â♣ ♠î✐ ❝õ❛ ❧✉➟♥ ✈➠♥
✲ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ tr➯♥ ♠→② t➼♥❤ ✤✐➺♥ tû✳


❈❤÷ì♥❣ ✶
❑✐➳♥ t❤ù❝ ❜ê trñ

✶✳✶✳ ❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✶✳ ✭❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✮ ▼ët ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥

(❤❛②

❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ ✤à♥❤ ❝❤✉➞♥) ❧➔ ❦❤æ♥❣ ❣✐❛♥ t✉②➳♥ t➼♥❤ X tr➯♥
tr÷í♥❣ P (P = R ❤♦➦❝ P = C) ❝ò♥❣ ✈î✐ ♠ët →♥❤ ①↕ X → R✱ ✤÷ñ❝ ❣å✐

❧➔ ❝❤✉➞♥ ✈➔ ❦þ ❤✐➺✉ ❧➔ . t❤ä❛ ♠➣♥ ❝→❝ t✐➯♥ ✤➲ s❛✉✿
1) (∀x ∈ X) x ≥ 0✱ x = 0 ⇔ x = θ❀
2) (∀x ∈ X) (∀α ∈ P ) αx = |α| x ❀
3) (∀x, y ∈ X) x + y ≤ x + y ✳
❙è x ❣å✐ ❧➔ ❝❤✉➞♥ ❝õ❛ ✈➨❝ tì x✳ ❚❛ ❝ô♥❣ ❦þ ❤✐➺✉ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥
❧➔ X ✳ ❈→❝ t✐➯♥ ✤➲ 1), 2), 3) ❣å✐ ❧➔ ❤➺ t✐➯♥ ✤➲ ❝❤✉➞♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✷✳ ✭❙ü ❤ë✐ tö tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✮ ❉➣② ✤✐➸♠
❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X ✤÷ñ❝ ❣å✐ ❧➔ ❤ë✐ tö tî✐ ✤✐➸♠ x ∈ X
♥➳✉ n→∞
lim xn − x = 0✳ ❑þ ❤✐➺✉ lim xn = x ❤❛② xn → x (n → ∞)✳
n→∞
{xn }

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✸✳ ✭❉➣② ❝ì ❜↔♥✮ ❉➣② ✤✐➸♠ {xn} tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤
❝❤✉➞♥ X ✤÷ñ❝ ❣å✐ ❧➔ ❞➣② ❝ì ❜↔♥ ♥➳✉ n,m→∞
lim

xn − xm

= 0✳

✣à♥❤ ♥❣❤➽❛ ✶✳✶✳✹✳ ✭❑❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✮ ❑❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥ X




ữủ ồ ồ ổ ồ ỡ tr X
ở tử

ử t ổ tỡ k Rk ợ ộ x Rk
x = (x1 , x2 , ..., xk ) tr õ xi R, i = 1, 2, .., k t x =

k

|xi |2

i=1

õ R ổ
k

ử ổ l2 ỗ tt ỳ số tỹ
ự x = (xn ) s ộ



2

|xn | ở tử ợ x =

n=1



|xn |2

n=1

ổ

ử ổ tỡ C[a,b] ợ số t ý x(t) C[a,b]
t t x = max |x(t)| õ C[a,b] ổ
[a,b]

ỵ
ổ tr M1 = (X, d1) ;
M2 = (X, d2 )

ổ M1 ổ M2 ồ
tỗ t số , 0 < 1 s
d2 (Ax, Ax ) d1 (x, x ) , x, x X
r ởt tứ X õ õ t õ ỳ
ừ õ trũ ợ õ ỳ ữ t tự ỳ

x s Ax = x ồ

t ở tr t t

ở ừ ởt õ ự ử tr t t
tr ỵ tt ữỡ tr r t


✻

✈➻ ♠ët ✤✐➸♠ x ❜➜t ✤ë♥❣ tr♦♥❣ →♥❤ ①↕ A ❝❤➼♥❤ ❧➔ ❧í✐ ❣✐↔✐ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ Ax = x

✣à♥❤ ❧þ ✶✳✷✳✶✳ ✭◆❣✉②➯♥ ❧þ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤✮ ▼å✐ →♥❤ ①↕ ❝♦ A →♥❤ ①↕
❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝ ✤õ ✭✤➛② ✤õ✮ M

❜➜t ✤ë♥❣ x∗ ❞✉② ♥❤➜t✳

= (X, d)

