(Luận văn thạc sĩ) không gian mêtríc nón và một số định lý điểm bất động

✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

▲➊ ❱❿◆ ▼■◆❍

❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆
❱⑨ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆ ❍➴❈

❚❍⑩■ ◆●❯❨➊◆ ✲ ✷✵✶✼


✣❸■ ❍➴❈ ❚❍⑩■ ◆●❯❨➊◆

❚❘×❮◆● ✣❸■ ❍➴❈ ❑❍❖❆ ❍➴❈

▲➊ ❱❿◆ ▼■◆❍

❑❍➷◆● ●■❆◆ ▼➊❚❘■❈ ◆➶◆
❱⑨ ▼❐❚ ❙➮ ✣➚◆❍ ▲Þ ✣■➎▼ ❇❻❚ ✣❐◆●
❈❤✉②➯♥ ♥❣➔♥❤✿ ❚♦→♥ ù♥❣ ❞ö♥❣
▼➣ sè✿ ✻✵✳✹✻✳✵✶✳✶✷

▲❯❾◆ ❱❿◆ ❚❍❸❈ ❙➒ ❚❖⑩◆

ữớ ữợ ồ

P Pì

✷✵✶✼


▼ư❝ ❧ư❝
▼Ð ✣❺❯
✶ ❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥

✶✳✶ ▼ð ✤➛✉ ✈➲ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✶ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ ✳ ✳ ✳ ✳ ✳
✶✳✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✶✳✷ ▼ët sè t➼♥❤ ❝❤➜t ✈➲ ❦❤æ♥❣ ♠➯tr✐❝ ♥â♥ ✳ ✳ ✳ ✳
✶✳✷✳✶ ❙ü ❤ë✐ tö tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥
✶✳✷✳✷ ◆❣✉②➯♥ ❧➼ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤

ỵ t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥
✷✳✶ ▼ët sè ♠ð rë♥❣ ừ ỵ
ỵ t ✳ ✳ ✳ ✳ ✳ ✳
✷✳✶✳✷ ▼ët sè ❞↕♥❣ ♠ð rë♥❣ ❦❤→❝ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷ ✣✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ →♥❤ ①↕ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✶ ▼ð ✤➛✉ ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝➦♣ ✳ ✳ ✳ ✳ ✳ ✳ ✳
✷✳✷✳✷ ❚r÷í♥❣ ❤đ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ t❤❛♠ ❦❤↔♦

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é
ỵ t ở ởt ✤➲ ♥❣❤✐➯♥ ❝ù✉ ❦❤→ ❝ì ❜↔♥
tr♦♥❣ ❝❤✉②➯♥ ♥❣➔♥❤ t♦→♥ ❣✐↔✐ t t ự ử ữủ t ợ
ổ tr➻♥❤ ❝õ❛ ❇r♦✇❡r✱ ❇❛♥❛❝❤✱ ♥❤ú♥❣ ✈➜♥ ✤➲ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ ❝→❝ →♥❤ ①↕ ✤ì♥ trà✱ ✤❛ trà tr♦♥❣ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ❦❤→❝ ♥❤❛✉
♥❣➔② ❝➔♥❣ t❤✉ ❤ót ✤÷đ❝ t ồ tr ữợ q
t ❝ù✉ ✈➔ ❝â ♥❤✐➲✉ ù♥❣ ❞ö♥❣ tr♦♥❣ ❝→❝ ❧➽♥❤ ✈ü❝ ❦❤→❝ ♥❤❛✉ ❝õ❛
t♦→♥ ❤å❝✿ t♦→♥ tè✐ ÷✉✱ ❝→❝ ❜➔✐ t♦→♥ ❦✐♥❤ t➳✳
❚❛ ♥❤➢❝ ❧↕✐ r➡♥❣✱ ✈ỵ✐ X ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✤➛② ✤õ✱ f ❧➔ ♠ët
→♥❤ ①↕ ❝♦✱ ◆❣✉②➯♥ ỵ ờ r r➡♥❣ f ❝â ❞✉②
♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣✳ ❱➲ s❛✉ ❝â r➜t ♥❤✐➲✉ ♥❤➔ t♦→♥ ❤å❝ ✤➣ q✉❛♥
t➙♠ ♥❣❤✐➯♥ ❝ù✉✱ t ỵ ợ ổ ❣✐❛♥ ❦❤→❝
♥❤❛✉✳
◆➠♠ ✷✵✵✼✱ ●✉❛♥❣ ✈➔ ❩❤❛♥❣ ✭❬✸❪✮ ✤➣ ❣✐ỵ✐ t❤✐➺✉ ♠ët ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥
♠ỵ✐✱ ❝→❝ t→❝ ❣✐↔ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ tr♦♥❣ ✤â ❝→❝ t→❝ ❣✐↔ ✤➣
t❤❛② t➟♣ sè t❤ü❝ tr♦♥❣ ✤à♥❤ ♥❣❤➽❛ ♠➯tr✐❝ ❜ð✐ ♠ët ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
♠➔ tr➯♥ ✤â ✤➣ ✤à♥❤ ♥❣❤➽❛ ♠ët q✉❛♥ ❤➺ t❤ù tỹ ở ỹ tr ởt
õ ữợ r ổ tr➻♥❤ ♥➔② ❝→❝ t→❝ ❣✐↔ ✤➣ ❝❤ù♥❣ ♠✐♥❤ ✤÷đ❝
♠ët sè t➼♥❤ ❝❤➜t t÷ì♥❣ tü ✈➲ ♠➯tr✐❝ tr➯♥ ♠➯tr✐❝ ♥â♥✱ ✤➦❝ t ự
ỵ tr♦♥❣ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥
✤➛② ✤õ✳ ❱➲ s❛✉✱ ❝â ♥❤✐➲✉ t→❝ ❣✐↔ ❦❤→❝ t✐➳♣ tö❝ ♣❤→t tr✐➸♥ ❝→❝ t➼♥❤ t
ừ ổ tr õ ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ ❣✐ú❛ ❝→❝
❦❤ỉ♥❣ ❣✐❛♥ ♥➔②✳
▼ư❝ ✤➼❝❤ ❝õ❛ ✤➲ t➔✐ ❧➔ ❣✐ỵ✐ t❤✐➺✉ ❧↕✐ ♠ët sè ❦➳t q✉↔ ♥❣❤✐➯♥ ❝ù✉ ❝õ❛
❝→❝ t→❝ ❣✐↔ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙② ✈➲ ổ tr õ ởt số
ỵ ❜➜t ✤ë♥❣✱ ❜➜t ✤ë♥❣ ❝➦♣ ❝õ❛ ❝→❝ →♥❤ ①↕ ❣✐ú❛ ❝→❝ ❦❤æ♥❣
❣✐❛♥ ♥➔②✳ ❈→❝ ❦❤→✐ ♥✐➺♠ ✈➔ ❦➳t q✉↔ tr➻♥❤ ❜➔② tr♦♥❣ ❧✉➟♥ ✈➠♥ ✤÷đ❝ ✈✐➳t
❞ü❛ tr➯♥ ❝→❝ ❜➔✐ ❜→♦ ❬✸❪✱ ❬✺❪✱ ❬✶❪✱ ❬✹❪✱ ❬✷❪ ✈➔ ❬✻❪✳
◆❣♦➔✐ ♣❤➛♥ ♠ð ✤➛✉✱ t ỗ ữỡ ữỡ
✤➛✉ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥ ❝❤ó♥❣ tỉ✐ tr➻♥❤ ❜➔② ♠ët sè ✈➜♥ ✤➲ ❝ì

❜↔♥ ✈➲ ♥â♥✱ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ t➼♥❤ ❝❤➜t ❝õ❛ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♥➔②✳
◆❣♦➔✐ r❛ ú tổ ợ t ỵ →♥❤ ①↕ ❣✐ú❛
❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♠❡tr✐❝ ♥â♥ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤
r ữỡ ỵ ✤✐➸♠ ❜➜t ✤ë♥❣ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠❡tr✐❝


