Bài toán đuổi bắt trong trò chơi tuyến tính với hạn chế tích phân trên thang thời gian

ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
-------------------------------

LÊ VĂN QUÝ

BÀI TOÁN ĐUỔI BẮT TRONG TRÒ CHƠI
TUYẾN TÍNH VỚI HẠN CHẾ TÍCH PHÂN
TRÊN THANG THỜI GIAN

LUẬN VĂN THẠC SĨ TOÁN HỌC

THÁI NGUYÊN - 2017


ĐẠI HỌC THÁI NGUYÊN
TRƯỜNG ĐẠI HỌC KHOA HỌC
-------------------------------

LÊ VĂN QUÝ

BÀI TOÁN ĐUỔI BẮT TRONG TRÒ CHƠI
TUYẾN TÍNH VỚI HẠN CHẾ TÍCH PHÂN
TRÊN THANG THỜI GIAN
LUẬN VĂN THẠC SĨ TOÁN HỌC
Chuyên ngành: Toán ứng dụng
Mã số
: 60 46 01 12

NGƯỜI HƯỚNG DẪN KHOA HỌC:
PGS.TS. Tạ Duy Phượng

THÁI NGUYÊN - 2017


▼ö❝ ❧ö❝
▼ð ✤➛✉
✶ ❑❤→✐ ♥✐➺♠ t❤❛♥❣ t❤í✐ ❣✐❛♥

✶
✹

✶✳✶

❚❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✶✳✷

❚æ ♣æ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✺

✶✳✸

❈→❝ ✤à♥❤ ♥❣❤➽❛ ❝ì ❜↔♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳

✻

✶✳✹

P❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✶✳✺

✹

✶✳✹✳✶

✣↕♦ ❤➔♠ ❍✐❧❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✶

✶✳✹✳✷

❚➼♥❤ ❝❤➜t ❝õ❛ ✤↕♦ ❤➔♠ ❍✐❧❣❡r ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✷

P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽
✶✳✺✳✶

❍➔♠ t✐➲♥ ❦❤↔ ✈✐ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✽

✶✳✺✳✷

P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✶✾

✶✳✻

❚➼♥❤ ❤ç✐ q✉② tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✶

✶✳✼

❍➔♠ ♠ô tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✸

✷ ❚rá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈î✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥
t❤❛♥❣ t❤í✐ ❣✐❛♥
✷✻
✷✳✶

❍➺ ✤ë♥❣ ❧ü❝ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✶✳✶

P❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣ tr➻♥❤ ✤ë♥❣ ❧ü❝ t✉②➳♥ t➼♥❤
❜➟❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✻

✷✳✶✳✷

❈æ♥❣ t❤ù❝ ♥❣❤✐➺♠ ❝õ❛ ♣❤÷ì♥❣ tr➻♥❤ ✈➔ ❤➺ ♣❤÷ì♥❣
tr➻♥❤ ✤ë♥❣ ❧ü❝ t✉②➳♥ t➼♥❤ ❜➟❝ ♥❤➜t ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✷✼

✷✳✶✳✸

❍➺ ✤ë♥❣ ❧ü❝ t✉②➳♥ t➼♥❤ ❝â ❤❛✐ t❤❛♠ sè ✤✐➲✉ ❦✐➸♥ ✳ ✳ ✳ ✸✶

✐


✐✐

✷✳✷

❚rá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈î✐ ❤↕♥ ❝❤➳ t➼❝❤ ♣❤➙♥ tr➯♥
t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✷

✷✳✸

❚rá ❝❤ì✐ ✤✉ê✐ ❜➢t t✉②➳♥ t➼♥❤ ✈î✐ t❤æ♥❣ t✐♥ ❝❤➟♠ ✈➔ ❤↕♥ ❝❤➳
t➼❝❤ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✳ ✸✽

❑➳t ❧✉➟♥
❚➔✐ ❧✐➺✉ tr➼❝❤ ❞➝♥

✹✹
✹✺


▼ð ✤➛✉
◆❤➡♠ t❤è♥❣ ♥❤➜t ♥❣❤✐➯♥ ❝ù✉ ❝→❝ ❤➺ ✤ë♥❣ ❧ü❝ ❧✐➯♥ tö❝ ✭❤➺ ♣❤÷ì♥❣ tr➻♥❤
✈✐ ♣❤➙♥✮ ✈➔ ❤➺ ✤ë♥❣ ❧ü❝ rí✐ r↕❝ ✭❤➺ ♣❤÷ì♥❣ tr➻♥❤ s❛✐ ♣❤➙♥✮✱ ❙t❡❢❛♥ ❍✐❧❣❡r
♥➠♠ ✶✾✽✽✱ tr♦♥❣ ❧✉➟♥ →♥ ❚✐➳♥ s➽ ❝õ❛ ♠➻♥❤✱ ✤➣ ✤÷❛ r❛ ❦❤→✐ ♥✐➺♠ t❤❛♥❣ t❤í✐
❣✐❛♥ ✭t✐♠❡ s❝❛❧❡✮✳ ❚ø ✤â ✤➳♥ ♥❛② ✤➣ ❝â ♠ët sè q✉②➸♥ s→❝❤✱ ❤➔♥❣ ❝❤ö❝ ❧✉➟♥
→♥ t✐➳♥ s➽ ✈➔ ❤➔♥❣ ♥❣➔♥ ❜➔✐ ❜→♦ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ❣✐↔✐ t➼❝❤ ✭♣❤➨♣ t♦→♥ ✈✐ ♣❤➙♥✱
t➼❝❤ ♣❤➙♥✮ ✈➔ ❤➺ ✤ë♥❣ ❧ü❝ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥✳
❚❤❛♥❣ t❤í✐ ❣✐❛♥ ❝â þ ♥❣❤➽❛ tr✐➳t ❤å❝ s➙✉ s➢❝✿ ❚❤❛♥❣ t❤í✐ ❣✐❛♥ ❝❤♦ ♣❤➨♣
♥❣❤✐➯♥ ❝ù✉ ❤❛✐ ♠➦t ❜↔♥ ❝❤➜t ❝õ❛ t❤ü❝ t➳✱ ✤â ❧➔ t➼♥❤ ❧✐➯♥ tö❝ ✈➔ t➼♥❤ rí✐ r↕❝✳
❚r♦♥❣ t♦→♥ ❤å❝✱ t❤❛♥❣ t❤í✐ ❣✐❛♥ ❝❤♦ ♣❤➨♣ ♥❣❤✐➯♥ ❝ù✉ t❤è♥❣ ♥❤➜t ♥❤✐➲✉
♠æ ❤➻♥❤ ❦❤→❝ ♥❤❛✉ ❞÷î✐ ❝ò♥❣ ♠ët ❦❤→✐ ♥✐➺♠ ✈➔ ❝æ♥❣ ❝ö✳
●✐↔✐ t➼❝❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❤➺ ✤ë♥❣ ❧ü❝ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✤❛♥❣
✤÷ñ❝ ♥❤✐➲✉ ♥❤â♠ ❝→❝ ♥❤➔ t♦→♥ ❤å❝ tr♦♥❣ ✈➔ ♥❣♦➔✐ ♥÷î❝ q✉❛♥ t➙♠✳ ✣➣ ❝â
♠ët sè ❜➔✐ ✈✐➳t ✈➲ ù♥❣ ❞ö♥❣ ❝õ❛ t❤❛♥❣ t❤í✐ ❣✐❛♥ tr♦♥❣ ♥❣❤✐➯♥ ❝ù✉ ❦✐♥❤ t➳
✈➽ ♠æ✱ ❤➺ s✐♥❤ t❤→✐✱ ❜➔✐ t♦→♥ tè✐ ÷✉✳
❇➔✐ t♦→♥ ✤✉ê✐ ❜➢t ❧➔ ♠ët tr♦♥❣ ♥❤ú♥❣ ❝→❝ ❜➔✐ t♦→♥ ❝ì ❜↔♥ ❝õ❛ ❧þ
t❤✉②➳t trá ❝❤ì✐✳ ❚r♦♥❣ ❜➔✐ t♦→♥ ✤✉ê✐ ❜➢t t❤➻ ♥❣÷í✐ ❝❤↕② ✭❣➢♥ ✈î✐ ❜✐➳♥ ✤✐➲✉