✈➔♦ ❝❤➼♥❤ ♥â ✤➲✉ ❝â ✤✐➸♠

❈❤ù♥❣ ♠✐♥❤✳ ❳❡♠ ❬✺❪

✶✳✸✳ ❚♦→♥ tû t✉②➳♥ t➼♥❤
❈❤♦ X ✱ Y ❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ✤à♥❤ ❝❤✉➞♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳✶✳ ✭❚♦→♥ tû t✉②➳♥ t➼♥❤✮ ▼ët t♦→♥ tû A : X → Y ❣å✐
❧➔ ♠ët t♦→♥ tû t✉②➳♥ t➼♥❤ ♥➳✉ t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿
1) (∀x, y ∈ X) A (x + y) = A (x) + A (y) ;
2) (∀x ∈ X) (∀α ∈ P ) A (αx) = αA (x) .
Ð ✤➙② ✤➸ ❝❤♦ ❣å♥ t❛ ✈✐➳t Ax t❤❛② ❝❤♦ A(x)✳ ◆➳✉ X ≡ Y t❛ ♥â✐ A ❧➔
t♦→♥ tû tr♦♥❣ X ✳
❚❛ ❦þ ❤✐➺✉ ✿

ImA = {y ∈ Y | y = Ax, ∀x ∈ X} ❧➔ ♠✐➲♥ ❣✐→ trà ❝õ❛ t♦→♥ tû A❀
KerA = {x ∈ X| Ax = 0} ❧➔ ❤↕❝❤ ✭❤↕t ♥❤➙♥✮ ❝õ❛ t♦→♥ tû A✳

❱➼ ❞ö ✶✳✸✳✶✳ ❈❤♦ A : Rn → Rm ①→❝ ✤à♥❤ ❜ð✐✿
n

A (x1 , x2 , ..., xn ) = (y1 , y2 , ..., ym ) ✈î✐ yi =

aij xj , i = 1, m
j=1

✭✶✳✶✮




tr õ aij ỳ số tr (aij )mìn ồ tr ừ t
tỷ A t (1.1) tờ qt ừ ồ t tỷ t t tứ

Rn Rm
k
ử X Y D[a;b]
ổ số õ

tử k tr [a; b]

Ax (t) = a0 x (t) + a1 x (t) + ... + ak x(k) (t)
tr õ a0 , a1 , ..., ak ỳ số ỳ số trữợ
k
t tỷ t t A ồ t tỷ
ừ t tở D[a;b]

ử X

Y C[a;b] Ax (t) =

b

K (t, s)x (s) ds tr õ

a

K (t, s) tử t t, s tr ổ a t, s b
A t tỷ t t ữủ ồ t tỷ t

tỷ tử sỷ X, Y ổ
tỷ A : X Y ồ tử t x0 X
{xn } X, xn x0 (n )

t Axn Ax0 (n )

tỷ A ồ tử tr X õ tử t ồ tở X

tỷ tỷ A : X Y ồ
tỗ t ởt số K > 0 s

Ax K x , (x X)

tỷ ữủ tỷ A ồ õ t tỷ ữủ

KerA = {} tự ữỡ tr Ax = 0 õ ởt
t x = ỵ A1




t A1 t tỷ t t tứ ImA X
1) (x X) A1 Ax = x;
2) (y ImA) AA1 y = y.

ừ t tỷ ố K ọ t tr

1.3.3 ồ ừ t tỷ A ỵ
1) (x X) Ax A x
2) (x X) Ax K x t A K

A

ữ

ỵ X, Y ổ tỷ t
t A : X Y õ s tữỡ ữỡ
1) A tử
2) A tử t x0 X
3) A
ự

ỵ A : X Y t tỷ t t t tỷ A
t

A = sup Ax





x 1

A = sup Ax

x =1

ự

ỵ tỷ t t A : X Y õ t tỷ ữủ A1

tử tỗ t số > 0 s
Ax x , (x X) .