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♥â♥✱ ❝❤ó♥❣ tỉ✐ ❣✐ỵ✐ t❤✐➺✉ ♠ët sè ❦➳t q✉↔ ✈➲ ỵ t ở
t ở ỳ ❝→❝ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤÷đ❝ ❝→❝ t→❝ ❣✐↔
●✉❛♥❣✱ ❩❤❛♥❣✱ ❘❡③❛♣♦✉r✱ ❍❛♠❧❜❛r❛♥✐✱ ❋✳ ❙❛❜❡t❣❤❛❞❛♠✱ ❍✳ P✳ ▼❛s✐❤❛
✈➔ ❆✳ ❍✳ ❙❛♥❛t♣♦✉r ❝❤ù♥❣ ♠✐♥❤ tr♦♥❣ t❤í✐ ❣✐❛♥ ❣➛♥ ✤➙②✳

▲✉➟♥ ✈➠♥ ♥➔② ✤÷đ❝ ❤♦➔♥ t❤➔♥❤ t↕✐ ❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝ ✲ ✣↕✐
❤å❝ ữợ sỹ ữợ t t ❝õ❛ t❤➛② ❣✐→♦
P●❙✳ ❚❙✳ ❍➔ ❚r➛♥ P❤÷ì♥❣✳ ❚→❝ ❣✐↔ ①✐♥ ❜➔② tä ❧á♥❣ ❜✐➳t ì♥ s➙✉ s➢❝ tỵ✐
t❤➛②✳
❚→❝ ❣✐↔ ①✐♥ ❝❤➙♥ t❤➔♥❤ ❝↔♠ ì♥ ❇❛♥ ❣✐→♠ ❤✐➺✉✱ ❑❤♦❛ ❚♦→♥✲❚✐♥
❚r÷í♥❣ ✣↕✐ ❤å❝ ❑❤♦❛ ❤å❝✱ ✣↕✐ ❤å❝ ❚❤→✐ ◆❣✉②➯♥ ✤➣ q✉❛♥ t➙♠ ✈➔ ❣✐ó♣
✤ï t→❝ ❣✐↔ tr♦♥❣ s✉èt t❤í✐ ❣✐❛♥ ❤å❝ t➟♣ t↕✐ ❚r÷í♥❣✳
◆❤➙♥ ❞à♣ ♥➔② ❡♠ ❝ơ♥❣ ①✐♥ ✤÷đ❝ ❣û✐ ❧í✐ ❝↔♠ ì♥ ❝❤➙♥ t❤➔♥❤ tỵ✐ ❣✐❛
✤➻♥❤✱ ❜↕♥ ❜➧ ✤➣ ❧✉ỉ♥ ❜➯♥ ❡♠✱ ❝ê ✈ơ✱ ✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï ❡♠ tr♦♥❣ s✉èt
q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ t❤ü❝ ❤✐➺♥ ❦❤â❛ ❧✉➟♥ tèt ♥❣❤✐➺♣✳
❇↔♥ ❧✉➟♥ ✈➠♥ ❦❤æ♥❣ t❤➸ tr→♥❤ ❦❤ä✐ ♥❤ú♥❣ t❤✐➳✉ sât✱ t→❝ ❣✐↔ r➜t ♠♦♥❣
♥❤➟♥ ✤÷đ❝ sü ❝❤➾ ❜↔♦ t➟♥ t➻♥❤ ừ t ổ ỗ
t❤→♥❣ ✾ ♥➠♠ ✷✵✶✼
❚→❝ ❣✐↔ ❧✉➟♥ ✈➠♥
▲➯ ❱➠♥ ▼✐♥❤

✸

❈❤÷ì♥❣ ✶

❑❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥
✶✳✶ ▼ð ✤➛✉ ✈➲ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥
✶✳✶✳✶ ◆â♥ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤

❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t r➡♥❣ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
t❤ü❝✳
✣à♥❤ ởt t ỗ P E ữủ ❣å✐ ❧➔ ♠ët ♥â♥ tr♦♥❣ E
♥➳✉ ♥â t❤ä❛ ♠➣♥ ❝→❝ ✤✐➲✉ ❦✐➺♥ s❛✉✿
✶✳ P ✤â♥❣✱ P = {∅}✱ P = {0} ;
✷✳ ❱ỵ✐ ♠å✐ a, b ∈ R, a, b 0, x, y ∈ P t❤➻ ax + by ∈ P ;
✸✳ ◆➳✉ x ∈ P ✈➔ −x ∈ P t❤➻ x = 0.
❇➙② ❣✐í t❛ ①❡♠ ①➨t ❦❤→✐ ♥✐➺♠ q✉❛♥ ❤➺ t❤ù tü tr➯♥ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
E ❧✐➯♥ q✉❛♥ ✤➳♥ ♥â♥✳ ❈❤♦ P ⊂ E ❧➔ ♠ët ♥â♥✳ ❚❛ ✤à♥❤ ♥❣❤➽❛ q✉❛♥ ❤➺
t❤ù tü ❜ë ♣❤➟♥ tr➯♥ E ♥❤÷ s❛✉✿
1. x y ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ y − x ∈ P ;
2. x < y ♥➳✉ x y ✈➔ x = y;
3. x
y ♥➳✉ y − x ∈ intP, tr♦♥❣ ✤â intP ❧➔ ❦➼ ❤✐➺✉ ♣❤➛♥ tr♦♥❣ ❝õ❛
♥â♥ P.
❚r♦♥❣ ❧✉➟♥ ✈➠♥ ♥➔② t❛ ❧✉æ♥ ❣✐↔ t❤✐➳t ♥â♥ P ❝â ♣❤➛♥ tr♦♥❣ intP = ∅✳


✹

▼➺♥❤ ✤➲ ✶✳✶✳ ❚ø ❦❤→✐ ♥✐➺♠ t❛ ❞➵ ❞➔♥❣ s✉② r❛✿
1. ◆➳✉ x

y t❤➻ x < y.

2. ◆➳✉ x

y✱ a

0 t❤➻ ax

ay.

✣à♥❤ ♥❣❤➽❛ ✶✳✷✳ ◆â♥ P ✤÷đ❝ ❣å✐ ❧➔ ✿
✶✳ õ t tỗ t ởt số K > 0 t❤ä❛ ♠➣♥✿ ✤✐➲✉ ❦✐➺♥
0 x y ❦➨♦ t❤❡♦ x
K y ✱ ✈ỵ✐ ♠å✐ x, y ∈ E. ❍➡♥❣ sè K > 0
♥❤ä ♥❤➜t t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ♥➔② ✤÷đ❝ ❣å✐ ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝✳
✷✳ ◆â♥ ♠✐♥✐❤❡❞r❛❧ ♥➳✉ sup(x, y) tỗ t ợ ồ x, y E.
◆â♥ ♠✐♥✐❤❡❞r❛❧ ♠↕♥❤ ♥➳✉ ♠å✐ t➟♣ ❝♦♥ ❜à ❝❤➦♥ tr➯♥ ❝õ❛ E ✤➲✉ ❝â
❝➟♥ tr➯♥ ✤ó♥❣✳
✹✳ ◆â♥ ✤➦❝ ♥➳✉ intP = ∅.
✺✳ ◆â♥ s✐♥❤ ♥➳✉ E = P − P.
✻✳ ◆â♥ ❝❤➼♥❤ q✉② ♥➳✉ ♠å✐ ❞➣② t➠♥❣ ❜à ❝❤➦♥ tr➯♥ ✤➲✉ ❤ë✐ tö✳ ◆❣❤➽❛ ❧➔
♥➳✉ {xn, n 1} ❧➔ ❞➣② tọ
x1

x2

ÃÃÃ

y

ợ y E t tỗ t x E t❤ä❛ ♠➣♥
lim

n−→∞

xn − x = 0.

▼➺♥❤ ✤➲ ✶✳✷✳ ▼å✐ ♥â♥ ❝❤➼♥❤ q✉② ✤➲✉ ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✳
❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû P ❧➔ ♥â♥ ❝❤➼♥❤ q✉② tr♦♥❣ E ♥❤÷♥❣ ❦❤ỉ♥❣ ♣❤↔✐ ❧➔
♥â♥ ❝❤✉➞♥ t➢❝✳ ❱ỵ✐ ♠é✐ n 1 t❛ ❝❤å♥ tn, sn ∈ P s❛♦ ❝❤♦ tn − sn ∈ P
✈➔ n2 tn < sn . ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ n 1, ✤➦t yn = ttn ✈➔ xn = st n ✳
n
n
2
❑❤✐ ✤â xn, yn ✈➔ y∞n − xn ✤➲✉ t❤✉ë❝ P ✱ yn = 1 ✈➔ n < xn ✈ỵ✐ ♠å✐
1
n 1. ❉♦ ❝❤✉é✐
yn ❧➔ ❤ë✐ tư ✈➔ P õ tỗ t y P s
n2



n=1



1
y=
yn .