❦✐➸♥ ❝õ❛ ♠➻♥❤✮ ❧✉æ♥ ❝è ❣➢♥❣ ❝❤↕② ❝➔♥❣ ♥❤❛♥❤✱ ❝➔♥❣ ①❛ ♥❣÷í✐ ✤✉ê✐ ❝➔♥❣
tèt✳ ❈á♥ ♥❣÷í✐ ✤✉ê✐ t❤➻ ❝è ❣➢♥❣ ✧♣❤→t r❛ ✧ ♥❤ú♥❣ ✤✐➲✉ ❦✐➸♥ ✤➸ t✐➳♥ ✤➳♥
♥❣÷í✐ ❝❤↕② ❝➔♥❣ ❣➛♥ ❝➔♥❣ tèt✳
◆❤÷♥❣ ✤➸ trá ❝❤ì✐ ❦➳t t❤ó❝ t❤➻ t❛ ♣❤↔✐ ✤➦t ❣✐↔ t❤✐➳t ❧➔ ♥❣÷í✐ ✤✉ê✐
♣❤↔✐ ❝â ❧ñ✐ t❤➳ ❤ì♥ ♥❣÷í✐ ❝❤↕② ♥❤÷ ❧➔ ❤↕♥ ❝❤➳ ✈➲ ♥➠♥❣ ❧÷ñ♥❣✱ ♥❣÷í✐ ✤✉ê✐
❧✉æ♥ ❜✐➳t ✤÷ñ❝ t❤æ♥❣ t✐♥ ✈➲ ❜✐➳♥ ✤✐➲✉ ❦✐➸♥ ❝õ❛ ♥❣÷í✐ ❝❤↕②✳✳✳❚r♦♥❣ ❧✉➟♥ ✈➠♥
♥➔② ❝❤ó♥❣ tæ✐ ♥❣❤✐➯♥ ❝ù✉ ✈➲ ✤✐➲✉ ❦✐➺♥ ✤õ ✤➸ ❦➳t t❤ó❝ trá ❝❤ì✐ ♥❤÷ ✈➟②✳

✶




ở ừ ự t ờ t tr
trỏ ỡ t t ợ t tr t tớ ữ r
t t tú ợ tọ t
ữủ
ở ừ ỗ ữỡ

ữỡ tr t tớ ỹ t

ởt số t t t ỡ t tớ
t tr t tớ ữủ tr ồ t
ự t trỏ ỡ ờ t t t tr t
tớ tr ữỡ

ữỡ tr ổ tự ừ ở ỹ trỏ ỡ ờ

t t t ợ t t trỏ ỡ ờ t t t

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tr ữỡ t q ừ t
ú ữủ tr tr
ữủ ỷ ớ ỡ s s tợ P Pữủ
ữớ t tớ ữợ t t t
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sốt q tr ồ t t trữớ
ữủ ỡ ổ ũ ỗ
tr rữớ tr ồ ờ tổ ữ t ữ ỡ tổ ổ
t t ồ tổ t ử ồ t
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trữớ ồ ổ ồ ũ ở t ú
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✸

❣✐❛ ✤➻♥❤✱ ✤ç♥❣ ♥❣❤✐➺♣ ✈➔ ♥❤ú♥❣ ♥❣÷í✐ ❜↕♥ ✤➣ t↕♦ ♠å✐ ✤✐➲✉ ❦✐➺♥ t❤✉➟♥ ❧ñ✐✱
✤ë♥❣ ✈✐➯♥✱ ❣✐ó♣ ✤ï tæ✐ tr♦♥❣ s✉èt q✉→ tr➻♥❤ ❤å❝ t➟♣ ✈➔ ❤♦➔♥ t❤✐➺♥ ❧✉➟♥ ✈➠♥✳

❚❤→✐ ◆❣✉②➯♥✱ ♥❣➔② ✶✵ t❤→♥❣ ✶✶ ♥➠♠ ✷✵✶✼
❍å❝ ✈✐➯♥

▲➯ ❱➠♥ ◗✉þ

❈❤÷ì♥❣ ✶
❑❤→✐ ♥✐➺♠ t❤❛♥❣ t❤í✐ ❣✐❛♥
❈❤÷ì♥❣ ♥➔② tr➻♥❤ ❜➔② ❦❤→✐ ♥✐➺♠ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ❉ü❛ t❤❡♦ ❬✺❪✱ ❬✻❪✱ ❬✽❪
✈➔ ♠ët sè t➔✐ ❧✐➺✉ ❦❤→❝✱ ❝→❝ ❦❤→✐ ♥✐➺♠✱ t➼♥❤ ❝❤➜t ❝ì ❜↔♥ ✈➲ t❤❛♥❣ t❤í✐ ❣✐❛♥
✈➔ ❝→❝ ✈➜♥ ✤➲ ✈➲ ❣✐↔✐ t➼❝❤ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✤÷ñ❝ tr➻♥❤ ❜➔②✳

✶✳✶ ❚❤❛♥❣ t❤í✐ ❣✐❛♥
✣à♥❤ ♥❣❤➽❛ ✶✳✶ ❚❤❛♥❣ t❤í✐ ❣✐❛♥ ✭t✐♠❡ s❝❛❧❡✮ ❧➔ t➟♣ ❝♦♥ ✤â♥❣ tò② þ ❦❤→❝

ré♥❣ tr♦♥❣ t➟♣ sè t❤ü❝ R. ❚❤❛♥❣ t❤í✐ ❣✐❛♥ t❤÷í♥❣ ✤÷ñ❝ ❦þ ❤✐➺✉ ❧➔ T.