õ

A1

1


ự

rt
X, Y ổ tỷ f : X Y ổ t
tt t t

x ởt ố tr ổ
X

tỷ f : X Y ồ t rt t x tỗ
t ởt t tỷ t t tử A : X Y s :
f (x + h) f (x) = A (h) + (x, h) , (h X)

tữỡ ữỡ hlim0 f (x+h)fh(x)A(h) = 0
A(h) ồ ởt ừ t tỷ f t x ỵ df (x, h)
tỷ A ồ ởt t rt ừ f t x ỵ
f (x) df (x, h) = f (x) .h
ú ỵ r f (x) ởt t tỷ ỵ õ tr
ừ t tỷ f (x) t h ổ tr t t [f (x)] (h)


(x,h)
h
h 0

lim

= 0

ỵ ởt t tỷ ữủ tr ởt t ừ

ởt ổ rt t ởt t õ tử
t õ
ự ởt t tr ổ X
tỷ f : Y x > 0 tọ x + h õ h <


✶✵

t❤➻ f (x + h) − f (x) = A (h) + Φ (x, h) → 0 ❦❤✐ h → 0✳ ✣✐➲✉ ♥➔②
❝❤ù♥❣ tä r➡♥❣ f ❧✐➯♥ tö❝ t↕✐ x✳

✣à♥❤ ❧þ ✶✳✹✳✷✳ ✭❚➼♥❤ ❞✉② ♥❤➜t ❝õ❛ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t✮ ◆➳✉ ♠ët t♦→♥ tû
❝â ✤↕♦ ❤➔♠ t❤➻ ✤↕♦ ❤➔♠ ✤â ❧➔ ❞✉② ♥❤➜t✳
❈❤ù♥❣ ♠✐♥❤✳ ❈❤♦ X, Y

❧➔ ❤❛✐ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✳ ❱î✐ ♠é✐ x ∈ X ✱ ❣✐↔

sû A, B ❧➔ ❤❛✐ t♦→♥ tû t✉②➳♥ t➼♥❤ ❧✐➯♥ tö❝ ❝ò♥❣ ❧➔ ✤↕♦ ❤➔♠ ❝õ❛ t♦→♥ tû

f : X → Y t↕✐ x✳ ❑❤✐ ✤â (∀h ∈ X) t❛ ❝â✿
f (x + h) − f (x) = A (h) + ΦA (x, h)
f (x + h) − f (x) = B (h) + ΦB (x, h)
❙✉② r❛

A (h) − B (h) ΦB (x, h) − ΦA (x, h)
=
→ θ ❦❤✐ h → 0
h
h
◆❤÷♥❣ (∀k ∈ X)✱ (∀ε > 0) t❛ ❝â✿

A(k)−B(k)
k

=

A(εk)−B(εk)
✳

εk

❑❤✐ ε → 0

t❤➻ εk → θ ♥➯♥ ✈➳ ♣❤↔✐ ❞➛♥ tî✐ θ ❞♦ ✤â A (k) = B (k) , ∀k ∈ X ❤❛②

A≡B

✣à♥❤ ❧þ ✶✳✹✳✸✳ ❈❤♦ X, Y, Z ❧➔ ♥❤ú♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝✳ ◆➳✉
❧➔ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x ∈ X ✈➔ f : Y → Z ❦❤↔ ✈✐ ❋r➨❝❤❡t
t↕✐ y = g (x) ∈ Y t❤➻ φ = f ◦ g ❝ô♥❣ ❦❤↔ ✈✐ ❋r➨❝❤❡t t↕✐ x ✈➔ φ (x) =
f (g (x)) g (x)✳

g :X →Y

❈❤ù♥❣ ♠✐♥❤✳ ❱î✐ x, h

∈ X ✱ t❛ ❝â✿ φ (x + h) − φ (x) = f (g (x + h)) −

f (g (x)) = f (g (x + h) − g (x) + g (x)) − f (g (x)) = f (d + y) − f (y)✱




tr õ d = g (x + h) g (x) õ (x + h) (x) f (y) d =

o ( d ) tr ừ d g (x) h = o ( h ) r
(x + h) (x) f (y) g (x) h = o ( h ) + o ( d )
õ g tử t x ỵ 1.3.1 t õ d = o ( h )

(x) .h = f (g (x)) g (x) .h

ử f : R R t rt trũ ợ
t tổ tữớ

ử t f : Rn R ợ x = (x1, x2, ..., xn) h = (h1, h2, ..., hn)
n

Rn rt ừ f t x df (x, h) =

i=1

ừ f t x f (x) =

f f
f
x1 , x2 , ..., xn

f (x)
xi

hi



ử ộ t tỷ fi (x1, x2, ..., xn) : Rn R, i = 1, m
t x = (x1 , x2 , ..., xn ) t t tỷ f = (f1 , f2 , ..., fm ) : Rn Rm
t x df (x, h) = (df1 (x, h) , df2 (x, h) , ..., dfm (x, h))
ừ f tr trữớ ủ ởt tr ù m ì n ợ
ỏ tự i fi (x) f (x) =