2
n=1 n
0

x1

ớ ú ỵ r

x1 +

1
x2
22

x1 +

1
1
x
+
x3
2
22
32

ÃÃÃ

y,

✺

❞♦ ✤â ❝❤✉é✐

∞

1
x
2 n
n=1 n

❤ë✐ tö ✈➻ P ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳ ❉♦ ✤â
1
xn = 0,
n−→∞ n2
lim

♠➙✉ t❤✉➝♥✳ ❱➟② P õ t

ổ tỗ t õ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝
K < 1.

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ P ❧➔ ♥â♥ ❝❤✉➞♥
t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K < 1. ❈❤å♥ ♠ët ♣❤➛♥ tû ❦❤→❝ ❦❤æ♥❣ x ∈ P
tũ ỵ 0 < < 1 s K < 1 − ε. ❑❤✐ ✤â
(1 − ε)x

x

♥❤÷♥❣

(1 − ε) x > K x .

✣➙② ❝❤➼♥❤ ❧➔ ♠➙✉ t❤✉➝♥✳

▼➺♥❤ ợ ộ M > 1 ổ tỗ t ♥â♥ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè

❝❤✉➞♥ t➢❝ K > M.

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû M > 1 ❧➔ ♠ët sè t❤ü❝ tũ ỵ t
E = ax + b : a, b ∈ R, x ∈ [1 − 1/k, 1] ,

❦❤✐ ✤â E ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✈ỵ✐ ❝❤✉➞♥ sup✳ ❑➼ ❤✐➺✉
P = {ax + b : a, b ∈ R, a

0, b

0}.

❑❤✐ ✤â P ❧➔ ♠ët ♥â♥ tr➯♥ E rữợ t t ự P õ
q ●å✐ {anx + bn, n 1} ❧➔ ♠ët ❞➣② t➠♥❣✱ tr tự tỗ t
ởt tỷ cx + d ∈ E s❛♦ ❝❤♦
a1 x + b1

a2 x + b 2

···

cx + d

✻

✈ỵ✐ ♠å✐ x ∈ [1 − 1/k, 1]. ❑❤✐ ✤â {an, n
t❤ü❝ t❤ä❛ ♠➣♥

1}✱ {bn , n

b1

b2

···

d

a1

a2

···

c,

1}

❧➔ ❤❛✐ ❞➣② sè

✈➔
❞♦ ✤â ❝→❝ ❞➣② {an, n 1}✱ {bn, n 1} ❤ë✐ tö✳ ●✐↔ sû n−→∞
lim an = a,

lim bn = b, ❦❤✐ ✤â lim an x + bn = ax + b✳ ❚ø ✤â s✉② r❛ P ❧➔ ♥â♥
n−→∞
n−→∞
❝❤➼♥❤ q✉②✳ ❚❤❡♦ ▼➺♥❤ ✤➲ ✶✳✷ t❛ s✉② r❛ P ❧➔ ♥â♥ ❝❤✉➞♥ t
tỗ t K 1 s ✤✐➲✉ ❦✐➺♥ 0 g f ❦➨♦ t❤❡♦ g K f
✈ỵ✐ ♠å✐ g, f ∈ E.
❇➙② ❣✐í t❛ ❝❤ù♥❣ ♠✐♥❤ K > M. ❚❛ t❤➜② f (x) = −M x + M ∈
P, g(x) = M ∈ P ✈➔ f − g ∈ P ✳ ❉♦ ✤â 0 g f ✱ ❦➨♦ t❤❡♦
M= g

K f = K.

▼➦t ❦❤→❝✱ t❛ ①➨t ❝→❝ ❤➔♠ sè
f (x) = −(M + 1/M )x + M,

t❤➻ f ∈ P, g ∈ P ✈➔ f − g ∈ P ✳ ❉♦ ✤â 0
M= g

❍ì♥ ♥ú❛
g =M

◆❤÷ ✈➟②

✈➔

g(x) = M

g

f✱

❦➨♦ t❤❡♦

K f .

f = 1 − 1/M + 1/M 2 .

M = g > M f = M + 1/M − 1.

◆❤÷ ✈➟②
M f < g

K f ,

❦➨♦ t❤❡♦ K > M.

▼➺♥❤ ✤➲ ✶✳✺✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ E t❛ ❧✉ỉ♥ ❝â
✐✮ ❱ỵ✐ ♠é✐ λ ∈ R, λ > 1 ổ tỗ t õ t ợ sè K > λ.


✼

✐✐✮ ◆â♥ P ❝❤➼♥❤ q✉② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ♠å✐ ữợ ở tử

ử E = Rn✱ t❛ ✤➦t

P = {(x1 , . . . , xn ) : xi

0, ∀i = 1, . . . , n}.

❑❤✐ ✤â P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✱ ♥â♥ s✐♥❤✱ ♠✐♥✐❤❡❞r❛❧✱ ♠✐♥✐❤❡❞r❛❧ ♠↕♥❤ ✈➔
✤➦❝✳
❱➼ ❞ö ✶✳✷✳ ❈❤♦ D ⊆ Rn ❧➔ ♠ët t➟♣ ❝♦♠♣❛❝t E = C (D) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥
❝→❝ ❤➔♠ sè ①→❝ ✤à♥❤ ✈➔ ❧✐➯♥ tư❝ tr➯♥ D✳ ❑➼ ❤✐➺✉
P = {f ∈ E |f (t)

0, ∀x ∈ D } .

❑❤✐ ✤â P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝✱ ♥â♥ s✐♥❤✱ ✤➦❝ ✈➔ ♠✐♥✐❤❡❞r❛❧ ♥❤÷♥❣ ❦❤ỉ♥❣
❧➔ ♥â♥ ♠✐♥✐❤❡❞r❛❧ ♠↕♥❤✱ P ❝ơ♥❣ ❦❤ỉ♥❣ ❧➔ ♥â♥ ❝❤➼♥❤ q✉②✳
1
❱➼ ❞ư ✶✳✸✳ ❑➼ ❤✐➺✉ E = C[0;1]
❧➔ ❦❤æ♥❣ ❣✐❛♥ ❝→❝ ❤➔♠ sè t❤ü❝ ❦❤↔ ✈✐
❝➜♣ ✶ tr➯♥ ✤♦↕♥ [0; 1] ✈ỵ✐ ❝❤✉➞♥
f = f

tr♦♥❣ ✤â

.

∞

+ f

∞

∞,

f ∈ E,

❧➔ ❝❤✉➞♥ max . ❑➼ ❤✐➺✉
P = {f ∈ E : f (t)

0}.

❉➵ ❞➔♥❣ ❝❤ù♥❣ ữủ P ởt õ ợ ộ k
x, g(x) = x2k ✱ ❦❤✐ ✤â
0

✈ỵ✐ ♠å✐ t ∈ [0; 1]✱ ❦➨♦ t❤❡♦ 0

g(t)
g

f✳

1

✤➦t f (x) =

f (t)

❚❛ t❤➜②

f = 2, g = 2k + 1.

❑➨♦ t❤❡♦ f < g ✳ ◆❤÷ ✈➟② k ❦❤ỉ♥❣ ♣❤↔✐ ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛
P ✈➔ P ❧➔ ♥â♥ ❦❤æ♥❣ ❝❤✉➞♥ t➢❝✳
✶✳✶✳✷ ❑❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥

✣à♥❤ ♥❣❤➽❛ ✶✳✸✳ ❈❤♦ X ❧➔ t➟♣ ❦❤→❝ ré♥❣ ✈➔ E ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥
❇❛♥❛❝❤ ✈ỵ✐ q✉❛♥ ❤➺ t❤ù tü ❜ë ♣❤➟♥

✤è✐ ✈ỵ✐ ♥â♥ P ✳ ❈❤♦ ❤➔♠ d :