❱➼ ❞ö ✶✳✶

✶✮ ❈→❝ t➟♣ R, Z, N, [0; 1] ∪ [2; 3] ❧➔ ❝→❝ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➻ ❝❤ó♥❣ ❧➔ ♥❤ú♥❣

t➟♣ ✤â♥❣ tr♦♥❣ R✳
✷✮ ❈→❝ t➟♣ Q, R\Q; [0, 1) ❦❤æ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➻ ❝❤ó♥❣ ❦❤æ♥❣
♣❤↔✐ ❧➔ t➟♣ ✤â♥❣ tr♦♥❣ R.
❚➟♣ ❝→❝ sè ❤ú✉ t➾ Q✱ t➟♣ ❝→❝ sè ✈æ t➾ R\Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥
✈➻ ❝❤ó♥❣ t✉② ♥➡♠ tr♦♥❣ R ♥❤÷♥❣ ❦❤æ♥❣ ✤â♥❣ tr♦♥❣ R✳
❚❤➟t ✈➟②✱ tr➯♥ Q ①➨t ❞➣② sè {xn }✿ ✶❀ ✶✱✹❀ ✶✱✹✶❀ ✶✱✹✶✹❀ ✳ ✳ ✳ ❚❛ t❤➜② xn ∈Q✱
√
♥❤÷♥❣ lim xn = 2 ∈
/ Q ♥➯♥ Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t➟♣ ❝♦♥ ✤â♥❣ tr➯♥ R. ❱➻
n→∞

✈➟② Q ❦❤æ♥❣ ♣❤↔✐ ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳
❚r➯♥ R\Q ①➨t ❞➣② sè

{xn } :

√

√

√
√
3 3
3
3;
;
;...;
;...
2 3
n
✹




t xn R\Q ữ lim xn = 0
/ R\Q R\Q ổ t
x

õ tr R. r R\Q ổ t tớ
tr R ổ t tớ
t ự C ổ t tớ C ổ
tr R, ũ õ t õ

ổ ổ tr t tớ
rữợ t t ởt tự ừ tổổ sỷ (X, ) ởt
ổ tổổ M X ởt t õ ổổ s M tr

M tứ ữủ ữ s
tr M tt t õ M = M U tr õ
M = {UM : UM = M U, U } ởt tổổ tr M
t t õ
X tở t = M, M = M M s r

M tở M .
sỷ V1 , V2 M t ủ t tự tỗ t U1 , U2 s
V1 = M U1 V2 = M U2 õ V1 V2 = (M U1 ) (M U2 ) =

M (U1 U2 ) U1 U2 s r V1 V2 M t
t M
sỷ {V }I ởt ồ t t tở M õ t õ

U ợ U I

(M U ) = M
I
I

V M

I

V =
I

U s r
I

ứ s r M ởt tổổ ồ tổổ s tứ tr M

(M, M ) ữủ ồ ổ tổổ s ừ ổ tổổ

(X, ) .
r t ổ tt r t tớ T ữủ
tr ởt tổổ s tứ tổổ tổ tữớ ừ t số tỹ tổổ
tổ tữớ tr t số tỹ R tổổ t ũ ợ


✻

❣✐❛♦ ❤ú✉ ❤↕♥ ✈➔ ❤ñ♣ ❜➜t ❦➻ ❝õ❛ ❝❤ó♥❣✮✱ ♥❣❤➽❛ ❧➔ ❝→❝ t➟♣ ♠ð ❝õ❛ T ❧➔
❣✐❛♦ ❝õ❛ ❝→❝ t➟♣ ♠ð tr♦♥❣ R ✈î✐ T. ❈→❝ ❦❤→✐ ♥✐➺♠ ❧➙♥ ❝➟♥✱ ❣✐î✐ ❤↕♥✱ ❧✐➯♥
tö❝✳✳✳✤÷ñ❝ ❤✐➸✉ ❧➔ ❧➙♥ ❝➟♥✱ ❣✐î✐ ❤↕♥✱ ❧✐➯♥ tö❝✳✳✳ tr♦♥❣ tæ♣æ ❝↔♠ s✐♥❤✳

✶✳✸ ❈→❝ ✤à♥❤ ♥❣❤➽❛ ❝ì ❜↔♥
✣à♥❤ ♥❣❤➽❛ ✶✳✷ ❈❤♦ T ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳

❚♦→♥ tû ♥❤↔② t✐➳♥ ✭❢♦r✇❛r❞ ❥✉♠♣✮ ❧➔ t♦→♥ tû

σ:T→T
✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝

σ(t) := inf{s ∈ T : s > t}.
❚♦→♥ tû ♥❤↔② ❧ò✐ ✭❜❛❝❦✇❛r❞ ❥✉♠♣✮ ❧➔ t♦→♥ tû

ρ:T→T
✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝

ρ(t) := sup{s ∈ T : s < t}.

◗✉② ÷î❝ inf ∅ = sup T, sup ∅ = inf T.

❙✉② r❛ σ(M ) = M ♥➳✉ M ❧➔ ♣❤➛♥ tû ❧î♥ ♥❤➜t ✭♥➳✉ ❝â✮ ❝õ❛ T;

ρ(m) = m ♥➳✉ m ❧➔ ♣❤➛♥ tû ♥❤ä ♥❤➜t ✭♥➳✉ ❝â✮ ❝õ❛ T.

❱➼ ❞ö ✶✳✷

✶✮ ❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = Z ✭t❤❛♥❣ t❤í✐ ❣✐❛♥ rí✐ r↕❝✮ t❤➻ σ(t) = t + 1 ✈➔

ρ(t) = t − 1 ✈î✐ ♠å✐ t ∈ T. ❳❡♠ ❍➻♥❤ ✶✳✶✭❜✮✳
✷✮ ❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = R ✭t❤❛♥❣ t❤í✐ ❣✐❛♥ ❧✐➯♥ tö❝✮ t❤➻

σ(t) = ρ(t) = t ✈î✐ ♠å✐ t ∈ T. ❳❡♠ ❍➻♥❤ ✶✳✶✭❛✮✳

❍➻♥❤ ✶✳✶


✼

✸✮ ❳➨t

∞

T=

[2k, 2k + 1]
k=0,k∈N

✰✮ ◆➳✉ t ∈ (2k, 2k + 1) t❤➻ σ(t) = inf{s ∈ T : s > t} = t;

ρ(t) = sup{s ∈ T : s < t} = t.
✰✮ ◆➳✉ t = 2k t❤➻ σ(t) = σ(2k) = 2k,

ρ(t) = ρ(2k) = 2k − 1.

✰✮ ◆➳✉ t = 2k + 1 t❤➻

σ(t) = σ(2k + 1) = 2k + 2, ρ(t) = ρ(2k + 1) = 2k + 1.
✹✮ ❈❤♦ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = {2z : z ∈ Z} ∪ {0}✳
◆➳✉ t ∈ T t❤➻ tç♥ t↕✐ z ∈ Z s❛♦ ❝❤♦ t = 2z s✉② r❛ σ(t) = 2t ✈➔ ρ(t) = 12 t.

✣à♥❤ ♥❣❤➽❛ ✶✳✸ ❈❤♦ T ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳

✣✐➸♠ t ∈ T ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ♣❤↔✐ ✭r✐❣❤t✲s❝❛tt❡r❡❞✮ ♥➳✉ σ(t) > t;
✣✐➸♠ t ∈ T ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝æ ❧➟♣ tr→✐ ✭❧❡❢t✲s❝❛tt❡r❡❞✮ ♥➳✉ ρ(t) < t;
✣✐➸♠ t ∈ T ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ✭✐♥s♦❧❛t❡❞✮ ♥➳✉ ρ(t) < t < σ(t).

✣à♥❤ ♥❣❤➽❛ ✶✳✹ ❈❤♦ T ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳

✣✐➸♠ t ∈ T ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ trò ♠➟t ♣❤↔✐ ✭r✐❣❤t✲❞❡♥❝❡✮ ♥➳✉ σ(t) = t.
✣✐➸♠ t ∈ T ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ trò ♠➟t tr→✐ ✭❧❡❢t✲❞❡♥❝❡✮ ♥➳✉ ρ(t) = t.

✣✐➸♠ t ∈ T ✤÷ñ❝ ❣å✐ ❧➔ ✤✐➸♠ trò ♠➟t ✭❞❡♥❝❡✮ ♥➳✉ ρ(t) = t = σ(t).