sỷ t tỷ f

fi
xj

tr ừ f

t ồ
tở t X ữ tr ởt
t tỷ t t tử tứ X Y tự f : L (X, Y ) õ
t tỷ f t x f t x tỗ t ởt
t tỷ t t tử Q : L (X, Y ) s ợ x, k õ
f (x + k) f (x) = Q (k) + (x, k) ợ (x,k)
0 k 0
k
ợ ồ h X t õ f (x + k) .h f (x) .h = Q (k) .h + (x, k) .h
: X Y


✶✷

❤❛② df (x + k, h) − df (x, h) = Q (k) .h + Φ (x, k) .h
✣➦t Q (k, h) = Q (k) .h✱ t❛ t❤➜② Q (k, h) ❧➔ t♦→♥ tû s♦♥❣ t✉②➳♥ t➼♥❤
❧✐➯♥ tö❝ tø X × X → Y ✳
❚♦→♥ tû Q ❣å✐ ❧➔ ✤↕♦ ❤➔♠ ❝➜♣ ❤❛✐ ❝õ❛ f t↕✐ x✱ ❦þ ❤✐➺✉ ❧➔ f (x)✳
Q (k, h) ❣å✐ ❧➔ ✈✐ ♣❤➙♥ ❋r➨❝❤❡t ❝➜♣ ❤❛✐ ❝õ❛ t♦→♥ tû f t↕✐ x✱ ❦þ ❤✐➺✉ ❧➔
d2 f (x; k, h)✳ ❱➟② d2 f (x; k, h) = f (x) . (k, h)✳

✶✳✺✳ P❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥

❈❤♦ A ❧➔ t♦→♥ tû tø ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✈➔♦ ❝❤➼♥❤ ♥â✳ P❤÷ì♥❣
tr➻♥❤ t♦→♥ tû✿

Ax = f

✭✶✳✺✮

x = λAx + f

✭✶✳✻✮

tr♦♥❣ ✤â f ∈ X ❝❤♦ tr÷î❝✱ λ ❧➔ ✈æ ❤÷î♥❣ tr➯♥ tr÷í♥❣ K(K = C ❤♦➦❝ K =

R)✳
P❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❧♦↕✐ ■✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮
✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❧♦↕✐ ■■✳

• ◆➳✉ A ❧➔ t♦→♥ tû t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ t❤➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮✱ ✭✶✳✻✮
❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤✳

• ◆➳✉ A ❧➔ t♦→♥ tû t➼❝❤ ♣❤➙♥ ♥❤÷♥❣ ❦❤æ♥❣ ❣✐↔ t❤✐➳t t✉②➳♥ t➼♥❤ t❤➻
♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✺✮✱ ✭✶✳✻✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ♣❤✐ t✉②➳♥ t➼♥❤✳


✶✸

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✶✳ P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣✿
x

K[x, t, y(t)]dt = f (x)

✭✶✳✼✮

a
x

y(x) =

K[x, t, y(t)]dt + f (x)

✭✶✳✽✮

a

tr♦♥❣ ✤â K[x, t, y(t)](t, s ∈ [a, b]) ❧➔ ❤➔♠ sè ❜❛ ❜✐➳♥ ❧✐➯♥ tö❝❀ y(x) ❧➔ ❤➔♠
sè ❝❤÷❛ ❜✐➳t✱ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b]❀ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
♣❤✐ t✉②➳♥ ❱♦❧t❡rr❛✱ t÷ì♥❣ ù♥❣ ❧♦↕✐ ■ ✈➔ ❧♦↕✐ ■■ ❞↕♥❣ ❯r②s♦❤♥✳ K[x, t, y(t)]
❧➔ ♥❤➙♥ ❝õ❛ t♦→♥ tû t➼❝❤ ♣❤➙♥✳

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳✷✳ P❤÷ì♥❣ tr➻♥❤ ❞↕♥❣✿
b

K[x, t, y(t)]dt = f (x)