X ì X E

tọ

0 d(x, y) ợ ♠å✐ x, y ∈ X ✱ d(x, y) = 0 ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ x = y✳
✷✳ d(x, y) = d(y, x) ✈ỵ✐ ♠å✐ x, y ∈ X ✳
✸✳ d(x, y) d(x, z) + d(z, y) ✈ỵ✐ ♠å✐ x, y, z ∈ X ✳
❑❤✐ ✤â d ✤÷đ❝ ❣å✐ ❧➔ ♠➯tr✐❝ ♥â♥ tr➯♥ X ✈➔ (X, d) ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣
❣✐❛♥ ♠➯tr✐❝ ♥â♥✳
❱➼ ❞ö ✶✳✹✳ ❈❤♦ E = R2✱ P = {(x, y) : x, y 0}✱ X = R ✈➔ d : X ×
X −→ E ①→❝ ✤à♥❤ ❜ð✐
d(x, y) = (|x − y| , α |x − y|),

tr♦♥❣ ✤â α 0 ❧➔ ♠ët ❤➡♥❣ sè✳ ❑❤✐ ✤â d ♠➯tr✐❝ ♥â♥ ✈➔ (X, d) ❧➔ ♠ët
❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳
▼ð rë♥❣ ❤ì♥✱ ❝❤å♥ E = Rn✱ P = {(x1, . . . , xn) : xi 0}✱ X = R ✈➔
d : X × X −→ E ①→❝ ✤à♥❤ ❜ð✐
d(x, y) = (|x − y| , α1 |x − y| , . . . , αn−1 |x − y|),

tr♦♥❣ ✤â α1, α2, . . . , αn−1 0 ❧➔ ❝→❝ ❤➡♥❣ sè✳ ❑❤✐ ✤â (X, d) ❝ơ♥❣ ❧➔ ♠ët
❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳
❱➼ ❞ö ✶✳✺✳ ❈❤♦ E = (CR [0; ∞) , . ∞)✱ P = {f ∈ E |f (t) 0}✱ (X, ρ)

❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ✈➔ X × X −→ E ✤÷đ❝ ①→❝ ✤à♥❤ ❜ð✐✿
d(x, y) = ρ(x, y).ϕ,

tr♦♥❣ ✤â ϕ : [0, 1] −→ R+ ❧✐➯♥ tö❝✳ ❑❤✐ ✤â P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➺
sè ❝❤✉➞♥ t➢❝ k = 1 ✈➔ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ❝❤✉➞♥ t➢❝✳
❚✐➳♣ t❤❡♦ t❛ ❣✐ỵ✐ t❤✐➺✉ ✈➲ ❝→❝ ✤✐➸♠ tæ♣æ tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝
♥â♥✳ ❈❤♦ E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ ✤➣ ✤÷đ❝ tr❛♥❣ ❜à q✉❛♥ ❤➺ t❤ù
tü ❜ë ♣❤➟♥ t❤❡♦ ♥â♥ ❝❤✉➞♥ t➢❝ P ✈➔ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝


✾

♥â♥✳ ❱ỵ✐ xo ∈ X ✈➔ r ∈ E, 0

r✳

❚➟♣

B(xo , r) = {x ∈ X : d(xo , x)

r}

❣å✐ ❧➔ ❤➻♥❤ ❝➛✉ ♠ð t➙♠ xo ❜→♥ ❦➼♥❤ r✳
❈❤♦ A ⊂ X ✱ ✤✐➸♠ xo ∈ A ✤÷đ❝ ❣å✐ ❧➔ tr ừ A tỗ t
r E, 0
r s❛♦ ❝❤♦ B(xo , r) ⊂ A✳ ❍✐➸♥ ♥❤✐➯♥✱ t❤❡♦ ✤à♥❤ ♥❣❤➽❛✱ ✤✐➸♠
tr♦♥❣ ❝õ❛ A ♣❤↔✐ t❤✉ë❝ t➟♣ ❤ñ♣ A✳
✣✐➸♠ xo ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ ❜✐➯♥ ❝õ❛ t➟♣ A ♥➳✉ ✈ỵ✐ ♠å✐ r ∈
E, 0
r✱ B(xo , r) ∩ A = ∅ ✈➔ B(xo , r) ∩ X\A = ∅✳ ❚➟♣ ❤ñ♣ t➜t ❝↔

❝→❝ ✤✐➸♠ ❜✐➯♥ ❝õ❛ A A ú ỵ r ừ A ❝â t❤➸
t❤✉ë❝ A ❤♦➦❝ ❦❤æ♥❣ t❤✉ë❝ A✳ ◆❣♦➔✐ r❛✱ t❛ ❝ơ♥❣ ❝â δA = δ(X\A)✳
✣✐➸♠ xo ∈ X ✤÷đ❝ ❣å✐ ❧➔ ✤✐➸♠ tư ❝õ❛ t➟♣ A ♥➳✉ ✈ỵ✐ ♠å✐ r ∈ E, 0 r✱
❤➻♥❤ ❝➛✉ B(xo, r) ❧✉æ♥ ❝❤ù❛ ✈ỉ sè ✤✐➸♠ ❝õ❛ A✳ ✣✐➸♠ tư ❝õ❛ A ❝â t❤➸
t❤✉ë❝ A ❤♦➦❝ ❦❤ỉ♥❣ t❤✉ë❝ A✳ ❚➟♣ ❝→❝ ✤✐➸♠ tư ❝õ❛ A ❦➼ ❤✐➺✉ ❧➔ Ad✳ ❈â
t❤➸ t❤➜② x ∈ Ad ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ✈ỵ✐ ♠å✐ r ∈ E, 0 r✱ ❤➻♥❤ ❝➛✉ B(xo, r)
❝❤ù❛ ➼t ♥❤➜t ♠ët ✤✐➸♠ ❝õ❛ A✳
❈❤♦ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ (X, d)✱ t➟♣ A ⊂ X ✤÷đ❝ ❣å✐ ❧➔ t➟♣ ♠ð
♥➳✉ ♠é✐ ✤✐➸♠ ❝õ❛ A ✤➲✉ ❧➔ ✤✐➸♠ tr♦♥❣ ❝õ❛ A✳ ❚➟♣ F ✤÷đ❝ ❣å✐ ❧➔ t➟♣
✤â♥❣ ♥➳✉ X\F ❧➔ t➟♣ ♠ð✳
❉➵ t❤➜② X ✈➔ t➟♣ ré♥❣ ❧➔ ♥❤ú♥❣ t➟♣ ♠ð✳ ❍➻♥❤ ❝➛✉ ♠ð B(xo, r) ❧➔
♠ët t➟♣ ♠ð✱ ✈➻ ✈ỵ✐ ♠å✐ x ∈ B(xo, r) ổ tỗ t 0 r1 = r d(xo, x)
s❛♦ ❝❤♦ B(x, r1) ⊂ B(xo, r)✱ tù❝ ❧➔ ♠å✐ ✤✐➸♠ ❝õ❛ B(xo, r) ✤➲✉ ❧➔ ✤✐➸♠
tr♦♥❣✳

✶✳✷ ▼ët sè t➼♥❤ ❝❤➜t ✈➲ ❦❤ỉ♥❣ ♠➯tr✐❝ ♥â♥
✶✳✷✳✶ ❙ü ❤ë✐ tư tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥

✣à♥❤ ♥❣❤➽❛ ✶✳✹✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳ {xn}n 1 ❧➔ ♠ët
❞➣② ❝→❝ ♣❤➛♥ tû ❝õ❛ X ✈➔ x ∈ X ✳ ❚❛ ♥â✐ ❞➣② {xn} ❤ë✐ tư tỵ✐ x ♥➳✉ ✈ỵ✐




ồ c E,


0

c tỗ t N N

s ❝❤♦ ✈ỵ✐ ♠å✐ n > N, d(xn, x)

c.

lim xn = x

❤♦➦❝ xn −→ x ❦❤✐ n −→ ∞.

n−→∞

▼➺♥❤ ✤➲ ✶✳✻✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ P

❧➔ ♥â♥
❝❤✉➞♥ t➢❝ tr♦♥❣ E ✈ỵ✐ ❤➺ sè ❝❤✉➞♥ t➢❝ K ✳ ❑❤✐ ✤â ❞➣② {xn } tr♦♥❣ X ❤ë✐
tư tỵ✐ x ∈ X ♥➳✉ ✈➔ ❝❤➾ ♥➳✉ d(xn , x) −→ 0 ❦❤✐ n −→ ∞.