❚❛ ❝â ❜↔♥❣ tâ♠ t➢t ✶✳✶

❇↔♥❣ ✶✳✶
❇↔♥❣ ✶✳✷ ❞÷î✐ ✤➙② ♠æ t↔ ❤➻♥❤ ↔♥❤ ❤➻♥❤ ❤å❝ ❝õ❛ ❝→❝ ✤✐➸♠


✽

❇↔♥❣ ✶✳✷
✣à♥❤ ♥❣❤➽❛ ✶✳✺ ❈❤♦ T ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥✳ ❍➔♠ ❤↕t ✭❣r❛✐♥✐❡ss✮ ❧➔
µ : T → [0; ∞)
✤÷ñ❝ ①→❝ ✤à♥❤ ❜ð✐ ❝æ♥❣ t❤ù❝ µ(t) := σ(t) − t.

✣à♥❤ ♥❣❤➽❛ ✶✳✻ ❈❤♦ T ❧➔ t❤❛♥❣ t❤í✐ ❣✐❛♥ ✈➔ ❤➔♠ f := T → R✳ ❚❛ ❦þ

❤✐➺✉ ❤➔♠

fσ : T → R
①→❝ ✤à♥❤ t❤❡♦ ❝æ♥❣ t❤ù❝ f σ (t) = f (σ(t)).

✣à♥❤ ♥❣❤➽❛ ✶✳✼ ❚➟♣ Tk ✤÷ñ❝ ①→❝ ✤à♥❤ ♥❤÷ s❛✉✳

◆➳✉ T ❝â ♣❤➛♥ tû ❧î♥ ♥❤➜t M ❧➔ ✤✐➸♠ ❝æ ❧➟♣ tr→✐ t❤➻ ✤➦t Tk := T\{M }

✈➔ Tk := T tr♦♥❣ tr÷í♥❣ ❤ñ♣ ❝á♥ ❧↕✐✳

❱➼ ❞ö ✶✳✸

✶✮ ❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = R t❤➻ σ(t) = ρ(t) = t, µ(t) = 0 ✈î✐ ♠å✐ t ∈ T.
▼å✐ ✤✐➸♠ t ∈ T ✤➲✉ ❧➔ ✤✐➸♠ trò ♠➟t✳

✷✮ ❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = Z t❤➻ σ(t) = t + 1, µ(t) = 1 ✈➔ ρ(t) = t − 1 ✈î✐
♠å✐ t ∈ T✳ ▼å✐ ✤✐➸♠ t ∈ T ✤➲✉ ❧➔ ✤✐➸♠ ❝æ ❧➟♣✳
✸✮ ❈❤♦ t❤❛♥❣ t❤í✐ ❣✐❛♥ T =

n
2

: n ∈ N0 ✈î✐ N0 ❧➔ t➟♣ ❝→❝ sè tü ♥❤✐➯♥ ✈➔

1
2

✈➔ µ(t) =

sè ✵✳
❚❛ ❝â σ(t) = t + 21 , ρ(t) = t −

1
2

✈î✐ ♠å✐ t > 0, t ∈ T.

✣✐➸♠ t = 0 ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ♣❤↔✐ ✈➔ ♠å✐ t ∈ T, t = 0 ✤➲✉ ❧➔ ✤✐➸♠ ❝æ ❧➟♣✳
✹✮ ❈❤♦ h > 0 ❧➔ ♠ët sè ❝è ✤à♥❤✳ ❳→❝ ✤à♥❤ t❤❛♥❣ t❤í✐ ❣✐❛♥ hZ ♥❤÷ s❛✉


✾

T = hZ = {hn : n ∈ Z} = {..., −3h, −2h, −h, 0, h, 2h, 3h, ...}. ❚❛ ❝â
σ(t) = t + h, ρ(t) = t − h, µ(t) = h ✈î✐ ♠å✐ t ∈ T✳ ❳❡♠ ❍➻♥❤ ✶✳✷✭❝✮✳ ❱➻

h > 0 ♥➯♥ ♠å✐ ✤✐➸♠ t ∈ T ✤➲✉ ❧➔ ✤✐➸♠ ❝æ ❧➟♣✳ ❈❤ó þ r➡♥❣ h > 0 ❝â t❤➸ ❧➔
√
sè ✈æ t➾✱ ✈➼ ❞ö h = 2.
✺✮ ❈❤♦ t❤❛♥❣ t❤í✐ ❣✐❛♥ T
∞

Pa,b =

[k (a + b) , k (a + b) + a].
k=0,k∈N

❳❡♠ ❍➻♥❤ ✶✳✷✭❞✮✳
✰✮ ◆➳✉ t ∈ (k(a + b); k(a + b) + a) t❤➻ σ(t) = t, ρ(t) = t ✈➔ µ(t) = 0.
▼å✐ t ∈ (k(a + b); k(a + b) + a) ✤➲✉ ❧➔ ✤✐➸♠ trò ♠➟t✳
✰✮ ◆➳✉ t = k(a + b) t❤➻ σ(t) = t, ρ(t) = t − b, ✈➔ µ (t) = 0.
❉➝♥ ✤➳♥ σ(t) = t, ρ(t) < t ♥➯♥ t = k(a + b) ❧➔ ✤✐➸♠ trò ♠➟t ♣❤↔✐✱ ✤ç♥❣
t❤í✐ ❧➔ ✤✐➸♠ ❝æ ❧➟♣ tr→✐✳
✰✮ ◆➳✉ t = k(a + b) + a t❤➻ σ(t) = t + b, ρ(t) = t ✈➔ µ (t) = b.
❉➝♥ ✤➳♥ σ(t) > t, ρ(t) = t ♥➯♥ t = k(a + b) + a ❧➔ ✤✐➸♠ trò ♠➟t tr→✐✱ ✤ç♥❣
t❤í✐ ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ♣❤↔✐✳

❍➻♥❤ ✶✳✷

❍➻♥❤ ✶✳✸


✶✵

✻✮ ❈❤♦ q > 1 ❧➔ ♠ët sè t❤ü❝ ❝è ✤à♥❤✱ ①→❝ ✤à♥❤ t❤❛♥❣ t❤í✐ ❣✐❛♥ q Z
♥❤÷ s❛✉ q Z ❂{q n : n ∈ Z} ∪ {0}❂ ..., q −3 , q −2 , q −1 , 0, 1, q, q 2 , q 3 , ... . ❚❛ ❝â

σ(t) = qt, ρ(t) =

t
q

✈➔ µ(t) = (q − 1) t. ❳❡♠ ❍➻♥❤ ✶✳✸✭❛✮

✼✮ ❈❤♦ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = N20 = n2 : n ∈ N0 . ❱î✐ t ∈ T t❤➻ tç♥ t↕✐ sè
√
n ∈ N0 s❛♦ ❝❤♦ t = n2 ❤❛② t = n.
√
√
2
t + 1 , µ(t) = 2 t + 1 ✈➔ ρ(t) =
❚❛ ❝â σ(t) = σ(n2 ) = (n + 1)2 =
√
2
ρ(n2 ) = (n − 1)2 =
t − 1 . ❳❡♠ ❍➻♥❤ ✶✳✸✭❜✮
√
✽✮ ❈❤♦ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = { n : n ∈ N0 }✳
√
◆➳✉ t ∈ T t❤➻ tç♥ t↕✐ sè n ∈ N0 s❛♦ ❝❤♦ t = n ❤❛② n = t2 ,
√
√
√
√
n − 1 = t2 − 1, n + 1 = t2 + 1.
√