✭✶✳✾✮

a
b

K[x, t, y(t)]dt + f (x)

y(x) =

✭✶✳✶✵✮

a

tr♦♥❣ ✤â K[x, t, y(t)](t, s ∈ [a, b]) ❧➔ ❤➔♠ sè ❜❛ ❜✐➳♥ ❧✐➯♥ tö❝❀ y(x) ❧➔ ❤➔♠
sè ❝❤÷❛ ❜✐➳t✱ ❧✐➯♥ tö❝ tr➯♥ ✤♦↕♥ [a, b]❀ ✤÷ñ❝ ❣å✐ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥
♣❤✐ t✉②➳♥ ❋r❡❞❤♦❧♠✱ t÷ì♥❣ ù♥❣ ❧♦↕✐ ■ ✈➔ ❧♦↕✐ ■■ ❞↕♥❣ ❯r②s♦❤♥✳
❚❛ ♥â✐ ♥❤➙♥ K[x, t, y(t)] ❝õ❛ t♦→♥ tû t➼❝❤ ♣❤➙♥ ❧➔ s✉② ❜✐➳♥ ♥➳✉
n

K[x, t, y(t)] =

gi (x)hi [t, y(t)]
i=1

❱➼ ❞ö ✶✳✺✳✶✳ P❤÷ì♥❣ tr➻♥❤ x(t) = λ

1

(t + s)x(s)ds+t2 ❧➔ ♣❤÷ì♥❣ tr➻♥❤

−1

t➼❝❤ ♣❤➙♥ t✉②➳♥ t➼♥❤ ✈î✐ ♥❤➙♥ s✉② ❜✐➳♥✳


✶✹

✶✳✻✳ P❤÷ì♥❣ ♣❤→♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❧✐➯♥ q✉❛♥
✶✳✻✳✶✳ P❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣
❳➨t ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮✱ ✤➦t T x = λAx + f ✳ ❚ ❧✐➯♥ tö❝ ▲✐♣s❝❤✐t③✿

Tx − Tx

= λA(x − x ) ≤ |λ| A

x−x

X ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✈➔ ♥➳✉ |λ| A < 1 t❤➻ t❤❡♦ ♥❣✉②➯♥ ❧þ →♥❤ ①↕
❝♦✱ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ❝â ♥❣❤✐➺♠ x∗ ❞✉② ♥❤➜t✱ ♥❣♦➔✐ r❛✱ ♣❤➨♣ ❧➦♣✿

xn+1 = λAxn + f (✈î✐ n ≥ 0)

✭✶✳✶✶✮

✈î✐ ♠å✐ ①➜♣ ①➾ ❜❛♥ ✤➛✉ x0 ✤➲✉ ❤ë✐ tö ✤➳♥ ♥❣❤✐➺♠ x∗ ✱ ❤ì♥ ♥ú❛✱ ❝❤ó♥❣ t❛
❝â ❝→❝ ÷î❝ ❧÷ñ♥❣ s❛✉✿

qn
x1 − x0
1−q
q
≤
xn − xn−1
1−q

xn − x∗ ≤

xn − x∗

q✉→ tr➻♥❤ t➻♠ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✻✮ ❜➡♥❣ ❝→❝❤ ①➙② ❞ü♥❣ ❞➣②
❧➦♣ ✭✶✳✶✶✮ ❣å✐ ❧➔ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣✳

❱➼ ❞ö ✶✳✻✳✶✳ ●✐↔✐ ♣❤÷ì♥❣ tr➻♥❤ s❛✉ ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣
x

[ty 2 (t) − 1]dt

y(x) =
0

●✐↔ sû ❝❤ó♥❣ t❛ ❝❤å♥ ①➜♣ ①➾ ❜❛♥ ✤➛✉ y0 (x) = 0 t❤➻
x

(−1)dt = −x

y1 (x) =
0


✶✺
x

1
(t3 − 1)dt = −x + x4
4

y2 (x) =

0
x

y3 (x) =

1 8 1 5
1
1
1 10
t − t + t2 − 1 dt = −x + x4 − x7 +
x
16
2
4
14
160

t
0

❱➟② ♥❣❤✐➺♠ ①➜♣ ①➾ ❧➔ y(x) = −x + 41 x4 −

1 7
14 x

+

1 10
160 x .