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {xn} ❤ë✐ tư tỵ✐ x✳ ❑❤✐ ✤â ✈ỵ✐ ♠é✐ ε > 0✱ t❛ ❝❤å♥
c ∈ E ✈ỵ✐ 0 = c ✈➔ K c < . õ tỗ t N > 0 s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐
n > N ✱ t❛ ❝â d(xn , x) < c. ❇ð✐ ✈➟②✱ ✈ỵ✐ n > N t❛ ❝â |d(xn , x)|
K |c| < ε✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ d(xn , x) −→ 0.
◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû d(xn, x) −→ 0 ❦❤✐ n −→ ∞✳ ❚ø ❣✐↔ t❤✐➳t t❛ ❝â
t❛ s✉② r❛✿ ✈ỵ✐ c ∈ E ợ 0 c tỗ t > 0 s ❦❤✐ x < δ
t❤➻ c − x ∈ intP. ❱ỵ✐ sè δ ✤â✱ tø ❣✐↔ t❤✐➳t d(xn, x) −→ 0 tỗ t
N N s ợ ồ n > N ✱ d(xn , x) < δ ✱ ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦
c − d(xn , x) ∈ intP ✱ ♥❣❤➽❛ ❧➔ d(xn , x)
c✳ ❙✉② r❛ {xn } ❤ë✐ tư tỵ✐ x✳

▼➺♥❤ ✤➲ ✶✳✼✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ P

❧➔ ♥â♥
❝❤✉➞♥ t➢❝ tr♦♥❣ E ✈ỵ✐ ❤➺ sè ❝❤✉➞♥ t➢❝ K ✈➔ {xn } ❧➔ ♠ët ❞➣② tr♦♥❣ X ✳
❑❤✐ ✤â ♥➳✉ {xn } ❤ë✐ tö tỵ✐ x ∈ X ✈➔ {xn } ❤ë✐ tư tỵ✐ y ∈ X t❤➻ x = y.
◆â✐ ❝→❝❤ ❦❤→❝✱ ❣✐ỵ✐ ❤↕♥ ❝õ❛ ♠ët ❞➣② tr♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥➳✉ ❝â ❧➔
❞✉② ♥❤➜t✳

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠ët ♣❤➛♥ tø c ∈ E tũ ỵ 0
tỗ t N N s ợ ♠å✐ n N t❛ ❝â
d(xn , x)

c,

d(xn , y)

c✱

c.

✣✐➲✉ ♥➔② s✉② r❛
d(x, y)

d(xn , x) + d(xn , y)

2c.

tø ❣✐↔ t❤✐➳t s✉② r❛


✶✶

❉♦ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ ♥➯♥
d(x, y)

2K c .

❉♦ c ồ tũ ỵ r ❦❤✐ d(x, y) = 0✱ tù❝ ❧➔ x = y.

▼➺♥❤ ✤➲ ✶✳✽✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ P

❧➔ ♥â♥
❝❤✉➞♥ t➢❝ tr♦♥❣ E ✈ỵ✐ ❤➺ sè ❝❤✉➞♥ t➢❝ K ✳ ❈❤♦ {xn } ✈➔ {yn } ❧➔ ❤❛✐ ❞➣②
tr♦♥❣ X ✳ ❑❤✐ ✤â ♥➳✉ {xn } ❤ë✐ tư tỵ✐ x ∈ X ✈➔ {yn } ❤ë✐ tư tỵ✐ y ∈ X
t❤➻ d(xn , yn ) s➩ ❤ë✐ tư tỵ✐ d(x, y) ❦❤✐ n −→ ∞.

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠å✐ ε > 0✱ ❝❤å♥ c ∈ E s❛♦ ❝❤♦ 0
c <

c

ε
.
4K + 2

❚ø ❣✐↔ t❤✐➳t xn −→ x ✈➔ yn y s r tỗ t N
ồ n N t❛ ❝â
d(xn , x)

c,

✈➔

d(yn , y)

∈ N∗

s❛♦ ❝❤♦ ✈ỵ✐

c.

✣✐➲✉ ♥➔② s✉② r❛
d(x, y)

d(xn , x) + d(xn , yn ) + d(yn , y)

d(xn , yn ) + 2c

d(xn , x) + d(x, y) + d(y, yn )

2c + d(x, y).

✈➔
d(xn , yn )

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦
0

d(x, y) + 2c − d(xn , yn )

4c.

❙✉② r❛
d(xn , yn ) − d(x, y)

d(x, y) + 2c − d(xn , yn ) + 2c
(4K + 2) c < ε.

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ d(xn, yn) −→ d(x, y).


✶✷

▼➺♥❤ ✤➲ ✶✳✾✳ ❈❤♦ {xn, n ∈ N∗} ✈➔ {yn, n ∈ N∗} ❧➔ ❤❛✐ ❞➣② tr♦♥❣

❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ (X, d)✳ ◆➳✉ xn

yn ✈ỵ✐ ♠å✐ n ∈ N∗ ✈➔

lim xn = x, lim yn = y

n−→∞

n−→∞

t❤➻ x y. ◆â✐ ❝→❝❤ ❦❤→❝✱ q✉❛♥ ❤➺ t❤ù tü ✤÷đ❝ ❜↔♦ t♦➔♥ q✉❛ ợ
tr ổ tr õ

ỵ ởt x ❧➔ ✤✐➸♠ tư ❝õ❛ t➟♣ ❤đ♣ A ❦❤✐ ✈➔ ❝❤➾ ❦❤✐ ❝â
♠ët ❞➣② ✤✐➸♠ ♣❤➙♥ ❜✐➺t {xn } ❝õ❛ A ❤ë✐ tư tỵ✐ x✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû x ❧➔ ✤✐➸♠ tư ❝õ❛ A. ❱ỵ✐ ♠é✐ n ∈ N∗✱ ❝❤å♥ rn ∈ E

s❛♦ ❝❤♦
1
0
rn ✈➔ K rn < .
n
❉♦ B(x, rn) ∩ A = ∅ ♥➯♥ t❛ ❝❤å♥ ✤÷đ❝ xn ∈ B(x, cn) ∩ A✳ ❉♦ B(x, rn)
❝❤ù❛ ✈æ sè ✤✐➸♠ ❝õ❛ A ♥➯♥ ❦❤æ♥❣ ♠➜t t➼♥❤ tê♥❣ q✉→t t❛ ❝â t❤➸ ❣✐↔
t❤✐➳t ❝→❝ xn ❧➔ ♣❤➙♥ ❜✐➺t✳ ❑❤✐ ✤â {xn} ⊂ A
0

d(xn , x)

rn ,

❦➨♦ t❤❡♦
d(xn , x)

K rn <

1
,
n

✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ n−→∞
lim xn = x✳
◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sỷ x X tỗ t ởt ❜✐➺t {xn} ⊂ A s❛♦
❝❤♦ n−→∞
lim xn = x✳ ❑❤✐ õ ợ ộ r E, 0
r tỗ t N ∈ N∗ s❛♦ ❝❤♦
✈ỵ✐ ♠å✐ n > N t❛ ❝â

d(xn , x)

r,

❦➨♦ t❤❡♦ d(x, xn) < r tù❝ ❧➔ xn ∈ B(x, r) ∩ A ✈ỵ✐ ♠å✐ n > N ✱ ❞♦ ✤â x
❧➔ ♠ët ✤✐➸♠ tö ❝õ❛ A.
❚✐➳♣ t❤❡♦ t❛ ❣✐ỵ✐ t❤✐➺✉ ❦❤→✐ ♥✐➺♠ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ừ
t rữợ t


✶✸

✣à♥❤ ♥❣❤➽❛ ✶✳✺✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ {xn}n 1 ❧➔ ❞➣②

tr♦♥❣ X ✳ ❉➣② {xn} ✤÷đ❝ ❣å✐ ❧➔ ♠ët ❞➣② ❈❛✉❝❤② ♥➳✉ ✈ỵ✐ ♠é✐ c ∈ E ợ
0
c tỗ t N N s ợ ♠å✐ m, n > N t❛ ❝â
d(xm , xn )

c.

▼➺♥❤ ✤➲ ✶✳✶✵✳ ❚r♦♥❣ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ ♠å✐ ❞➣② ❤ë✐ tö ✤➲✉ ❧➔
❞➣② ❈❛✉❝❤②✳

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû (X, d) ❧➔ ởt ổ tr õ ợ d : X ì
X −→ E, tr♦♥❣ ✤â E ❧➔ ❦❤æ♥❣ ❣✐❛♥ ❇❛♥❛❝❤ t❤ü❝ s➠♣ t❤ù tü ♠ët ♣❤➛♥
t❤❡♦ ♥â♥ P. ●✐↔ sû {xn, n ∈ N∗} ❧➔ ♠ët ❞➣② ❝→❝ ♣❤➛♥ tû ❝õ❛ X ❤ë✐ tư
✈➲ x ∈ X. ❑❤✐ ✤â✱ ✈ỵ✐ ♠é✐ c E, 0 c tỗ t N N s❛♦ ❝❤♦ ✈ỵ✐
♠å✐ n > N t❛ ❝â
d(xn , x)

c/2.