√
√
❚❛ ❝â σ(t) = t2 + 1✱ ρ(t) = t2 − 1✱ ✈➔ µ(t) = t2 + 1 − t ✈î✐ ♠å✐ t = 0✱

t ∈ T.
✣✐➸♠ t = 0 ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ♣❤↔✐✳ ▼å✐ ✤✐➸♠ t ∈ T✱ t = 0 ✤➲✉ ❧➔ ✤✐➸♠ ❝æ ❧➟♣✳
❚❛ ❝â ❜↔♥❣ tâ♠ t➢t ❝→❝ t❤❛♥❣ t❤í✐ ❣✐❛♥ t❤÷í♥❣ ❣➦♣

T

σ(t)

µ(t)

ρ(t)

Z

t+1

1

t−1

R

t

0

t

2N

2t

t

qt

(q − 1)t
√
2 t+1
√
t2 + 1 − t

t
2
t
q

q N (q > 1)
N20
√
N0

√
√

t+1

t2 + 1

2

√
√

t−1
t2 − 1

2


✶✶

✶✳✹ P❤➨♣ t➼♥❤ ✈✐ ♣❤➙♥ tr➯♥ t❤❛♥❣ t❤í✐ ❣✐❛♥
✶✳✹✳✶ ✣↕♦ ❤➔♠ ❍✐❧❣❡r
✣à♥❤ ♥❣❤➽❛ ✶✳✽ ●✐↔ sû f : T → R ✈➔ t ∈ Tk ✳ ❉❡❧t❛ ✤↕♦ ❤➔♠ ✭✤↕♦ ❤➔♠
❍✐❧❣❡r✮ ❝õ❛ f t↕✐ t ∈ Tk ❧➔ ♠ët sè ✭♥➳✉ tç♥ t↕✐✮✱ ❦➼ ❤✐➺✉ ❧➔ f ∆ (t), ♥➳✉ ✈î✐ ♠é✐

ε > 0 ❝❤♦ tr÷î❝ tç♥ t↕✐ ♠ët ❧➙♥ ❝➟♥ U ❝õ❛ t ✭♥❣❤➽❛ ❧➔ U = (t − δ; t + δ) ∩ T
✈î✐ δ > 0 ♥➔♦ ✤â✮✱ s❛♦ ❝❤♦
|[f (σ(t)) − f (s)] − f ∆ (t)[σ(t) − s]| ≤ ε|σ(t) − s| ✈î✐ ♠å✐ s ∈ U.

(∗)

✣à♥❤ ♥❣❤➽❛ ✶✳✾ ❍➔♠ f ✤÷ñ❝ ❣å✐ ❧➔ ∆ ❦❤↔ ✈✐ ✭♥❣➢♥ ❣å♥ ❧➔ ❦❤↔ ✈✐✮ tr➯♥
Tk ♥➳✉ ❝â ✤↕♦ ❤➔♠ t↕✐ ♠å✐ ✤✐➸♠ t ∈ Tk .

◆❤➟♥ ①➨t ✶✳✶

✶✮ ❳➨t t❤❛♥❣ t❤í✐ ❣✐❛♥ ❧✐➯♥ tö❝ T = R t❛ ❝â σ(t) = t ✈î✐ ♠å✐ t ∈ R. ❚❤❡♦
✭✯✮ t❛ ❝â

[f (σ(t)) − f (s)] − f ∆ (t).(σ(t) − s)| ≤ ε.|(σ(t) − s)|
⇔

f (t)−f (s)
t−s

− f ∆ (t) ≤ ε.

f (t) − f (s)
= f (t).
s→t.
t−s
❱➟② ✤↕♦ ❤➔♠ ❍✐❧❣❡r ❝❤➼♥❤ ❧➔ ✤↕♦ ❤➔♠ t❤æ♥❣ t❤÷í♥❣ ❦❤✐ T = R.
✷✮ ❳➨t t❤❛♥❣ t❤í✐ ❣✐❛♥ rí✐ r↕❝ T = Z t❛ ❝â σ(t) = t + 1 ✈î✐ t ∈ Z.
❱➻ ε ❜➜t ❦➻ ♥➯♥ s✉② r❛ f ∆ (t) = lim

❑❤✐ ➜② t❛ ❝â UT (t) := Uδ (t) ∩ T = {t} ✈➔ s ∈ UT (t) s✉② r❛ s = t ✭❦❤✐

δ ∈ (−1, 1)✮✳ ❱➟② ✭✯✮ ❝â ❞↕♥❣
|[f (t + 1) − f (t)] − f ∆ (t).(t + 1 − t)| ≤ ε.|(t + 1 − t)|
⇔ |[f (t + 1) − f (t)] − f ∆ (t)| ≤ ε ✈î✐ ♠å✐ ε.
❙✉② r❛ f ∆ (t) = f (t + 1) − f (t), ❤❛② f ∆ (t) ❝❤➼♥❤ ❧➔ s❛✐ ♣❤➙♥ t✐➳♥ ❝õ❛ f t↕✐

t ✈➔ f ∆ (t) = ∆f (t).

❱➼ ❞ö ✶✳✹

◆➳✉ f : T → R ✈➔ f (t) = C ✈î✐ ♠å✐ t ∈ Tk ✈➔ C ∈ R t❤➻ f ∆ (t) = 0.




t ợ ồ > 0 tỗ t t U s s U t õ

|[f ((t)) f (s)] f (t)[(t) s]| .|((t) s)|
|f (t).[(t) s]| .|((t) s)
|f (t)| ợ ồ .
f (t) = 0 ợ ồ t T.

ử f : T R f (t) = t ợ ồ t Tk t f (t) = 1.
t ợ ồ > 0 tỗ t t U s s U t õ

|[f ((t)) f (s)] f (t)[(t) s]| |(t) s|
|((t) s) f (t)((t) s) |(t) s|
|1 f (t)| ợ ồ .
f (t) = 1.

t ừ r
ỵ f : T R ợ ồ t Tk .
f t t t f tử t t.

f tử t t t ổ t f t t

f (t) =

f ((t)) f (t)

.
à(t)

t trũ t t f t t ợ
f (t)f (s)
ts
st,sT

lim

(s)
tỗ t ỳ õ t õ f (t) = lim f ((t))f
(t)s .
st

f t t t

f ((t)) = f (t) + à(t)f (t).