✶✳✻✳✷✳ P❤÷ì♥❣ ♣❤→♣ ♥❤➙♥ s✉② ❜✐➳♥
❳➨t ♣❤÷ì♥❣ tr➻♥❤ t➼❝❤ ♣❤➙♥ ✭✶✳✶✵✮
b

y(x) =

K[x, t, y(t)]dt + f (x)
a
n

✈î✐ ❤↕t ♥❤➙♥ s✉② ❜✐➳♥ K[x, t, y(t)] =

gi (x)hi [t, y(t)]
i=1

b

n

y(x) =

hi [t, y(t)]dt + f (x)

gi (x)
i=1

a

✣➦t

b

ci =

hi [t, y(t)]dt

✭✶✳✶✷✮

ci gi (x) + f (x)

✭✶✳✶✸✮

a

t❛ ❝â ✤÷ñ❝

n

y(x) =
i=1

t❤❛② tø ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✸✮ ✈➔♦ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✷✮ ❝❤ó♥❣ t❛ t❤✉ ✤÷ñ❝
❤➺ ♣❤÷ì♥❣ tr➻♥❤ t✉②➳♥ t➼♥❤
b

cj =

b

hi [t, y(t)]dt =

a

n

hi t,
a

ci gi (t) + f (t) dt
i=1

✭✶✳✶✹✮


✶✻

◆➳✉ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✹✮ ❝â ♥❣❤✐➺♠ cj t❤➻ ♥❣❤✐➺♠ ❝➛♥ t➻♠ ❝â ❞↕♥❣
n

y(x) =

ci gi (x) + f (x)
i=1

✈î✐ ci ❧➔ ♥❣❤✐➺♠ ❝õ❛ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✭✶✳✶✹✮

❱➼ ❞ö ✶✳✻✳✷✳ ⑩♣ ❞ö♥❣ ❣✐↔✐ ♣❤÷ì♥❣ tr➻♥❤
1

(x + t)y(t)dt + x2

y(x) =
−1

❑❤❛✐ tr✐➸♥ ❝❤ó♥❣ t❛ ❝â
1

y(x) =

1

ty(t)dt + x2

xy(t)dt +
−1

−1

= xc1 + c2 + x2
✈î✐

1

1

c1 =

y(t)dt, c2 =
−1

❚❛ ❝â ❤➺ ♣❤÷ì♥❣ tr➻♥❤✿


1


2

 c1 = (tc1 + c2 + t )dt
−1
1




 c2 =

2

t(tc1 + c2 + t )dt
−1

ty(t)dt
−1

⇔



 c1 = 2c2 +

 c = 2c

2
3 1

2
3

⇔



 c1 = −2

 c = −4
2
3

❚ø ✤â ❝❤ó♥❣ t❛ ❝â ✤÷ñ❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✤➣ ❝❤♦ ❧➔✿

y(x) = x2 − 2x −

4
3


✶✼

❑➳t ❧✉➟♥
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② ✤➣ tr➻♥❤ ❜➔② ✤à♥❤ ♥❣❤➽❛✱ ♠ët sè t➼♥❤ ❝❤➜t ❝ì ❜↔♥
❝õ❛ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤✱ t♦→♥ tû t✉②➳♥ t➼♥❤✱ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t✱ ❝→❝ ♣❤÷ì♥❣
♣❤→♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ❧✐➯♥ q✉❛♥ ✈➔ ♠ët sè ✈➼ ❞ö ♠✐♥❤ ❤å❛✳ ✣➙② ❧➔ ❝❤÷ì♥❣

r➜t ❝➛♥ t❤✐➳t ♥❤➡♠ ❤é trñ✱ ❜ê s✉♥❣ ♥❤ú♥❣ ❦✐➳♥ t❤ù❝ ❝ì ❜↔♥ ♣❤ö❝ ✈ö ❝❤♦
♥ë✐ ❞✉♥❣ ❝→❝ ❝❤÷ì♥❣ s❛✉✳ ◆ë✐ ❞✉♥❣ ❝❤÷ì♥❣ ✷ s➩ tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣
◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤ ✈➔ ♠ët sè ✤à♥❤ ❧þ ❝ì ❜↔♥ ❝õ❛ ♣❤÷ì♥❣ ♣❤→♣ ✤â✳
❚r♦♥❣ ❝❤÷ì♥❣ ♥➔② t→❝ ❣✐↔ t❤❛♠ ❦❤↔♦ ♥ë✐ ❞✉♥❣ ❝õ❛ ❝→❝ t➔✐ ❧✐➺✉ ❬✶❪✱ ❬✸❪✱
❬✺❪✱ ❬✻❪✱ ❬✶✶❪✳


❈❤÷ì♥❣ ✷
P❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤

●✐↔ sû P ❧➔ t♦→♥ tû t→❝ ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ X ✳ ❚r♦♥❣
❝❤÷ì♥❣ ♥➔② ❝❤ó♥❣ t❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤ t♦→♥ tû P (x) = 0 ✈➔ ❣✐↔✐ ❣➛♥
✤ó♥❣ ♣❤÷ì♥❣ tr➻♥❤ ♥➔② ❜➡♥❣ ♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤✳

✷✳✶✳ P❤÷ì♥❣ ♣❤→♣ ❧➔♠ trë✐
P❤÷ì♥❣ ♣❤→♣ ❧➔♠ trë✐ ✤â♥❣ ✈❛✐ trá q✉❛♥ trå♥❣ tr♦♥❣ ✈✐➺❝ ♥❣❤✐➯♥ ❝ù✉
♣❤÷ì♥❣ ♣❤→♣ ◆❡✇t♦♥ ✲ ❑❛♥t♦r♦✈✐❝❤✳ ❈❤➼♥❤ ✈➻ ✈➟②✱ tr♦♥❣ ♠ö❝ ♥➔② t❛ s➩
tr➻♥❤ ❜➔② ♣❤÷ì♥❣ ♣❤→♣ ❧➔♠ trë✐ ✈➔ ❝→❝ ♠ð rë♥❣ ❝õ❛ ♥â✳
❳➨t ♣❤÷ì♥❣ tr➻♥❤✿

x = A (x)

✭✷✳✶✮

tr♦♥❣ ✤â A ❧➔ t♦→♥ tû ①→❝ ✤à♥❤ tr♦♥❣ ❤➻♥❤ ❝➛✉ S (x0 , r) ❝õ❛ ❦❤æ♥❣ ❣✐❛♥
❇❛♥❛❝❤ X ✳ ❈ò♥❣ ✈î✐ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✱ t❛ ①➨t ♣❤÷ì♥❣ tr➻♥❤✿

u = ϕ (u)

✭✷✳✷✮

tr♦♥❣ ✤â ϕ (u) ❧➔ ❤➔♠ sè ①→❝ ✤à♥❤ tr➯♥ ✤♦↕♥ [u0 ; u ] , (u = u0 + r)✳


✶✾

✷✳✶✳✶✳ ❚♦→♥ tû ❦❤↔ ✈✐
✣à♥❤ ♥❣❤➽❛ ✷✳✶✳✶✳ ❚❛ ♥â✐ r➡♥❣ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❧➔♠

trë✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ♥➳✉ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉ ✤÷ñ❝ t❤ä❛ ♠➣♥✿
1) A (x0 ) − x0 ≤ ϕ (u0 ) − u0 ❀
2) A (x) ≤ ϕ (u) ♥➳✉ x − x0 ≤ u − u0 ✳
tr♦♥❣ ✤â A (x) ❧➔ ✤↕♦ ❤➔♠ ❋r➨❝❤❡t ❝õ❛ t♦→♥ tû A (x)✳
❈→❝ ①➜♣ ①➾ ❧✐➯♥ t✐➳♣ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ (2.1) ✈➔ (2.2) ✤÷ñ❝ ①➙② ❞ü♥❣
♥❤÷ s❛✉✿
xn = A (xn−1 ) , n = 1, 2, ...

✭✷✳✸✮

un = ϕ (un−1 ) , n = 1, 2, ...; u0 = u0

✭✷✳✹✮

❇ê ✤➲ ✷✳✶✳ ◆➳✉ ❤➔♠ sè ϕ (u)✱ γ1 ≤ u ≤ γ2 ❧✐➯♥ tö❝✱ ❦❤æ♥❣ ❣✐↔♠ ✈➔

♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠✳ ❑❤✐ ✤â✿
1) ◆➳✉ ϕ (γ1 ) ≥ γ1 t❤➻ ❞➣② {un } ❤ë✐ tö ✤➳♥ ♥❣❤✐➺♠ u∗ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✭✷✳✷✮
2) ◆➳✉ ϕ (γ2 ) ≤ γ2 t❤➻ ❞➣② {un } ❤ë✐ tö ✤➳♥ ♥❣❤✐➺♠ u∗ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✭✷✳✷✮❀