❑❤✐ ✤â ✈ỵ✐ ♠å✐ m, n > N t❛ ❝â
d(xn , xm )

d(xn , x) + d(x, xm )

c.

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ {xn, n ∈ N∗} ❧➔ ❞➣② ❈❛✉❝❤②✳ ▼➺♥❤ ✤➲ ✤÷đ❝ ❝❤ù♥❣
♠✐♥❤✳

▼➺♥❤ ✤➲ ✶✳✶✶✳ ❈❤♦ (X, d) ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ P ❧➔ ♥â♥ ❝❤✉➞♥

t➢❝ ✈ỵ✐ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ K. ❈❤♦ {xn , n ∈ N∗ } ❧➔ ♠ët ❞➣② ❝→❝ ♣❤➛♥
tû ❝õ❛ X ✳ ❑❤✐ ✤â ❞➣② {xn , n ∈ N∗ } ❧➔ ❞➣② ❈❛✉❝❤② ❦❤✐ ✈➔ ❝❤➾ ❦❤✐
d(xn , xm ) −→ 0 ❦❤✐ m, n −→ ∞.

❈❤ù♥❣ ♠✐♥❤✳ ●✐↔ sû {xn, n ∈ N∗} ❧➔ ❞➣② ❈❛✉❝❤②✱ ❦❤✐ ✤â ✈ỵ✐ ♠é✐ ε > 0✱
t❛ ❝❤å♥ c ∈ E s❛♦ ❝❤♦ 0 c ✈➔ K c < . õ tỗ t N N s❛♦
❝❤♦ ✈ỵ✐ ♠å✐ m, n > N t❛ ❝â d(xm, xn) c. ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ ✈ỵ✐ ♠å✐
m, n > N ✱ t❛ ❝â
d(xn , xm )

K c < ε.

❙✉② r❛ d(xn, xm) −→ 0 ❦❤✐ m, n −→ ∞.

✶✹

◆❣÷đ❝ ❧↕✐✱ ❣✐↔ sû d(xn, xm) −→ 0 ❦❤✐ m, n −→ ∞. ❑❤✐ ✤â ✈ỵ✐ ♠é✐
c ∈ E, 0
c✱ tỗ t > 0 s x < t❤➻ c − x ∈ intP. ❱ỵ✐ sè δ
♥➔② s➩ tỗ t N N s ợ ồ m, n > N t❛ ❝â
d(xn , xm ) < δ.

❑❤✐ ✤â c − d(xn, xm) ∈ intP, ❦➨♦ t❤❡♦ d(xn, x − n) c. ◆❤÷ ✈➟② {xn}
❧➔ ❞➣② ❈❛✉❝❤②✳
✣à♥❤ ♥❣❤➽❛ ✶✳✻✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ ♥➳✉ ♠å✐
❞➣② ❈❛✉❝❤② tr♦♥❣ X ✤➲✉ ❤ë✐ tö t❤➻ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝
♥â♥ ✤➛② ✤õ✳
✣à♥❤ ♥❣❤➽❛ ✶✳✼✳ ❈❤♦ (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✱ ♥➳✉ ♠å✐
❞➣② ✈æ ❤↕♥ ❝→❝ ♣❤➛♥ tû ❝õ❛ X ✤➲✉ tr➼❝❤ r❛ ✤÷đ❝ ♠ët ❞➣② ❝♦♥ ❤ë✐ tư
tr♦♥❣ X t❤➻ X ✤÷đ❝ ❣å✐ ❧➔ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ❝♦♠♣❛❝t ❞➣②✳
✶✳✷✳✷ ◆❣✉②➯♥ ❧➼ →♥❤ ①↕ ❝♦ ❇❛♥❛❝❤

◆❤÷ ✤➣ ♥â✐ tr♦♥❣ ♣❤➛♥ ♠ð ✤➛✉✱ ♥➠♠ ✷✵✵✼✱ ▲✲● ❍✉❛♥❣ ✈➔ ❳✳ ❩❤❛♥❣
✭❬✸❪✮ ✤➣ ❝❤ù♥❣ ởt số t q ỵ ❝♦ ❇❛♥❛❝❤ ✤è✐
✈ỵ✐ ❧ỵ♣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ❚r♦♥❣ ♣❤➛♥ ♥➔② ❝❤ó♥❣ tỉ✐ s➩
♣❤→t ❜✐➸✉ ✈➔ ❝❤ù♥❣ ♠✐♥❤ tt ỵ rữợ t t ợ t
ởt ✈➔✐ ❦❤→✐ ♥✐➺♠✳
✣à♥❤ ♥❣❤➽❛ ✶✳✽✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥✳ ⑩♥❤ ①↕ T :
X −→ X ✤÷đ❝ ồ tỗ t ởt sè k ∈ [0, 1) s❛♦
❝❤♦
d(T x, T y)

kd(x, y)

✈ỵ✐ ♠å✐ x, y ∈ X ✳ ⑩♥❤ ①↕ T ✤÷đ❝ ❣å✐ ❧➔ ❣✐↔ ❝♦ ♥➳✉
d(T x, T y) < kd(x, y)

✈ỵ✐ ♠å✐ x = y ∈ X ✳


✶✺

✣à♥❤ ♥❣❤➽❛ ✶✳✾✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✈➔ T : X −→ X

❧➔ ♠ët →♥❤ ①↕✳ ▼ët ✤✐➸♠ x0 ∈ X ✤÷đ❝ ❣å✐ ✤✐➸♠ ❜➜t ✤ë♥❣ ừ T
T (x0 ) = x0 .

ỵ ỵ (X, d) ổ

♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✱ P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ❇❛♥❛❝❤
t❤ü❝ E ✈ỵ✐ ❤➡♥❣ sè K ✈➔ T : X −→ X ❧➔ ♠ët →♥❤ ①↕ ❝♦✳ ❑❤✐ ✤â T ❝â
✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ❍ì♥ ♥ú❛ ✈ỵ✐ ♠é✐ x ∈ X, ❞➣② {T n x} ✤➲✉ ❤ë✐
tö ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ ✤â✳

❈❤ù♥❣ ♠✐♥❤✳ ▲➜② x0 ∈ X tũ ỵ t {xn} X
xn = T n x,

✈ỵ✐ ♠å✐ n ≥ 1.

❑❤✐ ✤â ✈ỵ✐ ♠å✐ n ≥ 1,
d(xn+1 , xn ) = d(T xn , T xn−1 )
k 2 d(xn−1 , xn−2 )

kd(xn , xn−1 )

...

k n d(x1 , x0 ).

❚ø ✤â s✉② r❛ ✈ỵ✐ n > m✱
d(xn , xm )

d(xn , xn−1 ) + d(xn−1 , xn−2 ) + ... + d(xm+1 , xm )
(k n−1 + k n−2 + ... + k m )d(x1 , x0 )
km
d(x1 , x0 ).
1−k

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦
d(xn , xm )

km
K d(x1 , x0 )
1−k

✈ỵ✐ ♠å✐ n > m✳ ❚ø ✤â t❛ ❝â
lim

n,m−→∞

d(xn , xm ) = 0.


✶✻

❙✉② r❛ {xn} ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ổ õ ừ
tỗ t x ∈ X s❛♦ ❝❤♦ n−→∞
lim xn = x∗ . ❍ì♥ ♥ú❛✱ tø ❜➜t ✤➥♥❣ t❤ù❝
tr➯♥ t❛ ❝â
d(T x∗ , x∗ )

d(T xn , T x∗ ) + d(T xn , x∗ )
kd(xn , x∗ ) + d(xn+1 , x∗ ),

s✉② r❛
d(T x∗ , x∗ )

K(k d(xn , x∗ ) + d(xn+1 , x∗ ) ) −→ 0.

✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ d(T x∗, x∗) = 0✳ ❉♦ ✤â T x∗ = x∗ ♥➯♥ x∗ ❧➔ ✤✐➸♠
❜➜t ✤ë♥❣ ❝õ❛ T ✳
●✐↔ sû tỗ t y X s T y = y∗✱ ❦❤✐ ✤â t❛ ❝â
d(x∗ , y ∗ ) = d(T x∗ , T y ∗ )
kd(x∗ , y ∗ ).

❙✉② r❛ d(x∗, y∗) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ x∗ = y∗. ❱➟② x0 ❧➔ ✤✐➸♠ ❜➜t
✤ë♥❣ ❞✉② ♥❤➜t ừ T. ỵ ữủ ự

q sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✱ P ❧➔ ♥â♥
❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè K. ❱ỵ✐ c ∈ E, 0

c ✈➔ x0 ∈ X ✱ ✤➦t

B(x0 , c) = {x ∈ X : d(x0 , x)

c} .

●✐↔ sû →♥❤ ①↕ T : X −→ X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ s❛✉
d(T x, T y)

kd(x, y)

✈ỵ✐ ♠å✐ x, y ∈ B(x0 , c), tr♦♥❣ ✤â k ∈ [0, 1) ❧➔ ♠ët ❤➡♥❣ sè ✈➔ d(T x0 , x0 )
(1 − k)c✳ ❑❤✐ ✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t tr♦♥❣ B(x0 , c)✳

❈❤ù♥❣ ♠✐♥❤✳ ❚❛ s➩ ❝❤ù♥❣ ♠✐♥❤ B(x0, c) ❧➔ ✤➛② ✤õ ✈➔
T : B(x0 , c) −→ B(x0 , c).


✶✼

❚❤➟t ✈➟② ❣✐↔ sû {xn} ❧➔ ♠ët ❞➣② ❈❛✉❝❤② tr♦♥❣ B(x0, c). ❑❤✐ ✤â {xn}
❝ô♥❣ ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X. X ừ tỗ t x X s ❝❤♦
lim xn = x. ▼➦t ❦❤→❝ t❛ ❝â
n−→∞
d(x0 , x)

d(xn , x0 ) + d(xn , x)

d(xn , x) + c

✈ỵ✐ ♠å✐ n ≥ 1. ❚ø n−→∞
lim xn = x t❛ s✉② r❛ lim d(xn , x) = 0. ❉♦ ✈➟②
n−→∞
d(x0 , x)

c ♥➯♥ x ∈ B(x0 , c), ✤✐➲✉ ♥➔② ❦➨♦ t❤❡♦ B(x0 , c) ❧➔ ✤➛② ✤õ✳
▼➦t ❦❤→❝ ✈ỵ✐ ♠é✐ x ∈ B(x0, c)✱ t❛ ❝â
d(x0 , T x)

d(T x0 , x0 ) + d(T x0 , T x)
(1 − k)c + kd(x0 , x)
(1 − k)c + kc = c.

❱➟② T x ∈ B(x0, c)✱ tù❝ ❧➔ T : B(x0, c) −→ B(x0, c). ⑩♣ ❞ö♥❣ ❝→❝❤
❝❤ù♥❣ ừ ỵ t s r T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉②
♥❤➜t tr♦♥❣ B(x0, c).

❍➺ q✉↔ ✶✳✷✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✈➔ P ❧➔ ♥â♥

❝❤✉➞♥ t➢❝ ✈ỵ✐ ❤➡♥❣ sè K ✳ ●✐↔ sû →♥❤ ①↕ T : X −→ X t❤ä❛ ♠➣♥
d(T n x, T n y)

kd(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X,

tr♦♥❣ ✤â n ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ k ∈ [0, 1) ❧➔ ♠ët ❤➡♥❣ sè✳ ❑❤✐
✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳

❈❤ù♥❣ ♠✐♥❤✳ ❚ø ỵ T n õ t ở ♥❤➜t x∗✳ ❱➻
T n (T x∗ ) = T (T n x∗ ) = T x∗ ♥➯♥ T x∗ ❝ô♥❣ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T n .
❱➟② T x∗ = x∗ ❞♦ T n ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ✣✐➲✉ ♥➔② s✉② r❛ x∗ ❧➔
♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❱➻ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ❝ô♥❣ ❧➔ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝õ❛ T n ♥➯♥ ✤✐➸♠ ❜➜t ✤ë♥❣ ừ T t

ú ỵ ỵ rở tỹ sỹ ừ ỵ t ở

ừ →♥❤ ①↕ ❝♦ tr♦♥❣ ❦❤ỉ♥❣ ❣✐❛♥ ♠➯tr✐❝ ❝❤♦ tr÷í♥❣ ❤đ♣ ❦❤æ♥❣ ❣✐❛♥




tr õ t ử ữợ s E = R2 ✈➔ ✤➦t
P = (x, y) ∈ R2 : x, y

0 .

❑❤✐ ✤â P ❧➔ ♥â♥ ❝❤✉➞♥ t➢❝ tr♦♥❣ E ✳ ❚❛ ❝❤å♥
X = (x, 0) ∈ R2 : 0

x

1 ∪ (0, x) ∈ R2 : 0

x

1 .

❱➔ ①➨t →♥❤ ①↕ d : X × X −→ E ①→❝ ✤à♥❤ ❜ð✐
4
d((x, 0), (y, 0)) = ( |x − y|, |x − y|),
3
2
d((0, x), (0, y)) = (|x − y|, |x − y|),
3
4
2

d((x, 0), (0, y)) = d((0, y), (x, 0)) = ( x + y, x + y).
3
3

❑❤✐ ✤â (X, d) ❧➔ ♠ët ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳
❳➨t →♥❤ ①↕ T : X −→ X ✈ỵ✐ T ((x, 0)) = (0, x) ✈➔ T ((0, x)) = ( 21 x, 0).
❑❤✐ ✤â T t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ ❝♦
d(T (x1 , x2 ), T (y1 , y2 ))

kd((x1 , x2 ), (y1 , y2 ))

✈ỵ✐ ♠å✐ (x1, x2), (y1, y2) ∈ X, ✈ỵ✐ ❤➡♥❣ sè k = 43 ∈ [0, 1)✳ ❉➵ t❤➜② T ❝â
❞✉② ♥❤➜t ♠ët ✤✐➸♠ ❜➜t ✤ë♥❣ (0, 0) ∈ X ✳ ❍ì♥ ♥ú❛✱ T ❦❤ỉ♥❣ ❧➔ →♥❤ ①↕
❝♦ t❤❡♦ ♥❣❤➽❛ ♠❡tr✐❝ ❊✉❝❧✐❞ t❤ỉ♥❣ t❤÷í♥❣✳




ữỡ

ỵ t ở tr
ổ tr õ
ởt số rở ừ ỵ
ỵ t

r r t ỵ
ỵ ừ ❝→❝ t→❝ ❣✐↔ ✤➣ ❜ä
✤÷đ❝ ✤✐➲✉ ❦✐➺♥ ❝❤✉➞♥ t➢❝ ❝õ❛ õ P tr ỵ ử t t
ự

ỵ (X, d) ổ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ ✈➔ T : X −→

X ❧➔ ♠ët →♥❤ ①↕ ❝♦✳ ❑❤✐ ✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳ ❍ì♥ ♥ú❛

✈ỵ✐ ♠é✐ x ∈ X, ❞➣② {T n x} ✤➲✉ ❤ë✐ tö ✈➲ ✤✐➸♠ ❜➜t ✤ë♥❣ õ

ự x0 X tũ ỵ t ✤à♥❤ ❞➣② {xn} ⊆ X ❜ð✐
xn = T n x, ợ ồ n 1.
ự ố ữ tr ỵ t õ ợ n > m
d(xn , xm )

❱ỵ✐ ♠é✐ c ∈ E,

0

c✱

km
d(x1 , x0 ).
1−k

t❛ ❝❤å♥ sè t❤ü❝ δ > 0 s❛♦ ❝❤♦
c + Nδ (0) ⊂ P,


✷✵

tr♦♥❣ ✤â Nδ (0) =m{y ∈ E : y < δ}. ❚❛ ❝❤å♥ N1 ∈ N∗ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐
k
m N1 t❛ ❝â

d(x1 , x0 ) ∈ Nδ (0). ❑❤✐ ✤â
1−k
km
d(x1 , x0 )
1−k

d(xn , xm )

c

✈ỵ✐ ♠å✐ m N1✳ ❙✉② r❛ {xn} ❧➔ ❞➣② ❈❛✉❝❤② tr♦♥❣ X ✳ ❱➻ X ổ
õ ừ tỗ t x ∈ X s❛♦ ❝❤♦ n−→∞
lim xn = x∗ . ❚❛ ❝❤å♥
N2 ∈ N∗ s❛♦ ❝❤♦ ✈ỵ✐ ♠å✐ n N2 t❛ ❝â d(xn , x∗ )
c/2. ❑❤✐ ✤â✱ tø ❜➜t
✤➥♥❣ t❤ù❝ tr➯♥ t❛ ❝â
d(T x∗ , x∗ )

d(T xn , T x∗ ) + d(T xn , x∗ )
kd(xn , x∗ ) + d(xn+1 , x∗ )
d(xn , x∗ ) + d(xn+1 , x∗ )
c/2 + c/2 = c.