ự sỷ f t t (0; 1) t
=

1 + 2à(t) + |f (t)|

(0; 1).

t õ
ợ ồ > 0 õ ởt U (t, ) ừ t s

[f ((t)) f (s)] f (t).((t) s)| .|((t) s)| ợ ồ s U (t, )

✶✸

✣➦t U ∗ = U (t, δ) ∩ U (t, ε∗ )✳ ▲➜② s ∈ U ∗ t❛ ❝â

|f (t) − f (s)| = |[f (σ(t)) − f (s) − f ∆ (t).(σ(t) − s)] − [f (σ(t)) − f (t)
− f ∆ (t).(σ(t) − t)] + (t − s).f ∆ (t)|
≤ ε∗ |σ(t) − s| + ε∗ |σ(t) − t| + |(t − s).f ∆ (t)|
≤ 2ε∗ (σ(t) − t) + ε∗ |t − s| + |t − s| |f ∆ (t)|
≤ε∗

≤ε∗

(✈➻ s ∈ U ∗ ⇒ |t − s| ≤ ε∗ )
≤ ε∗ .[1 + 2µ(t) + |f ∆ (t)] ≤ ε ✈î✐ ♠å✐ s ∈ U ∗ .
❉♦ ✤â lim[f (t) − f (s)] = 0 ⇔ lim f (s) = f (t) ✈î✐ ♠å✐ s ∈ U ∗ ✳
s→t

s→t

❱➟② f ❧✐➯♥ tö❝ t↕✐ t✳

✷✮ ●✐↔ f ❧✐➯♥ tö❝ t↕✐ t ∈ Tk ✈➔ t ❧➔ ✤✐➸♠ ❝æ ❧➟♣ ♣❤↔✐✳ ❚ø t➼♥❤ ❧✐➯♥ tö❝
❝õ❛ ❤➔♠ f t↕✐ t ∈ Tk . ❚❛ ❝â

f (σ(t)) − f (s) f (σ(t)) − f (t) f (σ(t)) − f (t)
=
=
.

s→t
σ(t) − s
σ(t) − t
µ(t)

lim

❱î✐ ε > 0, tr♦♥❣ ❧➟♥ ❝➟♥ U ❝õ❛ s t❛ ❝â

f (σ(t)) − f (s) f (σ(t)) − f (t)
−
≤ ε ∀s ∈ U
σ(t) − s
σ(t) − t
⇔ f (σ(t)) − f (s) −

f (σ(t))−f (t)
. [σ(t)
µ(t)

❚ø ✤â t❛ ❝â

f ∆ (t) =

− s] ≤ ε [σ(t) − s] ∀s ∈ U.

f (σ(t)) − f (t)
.
µ(t)

✸✮ ●✐↔ sû f ❦❤↔ ✈✐ t↕✐ t ∈ Tk ✈➔ t ❧➔ ✤✐➸♠ trò ♠➟t ♣❤↔✐✳
❈❤♦ ε > 0. ❱➻ f ❦❤↔ ✈✐ t↕✐ t ∈ Tk ♥➯♥ tr♦♥❣ ❧➙♥ ❝➟♥ ❝õ❛ t t❛ ❝â

[f (σ(t)) − f (s)] − f ∆ (t) [σ(t) − s] ≤ ε |σ(t) − s|

∀s ∈ U.

❱➻ σ(t) = t ♥➯♥ t❛ ❝â

[f (t) − f (s)] − f ∆ (t) [t − s] ≤ ε |t − s|

∀s ∈ U.


✶✹

⇔

[f (t) − f (s)]
− f ∆ (t) ≤ ε,
t−s

∀s ∈ U, s = t.

(s)
❙✉② r❛ f ∆ (t) = lim f (t)−f
t−s .
s→t

✹✮ ●✐↔ t❤✐➳t t❛ ❝â f ❦❤↔ ✈✐ t↕✐ t.

❚r÷í♥❣ ❤ñ♣ ✶ ◆➳✉ σ(t) = t ⇔ µ(t) = 0

(1.2) ⇔ f (t) = f (t).
f (σ(t)) − f (s)
=
s→t
σ(t) − s

❚r÷í♥❣ ❤ñ♣ ✷ ◆➳✉ σ(t) > t✳ ❉♦ f ❦❤↔ ✈✐ t↕✐ t f ∆ (t) = lim

f (σ(t)) − f (t)
σ(t) − t
⇔ f (σ(t)) = f (t) + µ(t).f ∆ (t).

❱➼ ❞ö ✶✳✻ ◆➳✉ f : T → R ✈➔ f (t) = t2. ❑❤✐ ➜② f ∆(t) = t + σ(t) ✈î✐ ♠å✐
t ∈ Tk .
❚❤➟t ✈➟②✱ ✈î✐ ♠å✐ ε > 0, s ∈ U t❤➻ |s − t| < ε. ❉♦ ✤â t❛ ❝â
[f (σ(t)) − f (s)] − f ∆ (t) [σ(t) − s] ≤ ε |σ(t) − s|
⇔ σ 2 (t) − s2 − f ∆ (t)(σ(t) − s) ≤ ε |σ(t) − s|
⇔ (σ(t) − s)(f ∆ (t) − (t + σ(t)) ≤ ε |σ(t) − s|
⇔ f ∆ (t) − (t + σ(t)) ≤ ε ✈î✐ ♠å✐ ε.
❱➟② ✈î✐ ♠å✐ t ∈ Tk t❛ ❝â f ∆ (t) = t + σ(t).
❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = R t❤➻ σ(t) ≡ t. ❉♦ ✤â f ∆ (t) = 2t = f (t).
❳➨t

n
: n ∈ N0 }.
2
❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = Z t❤➻ σ(t) ≡ t + 1. ❉♦ ✤â f ∆ (t) = 2t + 1 =

T := {

∆f (t) = f (t + 1) − f (t).
❳➨t
T := {
❚➼♥❤ f ∆ ✈î✐
❛✮ f (t) = t2 ;
√
❜✮ f (t) = t.

n
: n ∈ N0 }.
2


✶✺

❚❛ ❝â

n
n+1
2t + 1
1
σ( ) =
❤❛② σ(t) =
=t+
2
2
2
2

❛✮ ❱î✐ f (t) = t2

f (σ(t)) =
❱➻ ✈➟②

√

.

t t❤➻

1 √
− t
2
∆
f (t) =
=
1
(t + ) − t
2
√
❳➨t f (t) = t t❤➻ t❛ ❝â
t+

❱➼ ❞ö ✶✳✼

2

f (σ(t)) − f (t)

1
= 2t + .
σ(t) − t
2

f ∆ (t) =
❜✮ ❱î✐f (t) =

1
t+
2

f ∆ (t) =

1
σ(t) +

1
.
1 √
t+ + t
2

√ .
t

❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = R t❤➻ σ(t) = t ♥➯♥

1
f ∆ (t) = √ = f (t).

2 t
❱î✐ t❤❛♥❣ t❤í✐ ❣✐❛♥ T = N t❤➻ σ(t) = t + 1 ♥➯♥

f ∆ (t) = √

√
√
1
√ = t + 1 − t.
t+1+ t

❱➼ ❞ö ♥➔② ❝❤➾ r❛ r➡♥❣✱ ❞❡❧t❛✲✤↕♦ ❤➔♠ ♣❤ö t❤✉ë❝ ✈➔♦ ❤➔♠ ♥❤↔② σ(t) ❝õ❛
t❤❛♥❣ t❤í✐ ❣✐❛♥ T✱ tù❝ ❧➔ ♣❤ö t❤✉ë❝ ✈➔♦ ❝➜✉ tró❝ ❝õ❛ t❤❛♥❣ t❤í✐ ❣✐❛♥ T✳

✣à♥❤ ❧þ ✶✳✷ ❈❤♦ f

: T → R ✈➔ g : T → R ❧➔ ❝→❝ ❤➔♠ ∆ ✲ ❦❤↔ ✈✐ t↕✐

t ∈ Tk . ❑❤✐ ✤â t❛ ❝â
✶✮ ❍➔♠ f + g ❧➔ ∆ ✲ ❦❤↔ ✈✐ t↕✐ t ∈ Tk ✈➔

(f + g)∆ (t) = f ∆ (t) + g ∆ (t).