3) ◆➳✉ γ1 ≤ ϕ (γ1 ) ≤ ϕ (γ2 ) ≤ γ2 ✈➔ ✭✷✳✷✮ ❝â ♥❣❤✐➺♠ ❞✉② ♥❤➜t t❤➻ ❝→❝
❞➣② {un}✱ {un} ❤ë✐ tö tî✐ ♥❣❤✐➺♠ ✤â✳
❈❤ù♥❣ ♠✐♥❤✳ 1) ❚r÷î❝ ❤➳t t❛ ❝❤ù♥❣ ♠✐♥❤ r➡♥❣ un ≤ u∗✱ tr♦♥❣ ✤â u∗ ❧➔
♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✳ ❚ø ✤â s✉② r❛ un ❝â ♥❣❤➽❛ ✈î✐ ♠å✐ n✳
❚❤➟t ✈➟②✱ t❛ ❝â u0 ≤ γ1 ≤ u∗ ✳ ●✐↔ sû un−1 ≤ u∗ ✳ ❑❤✐ ✤â un =

ϕ (un−1 ) ≤ ϕ (u∗ ) = u∗ ✱ ❞♦ ✤â un ≤ u∗ , ∀n✳ ❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤
❞➣② {un } ❦❤æ♥❣ ❣✐↔♠✳ ❚❛ ❝â u1 = ϕ (u0 ) = ϕ (γ1 ) ≥ γ1 = u0 ✳ ●✐↔ sû


✷✵

un ≥ un−1 ✳ ❑❤✐ ✤â un+1 = ϕ (un ) ≥ ϕ (un−1 ) = un ✳ ❱➟② ❞➣② {un } ❦❤æ♥❣
❣✐↔♠✱ ❜à ❝❤➦♥ tr➯♥ ♥➯♥ ♥â ❤ë✐ tö✳ ●✐↔ sû lim un = u✳ ❈❤✉②➸♥ q✉❛ ❣✐î✐
n→∞

❤↕♥ tr♦♥❣ un = ϕ (un−1 ) , n = 1, 2, ... t❛ ✤÷ñ❝ u ❧➔ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣
tr➻♥❤ ✭✷✳✷✮✳ ❈❤✉②➸♥ q✉❛ ❣✐î✐ ❤↕♥ un ≤ u∗ t❛ ❝â u ❧➔ ♥❣❤✐➺♠ ❞÷î✐ ❝õ❛
♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮❀

2) ❈❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü 1)❀
3) ❚ø 1) ✈➔ 2) s✉② r❛ 3)✳

✣à♥❤ ❧þ ✷✳✶✳✶✳ ●✐↔ sû t♦→♥ tû A ❝â ✤↕♦ ❤➔♠ ❧✐➯♥ tö❝ tr♦♥❣ ❤➻♥❤ ❝➛✉

S (x0 , r)✱

❤➔♠ sè ϕ (u) ❦❤↔ ✈✐ tr♦♥❣ ✤♦↕♥ [u0; u ] ✈➔ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮
❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❧➔♠ trë✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮✳ ◆❣♦➔✐ r❛ ❣✐↔ sû r➡♥❣
♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ tr➯♥ ✤♦↕♥ [u0; u ]✳ ❑❤✐ ✤â

♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮ ❝â ➼t ♥❤➜t ♠ët ♥❣❤✐➺♠ x∗ ✈➔ x∗ − x0 ≤ u − u0✱
tr♦♥❣ ✤â u ❧➔ ♥❣❤✐➺♠ ❞÷î✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮✳ ◆❣❤✐➺♠ x∗ ❧➔ ❣✐î✐ ❤↕♥
❝õ❛ ❞➣② ①➜♣ ①➾ ✭✷✳✸✮✳ ◆❣♦➔✐ r❛
xn − x∗ ≤ u − un , n = 1, 2, ...

tr♦♥❣ ✤â un = ϕ (un−1) ,

✭✷✳✺✮

n = 1, 2, ...; u0 = u0 ✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ ❝❤ù♥❣ ♠✐♥❤ ❞➣② {xn} ⊂ S (x0, r) ✈➔ ♥â ❧➔ ❞➣② ❤ë✐ tö✳
❱➻ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✷✮ ❧➔ ♣❤÷ì♥❣ tr➻♥❤ ❧➔♠ trë✐ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✭✷✳✶✮
♥➯♥ x1 − x0 ≤ u1 − u0 = r✱ ❞♦ ✤â x1 ∈ S (x0 , r)✳ ●✐↔ sû x1 , x2 , ..., xn ∈

S (x0 , r) ✈➔
✭✷✳✻✮

xk+1 − xk ≤ uk+1 − uk
❑❤✐ ✤â xn+1 − xn = A (xn ) − A (xn−1 ) =

xn
xn−1

✣➦t x = xn−1 + τ (xn − xn−1 )

A (x) dx ✳