❈❤ù♥❣ ♠✐♥❤ q✉② t ữủ
d(T x , x )

c
m

ợ ồ m 1. ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ mc − d(T x∗, x∗) ∈ P ✈ỵ✐ ♠å✐ m 1.

❱➻ mc −→ 0 ❦❤✐ m −→ ∞ ✈➔ P ✤â♥❣ ♥➯♥ −d(T x∗, x∗) ∈ P ú ỵ r
d(T x , x ) P ✱ ❞♦ ✤â d(T x∗ , x∗ ) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ T x∗ = x∗ ✱ tø
✤â x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳
●✐↔ sû tỗ t y X s T y = y∗✱ ❦❤✐ ✤â t❛ ❝â
d(x∗ , y ∗ ) = d(T x∗ , T y ∗ )
kd(x∗ , y ∗ ).

❙✉② r❛ d(x∗, y∗) = 0✳ ✣✐➲✉ ♥➔② ❦➨♦ t❤❡♦ x∗ = y∗. ❱➟② x0 ❧➔ ✤✐➸♠ ❜➜t
✤ë♥❣ ❞✉② ♥❤➜t ừ T. ỵ ữủ ự
ử ỵ ✷✳✶ ✈➔ ❝❤ù♥❣ ♠✐♥❤ t÷ì♥❣ tü ♥❤÷ ❝→❝ ❤➺ q✉↔ ✶✳✶ ✈➔
✶✳✷ t❛ ❝â


✷✶

❍➺ q✉↔ ✷✳✶✳ ●✐↔ sû (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ❱ỵ✐ c ∈ E,
0

c ✈➔ x0 ∈ X ✱ ✤➦t

B(x0 , c) = {x ∈ X : d(x0 , x)

c} .

●✐↔ sû →♥❤ ①↕ T : X −→ X t❤ä❛ ♠➣♥ ✤✐➲✉ ❦✐➺♥ s❛✉
d(T x, T y)

kd(x, y)

✈ỵ✐ ♠å✐ x, y ∈ B(x0 , c), tr♦♥❣ ✤â k ∈ [0, 1) ❧➔ ♠ët ❤➡♥❣ sè ✈➔ d(T x0 , x0 )

(1 − k)c✳ ❑❤✐ ✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t tr♦♥❣ B(x0 , c)✳

❍➺ q✉↔ ✷✳✷✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ✤➛② ✤õ✳ ●✐↔ sû →♥❤

①↕ T : X −→ X t❤ä❛ ♠➣♥

d(T n x, T n y)

kd(x, y) ✈ỵ✐ ♠å✐ x, y ∈ X,

tr♦♥❣ ✤â n ❧➔ ♠ët sè ♥❣✉②➯♥ ❞÷ì♥❣ ✈➔ k ∈ [0, 1) ❧➔ ♠ët ❤➡♥❣ sè✳ ❑❤✐
✤â T ❝â ✤✐➸♠ ❜➜t ✤ë♥❣ ❞✉② ♥❤➜t✳

◆➠♠ ✷✵✵✼✱ ▲✲● ự ỵ
t ở ữ s

ỵ ✷✳✷✳ ❈❤♦ (X, d) ❧➔ ❦❤æ♥❣ ❣✐❛♥ ♠➯tr✐❝ ♥â♥ ❝♦♠♣❛❝t ❞➣②✱ P ❧➔

♥â♥ ❝❤➼♥❤ q✉② ✈➔ T : X −→ X ❧➔ →♥❤ ①↕ ❣✐↔ ❝♦✳ ❑❤✐ ✤â T ❝â ✤✐➸♠ ❜➜t
✤ë♥❣ ❞✉② ♥❤➜t✳

❈❤ù♥❣ ♠✐♥❤✳ ❱ỵ✐ ♠é✐ x0 ∈ X ✱ t❛ ①→❝ ✤à♥❤ ❞➣② {xn} ⊆ X ❜ð✐
xn = T n x0

ợ ồ n 1 tỗ t n s❛♦ ❝❤♦ xn+1 = xn✱ ❦❤✐ ✤â xn ❧➔ ✤✐➸♠ ❜➜t
✤ë♥❣ ❝õ❛ T ✈➔ t❛ ❝â ✤✐➲✉ ❝➛♥ ❝❤ù♥❣ ♠✐♥❤✳
❉♦ ✤â t❛ ❝❤➾ ❝➛♥ ①➨t tr÷í♥❣ ❤đ♣ xn+1 = xn ✈ỵ✐ ♠å✐ n✳ ✣➦t
dn = d(xn , xn+1 ).

❑❤✐ ✤â

dn+1 = d(xn+1 , xn+2 ) = d(T xn , T xn+1 ) < d(xn , xn+1 ) = dn .




{dn} ỡ ữợ 0 tr ổ
E tỗ t d ∈ E s❛♦ ❝❤♦
lim dn = d∗ .

n−→∞

▼➦t ❦❤→❝✱ ✈➻ X ổ t tỗ t {xn } ❝õ❛
{xn } ✈➔ x∗ ∈ X s❛♦ ❝❤♦ lim xn = x∗ ✳ ❑❤✐ ✤â t❛ ❝â
i−→∞
i

i

d(T xni , T x∗ ) < d(xni , x∗ ),

✈ỵ✐ ♠å✐ i = 1, 2 . . . . ✣✐➲✉ ♥➔② s✉② r❛
d(T xni , x∗ )

K d(xni , x∗ ) −→ 0

❦❤✐ i −→ ∞✱ tr♦♥❣ ✤â K ❧➔ ❤➡♥❣ sè ❝❤✉➞♥ t➢❝ ❝õ❛ P ✳ ✣✐➲✉ ♥➔② ❦➨♦
t❤❡♦ i−→∞
lim T xn = T x∗ ✈➔ lim T 2 xn = T 2 x∗ . ❚ø ▼➺♥❤ ✤➲ ✶✳✽ t❛ ❝â
i−→∞
i

i

lim d(T xni , xni ) = d(T x∗ , x∗ )

i−→∞

✈➔
lim d(T 2 xni , T xni ) = d(T 2 x∗ , T x∗ ).

i−→∞

✣✐➲✉ ♥➔② ❝❤ù♥❣ tä

d(T xni , xni ) = dni → d∗ = d(T x∗ , x∗ )

❦❤✐ i −→ ∞. ❇➙② ❣✐í ❝❤ù♥❣ ♠✐♥❤ T x∗ = x∗✳ ●✐↔ sû T x∗ = x∗✱ ❦❤✐ ✤â
d∗ = 0 ✈➔
d∗ = lim dni +1 = lim d(T 2 xni , T xni )
i−→∞

i−→∞

< d(T 2 x∗ , T x∗ ) = d(T x∗ , x∗ ) = d∗ ,

♠➙✉ t❤✉➝♥✳ ❱➟② T x∗ = x∗ ✈➔ x∗ ❧➔ ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ✳ ❚➼♥❤ ❞✉②
♥❤➜t ✤✐➸♠ ❜➜t ✤ë♥❣ ❝õ❛ T ❧➔ ❤✐➸♥ ♥❤✐➯♥✳
✷✳✶✳✷ ▼ët sè ❞↕♥❣ ♠ð rë♥❣ ❦❤→❝

❈→❝ ❦➳t q✉↔ s❛✉ ✤÷đ❝ ❝❤ù♥❣ ♠✐♥❤ ❜ð✐ ▲✲● ❍✉❛♥❣ ✈➔ ❳✳ ❩❤❛♥❣ tr♦♥❣

❬✸❪ ✈➔♦ ♥➠♠ ✷✵✵✼✳