✶✻

✷✮ ❍➔♠ f.g ❧➔ ∆ ✲ ❦❤↔ ✈✐ t↕✐ t ∈ Tk ✈➔

(f.g)∆ (t) = f ∆ (t).g(t) + f (σ(t))g ∆ (t) = f (t)g ∆ (t) + f ∆ (t).g(σ(t)).
✸✮ ◆➳✉ f (t)f (σ(t)) = 0 t❤➻

1
f

∆

∆

❧➔ ∆ ✲ ❦❤↔ ✈✐ t↕✐ t ∈ Tk ✈➔

f ∆ (t)
(t) = −
.
f (t)f (σ(t))

✹✮ ◆➳✉ g(t)g(σ(t)) = 0 t❤➻

f
g

1
f

f
g

❧➔ ∆ ✲ ❦❤↔ ✈✐ t↕✐ t ∈ Tk ✈➔

f ∆ (t).g(t) − f (t)g ∆ (t)
(t) =

.
g(t)g(σ(t))

❈❤ù♥❣ ♠✐♥❤ ●✐↔ sû f, g ❧✐➯♥ tö❝ t↕✐ t ∈ Tk ✳

✶✮ ❈❤♦ ε > 0✱ U1 , U2 ❧➔ ❧➙♥ ❝➟♥ ❝õ❛ t t❛ ❝â✿
ε
f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s) ≤ |σ(t) − s| ✈î✐ ♠å✐ s ∈ U1
2
✈➔

g(σ(t)) − g(s) − g ∆ (t)(σ(t) − s) ≤

ε
|σ(t) − s| ✈î✐ ♠å✐ s ∈ U2
2

▲➜② U = U1 ∩ U2 t❤➻ ✈î✐ ♠å✐ s ∈ U t❛ ❝â

(f + g)(σ(t)) − (f + g)(s) − f ∆ (t) + g ∆ (t) (σ(t) − s)
= f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s) + g(σ(t)) − g(s) − g ∆ (t)(σ(t) − s)
≤ f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s) + g(σ(t)) − g(s) − g ∆ (t)(σ(t) − s)
≤ 2ε |σ(t) − s| + 2ε |σ(t) − s|
= ε |σ(t) − s| .
❱➻ ✈➟② f + g ❦❤↔ ✈✐ t↕✐ t ✈➔ (f + g)∆ = f ∆ + g ∆ t↕✐ t ∈ Tk ✳
✷✮ ❈❤♦ ε ∈ (0, 1) . ✣➦t ε∗ = ε 1 + |f (t)| + |g(σ(t))| + g ∆ (t)

−1

❑❤✐ ➜② ε∗ ∈ (0, 1)✱ ✈➻ ✈➟② tr♦♥❣ ❧➙♥ ❝➟♥ U1 , U2 , U3 ❝õ❛ t t❤ä❛ ♠➣♥✿

f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s) ≤ ε∗ |σ(t) − s| ✈î✐ s ∈ U1 ;

.


✶✼

g(σ(t)) − g(s) − g ∆ (t)(σ(t) − s) ≤ ε∗ |σ(t) − s| ✈î✐ s ∈ U2 .
❚❤❡♦ ✣à♥❤ ❧þ ✶✳✶ ♣❤➛♥ ✶✮ t❛ ❝â |f (t) − f (s)| ≤ ε∗ ✈î✐ s ∈ U3 .
✣➦t U = U1 ∩ U2 ∩ U3 t❤➻ ✈î✐ s ∈ U t❛ ❝â✿

(f g)(σ(t)) − (f g)(s) − f ∆ (t)g(σ(t)) + f (t)g ∆ (t) (σ(t) − s
=

f (σ(t)) − f (s) − f ∆ (t)(σ(t) − s) g(σ(t))

+ g(σ(t)) − g(s) − g ∆ (t)(σ(t) − s) f (t)
✰ g(σ(t)) − g(s) − g ∆ (t)(σ(t) − s) [f (s) − f (t)]
✰ (σ(t) − s)g ∆ (t) [f (s) − f (t)]

≤ ε∗ |σ(t) − s| |g(σ(t))| + ε∗ |σ(t) − s| |f (t)|
+ε∗ ε∗ + |σ(t) − s| + ε∗ |σ(t) − s| g ∆ (t)
= ε∗ |σ(t) − s| |g(σ(t))| |f (t)| + ε∗ + g ∆ (t)
≤ ε∗ |σ(t) − s| 1 + |f (t)| + |g(σ(t))| + g ∆ (t)
= ε |σ(t) − s| .
❱➟② (f g)∆ (t) = f ∆ g (σ(t)) + f g ∆ (t) t↕✐ t ∈ Tk ✳
❚➼♥❤ ❝❤➜t ✷✮ ✤÷ñ❝ ❝❤ù♥❣ ♠✐♥❤✳
❚ø ❚➼♥❤ ❝❤➜t ✷✮ t❛ s✉② r❛ ❚➼♥❤ ❝❤➜t ✸✮ ✈➔ ❚➼♥❤ ❝❤➜t ✹✮✳
◆❤➟♥ ①➨t ❝→❝ t➼♥❤ ❝❤➜t tr➯♥ t÷ì♥❣ tü ♥❤÷ ❝→❝ t➼♥❤ ❝❤➜t ❝õ❛ ✤↕♦ ❤➔♠ t❤æ♥❣

t❤÷í♥❣✱ ♥❤÷♥❣ ✤➣ t❤➯♠ ②➳✉ tè ❤➔♠ ♥❤↔② t✐➳♥ σ(t) t❤❛♠ ❣✐❛ tr♦♥❣ ❝→❝ ❝æ♥❣
t❤ù❝✳
❚❛ ❝â ❜↔♥❣ s♦ s→♥❤
T ❜➜t ❦➻

T=R

T=Z

(k.f ) = k.f ∆

(k.f ) = k.f

∆(k.f ) = k.∆f

(f + g)∆ = f ∆ + g ∆

(f + g) = f + g

∆ (f + g) = ∆f + ∆g

∆

(f.g)∆ = f.g ∆ + f ∆ .g σ (f.g) = f.g + f .g ∆ (f.g) = f.∆g + ∆f.g(t + 1)
f
g

∆

=

f ∆ .g−f.g ∆
g.g σ

f
g

=

f .g−f.g
g2

✣↕♦ ❤➔♠ ❝õ❛ ✈❡❝tì ❤➔♠ ✈➔ ♠❛ tr➟♥ ❤➔♠

∆

f
g

=

∆f.g−f.∆g
g.g(t+1)

●✐↔ sû f : T → Rn ❧➔ ❤➔♠ ✈❡❝tì n ❝❤✐➲✉ ❤♦➦❝ A : T → Rn×m ❧➔ ❤➔♠
♠❛ tr➟♥ ❝➜♣ n × m t❤➻ ✤↕♦ ❤➔♠ ❝õ❛ ❤➔♠ ✈❡❝tì ✈➔ ❤➔♠ ♠❛ tr➟♥ ✤÷ñ❝ ✤à♥❤
♥❣❤➽❛ ♥❤÷ ❧➔ ✈❡❝tì ✈➔ ♠❛ tr➟♥ ❝õ❛ ✤↕♦ ❤➔♠ ❝õ❛ ❝→❝ ❤➔♠ t❤➔♥❤ ♣❤➛♥✳

P t t tr t tớ
t
số f : T R ữủ ồ q rt
ợ ừ õ tỗ t ỳ t ồ trũ t

tr T ợ tr ừ õ tỗ t ỳ t ồ trũ t
tr ừ T

f

: T R ữủ ồ r tử rts

ts õ tử t ồ trũ t tr T ợ
tr ừ õ tỗ t ỳ t trũ t tr tr T

ởt m ì n tr A(.) tr t tớ
T ữủ ồ r tử ộ tỷ ừ A(.) r tử

X ởt ổ f : TìX

X (t, x) f (t, x) ữủ ồ r tử tọ s
f tử t ộ (t, x) ợ t trũ t t =
maxT
ợ

lim
(s,y)(t,x),st

f (s, y) lim f (t, y) tỗ t t ộ

yx

(t, x) ợ t trũ t tr

ỵ r t f : T R t õ
f tử t f r tử

f r tử t f q
f q r tử t f := f ụ q
r tử
f tử g : T R q r tử t f g
ụ q r tử

ởt tử f : T R ữủ ồ t

rrt ợ D s ỗ tớ
ữủ tọ


✶✾

✶✮ D ⊂ Tk ;
✷✮ Tk \ D ❧➔ ❦❤æ♥❣ q✉→ ✤➳♠ ✤÷ñ❝ ✈➔ ❦❤æ♥❣ ❝❤ù❛ ✤✐➸♠ ❝æ ❧➟♣ ♣❤↔✐ ♥➔♦
❝õ❛ T;
✸✮ f ❦❤↔ ✈✐ t↕✐ ♠é✐ ✤✐➸♠ t ∈ D.

✣à♥❤ ❧þ ✶✳✹ ✭✣à♥❤ ❧þ ❣✐→ trà tr✉♥❣ ❜➻♥❤✮ ❬✻✱ ❚❤❡♦r❡♠ ✶✳✾❪ ❈❤♦ f ✈➔ g ❧➔

❝→❝ ❤➔♠ ♥❤➟♥ ❣✐→ trà t❤ü❝✱ ①→❝ ✤à♥❤ tr➯♥ T ✈➔ ❧➔ t✐➲♥ ❦❤↔ ✈✐ ✈î✐ ♠✐➲♥ ❦❤↔
✈✐ D ✳ ❑❤✐ ✤â✱ ♥➳✉

|f ∆ (t)| ≤ g ∆ (t) ✈î✐ ♠å✐ t ∈ D
t❤➻ |f (s) − f (r)| ≤ g(s) − g(r) ✈î✐ ♠å✐ r, s ∈ T ✈➔ r ≤ s.

✣à♥❤ ❧þ ✶✳✺ ❬✻✱ ❚❤❡♦r❡♠ ✶✳✷✺❪ ❈❤♦ f ❧➔ ♠ët ❤➔♠ ❝❤➼♥❤ q✉②✳ ❑❤✐ ✤â tç♥

t↕✐ ♠ët ❤➔♠ t✐➲♥ ❦❤↔ ✈✐ F ✈î✐ ♠✐➲♥ ❦❤↔ ✈✐ D s❛♦ ❝❤♦ F ∆ (t) = f (t) ✈î✐
♠å✐ t ∈ D.

✶✳✺✳✷ P❤➨♣ t➼♥❤ t➼❝❤ ♣❤➙♥
✣à♥❤ ♥❣❤➽❛ ✶✳✶✺

✶✮ ❍➔♠ F tr♦♥❣ ✣à♥❤ ❧þ ✶✳✺ ✤÷ñ❝ ❣å✐ ❧➔ ♠ët t✐➲♥ ♥❣✉②➯♥ ❤➔♠ ✭♣r❡✲

❛♥t✐❞❡r✐✈❛t✐✈❡✮ ❝õ❛ ❤➔♠ ❝❤➼♥❤ q✉② f.
✷✮ ❚➼❝❤ ♣❤➙♥ ❜➜t ✤à♥❤ ❝õ❛ ♠ët ❤➔♠ ❝❤➼♥❤ q✉② f ❧➔

f (t).∆t :=
F (t) + C tr♦♥❣ ✤â C ❧➔ ♠ët ❤➡♥❣ sè tò② þ ✈➔ F ❧➔ ♠ët t✐➲♥ ♥❣✉②➯♥ ❤➔♠
❝õ❛ ❤➔♠ f.
✸✮ ❚➼❝❤ ♣❤➙♥ ①→❝ ✤à♥❤ ❝õ❛ ♠ët ❤➔♠ ❝❤➼♥❤ q✉② f ❧➔
s

f (t)∆t := F (s) − F (r)
r

✈î✐ r, s ∈ T, ✈î✐ F ❧➔ ♠ët t✐➲♥ ♥❣✉②➯♥ ❤➔♠ ❝õ❛ ❤➔♠ f
✹✮ ▼ët ❤➔♠ F : T → R ✤÷ñ❝ ❣å✐ ❧➔ ♠ët ♥❣✉②➯♥ ❤➔♠ ✭❛♥t✐❞❡r✐✈❛t✐✈❡✮
❝õ❛ f : T → R ♥➳✉ F ∆ (t) = f (t) ✈î✐ ♠å✐ t ∈ Tk .

ỵ r ồ f

r tử õ

F ừ f ữủ
t

f ( )( ), t T, tr õ t0 T.

F (t) :=
t0

ứ s t sỷ ử

Crd Crd (T) Crd (T, R) t ủ r tử

ỵ r f Crd t Tk t

(t)
f ( )( )
t

=

à(t)f (t).
f

ự f r tử tỗ t ởt F ừ
(t)

õ

f (s)s = F ((t)) F (t).
t

ỵ t õ F ((t)) F (t) = à(t)F (t) = à(t)f (t).
(t)



f (s)s = à(t)f (t)
t

ỵ r a, b, c T, R f, g Crd t









b
b
b
a [f (t + g(t))]t = a f (t)(t) + a g(t)(t);

b
b
a (f )(t)t = a f (t)(t);
b
a
a f (t)t = b f (t)(t);
b
c
b
a f (t)t = a f (t)(t) + c f (t)(t);
b
b

a f ((t))g (t)t = (f.g)(b) (f.g)(a) a f (t)g(t)t;
b
b

a f (t)g (t)t = (f.g)(b) (f.g)(a) a f (t)g((t))t;
a
a f (t)t = 0;

|f (t)| g(t) ợ ồ t [a, b) t
f (t) 0 ợ a t b t

t

b
a f (t)t

ợ t tớ T = R t

tử

b
a f (t)t



b
a g(t)t;

0.

b
a f (t)t

=

b
a f (t)dt,

f




ợ t tớ T = Z t t õ à(t) = (t) t = t + 1 t = 1.
[a; b] = {a; a + 1; a + 2; ...; b 1; b}. õ

b1

f (t)
a < b;



t=a
b
f (t)t = 0
a = b;

a

a1



f (t)

t=b

ợ f : Z R ởt tũ ỵ

a T, sT = f r tử tr [a, )
t t t s rở


b

f (t)t := lim

b

a

f (t) t.
a

ợ tỗ t t õ t ở tử ữủ t õ t
ý

ỵ ờ ữợ t r sỷ v :

T R ởt t t T = v(T) ụ ởt t tớ

f : T R r tử v ợ v r
tử t ợ a, b T t õ
b

v(b)

(f v 1 )(s)s.



f (t)v (t)t =
a

v(a)

ỗ q tr t tớ
õ õ ữ s
ủ A ũ t ữủ ồ õ tọ
s
ồ x, y tở A t x y y x tở A;
ồ x, y, z tở A t (x y) z = x (y z);